adaptive voltage regulator design for static var systems

Control Engineering Practice 9 (2001) 759}767

Adaptive voltage regulator design for static VAR systems

Giuseppe Fusco*, Arturo Losi, Mario RussoUniversita% degli Studi di Cassino, via G. Di Biasio, 43 } 03043 Cassino, Italy

Received 20 September 2000; accepted 8 December 2000

Abstract

The performance of a static VAR system (SVS) strongly depends on the operating conditions of the power system at which the SVSis connected. Since these operating conditions often vary unpredictably, limitations to the performance of a SVS may be derived. Toovercome such drawbacks, a new SVS voltage regulation scheme based on the adaptive control theory is proposed. The design of theadaptive voltage regulator is described in detail. The results obtained by numerical simulations show the e!ectiveness of the proposedcontrol scheme. � 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Adaptive control; Discrete time; Output regulation; Power system control; Power system voltages

1. Introduction

The aim of keeping an adequate nodal voltage pro"lein power systems is generally pursued by complex con-trol systems. Various control architectures have beenproposed in literature and implemented in real systems(Cigre TF 39/02, 1992; IEE Colloquium, 1993); they di!erin terms of the degree of function automation and ofcentralization/decentralization. Nevertheless, all thesearchitectures rely on hierarchical control structures inwhich the primary level is performed by devices whichdirectly and locally regulate the voltage amplitude of thebusbar at which they are connected.

Among such devices, static VAR systems (SVSs) arewidely adopted. Their primary task is fast control ofnodal voltage in transmission networks (IEEE SpecialStability Working Group, 1994); in addition, they canhelp to improve power system transient stability and todamp power system inter-area oscillations (O'Brien &Ledwich, 1987; Larsen et al., 1996). SVS technology isnowadays a mature one, mainly adopting thyristor-basedswitching and control of shunt capacitors and reactors.

In practice, the performance of a SVS depends on theoperating conditions of the power system at which theSVS is connected. In literature (Larsen et al., 1996;

*Corresponding author. Fax: #39-0776-299-707.E-mail addresses: [email protected] (G. Fusco), [email protected] (A. Losi),

[email protected] (M. Russo).

Sybille, Giroux, Dellwo, Mazur, & Sweezy, 1996) and inpractical applications, the SVS control scheme isdesigned assuming that the power system model para-meters are known and time-invariant. But this assump-tion may be inadequate, because the power system isoften subject to unexpected changes of its operatingconditions and to structural modi"cations. Then, anadaptive approach is needed for the design of the SVScontrol.

2. SVS con5guration and standard voltage regulation

The SVSs use conventional thyristors to achieve fastcontrol of shunt connected capacitors and reactors.Many di!erent con"gurations have been proposed andadopted in practical applications. In the following a "xedcapacitor}thyristor controlled reactor (FC-TCR) con"g-uration is referred to (Fig. 1), but all the considerationson the control scheme can be easily extended to otherSVS con"gurations.

The SVS control must take into account di!erentperturbations that may occur to the power system oper-ating conditions.

During normal operation of the electrical power sys-tem, perturbations are caused by changes of the loads, ofthe powers injected by generators and of the topology ofthe network. When such perturbations take place, theprimary task of the SVS is to support the voltage of thebus at which it is connected, in this way, supporting the

0967-0661/01/$ - see front matter � 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 9 6 7 - 0 6 6 1 ( 0 1 ) 0 0 0 2 9 - 6

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Fig. 1. FC-TCR con"guration for a SVS.

Fig. 2. SVS voltage}current operating characteristic.

Fig. 3. Block scheme of the SVS standard voltage regulation.

Fig. 4. Power system and SVS equivalent electrical circuit.

voltage pro"le of the whole power system. To this aim,standard voltage regulators are adopted.

During abnormal operation of the power system, suchas in the case of short circuits, the standard voltageregulator is blocked and the SVS operation is switched toa nonlinear ON/OFF control.

In the following only normal operating conditions ofthe power system are considered.

