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An Adaptive Regulator for the Excitation of Synchronous Generators
Copyright © I FAC Po wer Systems Modelling and Control Apphcallons. Brussels. Belgium 1988
AN ADAPTIVE REGULATOR FOR THE EXCITATION OF SYNCHRONOUS GENERATORS K. Fork* and C. Schreurs** *Siflllens. C3-/32. Halllm nbacherstraJ3e 12u. 14, 85 20 Er/angen. FRG ** DS M , EE -ETI.\IS2. p.a . B ox 603 , 6 160 MH Ge/eell . Netherlands
Abstracl . This paper describes the developmenl of an adaplive regulalor for the excilalion of synchronous generators which should realise identical and defined oscillating behaviour under different operational conditions . The regulator uses the state feedback principle . and adapts itself to the different operating points of the generator using a modified recursive least-squares algorilhm for parameter identification . The algorithm has some special features which allow of idenlification in a closed loop syslem using normal syslem nuclualions . No extra excitation signals (lIke PRBS) are needed . Also. the algorithm Identifies the n x n parameters of the n x n system matrix directly in a matrix-form with hardly more calculating effort than required for n parameters of a single matrix row . With the identified parameters. the parameters of the state-feedback regulator are determined using a pole-placement technique . In digital and analog simulalions. the regulator works very satisfactorily. but the problems met during the design made clear that it will still take a lot of work before the regulator can be applied to a real machine . Keywords . Adaptive control ; stale-space methods ; pole placement; identification; stabilisers; synchronous generators; excitation systems . according to a defined degree of damping (damping only will be "better"). Qecent developments in control theory seem to promise some new solutions to these problems . Therefore. it was decided to start research on a new type of "integrated" regulator for VOltage regulating ana stabilizing without the disadvantages of the older conventional design . The final goal was to find a regulator which would realise identical and defined oscillating behaviour under di ff erent operational conditions of the generator . A full description of this research can be found at Schreurs (1966) .
INTRODUCTION It has been known for several years that power-oscillations can severely disturb the operation of synchronous generators and power systems (IEEE. 1960; Dandeno. 1962). Two oscillation modes can occur . The first one results from L~e synchronizing torque. which acts on the rotor. and the rotating masses of the rolor and the turbine . This rotating spring-mass system has a typical oscillating frequency of I Hz . The other oscillation mode occurs between several groups of generators This is called a power- or inter-system oscillation . with frequencies even below 0.5 Hz .
In the conventional design of PSS and voltage regulator. two signals are used to control generator behaviour . namely terminal voltage and electric power (or speed) . It seems logical that more information from the system to be controlled will lead to better control. After a thorough investigation of the several possibilities. our thoughts were direcled to a state feedbaCK regulator. with parameters calculated using a pole-placement technique . This principle was also chosen because of its simple structure By means of the pole-placement. oscillations would get the defined behaviour . The requirement of identical behaviour under different operating conditions also means that the regulator should be adaptive.
Several authors have shown. that a fast acting voltage regulator of conventional des ign increases the transient stability of the generator. but deteriorates the static stability . A thorough analysis of this problem can be found In the paper of Demello and Concordia (1969) This connict was the reason why Power-System-Stabilizers (PSS) were developed . These devices compensate for the destabilizing innuence of the voltage regulator - without affecting the regulating action itself - and even provide extra damping .
For a state feedbaCK regulator we need a state-space model of In oscillating generator. The first part of the work therefore was to find such a model. For calculating reasons. it should be IS simple as possible . Based (again) on the work of Demello and Concordla (1969) a relatively simple model could be found. using only :3 state variables . Experiments on a real 770 MVA turbogenerator showed thlt this model was sufficiently Iccurate.
Siemens started the development of her own PSS in 1975 (see Fork . 1981). This stabilizer is now in operation in several power plants . The first experience with this PSS showed that it dampened the 1 Hz mode fairly well. but slow changes of electric power at the lower (Inter-system) OSCillating frequencies brought relatively big changes in lerminal voltage and reactive power . Also. the stabilizing effect of the PSS was strongly dependent on the operating point of the generator . By carefully tuning the PSS. these problems could greatly be reduced. and the PSS now operales very satisfactorily.
For the Idaptation of the regulator to different operationll conditions. the relevant parameters are directly found In a stataspace form by means of a modified recursive least-squares algorithm . This algorithm does not need any excitation Signals to the system. Identification is done using the normal nucluations of the state variables during a power oscillation.
Still the problem remains that the effect of the PSS is dependent on the operating point. Also. it is very difficult to tune the PSS
43
44
K. Fo rk and C. Schreurs
The algorithm also has some features which eliminate partly the innuence of dislurbing signlls from the power syslem Ind of measuring errors. Then. using these parameters and the poleplllC&menllechnique. the slale feedblck regul.tor is delermined .
In digilal Ind Inllog simulllions the regul.lor works very satisfactorily . This shows that the adaptive regulating method has gr8ll possibilities. However the problems mel in developing the algorithm also made clear that implementing it on real machines will slill require a lol of work .
The Park.-equetions and the OeCo-model describe the behaviour of • generltor cOMllcled by means of an extern.1 (line-) rellCt..nce xe to an Infinitely strong pow8r system with voltage e8 (Fig. 3).
For the conditions during this experiment (S88 appendix 2). we find the following theoreticel system and input mltrix of the DeComodel :
Ac =
THE STATE-SPACE REPRESENTATION a: AN OSCILLA TlNG GENERA TOR
Bc =
For the appliclltion of 11 slate feedback regullltor. we need 11 good state-space represenlation of an oscillating generator . Based on the work of Demello and Concordia (for a full treatment. see Appendix 1). we can find a model with three state variables :
(1)
With: Aeq : Deviation of vollage behind synchronous reactance Ill.) : Deviation of roLor speed Am : Deviation of electrical torque e The input-variables are Aaf : Deviation of field vollage Am : Deviation of driving torque a We will henceforth call thi5 model on the Demello-Concordia (DeCo-) model. All variables and D3rametFs in themodel are given as per-unit values . The model only describes the dynamic behaviour of a generator in the frequency-range from 0 to ca. 10 Hz. In thi5 range. the devilltion of voltage behind synchronous reactance Ae q is equal to the change of field current. Also. the deviation of electrical torque Ame is equal to the change of electric power output. In other words. all slate variables of the model are measurable . We first hllV8 to define the meaning of the deviations A. The DeComodel is found by linearizing the full Park-equations (Park. 1929. 1933) around a stationary operating point of the machine. 50 A means a deviation of a signal from the constant value Cof speed. power. voltage and 50 on). Now it is time to look at the behaviour of a real generator . Experiments were performed on two 770 MVA units of the "Clsus" power plant in Maasbracht. the Netherlands . (588 Schreurs and Simons. 1965). The parameters of the units are given in Appendill 2 . During these ellperimenu. "stationary" operation of a generltor was carefully recorded . Fig . 1 shows a plot of several signals. menured for 700 s with I zer01Ullprll5sion to see the deviations only . A raal stationary point does not seem to exist. The signlls look very "noisy".
pe.
