A robust H∞ power system stabilizer design using reduced-order models

A robust HN power system stabilizer design using reduced-order models

Hardiansyah, Seizo Furuya *, Juichi Irisawa

Department of Electrical Engineering, Nagaoka University of Technology, 1603-1 Kamitomiokamachi, Nagaoka, Niigata 940-2188, Japan

Received 18 December 2003; received in revised form 23 June 2005; accepted 26 September 2005

Abstract

This paper deals with a robust HN power system stabilizer (HPSS) design using reduced-order models to improve the damping oscillation in

power systems. The stabilizer is dynamic, low order and robust. In order to obtain a reduced-order controller, the method of balanced truncation is

used. Sufficient conditions in the form of two algebraic Riccati equations (AREs) and an upper bound explicitly characterize an HN controller of

lower dimensions. Furthermore, the bilinear transformation has been used to the design to prevent the pole-zero cancellation of the poorly damped

poles and to improve the control system performance. The proposed technique is illustrated with applications to the design of stabilizer for a multi-

machine power system. Simulation results under various operation conditions are given which show that the proposed HPSS damps the low-

frequency oscillation in an efficient manner.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: HN controller; Power system stabilizer design; Reduced-order models; Multi-machine power systems

1. Introduction

Power systems are complex non-linear systems and often

exhibit low-frequency electromechanical oscillations due to

insufficient damping caused by adverse operating conditions.

Power system stabilizer (PSS) units have long been regarded as

an effective way to enhance the damping of electromechanical

oscillations in power system [1]. As a supplementary control to

provide extra damping for synchronous generators, power

system stabilizer (PSS) have been widely used in the electric

power industry. Studies have shown that a well-tuned PSS can

improve power system dynamic stability effectively. Over the

past two decades, various control methods have been proposed

for PSS design to improve overall system performance. Among

these, conventional PSS of the lead-lag compensation type

[1,2] have been adopted by most utility companies because of

their simple structure, flexibility and easy of implementation.

However, the performance of these stabilizers can be

considerably degraded with the changes in the operating

condition during normal operation. Most methods developed in

recent years are based on well-developed modern control

theory. These include: pole assignment [3–6], optimal control

0142-0615/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijepes.2005.09.002

* Corresponding author. Tel.: C81 258 47 9539; fax: C81 258 47 9500.

E-mail address: [email protected] (S. Furuya).

[7], self-tuning and adaptive control [6], variable structure

control [8,9], rule-based and neural network based control

[10–12]. The reduced order techniques [13,14] have been

applied to the PSS design problem. The first one based on the

LQR technique and the second one based on the iterative

perturbation scheme. One of the principal disadvantages of

these methods themselves is the lack of robustness.

In recent years, the standard HN control problem has

received increasing attention: for a given g O0, find all

controllers such as that the HN norm of the closed-loop transfer

function is less than g [21]. Practical power systems with PSS

must be robust over a wide range of operating conditions. The

developed HN and related design methods lead to a fixed-

structure and fixed-parameter, yet robust controller. Some

research on applying HN methods to PSS design is also

presented in some publications [15–20] where the importance

and the difficulties in the selection of weighting functions are

reported. And also, the standard optimal HN control method is

known to obtain controllers of the same order as that of the

open loop system [21]. Sufficient conditions in the form of two

algebraic Riccati equations (AREs) and an upper bound

explicitly characterize an HN output feedback controller of

lower dimensions.

In this paper, we present a robust reduced-order controller

based power system stabilizer design to improve the damping

oscillation in power systems. Then, the bilinear transformation

is applied to the plant model to prevent the pole-zero

Electrical Power and Energy Systems 28 (2006) 21–28

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Hardiansyah et al. / Electrical Power and Energy Systems 28 (2006) 21–2822

cancellation and to deal with the ill conditioning problems that

arise in the design process. The design method is implemented

to a multimachine power system. The performance of the

proposed controller is examined and compared with both the

CPSS and the standard HN PSS (standard HPSS). Thus, the

design of the proposed controller is simple and it is easy to

implement.

The rest of the paper is organized as follows. A detailed

description of the proposed design procedure is given in

Section 2. The studied power system is given in Section 3.

Simulation results are presented in Section 4 to demonstrate the

effectiveness of the proposed method. Conclusions are drawn

in Section 5.