The SVS steady-state operating characteristic is shownin Fig. 2: on the axes the rms injected current (positivewhen inductive) and the rms bus voltage are reported.The straight line between points A and B represents thelinear characteristic in the regulating range of the SVS:the thyristor controlled reactors are switched o! at pointA and fully switched on at point B, whilst the "xedcapacitor banks are always connected. In between pointsA and B the reactors are switched on partially andbehave as a controllable reactance. The slope of thecharacteristic determines the SVS current variation inresponse to the bus voltage variation. Outside the regula-ting range, the control is saturated and the SVS behaveslike a "xed shunt capacitor or reactor.

In addition to the voltage support task, SVSs are oftenrequired also to track a changing reference voltage signal.In fact, in many practical cases (O'Brien & Ledwich,1987; Larsen et al., 1996), the small signal stability analy-

sis of power systems has shown that adequate modula-tion of the voltage reference to the SVS improves thedamping of the system electromechanical oscillations. Inthese cases, additional stabilizing signals are added to thevoltage reference input causing its modulation at lowfrequencies (at most few hertz).

As a "nal, yet trivial, requirement, the SVS controlmust be stable when the operating conditions of thepower system change.

To meet these requirements, the standard controlscheme of Fig. 3 is usually adopted. The voltage error isgenerated on the basis of the voltage reference <

��, the

optional stabilizing signals, the actual rms value of thevoltage <

��and an additional signal proportional to

the actual rms value of the current. The voltage regulatoris usually arranged in a PI (at minimum simply I) con"g-uration, with a saturating integrator to account for theextreme values of the linear characteristic (points A andB in Fig. 2). The "ring angle control block performs thecompensation of the FC-TCR nonlinear law B

��(�). The

gain K�in the current feedback loop assumes usually

very small values and represents the slope of the SVSsteady-state characteristic of Fig. 2.

Concerning the power system, it can be easily modeledby the no-load voltage <M

�and the Thevenin equivalent

impedance ZQ��

as seen from the SVS bus at fundamentalfrequency. The equivalent electrical circuit is shown inFig. 4 and the model is described by phasor equations;then, a nonlinear relationship stands between the SVSequivalent susceptance B

��and the rms value of the bus

voltage<M . In addition, the values of the equivalent electri-cal circuit parameters<M

�,ZQ

��change when perturbations

happen in the power system.

760 G. Fusco et al. / Control Engineering Practice 9 (2001) 759}767

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Fig. 6. Block scheme of the adaptive voltage regulator.

Fig. 5. Block scheme of the SVS adaptive voltage regulation.

The design of the PI voltage regulator is typicallyperformed adopting a `worst-casea conservative ap-proach. A range of possible values for the short-circuitpower at the SVS bus is supposed and the design isperformed assuming the corresponding smallest value,that is, assuming the highest value of the power systemequivalent gain.

The standard voltage regulation has two drawbacks(Romegialli & Beeler, 1981). When the power systemequivalent gain changes and assumes values that aresmaller than the one assumed in the control design, theSVS performance worsens in terms of quickness of re-sponse and of reference tracking capability. In addition,it is always possible that unpredicted changes in thepower system con"guration, which have not been con-sidered in the design phase, may cause the voltage regula-tor to become unstable, when the value of the powersystem equivalent gain becomes higher than the oneassumed in the control design.

3. Adaptive voltage regulation scheme

To overcome the limitations of the standard control,a new adaptive voltage regulation scheme is presentedbelow.

The adaptive control scheme accounts for the nonlin-ear model of the power system and for the changes of theequivalent circuit parameter values so as to meet theclosed loop requirements. The latter ones are usuallyexpressed in terms of steady-state error between the de-sired reference input and the system output, character-istics of the transient response and robustness withrespect to parameter variations.

The proposed adaptive control scheme is outlined inFig. 5. The voltage reference <

��, the additional stabiliz-

ing signals and the actual value of the rms bus voltage<��

are fed to the adaptive voltage regulator, whichdetermines the "ring angle �. To take into account thechanges of the power system con"guration, an identi"ca-tion block estimates the values of the parameters of thepower system Thevenin equivalent circuit (Fig. 4) usingthe on-line measurements of the current derived from theSVS and of the bus voltage. The estimated values<K

�,ZQK

��are fed to the adaptive regulator.