A spectral analysis of the electric power. fig . 2. shows a distinct Itl.16 Hz . Furthlr peaks lie It higher frequencies . At15 Hz. the first torsional oscillating frequency of the lurbinegenerator shaft can be detected (theoretical vllue 14.97 Hz). Below 1.16 Hz. WI still SII high amplitudes. clearly showing that Ilso lower oscillating frequencies are present.
C'5852
0.0000 -0 .1741
2.4271 0.0000 1.2584
( 0 .5852 0 .0000 0.1741
0.0000 43 .6158 0.0000
0.0000 -43 .8158 0.0000
)
)
(2)
The eigenvalues of the matrix Ac are: 11 = -0 .2498 ~ = -0 .1677 + j 7.4179 13 = -0 .1677 - j 7.4179
The poles ~ and 13 show. damped oscillalion with a frequency of 1.16 Hz . identical with the peak seen in fig . 2. We can identify this frequency as the oscillaling frequency of the rotating springmass system of rotor and synchronizing torque already mentioned in the introduction. From this experiment it follows thal the DeCo-model describes the rotor oscillations of the generator fairly well. However more oscillating frequencies Ire present. EIK trlcPowtr (MW I
,.... ('5 ·
I .lo- l )
hrmlna l Volt.
i
.,
i
r .-,, .,
( ~ of nominal volt.)
. RPKt lv pPo wfr (MVAr)
FreqJtnCy
i
.,
.,
i
hnHI l
100
100
lOO
-,
lOO
600 sec 100
Fig . 1 The "noise" of. g.ner.tor 118 Hi Osc illating frtqAf\Cy
1
0'
Gtntr.tor ~ rot or
F_
~rP------~------~~---------------------'
i:
1 Tor'IOf\II
t
~Ii~
______________, -____________
' 10.1
~
____
10 '
~~~
HI
---+
Fig. 2 The spectral analysis of ,Iectric 10811
FrtO*'lCY
Adaptive Regulator for Excitation of Synchronous Generators In the phasor-diagrlllTl of Fig . :5 of a generalor connected 10 a power system, W8 can 8Mplaln the other os;cillallng frequentl .. only by dev1atlons In voltage Aee aOllload-angle Ay In tile 'Infinitely slrong' power syslem . These devialions have "noisy' properties, and are caused by the dynamic behaviour of lhe loads In the power system or of generators In lhe neighbourhood . For very low frequencies, generalor and power syslem will swing together, and In this case the changes of phasor angle will be: lilt.
= Ay
(:5)
This Is lhe crllerlon for a "stallonary" polnl in the DeCo-model; no !:!l!lli!. change of load angle :
(4)
M= IIIt.-Ay=O
Il is expecled thal eq . 4 applies 10 frequencies below ca . 0 .1 Hz . Oscillations wilh frequencies lower lhan lhis value are regarded 85 stationary In the DeCo-model. This means thal for measuring lhe slale variables hlgh-pass fillers should be used wilh a cul-off frequency of ca . 0 .1 Hz and zero phase shifl from ca . 0.2 Hz upward . For lhis work, second order high-pass fillers wilh double real poles were used.
THE DEVELOPMENT Of THE IDENTIFICATION ALGCRITHM The DeCo-model is a linearized model. Its parameters (within the matrices Ae and Bel therefore will change from operating point to operating point. As follows from the theoretical analysis (see Appendix I) the~e parameters are delarmined by some known system constants (like the synchronous reactance xd)' but also by some unknown con5lanl5 . A special unknown con5lanl is lhe exlernal reaelance xe' This is the reactance between the generator terminals and an imaginary poinl "somewhere" in lhe power syslem, which is called 'infinilely slrong·. 11 is a lheorelical hypothesis lhallhis poinl exisls, bul its actual posilion can never be known beforehand .
45
For lhe calculation of the paramelers of the regulaling matrix l the actual vIlues of paramelers of lhe DeCo-model ar, needed .
hIVe to be determIned by an Identlncatlon algorithm. In devaloping this Ilgorilhm. we have to solve the following problems : Tllese
Ildenlificalion musl be performed in a closed regulaling syslem . INo extra excilalion signals (PRBS or while noise) lo field vollage or driving lorque are allowed . This would dlslurb the normal operalion of lhe generalor, especially when excitalion signals are fed to the turbine regulalor in order 10 get changes of driving torque . It should be lried to take lhe normal nuclualions of the signals, as shown In Fig . 1. -E)(lernal perlurbalions - modelled by changes of 40e e and 40y hIVe unknown "noisy" properties, bul may also hIVe "step""pulse' and "sinus' -like changes . IAlso parametriC changes (like a change in external reactance )(e' caused by a swilching of lines wllhin the power syslem) will produce oscillations and musl be identified . -The accurlcy of signll measuremenl is limiled . Normel power planllnstrumentatlon is characterized by a series connection of many elemenls (measuring lranformers, lransducers, fillers) The accuracy of lhe whole measuring circuit will hardly be beller lhan 1 ~. A typical value is 5 ~ . Also lime lags and pilaseshifls will be inlroduced . For the applicalion of IIn identificalion method with a sampling lime T, we hIVe 10 develop a discrete-time representalion of the DeCo-model: x(k + 1) = Ar x(k) + Br u(k)
(5)
Ar IInd Br lire the discrete time matrices, It the time step . AT can be found using the continuous time mlltrix Ae as follows (sea for eXlmple Kuo 1980):
= e Ae T = I + A T +
(6)
e
and BT is given by BT =Ae -1[A T - I J Be
For lhis represenlalion, the slale feedbaclt conlroller will be :
Aa
w=1
Xd !d.
u(k) =- l .(It)
(8)
The closed conlrolled syslem is described by : .(It + 1) = (AT - BTl) .(k) =A .(k)
(9)
Now lhe queslion is, whal Slmpling lime should be used . The hlghesl value of the sllmpllng lime is givan by the Shannon lheorem. : ( 10)
A~
.~.....'
Fig . :5 The phllor-di8C)l'1/n of III OKilllling
generltor P.5.-<:
with fc the highesl frequency thll is to be det.ect.ed . For I lypical oscillating frequency or 1 Hz, this would mean a sampling lime or 500 ms or less . Looking III r ..1generllor, also short circuits with I fault clearing lime of 100 ms should be deteclad. Therefore, a vllue of 50 ms is regarded IS the highest possibla . Also, with I long sampling lime It would u.ke rellllvely long before enough samples ar, t.ektn to perform the idenlificllion . Ilsaems then, that the sampling lime should be IS short IS possible, bullhen the identified mltrilC will be cloll to the identily-matrilC I (Ha Iq . 6) Ind glvas no Informlllon about the orlglnll system. As. compromill. I sampling lime of 20 ms WH chosen.
