a robust decentralized sliding-mode control for a … · international journal of mechanical &...

International Journal of Mechanical & Mechatronics Engineering IJMME / IJENS 148

164201-3737-IJMME-IJENS © February 2016 IJENS

A ROBUST DECENTRALIZED SLIDING-MODE CONTROL FOR A

SMIB-SVC SYSTEM

Alfonso López-Martíneza, Marco T. Mata-Jiménez

b, Daniel Alaniz-Lumbreras

a, Manuel A. Andrade

b,

Vianey Torres-Argüellesc and Víctor M. Castaño

d*

aGraduate School of Engineering Sciences, Autonomous University of Zacatecas, Carretera a la Bufa,

98000 Zacatecas, Zac., México.

bFacultad de Ingeniería Mecánica y Eléctrica, Universidad Autónoma de Nuevo León, Avenida

Universidad S/N, Ciudad Universitaria, C.P. 66455, San Nicolás de los Garza, NL, México.

cDepartamento de Ingeniería Industrial y Manufactura, Instituto de Ingeniería y Tecnología, Universidad

Autónoma de Ciudad Juárez, Av. Del Charro 450 N, C.P. 32310 Juárez, Chih., México.

d*Centro de Física Aplicada y Tecnología Avanzada, Universidad Nacional Autónoma de México, Blvd.

Juriquilla 3001, C.P. 76230, Querétaro, México. Corresponding author e-mail: [email protected]

ABSTRACT

An observer-based output feedback controller is presented in order to enhance the transient

stability and voltage regulation of a SMIB-SVC (Single Machine Infinite Bus-Static VAR Compensator)

system. The generator excitation and the SVC regulator controllers consider the uncertainties of the

nonlinear structure and the interconnection between subsystems as perturbations, requiring only local

measurements. A control scheme using high-order sliding-mode differentiator is designed obtaining

robustness in presence of uncertainties and parameter variations. Simulations are used to validate the

performance of the proposed scheme. Numerical results show that transient stability can be effectively

improved under fault scenarios and the effect on power quality.

KEY WORDS

SVC regulator, SMIB system, nonlinear system, transient stability, high-order sliding-mode,

robustness, interconnected systems, output feedback control, power quality.

mailto:[email protected]



1 Introduction

In recent years, power systems have been increasingly operating close to their power transfer

limits, becoming complex structures due to both the use of power electronics and the operation in global

markets. Furthermore the expansion of these systems in generation and transmission sections has been

limited due to limited resources and environmental reasons. As consequence of these operative

conditions, some transmission lines are heavily loaded, affecting both the operational system economics

and security [1]. To respond to these requirements, control and power system communities are designing

structures in order to allow the generation and distribution of electrical energy more efficiently [2].

To improve the performance of power systems several control approaches have been used [3, 4]. Linear

control techniques as automatic voltage regulator/power system stabilizer (AVR/PSS) with their

inherent restrictions [3] are widely used in power systems operation.

In addition FACTS (Flexible AC Transmission Systems) devices are being employed as alternatives in

order to increase flexibility of AC systems [5, 6]. A widely used FACTS device is the static VAR

compensator (SVC). This regulation device absorbs or supplies the reactive power needed by the

electrical network allowing a greater control of high-voltage transmission lines [2, 7]. However, these

devices add complexity to the power systems, requiring new techniques to enhance the dynamic

performance. Research in this field has evolved from the optimum placement of compensation devices

[8, 9] to the analysis and characterization of the dynamic oscillations due to the interactions between

those devices and the power system [10–12].

Despite the above, in cases such as highly nonlinear systems in increasing demand for quality energy as

the power systems, these techniques may not guarantee performance. To enhance the fulfillment

techniques like backstepping [13], passivity-based approach [14, 15], adaptive control [16, 17], feedback

linearization and robust control theory approach to design a feedback controller [18] have been

successfully applied to achieve high dynamic performance under unexpected contingencies. However



they have a common disadvantage, the controller design requires knowing the system structure and the

system parameters accurately.

In this condition of uncertainties, the high-order sliding-mode (HOSM) technique has been successfully

applied to achieve high performance despite the nonlinear behavior of power systems. This approach has

demonstrated robustness under parameter uncertainties and external disturbances [19]. The drawback of

the standard approach is the presence of the so-called chattering effect [20], this problem was faced in

[21] by Levant, as a result this effect is significantly reduced. In [22], an output feedback controller is

combined with this technique as a solution to the problem of transient stabilization of a multi-machine

power system.