The identi"cation procedure is based on a constrainedweighted recursive least-squares algorithm with variableforgetting factor, whose characteristics have been de-scribed in detail in Fusco, Losi, and Russo (1998, 2000).

The block scheme of the adaptive voltage regulator isshown in Fig. 6. The desired rms voltage <

�(evaluated

starting from the voltage reference, the additionalstabilizing signals and the voltage slope signal) is fed tothe voltage error generator. The latter generates a volt-age error �

�starting from the values of <

�and of the

actual rms voltage <��

. The voltage error is fed toa saturated integrator, whose linear range is adaptivelychanged on the basis of the estimated values <K

�,ZQ K

��; the

output of the integrator is the command signal u�. To

obtain the desired value of the SVS equivalent admit-tance B

��from u

�, an adaptive nonlinear block is used,

which compensates for the power system nonlinearmodel using the estimated values<K

�, Z�)

��. Finally, a block

G. Fusco et al. / Control Engineering Practice 9 (2001) 759}767 761

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Fig. 7. Simpli"ed block scheme adopted in the voltage error generatordesign.

compensating for the FC-TCR nonlinear law B��(�) is

used to generate the thyristor "ring angle �. The voltageslope signal generator uses the value of the rms currentnumerically evaluated from the values of B

��and <

��.

The rms current value is fed to a block with gain equal toK

�, which is the required slope of the steady-state charac-

teristic.In the following, designs of the voltage error generator,

adaptively saturated integrator and nonlinearity com-pensating blocks are described in detail.

3.1. Voltage error generator design

The design of the voltage error generator is carried outunder the following assumptions:

(i) the voltage slope signal amplitude is negligible withrespect to the amplitudes of <

��and of the addi-

tional stabilizing signals;(ii) the power system nonlinear model is fully compen-

sated (see Section 3.2);(iii) the integrator in Fig. 6 is not saturated.

Assumption (i) stands because the value of the K�gain

is small and the "lter D(z) ensures that the time responseof the voltage slope signal generator is higher than thetime response of the voltage error generator.

The simpli"ed block scheme adopted for the design ofthe voltage error generator is shown in Fig. 7. The trans-fer functionP(s) represents the overall transfer function ofthe system composed by the cascade of the TCR and rmsvoltmeter. Since the design of the voltage error generatoris performed in discrete time, it is necessary to introducea zero-order hold (ZOH).

To determine the transfer function P(s), the models ofthe TCR and rms voltmeter have to be introduced.Di!erent models of the TCR have been presented inliterature (IEEE Working Group, 1994; Kundur, 1994);in the remainder the following second order model isassumed:

TCR(s)"e��

(1#s¹��

), (1)

where ¹��

and ¹�are the time constant and the delay

time of the TCR, respectively; typical values for 50Hzfundamental frequency are ¹

��"5ms and ¹

�"

3.33ms. The time delay can be expressed in terms of aninteger k, multiple of the sampling time ¹

�, and a partial

time delay m¹�, as: ¹

�"k¹

�!m¹

�, where 0)m)1.

Then, (1) can be written as

TCR(s)"e��

(1#s¹��

). (2)

The rms voltmeter is represented by the following secondorder model:

TR(s)"1

(1#s¹��), (3)

in which the time constant ¹��

is equal to 3}4ms.The voltage error regulator design is performed by the

pole assignment technique.Let M(z��), de"ned as

M(z��),B(z��)

A(z��)

,

z��(��#�

�z��#�

z�#�

z� #�

�z��)

(1#��z��#�

z�#�

z� #�

�z��#�

�z��)

, (4)

denote the discrete transfer function from <��

to thevoltage error �

�(Fig. 7); it is obtained by the following

Z-transformation:

M(z��)"Z�ZOH(s)P(s)�

(1!z��)"Z�

P(s)

s �" z��Z�

e��

s(1#s¹��

)(1#s¹��)�

" z��Z�e��

s#

R�e��

(1#s¹��

)#

Re��

(1#s¹��

)

#

R e��

(1#s¹��)#

R�e��

(1#s¹��)�, (5)

where ZOH(s) is the transfer function of the zero-orderhold. In Appendix the residuals R

�and the coe$cients

��and �

�are determined.