K. Fork and C. Schreurs
46
For the example given In eq. 2. we then find with eqs . 6 and 7 the following time discrete metrices Ar and Br :
By constructing differences :
d(k+ I): xlk+ I rl(k) 0 .9884 AT· 0 .0015 ( -0.0034
0 .0481
0.0116 BT· -0.0015 ( 0.0034
0 .0211 ) 0 .6731 0 .0110
0 .9690 0 .0250
tl(k+2): .(k+2)-.(k+ 1)
-0 .0211 ) -0 .6731 0 .9890
any constanl signal parl is eliminated. Apply theSl to a r.cursi .... 1.lsl squar.s llgorithm in the following way:
( III
First trials showed that the identified parameters had an error of Ipproximately O. I pu caused by the measuring errors and the "noise" of the external perturbations . Theoretical values of several parameters in the discrete time melrix AT are. however. smaller than this error of 0.1 pu. so the idenltncation yields no significanl informelion aboul the real system . Other experiments with sampling times up to 50 ms brought no solution. 50 another way out of these problems had to be found. The nexl problem concerned the inforlTl8tion that is needed for the identification. For an identification. the system must be oscillaling. caused by some external perturbation. Pulse changes of phasor angle Ay and external voltage Ae e are no problem. because immediately after the perturbation the system is described by eq. 9. and the parameters of this equation can be identified. The new stationary point is equal to the old one . Step changes of phasor angle Ay and external voltage Ae e • and also parametriC changes. bring the system to 11 new stationary point different from the old one. INe now want to study what happens with the identification . Starting at eq . 9. we put the step changes together in a constanl perturbation vector nO ' Immediately after the application of the step. system behaviour is described by: (12)
x(k+1) = A x(k) + nO
The veclor .0 is nol known to the identificalion algorithm. which tries to find a mathematically exact equation like : x(k+ Il = A
(13)
The constant vector nO is "packed" into the identified parameters and brings an error. So we have to eliminate this constent vector . However. It was experienced that the 0 .1 Hz hlgh-pass nlters used for measuring the stete vlriables did not filter out the constent signal part fast enough. High-pass filters which eliminate constant signals faster have a cut-off frequency above 0 . 1 Hz and bring a greater phase-shift at I Hz. For the slate feedback regulltor. phase-shifting should. however. be as small IS possible . Also. the dynamics of the filter Itself would disturb the identification . A way of solving this problem would be to identify the constant vector nO like a parameter or to integrate a model of the filters in the Ilgorithm . But this would meln In increase of calculating effort; a filter of order n brings n new parameters . To solve the problem of identification error as well as that of eliminltion of constent signll parts. we propose the following algorithm:
a,' •
d (k.+2}-d , (k.+ 1) l + T (k)P(k)d(k+ 1) ( ~---
T P(k+ I) = [P(k) - y(k)P(k)tI(k+ 1) cI ' (k+ 1) P(k») 1 p y(k) = [cI ' (k+1) P(k)cI(k+1) + p)-1
•
(Ill • al2 •. ..... • Ilm); these parameters haw to be identified . k : the number of the time step . p : the "forgetUng"-factor. with a value between 0 and 1. P : the covrilllCe-matrix y : the "weighting"-factor
(14)
This algorithm musl be applied to every row . The advantage of this method is that for updating all matrix-paramelers. matrix P and factor y need lo be calculat.d only once per;. recursion (Prager and INellstead. 1980). The identified matrix A has the following properlies :
•
(cI(k+2}-d(k+ 1))/T = A d(k.+ 1) =( A x(k+ 1}-.(k+ 1))/T -( A .(lc.}-I(k»)lT ="T(A-' )(x(k+1)-I(lc.))
=
'1 T( A -, ld(k +
A • = 11 T[ A - I
(IS)
1)
Thus, the idenlified matrix A
• is idenlical lo :
I = I 1 T[ AT - BrK - '1
= "T (AT - J] J /T BrK For lhis we wrile : A· =A
• x(k)
_, : The vector of row I In a matrix A
a,(H Il = _,(k)
Q
(16)
- K BQ
•
INe cIII A : the qUlsi-conlinuous matrix of the closed syslem Aa = '1 T [AT -
'1 : the quasl-conllnuous system
matrIx
Ba = I, T B T : the Quasi-continuous input matrix From eqs . 6 and 7 it follows:
(17) By choosing a small sampling time. we approximate in this way the continuous time mltrices. There is no danger anymore. so w. hope. of too small parameler values . Immedlataly after the detection of a disturbance. the identification wlits a short waiting time of 0.1 5 before sterting . This is done to avoid problems with sub-transient and transient behaviour of the generator. which is not described by the DeComodel. After 0 .1 s a ·pure· oscillation exists. which delivers useful information to the identification algorithm .
47
Adaptive Regulator for Excitation of Synchronous Generators THE DETERMINATION Cf THE REGULATING MATRIX
For the example given in eQ . 11. we find with eQ. 16 the following qull5i-1:ontin0u5 IllIIItrices: AQ =
BQ =
-
The algorithm was first tested by digital simulation (see Schreurs. 1966). After these tests. the Illgorithm was implemented on a microprocessor system and subjected to tests with an analog-1:omputer simulating a generator. The analog generator model (see Waldmann. 1'972) behaves like a real machine. and enables testing the whole device in a relllistic manner . After a switching operation in the power system. Fig . 4. resulting in a doubling of the externlll reliC lance xe' the electric power output starts to oscillate . The theoretical continuous time system matrix for this clIse is: -
2.4041 -1.0574 ) -0.5491 -43 .6550 1.2496 -
Ac =
0
( -
1.7206 -0 .3231 1.4129
-29. 63~7 -
)
with eigenvelues:
0.:5606 1.0574) -0 .0758 43 .6552 ( 0 .1724 0.5491
(16)
From this example. and also from the theoretical analysis. it follows that the first colUlTVl and the last column of matrix AQ are opposite to the first and the last column of matrix BQ . So the problem reduces to 9 equetiDM with 9 unknown perameters. which can be solved . Concerning the regulating matrix 1(. we must accept that it is not !msible to influence the driving torque directly . The turbine regullltor. the VIIIVIIs and the turbine have seVllrll1 time lags. Also. influencing the driving torque directly is not IIlIowed. because this would lead to VIIry unquiet motion of turbine valves. with corresponding changes of steam pressure and flow. This would ClYse extra thermal and mechanical stress for the boiler or the reactor. the high-pressure steam pipes. the hydraulic valve drives and the turbine. Therefore. the matrix K becomes:
(
TEST Cf THE ALGORITHM
I(
For the determination of the parameters of the regulating matrix K. w. need to recDMtruct the parameters of the (quasicontinuo~ system matrix AQ from the identified closed-loop ""trix A . Normally. this problem camot be 501ved. see for eXJlTll)le Isermann (1977). There,.are 9 equations (every identified parameter of matrix A ) with 15 unknown variabl85 (9 p..ameters of matrix AQ and 6 of matrix BQ)' Fortunately. however. the DeCo-model has some properties which enable us to solve the problem.