2. The proposed controller design

2.1. Problem statements

Let us consider a linear system described by the state-space

equations of the following form,

_x Z Ax CBu (1)

y Z Cx CDu

where x2Rn the state, u2Rm the control signal, y2Rq the

output signal; and A, B, C and D are real matrices of

appropriate dimensions. It is further assumed that given system

is stabilizable and detectable, i.e. minimal [21].

We are interested in designing a dynamic output feedback

controller of the form

_xK Z AKxK CBKy (2)

v Z CKxK

where xK2Rr the state, n2Rm the output signal for the

controller; AK, BK and CK are real matrices of the appropriate

dimensions.

The model-order reduction problem consists of approxi-

mating a high-order system G by lower-order system Gr

according to some given criterion. The structure of the overall

system is given in Fig. 1, where Gr is reduced-order model, Gg0

the modeling error and GK the desired controller. The reduced-

order model is unique from the input–output behavior point of

view. As it is assumed that the stable part is an upper bounded,

i.e. jjGg0jjN%g0, therefore, HN-control tools can be used to

rΣ

u

v

w

y+

+

+

–

Gr

GK

Gg 0

Σ

Fig. 1. Configuration of the closed-loop system.

find a stabilizing controller. Here the standard definition for

HN-norm is given by

jjGg0jjN Z supusmax½GðjuÞKGrðjuÞ� (3)

where smax($) is the maximum singular value.

2.2. Reduced-order model formulation

One way of obtaining a low order controller is to work with

a low order system (plant). The standard optimal HN control

method is known to obtain controllers of the same order as that

of the open loop system. Thus, if the full model of the system is

used, the optimal HN controller order will be unacceptably

high. In order to reduce the order of the controller, we make use

of the method of balanced truncation. We will present a brief

outline of this procedure. The details of the balanced truncation

algorithm can be found in [23–25].

Let (A,B,C,D) be an nth order stable system, but not

necessarily minimal, state-space realization of the transfer

function GðsÞZDCCðsIKAÞK1B. The controllability and

observability Grammians are defined as:

�P Z

ðN0

eAtBBT eAT tdt (4)

�Q Z

ðN0

eAT tCT CeAtdt

The Hankel singular values S are defined as SZffiffiffiffiffiffiffiffiffiffiffiffiffilðPQÞ

pwhere lðPQÞdenotes the eigenvalue of PQ. Let �T be a

transformation for balanced realization with xðtÞZ �Txb. Then

the state space of balanced system can be expressed as:

_xb Z Abxb CBbu (5)

yb Z Cbxb CDbu

We partition the state vector xb into two parts [xb1 xb2]T where

xb2 is the vector of the states that we wish to eliminate. Thus,

Eq. (5) becomes:

_xb1

_xb2

" #Z

Ab11 Ab12

Ab21 Ab22

" #xb1

xb2

� �C

Bb1

Bb2

" #u (6)

yb Z Cb1 Cb2

� xb1

xb2

� �CDbu

The controllability and observability Grammians of the

balanced truncation system are diagonal and satisfy the

following equation:

Ab11 Ab12

Ab21 Ab22

" #SCS

Ab11 Ab12

Ab21 Ab22

" #T

CBb1BT

b1 Bb1BTb2

Bb2BTb1 Bb2BT

b2

" #Z0

(7)

Hardiansyah et al. / Electrical Power and Energy Systems 28 (2006) 21–28 23

Ab11 Ab12

Ab21 Ab22

" #T

SCSAb11 Ab12

Ab21 Ab22

" #C

CTb1Cb1 CT

b1Cb2

CTb2Cb1 CT

b2Cb2

" #Z0

(8)

where SZdiag S11 S22

� is the matrix of Hankel singular

values arranged in decreasing order on the diagonal. The model

reduction scheme of balanced truncation consists of removing

states xb2 from xb. The reduced model is given by:

_xb1 ZAb11xb1 CBb1u (9)

yb ZCb1xb1 CDbu

Glover [24] proved that the reduced model Gr enjoys the

attractive frequency-domain LN-norm error bound

�sðGðjuÞKGrðjuÞÞ%2Xn

iZkC1

si cu (10)

At this end, use the corresponding above formulas to obtain a

low-frequency reduced-order realization (Ar,Br,Cr,Dr). The

reduced-order model can be rewritten of the form

GrðsÞZAr Br

Cr Dr

" #(11)

where Ar2Rr!r, Br2Rr!m, Cr2Rq!r, and Dr2Rq!m.