With the pole assignment technique, the controllerpolynomials F(z��) and G(z��) (see Fig. 7) are assignedso as to shift the closed loop poles in some speci"edlocations such that the desired characteristics, expressedin terms of control performance, are satis"ed. For in-stance, if a second order response is required with a de-sired settling time ¹

and with an assigned damping

factor �, which determines the overshoot in the stepresponse, the closed loop desired pole set consists of thetwo roots of the second order polynomial:

¹(z��)"1#��z��#�

z� (6)

where

t�"!2e�� cos�

4¹�

¹�1!�, t

"e�� .


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In practice, to obtain numerically stable polynomialsF(z��) and G(z��), the closed loop pole set may includesome additional high frequency poles; the degree of¹(z��) is indicated by n

�.

Once the closed loop poles are de"ned and the coe$-cients of the polynomial ¹(z��) are "xed, the pole assign-ment techniques requires the following Diophantineequation to be solved (Astrom & Wittenmark, 1989;Wellstead & Zarrop, 1991):

A(z��)F(z��)#B(z��)G(z��)"¹(z��), (7)

in which F(z��) and G(z��) are the unknown poly-nomials.

To allow a faster solution of (7), it is useful to cancel thestable zeros of the polynomial B(z��). Since the TCRintroduces a time delay ¹

�, the time-discrete system

M(z��) is a non-minimum phase system and the poly-nomial B(z��) has some of its zeros outside the unitcircle, which cannot be cancelled by the control. Toovercome this problem, B(z��) is factorized as: B(z��)"z��B�(z��)B�(z��), where the polynomials B�(z��) andB�(z��) contain the n

�� roots of B(z��) inside the unit

circle and the n�

� roots of B(z��) outside the unit circle,respectively (it stands n

��#n

��"4, see (4)). Consequently,

the following equation substitutes for (7):

A(z��)F(z��)#z��B�(z��)B�(z��)G(z��)

"¹(z��)B�(z��). (8)

Eq. (8) can be rewritten as

A(z��)FI (z��)#z��B�(z��)G(z��)"¹(z��), (9)

being

F(z��)"FI (z��)B�(z��). (10)

The coe$cients of FI (z��) and G(z��) are obtainedfrom (9), which has a unique solution if A(z��) andB�(z��) have no common roots and the degrees of thepolynomials FI (z��), G(z��), A(z��), B(z��), and ¹(z��)satisfy the following constraints (Astrom & Wittenmark,1989; Wellstead & Zarrop, 1991):

n�I "k!1#n

�� ,

n

"n

!1, (11)

n�

)n

#n�

�#k!1,

where n�I , n

and nare the degrees of FI (z��), G(z��) and

A(z��), respectively (it is n

"5, see (4)).Equating the coe$cients of z�� in (9), for each value of

i, yields a set of linear equations:

��"b, (12)

where matrix � and vector b are known and the vector� contains the coe$cients of the polynomials FI (z��) andG(z��). The solution of (12) yields the coe$cients ofFI (z��) and G(z��).

Finally, the voltage error generator is represented bythe following equation (see Fig. 7):

F(z��)��"!G(z��)<

��#H(z��)<

�, (13)

where H(z��) is given by

H(z��)"h�"

¹(z��)

B�(z��) ��

. (14)

Eq. (14) ensures a unitary steady-state gain for the closedloop transfer function:

B�(z��)H(z��)/¹(z��).

3.2. Adaptively saturated integrator and compensations ofmodel nonlinearities

In the adaptive voltage regulator scheme shown inFig. 6, the estimated values <K

�,ZQK

��of the equivalent

circuit parameters are used:

(i) to determine the saturation levels of the integrator,(ii) to perform the compensation of nonlinearities.

To analyze how the adaptive actions are embedded, letus refer to the equivalent electrical circuit shown in Fig. 4;it can be easily solved yielding the following equation:

<"�<M

�1#jZQ

��B��. (15)

Since the actual SVS presents losses in both capacitorsand reactances, in practice it is better to consider thefollowing more general equation:

<"�<M

�1#ZQ

��>Q

��, (16)

where the total SVS equivalent admittance >Q��

substi-tutes for the susceptance jB

��.