)
(19)
1.1 = -0 .3203 1.2· -0 .2653 - j 6.4678 l.:s = -0 .2653 + j 6.4678 which point to a poorly damped oscillation of about 1.0 Hz. In Fig. Cl . 1.0 Hz a15O .
4. the oscilllltion frequency is
Under the sama conditions. the slate splice regullltor WIIS put into operation. see Fig . 5. The algorithm needs some time before the parllmeters ere identified. 6ut immedilltely IIfter the identification and calculation of the parameters of the regulator l the oscillation is damped promptly . To test whether the IIlgorithm delivered consistent parameters. the experiment was repeated 4 times . The mean values of the identified paramaters of the (quIsi-1:ontinuous tima) system matrix are: AQ =
en,
-1 Cl; - - - - -
0.16 0.60 ( 0.61
(20)
0 .07 0 .01 0.01
0.24 ) 0 .005 0.12
From this it follows that the values of the parameters aq21 .a q31 and aq 13 are not consistently identified. From the mean VIIlues of the identified matrix AQ we can determine the continuous time system matrix Ac : Ac =
T
1.4211 9.0215) -0 .7320 -29 .2640 1.3568 -0 .4558
with the following normalized standllrd deviations:
So w. have to determine only three parameters for the regulator .
For using the pol.-plllcementtechnique. w. must remember that we are working in the quasi'"1:ontinuous time repr.,.,tation. We want the three pol.s Ql' ~ and Q3 of the matrix A of the closed system to lie at defined places in the complex plane. The r.lationship with the continuous time poles 1 1, '2 and 13 is then given by :
-6 .1400 -
-8.8772 -
1.5076 -0 .3203 1.3668
3.7651) -29 .4040 -0.0518
This is done by reverting eq . 6. The eigenvalues are: For a sman sampling time T. the Quasi'"1:ontinuous poles Qi approech the continuous ti"" poles 1; .
0'
Given the three poles wanted. and the parameters matrix Aa and BQ reconstructed from the identified matrix A • we can delermine the peremeters of reguleting metriK I from : (21)
by substituting wery pole wantad Cl; and soMng for kl' 1:2 and k3 .
11 - -8.8001 ~ = ~.2246 - j 6.3486 l.:s - -
48
K. Fork and C. Schreurs
Sy switching back to the normal value of the external reactance, identification in the closed loop system could be tested. This
experiment was performed :5 times. Prescribed poles are: Ql =-5
q2 = -5 - j8 IQ = -5 + j8 The mean Identlned matrix of the closed system Is:
• (21.3057 A· 0.0:50:5 1.0329
91300 9 .4742) -2.6731 -25 .3103 4.5018 -8 .6162
The eigenvalues of the identified matrixA • are:
greater discrepancies between expected and identified prlllll8ters. The main problem here is the accurate measuring of small dev1ations of the state-space values and correct elimination of step changes . Also, it is e)(pected that stochastic variations in power system voltage Ae e end load angle Ay will produce bias in the identified parameters, beClUse not all conditions for • bilSfree ldentlncatlon of parameters using a least-squares algorithm Ire fulfilled (Eykhoff, 1974). Perhaps the identification algorithm needs some extensions in order to Identify some parameters of a slocMslic disturbance modelllld to incorporate a model for the high-pass ",ters. Practical application on real machines will therefore require still a lot of work . But it is expected that a novel regulating concept like the state-space regulator in principle hiS better possibilities to increlse the dynamic stability of generators than a PSS of conventional design . SYMBOlS
q I = -22.43 q2 = -5 .08 - j 10.66 Cb = -5.08 + j 10.66
voltage behind synchronous reactance external (power system) voltage field voltage electrical torque driving torque terminal voltage output of regulator synchronous reaelance transient synchronous reactance external reactance Coefficient of mechanical damping Coemclent of electrical damping
These poles do not lie e)(actly at the desired places, but the real part of the ·oscillation-poles· Q2 and Q:5 is practically equal to the desired value and the oscillation is damped much better.
COOCLUSIOOS The algorithm works very satisfactorily in digital simulations . However, practical application on an analog model shows some -0,13
°
t!.e
r
Ae
r
pu.
° p.u.
0,13 -0,63
-0,63
0
Dc.:>
~
t!.c.:>
~
rad/s
0 rad/s
0.63
0,63
0,28
0,28
t
t
p.u.
t!. m e
t!.me
0
p.u.
0
-0,28 -0,3
A
~
U
r
0
p.u. 0,3
0
4
8
12
20
10 t
s
----+
Fig. 4 Tut on IfIIlog model Oscillations without state-sp8Ce regulator
Fig. 5 Damped oscillations after action of regulltor
Adaptive Regulator for Excitation of Synchronous Generators Td' TclO' T.
Idual field lime conslanl no-load fi"ld lime cons;lanl inerlia lime constant of generator-rotor a load-angle Index 0 stationary state s laplace-operator Except for the time constants and the angles, all quantities per-unit values as given by IEC 34-10
49
~3=----
T. The parameters of matrix 8 are : b ll
er.
=
1/ Td'
b 12 = 0 b21 = 0
APPENDIX 1 The Demello-Concorclla-moclel In the following, a short description of lhe basic formulas of the model of Demello and Concordia is given . For a full description, see Demallo and Concordia (1969), Anderson and Fouad (1977) or Schreurs (1986).