2.3. The bilinear transformation

If the plant has ju-axis poles or zeros, the controller cannot

be computed directly because of ill conditioning problems.

Several methods can be used to overcome this situation [20,26]

. The bilinear transformation method has been applied to the

plant in our design. Therefore, we have transformed the

original s-plane into a new �s-plane using the following

transformation:

s Zð�s Cp1Þ

ð1 C �s=p2Þ(12)

where p1 and p2 are the endpoints of the diameter of circle in

the left s-plane.

By setting the parameters of the bilinear transformation

p1!0 and p2!0 appropriately, the critical zeros and also the

poorly damped poles are moved away from the ju-axis and

settled in the right-half plane. It requires special attention for

the choice of the parameters of bilinear transformation. After

the controller is computed, the inverse bilinear transformation

is used to map the controller back to the original s-plane. It

should be noted that the resulting controller is sub-optimal for

the original system. However, the result shows that it can

improve the dynamic stability and provide good damping.

2.4. Low order controller design

In this section, we gives conditions for the existence of a

controller GK such that the closed-loop system is stabilized

when (i) Gg0does not have unstable eigenvalues, and (ii) jj

Gg0jjN is less than or equal to some a priori number g0. It

should be noted that condition on the stability of Gg0is needed

since otherwise there are always arbitrary small perturbations

that destabilize the closed-loop system. The following theorem

gives the condition for all stabilizable controllers [21,22].

Theorem 1. Let the controller GK of the form Eq. (2) be given.

If GK internally stabilizes Gr then ICGrGK is invertible as a

proper rational matrix and the closed-loop system is well-posed

and internally stable when jjGKðICGrGKÞK1jjN%gK1

0 .

Proof. It is easy to verify that a controller GK applied to Gr

yields a closed-loop system that is well-posed and internally

stable. If the same controller is applied to G(s) it yields a

closed-loop system which is well-posed and internally stable

when jjGKðICGrGKÞK1jjN%gK1

0 . By the small-gain theorem

[21], the above is true for all stables Gg0with HN-norm less or

equal to some a priori number g0.

Let us define XN and YN as the unique positive semi-

definite matrices satisfying the following two algebraic Riccati

equations (AREs):

XNAr CATr XN Z XNBrB

Tr XN (13)

YNATr CArYN Z YNCT

r CrYN

such that the matrices Ar KBrBTr XN and Ar KYNCT

r Cr are

asymptotically stable. The existence and uniqueness of XN and

YN is guaranteed by standard linear quadratic control. The

solution to the HN-control problem is given by the following

theorem.

Theorem 2. There exists an admissible controller GK(s) such

that jjGKðICGrGKÞK1jjN%gK1

0 if and only if (i) XNR0, (ii)

YNR0, and (iii) rðXNYNÞ!gK20 where r($) is the spectral

radius. Moreover, when the above conditions hold, one such

controller of the form Eq. (2) is described by

CK Z BTr XN (14)

BK Z IKg2YNXN

� �K1YNCT

r

AK Z Ar KBrBTr XNKBK Cr KDrB

Tr XN

� �Proof. Let us define a new set of equations in the HN

framework from Fig. 1:

_xr Z Arxr CBru (15)

z Z u

y Z Crxr Cw

from which we would like to find a controller such that

jjTzwjjN%gK10 . Then by following [21] the set of stabilizing

controllers as given by Eq. (14) is obtained for different value

of g.

G1

G2 G3

Load A Load B

Load C

1

2 3

4

5 6

78

9

Fig. 2. Three-machine nine-bus power system.


Our goal is to design a low order controller such that the

conditions above are not violated. Thus, for every g smaller

than the bound [r(XNYN)]K1/2and greater than jjGg0jjN we can

find a suitable controller described by Eq. (14).

The procedure for solving the low order controller using the

proposed approach implicitly described above suggests the

following algorithm:

(1) Reduced-order model: G/Gr (r!n)

(2) Bilinear transformation: Ar)ArKpI; (p!0)

(3) Solve ARE Eq. (13) for XN and YN

(4) Set g%[r(XNYN)]K1/2

(5) Solve Eq. (14) for (AK, BK, CK)

(6) Inverse bilinear transformation: AK)AKCpI

3. The studied power system

A three-machine nine-bus power system [29] shown in

Fig. 2 is used to illustrate the effectiveness of the proposed

HPSS. Each synchronous machine is described by a nonlinear

Table 1

Loading condition (in p.u)

Heavy Nominal

P Q P

Generator

G1 1.330 0.630 0.716

G2 1.900 0.361 1.630

G3 1.200 0.120 0.850

Load

A 1.750 0.700 1.250

B 1.200 0.400 0.900

C 1.400 0.500 1.000

fourth-order model as given in the Appendix A. Details of the

system data are given in Appendix B.