The value of >Q��

is composed of two terms: a "xedterm >Q

��related to the FC branches and a variable term

>Q��

related to the TCR branches (see Fig. 1), whosevalue depends on the thyristor "ring angle �; then, it canbe written as

>Q��

">Q��

#>Q��

with>Q��

"x(�)>Q�, (17)

where >Q�

is the total value of the admittance of thereactor branch when no partialization is performed andthe partialization term x(�) assumes values in the range[0,1].

The "rst adaptive action (i), that refers to the satura-tion limits of the integrator, is easily obtained: the satura-tion limits of the command signal u

�(see Fig. 6) are

obtained by evaluating (16) assuming, respectively,>Q

��">Q

��for the upper limit and >Q

��">Q

��#>Q

�for the lower limit.

The second adaptive action (ii) refers to the powersystem nonlinearity compensating block in Fig. 6. In thisblock, the required value of >Q

��is evaluated for a given


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Fig. 8. Simulated power system.

Table 1Step response characteristics of the SVS adaptive voltage regulationscheme

Overshoot (%) Settling time (s) Voltage slope (%)

Design 4.3 0.140 0.50Case 1 4.5 0.140 0.48Case 2 4.4 0.150 0.49

value of the command signal u�, that is, the following

problem is solved (see (16) and (17)):

"ndx3[0,1] such that: u�"

<K�

1#ZQK��(>Q

��#x>Q

�). (18)

The existence of the solution of (18) is assured by theadaptive saturation action on the integrator that deter-mines u

�. The solution can be trivially obtained in closed

analytical form.Once the value of x is determined, the output B

��of the block (see Fig. 6), that is the desired value ofthe SVS equivalent admittance, can be directly evaluatedby

B��

"Im(>Q��)#xIm(>Q

�), (19)

where operator Im(>Q ) represents the imaginary part of>Q .An additional block is needed to compensate for the

FC-TCR nonlinearity (see Fig. 6), but no adaptivity isneeded in its action. The task is to determine the "ringangle � for a given value of B

��in (19). To this aim, the

partialization law x(�) must be known. If, for sake ofsimplicity, the e!ect of the losses on the time length of theswitching periods is neglected, the following well-knownequation (Kundur, 1994) can be assumed for the partial-ization of the TCR branches:

x"2!

2��

#

sin(2�)�

, (20)

where the "ring angle � is measured starting fromthe zero-crossing of the phase-to-phase voltage and�/2)�)�.

In conclusion, the FC-TCR nonlinearity compensatingblock "rstly evaluates x from (19) for a given value ofB��

. Then, to determine the value of � starting from theobtained value of x, a numerical procedure is adopted,since (20) cannot be solved analytically. This procedureis based on a look-up table and linear interpolationbetween subsequent points of the table.

Finally, the output of the FC-TCR nonlinearity com-pensating block is the "ring angle � that is sent to theZOH (see Fig. 6).

4. Case study

The performance of the adaptive voltage regulationscheme has been tested by means of time-domain simula-tions of the power system shown in Fig. 8. The simulationis performed in (MATLAB� Reference guide, 1999). Thethree-phase 132kV}50Hz system is assumed to be bal-anced in all its components. The load L

and L

�are

equal to 100 and 67MW, respectively, both with a lag-ging power factor equal to 0.9; loads are represented bymeans of shunt resistors and reactances. A 100 MVARSVS is connected to the bus b

�; it is represented with the

con"guration shown in Fig. 1. Details about the electricalline representation are reported in (Fusco et al., 1998).

The voltage and current measurements, necessary forthe identi"cation procedure (see Fig. 5), are a!ected bynoises that are assumed to be white with standard devi-ations equal, respectively, to 1.4 kV and 14A. Detailsabout the performance of the Kalman "lter and theidenti"cation algorithms, adopted for the identi"cationprocedure, are reported in Fusco et al. (2000).