b = 22
IIln/
Ta
-----5sin &0 d xd + xe
T '
For small changes of voltage behind synchronous reactance u q and load angle AoS , the change of electrical torque is given by: eqOeeO Am e
:
eqO
~os
xd + xe
30 AoS+
--51n
xd + xe
30 u q +sDeAoS
The change of voltllge behind synchronous rellctllnce is given by : Xd - xd' Ae q :~Aef+ sTd' ~eO sin I + sTd' xe + xd'
3o AoS )
And lhe motion of lhe geneI'll tor-rotor Is described by :
Am a
:
Am
2 Ta e+ sDmAoS + 5 --AoS
The actual field time constant Td' is :
From this, the state-space equation 1 can be derived with the following parllmeters: 1
/ Td'
xd - xd'
a12 :
eeO sin
30
xe + xd' a13: 0 a21 : 0 a22 = - OmiT
a
8 eO 131 = - -----5sin 80 Td' xd + xe
Xd - xd'
'qO'.o xd + x,
Xd' : 0.36 pu Un = 21 ,000 V TdO': 7.37 s
Operating point: P = 495 MW U : 21.000 V
Q
,,207 MVA R
xe" 0.21 pu
Anderson, P. M. and A. A. Foulld (1977). Power system control and stability . Iowa State University Press, Ames, lo.wa, . USA. Dandeno, P. L. (1982) . General overview of steady-state (small signal) stability in bulk electricity systems : a Norlh American perspective . 1nl. J. of Electrical Power ~ Eneray Systems , ~ , 253-264. Demello, F. P and Ch . Concordia, (1969) . Concepts of synchronous mllchine stllbility 115 IIffected by excitation control . IEEE Transactions on Power Apparatys and Syst.ms .. PAS-88 , 316-329. Eykhoff, P (1974). System Identification, Parameter and State Estimation . Wiley, London. Fork, K., W. Juling and W. Kaufhold (1981). Dimpfung van Leistungspendelungen durch Beeinfliissung der G.neralorerregung . Si.mens En.rai.l.chnik ,~, 9-12. IEEE (1980). Power system stabilizlltion by .xcitation control. IEEE Tulorial Course No . 81 EHO 175-0 PINR. Kuo, B. C. (1980). Digital Control SystelTl5. Holt, Reinehrt 8. INinston, New York. Park, R. H. (1929). Two reaction theory of synchronous machines I. AIEE Transaclions .. ~, 716-730 . Perk, R. H. (1933). Two reaction theory of synchrOllOU! machines 11. AIEE Transactions • ~ .. 352-355. Prager, D. l. IInd P. E. Wellstead (1980). Mullivarilble poleIssignment s.lf-tuning regUlators. ~. 9-182 . Schreurs. C. end J . Simons (1985). Messungen in der "CllUScentrll.". Elektrotachniek . M. 291-300 . Schreurs. C. (1986). A.tive Regelverflhren fUr die EIT!C!!!!Q yoo SyocbrOO!!lll!!Cltoreo. Eiodhoven University of Technology. Dissertation. Eindhoven. Netherlands. Wlldmlllll. H.• M. Weibelzihl IOd J . Wolf (1972). Ein elektroni!IChH I10deIl der Synchronml!lChine . ~ forsclMlgs UOd Enlwicklyngsberjchle • .1. 157-166.
m.
-----5sln &0 --~e,O sin xd + x, x, + xd'
+
Xd = 2.25 pu . Sn z 770 MVA Pn : 600.6 MW T. =7 .175
REFERENCES
~n
1111 : -
APPENDIX 2 Paramelers of ·Claus· -unit
D,Dm~n
cos·OTa
30
AN ADAPTIVE REGULATOR FOR THE EXCITATION OF SYNCHRONOUS GENERATORS K. Fork* and C. Schreurs** *Siflllens. C3-/32. Halllm nbacherstraJ3e 12u. 14, 85 20 Er/angen. FRG ** DS M , EE -ETI.\IS2. p.a . B ox 603 , 6 160 MH Ge/eell . Netherlands
Abstracl . This paper describes the developmenl of an adaplive regulalor for the excilalion of synchronous generators which should realise identical and defined oscillating behaviour under different operational conditions . The regulator uses the state feedback principle . and adapts itself to the different operating points of the generator using a modified recursive least-squares algorilhm for parameter identification . The algorithm has some special features which allow of idenlification in a closed loop syslem using normal syslem nuclualions . No extra excitation signals (lIke PRBS) are needed . Also. the algorithm Identifies the n x n parameters of the n x n system matrix directly in a matrix-form with hardly more calculating effort than required for n parameters of a single matrix row . With the identified parameters. the parameters of the state-feedback regulator are determined using a pole-placement technique . In digital and analog simulalions. the regulator works very satisfactorily. but the problems met during the design made clear that it will still take a lot of work before the regulator can be applied to a real machine . Keywords . Adaptive control ; stale-space methods ; pole placement; identification; stabilisers; synchronous generators; excitation systems . according to a defined degree of damping (damping only will be "better"). Qecent developments in control theory seem to promise some new solutions to these problems . Therefore. it was decided to start research on a new type of "integrated" regulator for VOltage regulating ana stabilizing without the disadvantages of the older conventional design . The final goal was to find a regulator which would realise identical and defined oscillating behaviour under di ff erent operational conditions of the generator . A full description of this research can be found at Schreurs (1966) .
INTRODUCTION It has been known for several years that power-oscillations can severely disturb the operation of synchronous generators and power systems (IEEE. 1960; Dandeno. 1962). Two oscillation modes can occur . The first one results from L~e synchronizing torque. which acts on the rotor. and the rotating masses of the rolor and the turbine . This rotating spring-mass system has a typical oscillating frequency of I Hz . The other oscillation mode occurs between several groups of generators This is called a power- or inter-system oscillation . with frequencies even below 0.5 Hz .
In the conventional design of PSS and voltage regulator. two signals are used to control generator behaviour . namely terminal voltage and electric power (or speed) . It seems logical that more information from the system to be controlled will lead to better control. After a thorough investigation of the several possibilities. our thoughts were direcled to a state feedbaCK regulator. with parameters calculated using a pole-placement technique . This principle was also chosen because of its simple structure By means of the pole-placement. oscillations would get the defined behaviour . The requirement of identical behaviour under different operating conditions also means that the regulator should be adaptive.
Several authors have shown. that a fast acting voltage regulator of conventional des ign increases the transient stability of the generator. but deteriorates the static stability . A thorough analysis of this problem can be found In the paper of Demello and Concordia (1969) This connict was the reason why Power-System-Stabilizers (PSS) were developed . These devices compensate for the destabilizing innuence of the voltage regulator - without affecting the regulating action itself - and even provide extra damping .
For a state feedbaCK regulator we need a state-space model of In oscillating generator. The first part of the work therefore was to find such a model. For calculating reasons. it should be IS simple as possible . Based (again) on the work of Demello and Concordla (1969) a relatively simple model could be found. using only :3 state variables . Experiments on a real 770 MVA turbogenerator showed thlt this model was sufficiently Iccurate.
Siemens started the development of her own PSS in 1975 (see Fork . 1981). This stabilizer is now in operation in several power plants . The first experience with this PSS showed that it dampened the 1 Hz mode fairly well. but slow changes of electric power at the lower (Inter-system) OSCillating frequencies brought relatively big changes in lerminal voltage and reactive power . Also. the stabilizing effect of the PSS was strongly dependent on the operating point of the generator . By carefully tuning the PSS. these problems could greatly be reduced. and the PSS now operales very satisfactorily.
For the Idaptation of the regulator to different operationll conditions. the relevant parameters are directly found In a stataspace form by means of a modified recursive least-squares algorithm . This algorithm does not need any excitation Signals to the system. Identification is done using the normal nucluations of the state variables during a power oscillation.
Still the problem remains that the effect of the PSS is dependent on the operating point. Also. it is very difficult to tune the PSS
43
44
K. Fo rk and C. Schreurs
The algorithm also has some features which eliminate partly the innuence of dislurbing signlls from the power syslem Ind of measuring errors. Then. using these parameters and the poleplllC&menllechnique. the slale feedblck regul.tor is delermined .