If it is considered that the proper coordination of all control devices requires the fulfillment of criteria,

such as being controllable and observable (concepts well developed for linear systems), one can agree

that in large-scale nonlinear systems such as interconnected power systems, much research remains to be

done [23].

A characteristic of the decentralized state feedback control schemes [24–26] is that the state variables

may not be available (or can be costly to measure them). Therefore, it is necessary to reconstruct the

states of each subsystem. In this sense, considerable attention has been focused on state observers of

interconnected systems as a way to solve this problem and simultaneously avoid the increase of

complexity of the nonlinear system, combining the controller with a robust differentiator.

Several methods to reconstruct the states in nonlinear systems have been suggested [27–29], being the

main difficulty in differentiator designs to obtain robustness to measurement errors. In [28], a

differentiator based on sliding-mode algorithms provides insensitivity with respect to internal and

external disturbances with exact finite-time convergence. Furthermore, due to the finite-time

convergence the separation principle holds [22], so the controller and the observer can be designed

independently.



In this paper a solution to the problem of simultaneous control in the transient stability and the voltage

regulation of a SMIB-SVC system is proposed. An output-feedback control scheme that combines

controllers and differentiators based in high-order sliding-mode is designed. The resulting controllers

require only a local variable respectively, achieving insensitivity to changes in both topology and

parameters.

2 Preliminaries of the Control Technique

In the following sections the necessary concepts related to the high-order sliding-mode control

strategy are introduced.

2.1 HOSM Technique

The quasi-continuous high-order sliding-mode (QCHOSM) is a type of finite-time convergent

controller where a tracking error σ follows a nonlinear surface

with a control depending only on the parameters . is an equilibrium point of the system.

To describe briefly the technique, consider a nonlinear system class represented by

(1)

where, , , is the state vector, n is the dimension of the system, is the

input control vector, denotes a closed and bounded subset, the vectorial field and are supposed

bounded and their components are smooth functions of [30], and are unknown smooth

functions. The relative degree of the system is constant and known, allowing the control to appear in the

-th total time derivative of . If there is a smooth output function

(2)

where , and

, then it is assumed that for , and



(3)

This is always true at least locally, so trajectories of (1) are supposed to be infinitely extendible in time

for any Lebesgue-measurable bounded control .

From (2) and (3)

(4)

The closed differential inclusion is understood here in the Filippov sense [21, 30].

To design the high-order sliding mode control it is necessary to consider the nonlinear surface

(5)

where every function is a function [21] such that . As are continuous functions

in the state spaces of the system, the relationship

(6)

is called r-sliding mode [21].

So, if , and

(7)

(8)

(9)

(10)

where are positive numbers, then the controller

(11)

is -sliding homogeneous and provides stability in a finite time. Each choice of parameters

determines a controller family applicable to all systems of relative degree .

2.2 Differentiator-Observer



To implement the controller (11), it is necessary to know the real time exact calculation or direct

measurement of , yet in order to reduce the number of sensors, it is assumed that the only

measurable signal in the system is .

Combining the controller (11) and the homogeneous differentiator described in [21]

.

. (12)

.

can be obtained for , where are estimates of the −th derivatives of .

In (12) the parameters of the differentiators are chosen according to the condition . is to satisfy

.

3 SMIB and SVC Regulator Models

In the following sections are described the SMIB-SVC system and the output feedback control

strategy.

A simplified model for a generator and an SVC system which forms the three-bus system shown in Fig.

1 is considered.



Fig. 1. SMIB-SVC system.

3.1 Synchronous Generator Model

In order to design the excitation controller, a single synchronous generator, where the dynamic of

the fluxes and damping windings are neglected [31], it is connected to an infinite-bus system with an

SVC connected at the transmission line. The dynamic equations and the electrical relationships of the

generator can be written as

(13)

(14)

(15)

and

(16)

(17)

(18)

(19)

where is the power angle of synchronous generator; , the relative speed; , the synchronous

machine speed; , the mechanical power of entry; , the active electrical power delivered by the

generator; , the damping constant; , the inertia constant; , the transient EMF in the quadrature

axis of the generator; , the EMF in the quadrature axis; , the equivalent EMF in the excitation



winding of the generator; , the direct axis transient open circuit time constant, the gain of the

generator excitation; the control input of the generator; , the direct axis reactance; , the direct

axis transient reactance; , the external bus voltage and is the equivalent transient reactance seen

from the generator.

The power system has an SVC, so the equivalent reactance seen from generator is time-varying even

if no perturbations are present in the system.

The differential equations described above represent a nonlinear third order system which retains the

main characteristics of the synchronous generator.