Concerning the closed loop characteristics of the adap-tive voltage regulation scheme, the desired values of thedamping factor and of the settling time ($2%) arechosen equal to: �"0.707 (s%"4.32) and ¹

"0.140 s,

respectively; moreover the desired value of the voltageslope is set equal to 0.5%. These values are reported inthe "rst row of the Table 1. The sampling time ¹

�has

been set equal to 1ms.Four di!erent perturbations have been applied to the

system. In all simulated cases the power system is as-sumed in steady-state condition before the perturbationoccurs at the instant t"0.5 s.

In the "rst simulation (case 1) a positive step variation(�<

��"#10.0%) of the voltage reference occurs. In the

second simulation (case 2) the voltage reference stepvariation of <

��is assumed negative (�<

��"!5.0%);

in addition, the power system operating conditions aredi!erent from the ones of the "rst case. Figs. 9 and 10show the time evolution of the controlled output <

��at

bus b�in cases 1 and 2, respectively. The transient re-

sponse is remarkably very close to desired one: it isevidenced by comparing in Table 1 the values of theovershoot and of the settling time obtained in cases 1 and


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Fig. 9. Time evolution of the controlled nodal voltage (case 1).




2 to the design values. Also the steady-state performanceis very close to the desired one: it is apparent from theactual values of the voltage slope obtained in cases 1 and2, as shown by Table 1.

The third simulation (case 3) considers a topologicalpower system variation caused by a 50.0% step increaseof the active and reactive powers absorbed by the loadL at the bus b

; the reference signal remains unchanged

(<��

"0.76 pu). Fig. 11 shows the time evolution of thecontrolled output <

��at bus b

�in case 3. Also in the

case of system parameter variation, the proposed adap-tive regulator still guarantees a settling time close to thedesired one (equal to 0.142 s). For sake of comparison,the time response of a standard nonadaptive regulator isalso shown; the parameters of such a regulator are deter-mined so as to assure in the worst case the same values ofthe overshoot and of the settling time as the ones of theadaptive regulator; the worst case is assumed to be theno-load operating condition of the power system. Fromthe analysis of Fig. 11, it is apparent that the standard

regulator transient response proves to be much slowersince the operating conditions of the power system are farfrom the worst case assumed in the design.

Finally, in the last simulation (case 4) a severe powersystem contingency is considered, which is determined bythe shedding of the load L

; the reference signal remains

unchanged (<��

"0.76 pu). Fig. 12 shows the time evolu-tion of the controlled output <

��at bus b

�in case 4. In

real applications such a severe contingency would in-volve the action of other voltage regulating devices in thepower system. In case 4 the voltage regulating action byother devices has been intentionally neglected so as toevidence the goodness of the proposed adaptive voltageregulator which is able to assure closed loop stability andconvergence towards the desired reference values, also inpresence of wide perturbations of the power system oper-ating conditions (Wellstead & Sano!, 1981; Osorio, Cor-dero &Mayne, 1981). It should be noted that even in case4 the proposed adaptive regulator still guarantees a sett-ling time close to the desired one (equal to 0.146 s).


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5. Conclusions

A new voltage regulation scheme for a static VARsystem (SVS) has been designed according to the adap-tive control theory. The voltage regulator counteractsthe e!ects of parameter variations due to changes in thepower system operating conditions, while assuringthe required closed loop performance. The adaptivevoltage regulation scheme has been tested by numericalsimulations, considering both SVS voltage referencevariations and changes of the power system operatingconditions.

Appendix

The residuals R�(i"1,2, 4) in (5) have the following

expressions:

R�"

1

¹��

1

¹��

!

1

¹��#

2

¹��

¹��

�1

¹��

!

1

¹��

,

R"!

1

¹��

¹��

�1

¹��

!

1

¹��

,

R "

!

1

¹��

1

¹��

!

1

¹��#

2

¹��

¹��

�1

¹��

!

1

¹��

,

R�"!

1

¹��

¹��

�1

¹��

!

1

¹��

.