In digilal Ind Inllog simulllions the regul.lor works very satisfactorily . This shows that the adaptive regulating method has gr8ll possibilities. However the problems mel in developing the algorithm also made clear that implementing it on real machines will slill require a lol of work .
The Park.-equetions and the OeCo-model describe the behaviour of • generltor cOMllcled by means of an extern.1 (line-) rellCt..nce xe to an Infinitely strong pow8r system with voltage e8 (Fig. 3).
For the conditions during this experiment (S88 appendix 2). we find the following theoreticel system and input mltrix of the DeComodel :
Ac =
THE STATE-SPACE REPRESENTATION a: AN OSCILLA TlNG GENERA TOR
Bc =
For the appliclltion of 11 slate feedback regullltor. we need 11 good state-space represenlation of an oscillating generator . Based on the work of Demello and Concordia (for a full treatment. see Appendix 1). we can find a model with three state variables :
(1)
With: Aeq : Deviation of vollage behind synchronous reactance Ill.) : Deviation of roLor speed Am : Deviation of electrical torque e The input-variables are Aaf : Deviation of field vollage Am : Deviation of driving torque a We will henceforth call thi5 model on the Demello-Concordia (DeCo-) model. All variables and D3rametFs in themodel are given as per-unit values . The model only describes the dynamic behaviour of a generator in the frequency-range from 0 to ca. 10 Hz. In thi5 range. the devilltion of voltage behind synchronous reactance Ae q is equal to the change of field current. Also. the deviation of electrical torque Ame is equal to the change of electric power output. In other words. all slate variables of the model are measurable . We first hllV8 to define the meaning of the deviations A. The DeComodel is found by linearizing the full Park-equations (Park. 1929. 1933) around a stationary operating point of the machine. 50 A means a deviation of a signal from the constant value Cof speed. power. voltage and 50 on). Now it is time to look at the behaviour of a real generator . Experiments were performed on two 770 MVA units of the "Clsus" power plant in Maasbracht. the Netherlands . (588 Schreurs and Simons. 1965). The parameters of the units are given in Appendill 2 . During these ellperimenu. "stationary" operation of a generltor was carefully recorded . Fig . 1 shows a plot of several signals. menured for 700 s with I zer01Ullprll5sion to see the deviations only . A raal stationary point does not seem to exist. The signlls look very "noisy".
pe.
A spectral analysis of the electric power. fig . 2. shows a distinct Itl.16 Hz . Furthlr peaks lie It higher frequencies . At15 Hz. the first torsional oscillating frequency of the lurbinegenerator shaft can be detected (theoretical vllue 14.97 Hz). Below 1.16 Hz. WI still SII high amplitudes. clearly showing that Ilso lower oscillating frequencies are present.
C'5852
0.0000 -0 .1741
2.4271 0.0000 1.2584
( 0 .5852 0 .0000 0.1741
0.0000 43 .6158 0.0000
0.0000 -43 .8158 0.0000
)
)
(2)
The eigenvalues of the matrix Ac are: 11 = -0 .2498 ~ = -0 .1677 + j 7.4179 13 = -0 .1677 - j 7.4179
The poles ~ and 13 show. damped oscillalion with a frequency of 1.16 Hz . identical with the peak seen in fig . 2. We can identify this frequency as the oscillaling frequency of the rotating springmass system of rotor and synchronizing torque already mentioned in the introduction. From this experiment it follows thal the DeCo-model describes the rotor oscillations of the generator fairly well. However more oscillating frequencies Ire present. EIK trlcPowtr (MW I
,.... ('5 ·
I .lo- l )
hrmlna l Volt.
i
.,
i
r .-,, .,
( ~ of nominal volt.)
. RPKt lv pPo wfr (MVAr)
FreqJtnCy
i
.,
.,
i
hnHI l
100
100
lOO
-,
lOO
600 sec 100
Fig . 1 The "noise" of. g.ner.tor 118 Hi Osc illating frtqAf\Cy
1
0'
Gtntr.tor ~ rot or
F_
~rP------~------~~---------------------'
i:
1 Tor'IOf\II
t
~Ii~
______________, -____________
' 10.1
~
____
10 '
~~~
HI
---+
Fig. 2 The spectral analysis of ,Iectric 10811
FrtO*'lCY
Adaptive Regulator for Excitation of Synchronous Generators In the phasor-diagrlllTl of Fig . :5 of a generalor connected 10 a power system, W8 can 8Mplaln the other os;cillallng frequentl .. only by dev1atlons In voltage Aee aOllload-angle Ay In tile 'Infinitely slrong' power syslem . These devialions have "noisy' properties, and are caused by the dynamic behaviour of lhe loads In the power system or of generators In lhe neighbourhood . For very low frequencies, generalor and power syslem will swing together, and In this case the changes of phasor angle will be: lilt.
= Ay
(:5)
This Is lhe crllerlon for a "stallonary" polnl in the DeCo-model; no !:!l!lli!. change of load angle :
(4)
M= IIIt.-Ay=O
Il is expecled thal eq . 4 applies 10 frequencies below ca . 0 .1 Hz . Oscillations wilh frequencies lower lhan lhis value are regarded 85 stationary In the DeCo-model. This means thal for measuring lhe slale variables hlgh-pass fillers should be used wilh a cul-off frequency of ca . 0 .1 Hz and zero phase shifl from ca . 0.2 Hz upward . For lhis work, second order high-pass fillers wilh double real poles were used.
THE DEVELOPMENT Of THE IDENTIFICATION ALGCRITHM The DeCo-model is a linearized model. Its parameters (within the matrices Ae and Bel therefore will change from operating point to operating point. As follows from the theoretical analysis (see Appendix I) the~e parameters are delarmined by some known system constants (like the synchronous reactance xd)' but also by some unknown con5lanl5 . A special unknown con5lanl is lhe exlernal reaelance xe' This is the reactance between the generator terminals and an imaginary poinl "somewhere" in lhe power syslem, which is called 'infinilely slrong·. 11 is a lheorelical hypothesis lhallhis poinl exisls, bul its actual posilion can never be known beforehand .