To identify the optimum locations of PSSs, the participation

factor method [27] was used. The result of this method

indicates that generators G2 and G3 are the optimum locations

for installing PSSs to damp out the electromechanical modes of

oscillations. Therefore, the generators G2 and G3 are equipped

with two of the proposed HPSS.

4. Simulation results

To design the proposed HPSS, three operation conditions,

i.e. a heavy loading condition, a nominal loading condition,

and a light loading condition, are considered as shown in

Table 1. The open loop eigenvalues (dominant eigenvalues) of

the study system for three operating conditions are given in

Table 2. As each pair of conjugate eigenvalues corresponds to

an oscillation mode, there are two modes in this study system.

Modes #1 and #2 are the rotor oscillation modes (the

electromechanical modes). It can be seen that the damping of

the rotor oscillation modes for all the operating conditions are

poor. In the power systems, a damping ratio (z) of at least 10%

and the real part of eigenvalue (s) not greater than K0.5 for the

troublesome low frequency electromechanical mode, guaran-

tees that the low frequency oscillations, when excited, will die

down in a reasonably short time [28].

In order to improve the damping of electromechanical

modes, a decentralized controller was designed for generators

G2 and G3 at the nominal loading conditions based on the

proposed design technique. The locally measured output signal

Du is feedback at the AVR reference input of each generator

after multiplication by suitable feedback gains. The design was

carried out using the method previously presented. The

numerical procedure described in Section 2 has been

implemented in MATLAB. After careful parameter tuning,

the setup values of g are displayed in Table 3.

The second order controller was found using the proposed

design technique. These controllers are reported below in the

transfer function form.

For generator 2:

GK2ðsÞ ZK114:8s C1031

s2 C7:398s C72:72(16)

Light

Q P Q

0.270 0.362 0.162

0.066 0.800 -0.109

-0.109 0.450 -0.204

0.500 0.650 0.550

0.300 0.450 0.350

0.350 0.500 0.250

Table 2

Open loop eigenvalues of the study system

Modes Heavy Nominal Light

1 K0.158Gj 7.60 K0.159Gj 7.63 K0.377Gj 7.44

2 K0.924Gj 13.2 K0.795Gj 13.3 K0.983Gj 13.40

Table 3

Setup values of g

Machine# jjGg0jjN g [r(XNYN)]K1/2

2 4.8355!10K3 1.0!10K2 2.2988!10K2

3 4.2872!10K3 5.2!10K3 1.6933!10K2


For generator 3:

GK3ðsÞ ZK62:78s C2712

s2 C7:448s C140:4(17)

It is well known that the effectiveness of the PSS in damping

oscillation depends upon its location, and the controller design

results depend upon the unstable or lightly damped electro-

mechanical modes of oscillations (the open loop system). It can

be noted that the controller are designed for each machine

separately. Its make the proposed design very simple and

guarantee the stability of all the systems.

In order to facilitate comparison with CPSS, the design and

tuning of CPSS for this multi-machine power system the

method in [30] were used. In this paper, a CPSS with transfer

function

GðsÞ Z Kc

sTw

1 CsTw

ð1 CsT1Þ2

ð1 CsT2Þ2

(18)

was used and the parameters of stabilizer have been tuned to

provide an adequate amount of damping for mode of

oscillation. The CPSS parameters for G2 and G3 are given in

Table 4. The comparison is also made between proposed HPSS

and the standard HPSS obtained from the mixed-sensitivity

approach [26], which has been designed at nominal loading

condition. Its transfer function is given as follows:

Table 4

CPSS parameters

Machine# Kc Tw T1 T2

2 2.41724 8.0 0.27085 0.04834

3 1.73439 8.0 0.19098 0.03403

Table 5

Closed-loop eigenvalues of the study system

Modes Heavy Nominal Light

1 K1.380Gj 6.85 K1.940Gj 8.60 K2.130Gj 7.87

2 K2.130Gj 14.60 K1.940Gj 14.4 K1.690Gj 14.80

Gh2 Z0:18s3 C29:1s2 C22:92s C1749

s3 C135:7s2 C3985s C7166(19)