Taking the z-transform of each single term in (5), thefunction M(z��) is obtained:

M(z��)"z�

1!z��#

M�z�

(1!e�� z��)

#

Mz�

(1!e�� z��)#

M z�

(1!e�� z��)

#

M�z�

(1!e�� z��),

where

M�"

R�

¹��

, M"R

¹�

¹��

e�� , M "

R

¹��

,

M�"R

�

¹�

¹��

e�� .

With trivial manipulations, the expressions of the coe$-cients �

�(i"1,2,5) and �

( j"0,2,4) of the poly-

nomials A(z��) and B(z��) which appear in (4), can beeasily derived; they are given by

��"!(1#2e�� #2e�� ),

�"e�� #e�� #2(e�� #e�� )

#4e�� ,

� "e�� (1!e�� )

!e�� (1!2e�� )

#4e�� ,

��"e�� e��

#2e�� e��

#2e�� ,

��"!e�� ;

��"1#M

�#M

,

��"!2(e�� #e�� )!M

�(1#e��

#2 e�� )#M!M

(1#2 e��

#e�� )#M�,

�"e�� #e�� #4 e��

#M�(e�� #2 e�� #2 e��

#e�� )#M ( 2 e�� #e��

#e�� #2 e�� )

#M(1#2 e�� )#M

�(1!2 e�� ),

� "!2 e�� (e�� #e�� )

!2M�e�� ( 2 e�� #e��

#e�� )#Me��

�(2#e�� )#M�e�� (2#e�� )

!M e�� (e�� #2 e��

#e�� ),

��"e�� #M

�e#��

!Me�� #M

e��

!M�e�� .

References

Astrom, K., & Wittenmark, B. (1989). Adaptive control. New York,USA: Addison-Wesley Publishing Company.

Cigre TF 39/02 (1992). Voltage and reactive power control. CigreMeeting, Paper 39}203, Paris, F.

Fusco, G., Losi, A., & Russo, M. (1998). Parameter tracking techniquesapplied to harmonic equivalent circuit identi"cation in power sys-tems. Proceedings of the IEEE international conference on controlapplications (pp. 1388}1393), Trieste, I.


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15161718192021222324252627

Fusco, G., Losi, A., & Russo, M. (2000). Constrained least squaremethods for parameter tracking of power system steady-state equiv-alent circuits. IEEE Transactions on Power Delivery, 15(3), 1073}1080.

IEE Colloquium (1993). International practices in reactive power control.IEE Digest No. 079, London, UK.

IEEE Special Stability Controls Working Group (1994). Static varcompensator models for power #ow and dynamic performancesimulation. IEEE Transactions on Power Systems, 9(1), 229}240.

Kundur, P. (1994). Power system stability and control. New York, USA:McGraw-Hill, Inc.

Larsen, E. V., Clark, K., Hill, A. T., Piwko, R. J., Beshir, M. J., Bhuiyan,M., Hormozi, F. J., & Braun, K. (1996). Control design for SVS's onthe Mead}Adelanto andMead}Phoenix transmission project. IEEETransactions on Power Delivery, 11(3), 1498}1506.

MATLAB� Reference Guide (1999). Version 5.3. The MathWorks Inc.,Natick (MA), USA.

O'Brien, M., Ledwich, G. (1987). Static reactive-power compensatorcontrols for improved system stability. IEE Proceedings, Part C,134(1), 38}42.

Osorio Cordero, A., Mayne, D.Q. (1981). Deterministic convergence ofa self-tuning regulator with variable forgetting factor. IEE Proceed-ings, Part D, 128 (1), 19}23.

Romegialli, G., Beeler, H. (1981) Problems and concepts ofstatic compensator control. IEE Proceedings, Part C, 128(6),382}388.

Sybille, G., Giroux, P., Dellwo, S., Mazur, R., & Sweezy, G. (1996).Simulator and "eld testing for Forbes SVS. IEEE Transactions onPower Delivery, 11(3), 1507}1514.

Wellstead, P. E., & Sano!, S. P. (1981). Extended self-tuning algorithm.International Journal of Control, 34(3), 433}455.

Wellstead, P. E., & Zarrop, M. B. (1991). Self-tuning systems control andsignal processing. Chichester, UK: John Wiley & Sons, Ltd.


adaptive voltage regulator design for static var systems

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