45
For lhe calculation of the paramelers of the regulaling matrix l the actual vIlues of paramelers of lhe DeCo-model ar, needed .
hIVe to be determIned by an Identlncatlon algorithm. In devaloping this Ilgorilhm. we have to solve the following problems : Tllese
Ildenlificalion musl be performed in a closed regulaling syslem . INo extra excilalion signals (PRBS or while noise) lo field vollage or driving lorque are allowed . This would dlslurb the normal operalion of lhe generalor, especially when excitalion signals are fed to the turbine regulalor in order 10 get changes of driving torque . It should be lried to take lhe normal nuclualions of the signals, as shown In Fig . 1. -E)(lernal perlurbalions - modelled by changes of 40e e and 40y hIVe unknown "noisy" properties, bul may also hIVe "step""pulse' and "sinus' -like changes . IAlso parametriC changes (like a change in external reactance )(e' caused by a swilching of lines wllhin the power syslem) will produce oscillations and musl be identified . -The accurlcy of signll measuremenl is limiled . Normel power planllnstrumentatlon is characterized by a series connection of many elemenls (measuring lranformers, lransducers, fillers) The accuracy of lhe whole measuring circuit will hardly be beller lhan 1 ~. A typical value is 5 ~ . Also lime lags and pilaseshifls will be inlroduced . For the applicalion of IIn identificalion method with a sampling lime T, we hIVe 10 develop a discrete-time representalion of the DeCo-model: x(k + 1) = Ar x(k) + Br u(k)
(5)
Ar IInd Br lire the discrete time matrices, It the time step . AT can be found using the continuous time mlltrix Ae as follows (sea for eXlmple Kuo 1980):
= e Ae T = I + A T +
(6)
e
and BT is given by BT =Ae -1[A T - I J Be
For lhis represenlalion, the slale feedbaclt conlroller will be :
Aa
w=1
Xd !d.
u(k) =- l .(It)
(8)
The closed conlrolled syslem is described by : .(It + 1) = (AT - BTl) .(k) =A .(k)
(9)
Now lhe queslion is, whal Slmpling lime should be used . The hlghesl value of the sllmpllng lime is givan by the Shannon lheorem. : ( 10)
A~
.~.....'
Fig . :5 The phllor-di8C)l'1/n of III OKilllling
generltor P.5.-<:
with fc the highesl frequency thll is to be det.ect.ed . For I lypical oscillating frequency or 1 Hz, this would mean a sampling lime or 500 ms or less . Looking III r ..1generllor, also short circuits with I fault clearing lime of 100 ms should be deteclad. Therefore, a vllue of 50 ms is regarded IS the highest possibla . Also, with I long sampling lime It would u.ke rellllvely long before enough samples ar, t.ektn to perform the idenlificllion . Ilsaems then, that the sampling lime should be IS short IS possible, bullhen the identified mltrilC will be cloll to the identily-matrilC I (Ha Iq . 6) Ind glvas no Informlllon about the orlglnll system. As. compromill. I sampling lime of 20 ms WH chosen.
K. Fork and C. Schreurs
46
For the example given In eq. 2. we then find with eqs . 6 and 7 the following time discrete metrices Ar and Br :
By constructing differences :
d(k+ I): xlk+ I rl(k) 0 .9884 AT· 0 .0015 ( -0.0034
0 .0481
0.0116 BT· -0.0015 ( 0.0034
0 .0211 ) 0 .6731 0 .0110
0 .9690 0 .0250
tl(k+2): .(k+2)-.(k+ 1)
-0 .0211 ) -0 .6731 0 .9890
any constanl signal parl is eliminated. Apply theSl to a r.cursi .... 1.lsl squar.s llgorithm in the following way:
( III
First trials showed that the identified parameters had an error of Ipproximately O. I pu caused by the measuring errors and the "noise" of the external perturbations . Theoretical values of several parameters in the discrete time melrix AT are. however. smaller than this error of 0.1 pu. so the idenltncation yields no significanl informelion aboul the real system . Other experiments with sampling times up to 50 ms brought no solution. 50 another way out of these problems had to be found. The nexl problem concerned the inforlTl8tion that is needed for the identification. For an identification. the system must be oscillaling. caused by some external perturbation. Pulse changes of phasor angle Ay and external voltage Ae e are no problem. because immediately after the perturbation the system is described by eq. 9. and the parameters of this equation can be identified. The new stationary point is equal to the old one . Step changes of phasor angle Ay and external voltage Ae e • and also parametriC changes. bring the system to 11 new stationary point different from the old one. INe now want to study what happens with the identification . Starting at eq . 9. we put the step changes together in a constanl perturbation vector nO ' Immediately after the application of the step. system behaviour is described by: (12)
x(k+1) = A x(k) + nO
The veclor .0 is nol known to the identificalion algorithm. which tries to find a mathematically exact equation like : x(k+ Il = A
(13)
The constant vector nO is "packed" into the identified parameters and brings an error. So we have to eliminate this constent vector . However. It was experienced that the 0 .1 Hz hlgh-pass nlters used for measuring the stete vlriables did not filter out the constent signal part fast enough. High-pass filters which eliminate constant signals faster have a cut-off frequency above 0 . 1 Hz and bring a greater phase-shift at I Hz. For the slate feedback regulltor. phase-shifting should. however. be as small IS possible . Also. the dynamics of the filter Itself would disturb the identification . A way of solving this problem would be to identify the constant vector nO like a parameter or to integrate a model of the filters in the Ilgorithm . But this would meln In increase of calculating effort; a filter of order n brings n new parameters . To solve the problem of identification error as well as that of eliminltion of constent signll parts. we propose the following algorithm:
a,' •
d (k.+2}-d , (k.+ 1) l + T (k)P(k)d(k+ 1) ( ~---
T P(k+ I) = [P(k) - y(k)P(k)tI(k+ 1) cI ' (k+ 1) P(k») 1 p y(k) = [cI ' (k+1) P(k)cI(k+1) + p)-1
•
(Ill • al2 •. ..... • Ilm); these parameters haw to be identified . k : the number of the time step . p : the "forgetUng"-factor. with a value between 0 and 1. P : the covrilllCe-matrix y : the "weighting"-factor
(14)
This algorithm musl be applied to every row . The advantage of this method is that for updating all matrix-paramelers. matrix P and factor y need lo be calculat.d only once per;. recursion (Prager and INellstead. 1980). The identified matrix A has the following properlies :
•
(cI(k+2}-d(k+ 1))/T = A d(k.+ 1) =( A x(k+ 1}-.(k+ 1))/T -( A .(lc.}-I(k»)lT ="T(A-' )(x(k+1)-I(lc.))
=
'1 T( A -, ld(k +
A • = 11 T[ A - I
(IS)
1)
Thus, the idenlified matrix A
• is idenlical lo :
I = I 1 T[ AT - BrK - '1
= "T (AT - J] J /T BrK For lhis we wrile : A· =A
• x(k)
_, : The vector of row I In a matrix A
a,(H Il = _,(k)
Q
(16)
- K BQ
•
INe cIII A : the qUlsi-conlinuous matrix of the closed syslem Aa = '1 T [AT -
'1 : the quasl-conllnuous system
matrIx
Ba = I, T B T : the Quasi-continuous input matrix From eqs . 6 and 7 it follows:
(17) By choosing a small sampling time. we approximate in this way the continuous time mltrices. There is no danger anymore. so w. hope. of too small parameler values . Immedlataly after the detection of a disturbance. the identification wlits a short waiting time of 0.1 5 before sterting . This is done to avoid problems with sub-transient and transient behaviour of the generator. which is not described by the DeComodel. After 0 .1 s a ·pure· oscillation exists. which delivers useful information to the identification algorithm .