3.2 SVC Regulator Model

In Fig. 2 is drawn the three-bus SVC system configuration employed, where the voltage

regulation in VB is performed varying the susceptance of the inductor.

Fig. 2. Equivalent electrical circuit.

The SVC regulator dynamic is defined as follows

(20)

where is the variable susceptance of the SVC regulator, the initial susceptance of the SVC,

the time constant of the SVC regulator, the gain of the SVC regulator and the input.

In order to maintain the SVC-bus voltage , the following equation is established [18] from Fig. 2



(21)

where , is the SVC bus voltage at the operating point, the fixed susceptance of

the SVC regulator and is the inductor current in the SVC. The term can be treated as an

external disturbance and is described [18] by

(22)

where , and are the phase angles of the sending bus, the receiving bus and the SVC bus at the

operating point respectively.

4 Interaction and Feedback Control of the System

In the power system the interrelated parameter is established [32] from Fig. 2 as

(23)

this variable reactance is considered as an uncertain parameter by the excitation controller (even in

absence of disturbances) and represents the effects of the SVC on the generator.

4.1 Interconnection Terms of the System

The described SMIB-SVC system can be represented by two interconnected SISO subsystems.

Defining , from (13)–(15) and

, from (20)–(21). The state representation of the subsystems can be written as

follows

(24)

From (13)–(15)



(25)

represents the dynamics of the interconnected generator, where can be written as

and from (20)–(21)

(26)

represents the dynamics of the interconnected SVC regulator, where

In , the parameter is updated with

where and are susceptances of the transmission lines.

As it can be seen in (25)–(26), the SMIB-SVC system contains interconnection terms, so and

subsystems are coupled. Despite of this, the proposed controllers require only local variables and, of the

references to control the whole system.

In the design of the controllers, each subsystem control is based on the dynamic errors defined as

and , where and are the reference values of the system. The complete



dynamic of the system can be expressed as follows

(27)

where , are smooth measurable outputs; and , are the control inputs of the system.

The interactions between the generator and the SVC described in (25)–(26) are considered as

perturbations, and are compensated by the robust controllers.

4.2 Excitation and SVC Regulator Controllers

The control objective is to incorporate a controller for (24) from the integration of a controller

for each subsystem using local information. The whole closed-loop system is shown in Fig. 3.

Fig. 3. Controllers, differentiators and SMIB-SVC system interaction.

In the first subsystem (the interconnected synchronous generator) the relative degree is . With

and available (dashed loop), the excitation controller equation is defined as

(28)

In the second subsystem (the interconnected SVC regulator) the relative degree is . With and



both available (dashed loop), the voltage regulation controller equation is

(29)

In order to reduce the number of sensors, it is assumed that the only measurable signals are and

and are both available), then homogeneous differentiators (12) for the dynamic errors and are

given by

(30)

and

(31)

respectively. In this way the estimated values and are applied in the control law (12).

Finally, the controllers for each subsystem (solid loops) are

(32)

and

(33)

5 Simulation Study

Simulation results are used to validate the performance of the proposed oserver-based output feedback



controller. The schemes (QCHOSM control systems with known and estimated (ˆ) modes) are tested and

compared using algorithms referred to the system of Fig. 1, which consists of a 500 MVA, 22 kV system

connected to an infinite bus, with parameters taken from Table 1.

The generator operating point considered is , p.u. and p.u. In the system it is

assumed that and are available and that is known. The computed-tested values , , and

are taken from [22], only , , and are tuned during the tests.

Table 1. Parameters of the Study System

Parameter Value Parameter Value

377 rad/s D 5.0 p.u.

H 4.0 s 1.863 p.u.

0.675 p.u.

6.9 s

1.0 0.1213 p.u.

10 s 0.75 p.u.

1.0 p.u. 500

700 35

1.1 1.5

2.0 -450

-20

In order to show the effectiveness of the nonlinear control scheme, numeric results are presented in a)

two cases of 3-phase fault, b) two cases of changes in the operating point, and c) a line-outage

disturbance.

Simulation results are presented from Fig. 4 to Fig. 14, where the performance of the decentralized

control scheme can be seen.



The first fault is at the sending end of the selected line , and the second one is at the middle point

of the transmission line. The faults considered in the simulations are symmetrical 3-phase short

circuits which occur in one of the transmission lines connected to the generator bus (see Fig. 1) with the

sequence:

i. The system is in a pre-fault steady-state.

ii. A fault occurs at s.

iii. The fault is removed by opening the breaker of the faulted line at s.

iv. The transmission line is restored with the fault cleared at s.

v. The system is in a post-fault state.