Gh3 Z0:045s3 K111:8s2K70:04sK17290

s3 C21:75s2 C166:8s C189:7(20)

With the proposed HPSS, the closed-loop eigenvalues

are given in Table 5. It is quite clear that the system

Fig. 3. The dynamic responses under nominal loading condition.


eigenvalues associated with the electromechanical modes have

been successfully shifted to the left of sZK0.5 line and

the damping ratio greater than 10%. This demonstrates that the

system damping with the proposed HPSS is greatly enhanced.

To demonstrate the capability of the proposed HPSS to

enhance system damping over a wide range of operating

Fig. 4. The dynamic responses under heavy loading condition.

conditions, three different loading conditions were considered.

A 10% step change in the reference voltage (Vref) of G3 was

applied.

Fig. 3(a)–(c) show the speed deviations of G2, G3, and

G2–G3, respectively, under nominal loading condition. It is

obvious that the system performance with the proposed

Fig. 5. The dynamic responses under light loading condition.

Table B-1

Generator and exciter data

Parameter Generator

G1 G2 G3

M 47.2800 12.8000 6.0200

xd 0.1460 0.8958 1.3125

xq 0.0969 0.8645 1.2578

x0d 0.0608 0.1198 0.1813

D 0.0000 0.0000 0.0000

T 0d0 8.9600 6.0000 5.8900

Ka 200.00 200.00 200.00

Ta 0.01 0.01 0.01


HPSS is better than both the CPSS and the standard HN

controller (standard HPSS). For cases under heavy loading

condition, the simulation results are shown in Fig. 4(a)–(c),

respectively. The results here show the superiority of the

proposed HPSS to the CPSS and standard HPSS. It can be

concluded that the proposed HPSS achieves robust perform-

ance and damp the oscillations very well over a wide range

of operating conditions. The simulation results are shown in

Fig. 5(a)–(c), respectively, the speed deviations of generators

under light loading conditions. It is clear that the proposed

HPSS provide good damping characteristics to low-frequency

oscillations and greatly enhance the dynamic stability of

power system.

Table B-2

Line data (in p.u)

Line R X B/2

1–4 0.0000 0.0576 0.0000

2–7 0.0000 0.0625 0.0000

3–9 0.0000 0.0586 0.0000

4–5 0.0100 0.0850 0.0880

4–6 0.0170 0.0920 0.0790

5–7 0.0320 0.1610 0.1530

6–9 0.0390 0.1700 0.1790

7–8 0.0085 0.0720 0.0745

8–9 0.0119 0.1008 0.1045

Table B-3

Bus data (nominal loading condition)

Bus Type P Q V

1 Slack – – 1.040:08

5. Conclusions

This paper has developed a reduced-order HN power

system stabilizer (HPSS) design to improve the damping

oscillation in power system. By solving two algebraic

Riccati equations (AREs), then a lower-order dynamic

output feedback controller is constructed. The bilinear

transformation has been used to deal with the ju-axis

poles and zeros and to prevent the pole-zero cancellation.

The resulting reduced-order controller is guaranteed to

stabilize the closed-loop system. Performance of the

proposed controller has been performed in multi-machine

power system. The results show that the proposed HPSS is

very effective and gives promising result for robustness and

stability in damping of low frequency oscillation at different

loading conditions.
2 PV 1.63 – 1.025
3 PV 0.85 – 1.025

4 PQ 0.00 0.00 –

5 PQ 1.25 0.50 –

6 PQ 0.90 0.30 –

7 PQ 0.00 0.00 –

8 PQ 1.00 0.35 –

9 PQ 0.00 0.00 –

Data is on system base: 100.00 MVA.

Appendix A

Machine models

_ui ZTmi KTeiKDiðuiK1Þ

Mi

(A.1)

_di Z ubðuiK1Þ (A.2)

_E’qi Z

EfdiKðxdiKx’diÞidi KE’

qi

T’d0i

(A.3)

_Efdi ZKaiðVrefiKVi CUiÞKEfdi

Tai

(A.4)

Tei Z E’qiiqiKðxqiKx’

diÞidiiqi (A.5)

Appendix B

Power system parameters (Table B-1–B-3).

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A robust H∞ power system stabilizer design using reduced-order models

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