Fig. 4 to Fig. 6 show the system responses when the fault occurs at position. Fig. 4 shows and

, both are damped out and restored by their control structures. The reaction of the controller to

variations is also shown.

Fig. 4. Response of (dashed) and (solid). Below the reaction of the controller input to

variations.

Fig. 5 shows the fast approximations between , , and their estimated modes.



Fig. 5. , , and their estimations (dotted).

In an effort to gain insight into the underlying dynamics of the controllers several frequency

representations were computed. Spectral magnitudes were computed using 9.5 s (0.5–10 s) and 5.0 s

(5.0–10 s) windows for the fault and the change of operating point tests respectively.

In Fig. 6 the spectral plot of , , and show that despite the highly nonlinear characteristics of the

structure there is not a significant harmonic content in these signals, having predominance the system

modes.

Fig. 6. Frequency (Magnitude) spectrum of , (solid) and (dashed) signals when the fault occurs

at the generator bus.



Fig. 7 and Fig. 8 show the system responses when the fault occurs at position. In Fig. 7, and

are compared, both are restored and damped out by their control structures in a time less than 1 s. The

reaction of the controller to variations is also shown.

Fig. 7. (dashed) and (solid), and the controller input .

As shown in Fig. 8 the controller modes have no significant effect in the system dynamics.

Fig. 8. Magnitude spectrum of , (solid) and , (dashed) when the fault occurs in the middle of

the line.



Fig. 9 and Fig. 10 show the system responses when steps in are applied (0.6914 and 0.7278 rad in 0.2

and 5.0 s respectively). In Fig. 9 the responses of , , (rotor speed) and , , show the

performance of the feedback system, all the variables converge to the equilibrium point within 2.0 s

approximately, whilst the oscillations are damped out effectively.

Fig. 9. Response of , , (solid) and , , (dashed) under steps.

In Fig. 10 it is shown that their spectral responses are stronger near the system operation modes, so the

power systems signal almost have no distortions despite the sudden changes in the operating point.



Fig. 10. Magnitude spectrum of , , (solid) and , , (dashed) when the change in the operating

point is due to step.

Fig. 11 and Fig. 12 show the system responses when steps in are applied (0.7 and 0.8 p.u. in 0.2 and

5.0 s respectively). In Fig. 11 reaches the reference almost instantaneously while oscillates

(around ) in a narrow band.

Fig. 11. and response under steps.



In Fig. 12 it can be seen that the spectral plot has uniform magnitude, this behavior corresponds with

amplitude. The system and controller modes can be seen in the spectral plots. In this interesting

case the combination controller-differentiator presents modes with significant magnitudes, this is

reflected in the oscillations.

Fig. 12. Magnitude spectrum of and when the change in the operating point is due to step.

Fig. 13 and Fig. 14 show the system response in a line-outage disturbance. In this severe disturbance the

transmission line indicated in Fig. 1 is permanently tripped-out at s In Fig. 13 the generator

signals are quickly restored to the steady value.



Fig. 13. ,

and , when a transmission line is permanently tripped out.

In Fig. 14 the SVC regulator current returns to the steady value slowly, whilst the regulator voltage is

almost constant.

Fig. 14. , and of , when the transmission line is permanently tripped out.



From Fig. 4 to Fig. 14, it is possible to see how the excitation and voltage regulation controllers

presented can damp out the transient oscillations effectively regardless of the operation conditions. So

the system state can be replaced by its estimate in the control law of the closed-loop system, with a low

impact on the power quality.

6 Conclusions

In the simulation results, the transient stability and the voltage regulation differentiator-based are

obtained in finite time despite the presence of severe faults, parameter variations and the existence of

uncertainties.

The method is suitable to design nonlinear decentralized controllers due to the fact that they only require

local measurements and reject interaction between subsystems allowing the simplification and

separation of the excitation and the SVC controllers. It is also appropriate where subsystems are highly

nonlinear as it is not required any compensation or transformation.

The nonlinear control structure presented can be applied without degradation of power quality in the

power system.

It is concluded that the sliding-mode structure proposed is robust regardless of the replacement of the

state by its estimate in the control law, the parameter variations and the presence of uncertainties in the

SMIB-SVC system.

Acknowledgment

The authors would like to thank to the FOMIX-CONACyT Government of the State of Zacatecas Fund,

Zacatecas, México, under grants ZAC-2007-CO1-82136 and ZAC-2010-CO4-149908, for the invaluable

support that made this research possible.



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