47
Adaptive Regulator for Excitation of Synchronous Generators THE DETERMINATION Cf THE REGULATING MATRIX
For the example given in eQ . 11. we find with eQ. 16 the following qull5i-1:ontin0u5 IllIIItrices: AQ =
BQ =
-
The algorithm was first tested by digital simulation (see Schreurs. 1966). After these tests. the Illgorithm was implemented on a microprocessor system and subjected to tests with an analog-1:omputer simulating a generator. The analog generator model (see Waldmann. 1'972) behaves like a real machine. and enables testing the whole device in a relllistic manner . After a switching operation in the power system. Fig . 4. resulting in a doubling of the externlll reliC lance xe' the electric power output starts to oscillate . The theoretical continuous time system matrix for this clIse is: -
2.4041 -1.0574 ) -0.5491 -43 .6550 1.2496 -
Ac =
0
( -
1.7206 -0 .3231 1.4129
-29. 63~7 -
)
with eigenvelues:
0.:5606 1.0574) -0 .0758 43 .6552 ( 0 .1724 0.5491
(16)
From this example. and also from the theoretical analysis. it follows that the first colUlTVl and the last column of matrix AQ are opposite to the first and the last column of matrix BQ . So the problem reduces to 9 equetiDM with 9 unknown perameters. which can be solved . Concerning the regulating matrix 1(. we must accept that it is not !msible to influence the driving torque directly . The turbine regullltor. the VIIIVIIs and the turbine have seVllrll1 time lags. Also. influencing the driving torque directly is not IIlIowed. because this would lead to VIIry unquiet motion of turbine valves. with corresponding changes of steam pressure and flow. This would ClYse extra thermal and mechanical stress for the boiler or the reactor. the high-pressure steam pipes. the hydraulic valve drives and the turbine. Therefore. the matrix K becomes:
(
TEST Cf THE ALGORITHM
I(
For the determination of the parameters of the regulating matrix K. w. need to recDMtruct the parameters of the (quasicontinuo~ system matrix AQ from the identified closed-loop ""trix A . Normally. this problem camot be 501ved. see for eXJlTll)le Isermann (1977). There,.are 9 equations (every identified parameter of matrix A ) with 15 unknown variabl85 (9 p..ameters of matrix AQ and 6 of matrix BQ)' Fortunately. however. the DeCo-model has some properties which enable us to solve the problem.
)
(19)
1.1 = -0 .3203 1.2· -0 .2653 - j 6.4678 l.:s = -0 .2653 + j 6.4678 which point to a poorly damped oscillation of about 1.0 Hz. In Fig. Cl . 1.0 Hz a15O .
4. the oscilllltion frequency is
Under the sama conditions. the slate splice regullltor WIIS put into operation. see Fig . 5. The algorithm needs some time before the parllmeters ere identified. 6ut immedilltely IIfter the identification and calculation of the parameters of the regulator l the oscillation is damped promptly . To test whether the IIlgorithm delivered consistent parameters. the experiment was repeated 4 times . The mean values of the identified paramaters of the (quIsi-1:ontinuous tima) system matrix are: AQ =
en,
-1 Cl; - - - - -
0.16 0.60 ( 0.61
(20)
0 .07 0 .01 0.01
0.24 ) 0 .005 0.12
From this it follows that the values of the parameters aq21 .a q31 and aq 13 are not consistently identified. From the mean VIIlues of the identified matrix AQ we can determine the continuous time system matrix Ac : Ac =
T
1.4211 9.0215) -0 .7320 -29 .2640 1.3568 -0 .4558
with the following normalized standllrd deviations:
So w. have to determine only three parameters for the regulator .
For using the pol.-plllcementtechnique. w. must remember that we are working in the quasi'"1:ontinuous time repr.,.,tation. We want the three pol.s Ql' ~ and Q3 of the matrix A of the closed system to lie at defined places in the complex plane. The r.lationship with the continuous time poles 1 1, '2 and 13 is then given by :
-6 .1400 -
-8.8772 -
1.5076 -0 .3203 1.3668
3.7651) -29 .4040 -0.0518
This is done by reverting eq . 6. The eigenvalues are: For a sman sampling time T. the Quasi'"1:ontinuous poles Qi approech the continuous ti"" poles 1; .
0'
Given the three poles wanted. and the parameters matrix Aa and BQ reconstructed from the identified matrix A • we can delermine the peremeters of reguleting metriK I from : (21)
by substituting wery pole wantad Cl; and soMng for kl' 1:2 and k3 .
11 - -8.8001 ~ = ~.2246 - j 6.3486 l.:s - -
48
K. Fork and C. Schreurs
Sy switching back to the normal value of the external reactance, identification in the closed loop system could be tested. This
experiment was performed :5 times. Prescribed poles are: Ql =-5
q2 = -5 - j8 IQ = -5 + j8 The mean Identlned matrix of the closed system Is:
• (21.3057 A· 0.0:50:5 1.0329
91300 9 .4742) -2.6731 -25 .3103 4.5018 -8 .6162
The eigenvalues of the identified matrixA • are:
greater discrepancies between expected and identified prlllll8ters. The main problem here is the accurate measuring of small dev1ations of the state-space values and correct elimination of step changes . Also, it is e)(pected that stochastic variations in power system voltage Ae e end load angle Ay will produce bias in the identified parameters, beClUse not all conditions for • bilSfree ldentlncatlon of parameters using a least-squares algorithm Ire fulfilled (Eykhoff, 1974). Perhaps the identification algorithm needs some extensions in order to Identify some parameters of a slocMslic disturbance modelllld to incorporate a model for the high-pass ",ters. Practical application on real machines will therefore require still a lot of work . But it is expected that a novel regulating concept like the state-space regulator in principle hiS better possibilities to increlse the dynamic stability of generators than a PSS of conventional design . SYMBOlS
q I = -22.43 q2 = -5 .08 - j 10.66 Cb = -5.08 + j 10.66
voltage behind synchronous reactance external (power system) voltage field voltage electrical torque driving torque terminal voltage output of regulator synchronous reaelance transient synchronous reactance external reactance Coefficient of mechanical damping Coemclent of electrical damping
These poles do not lie e)(actly at the desired places, but the real part of the ·oscillation-poles· Q2 and Q:5 is practically equal to the desired value and the oscillation is damped much better.
COOCLUSIOOS The algorithm works very satisfactorily in digital simulations . However, practical application on an analog model shows some -0,13
°
t!.e
r
Ae
r
pu.
° p.u.
0,13 -0,63
-0,63
0
Dc.:>
~
t!.c.:>
~
rad/s
0 rad/s
0.63
0,63
0,28
0,28
t
t
p.u.
t!. m e
t!.me
0
p.u.
0
-0,28 -0,3
A
~
U
r
0
p.u. 0,3
0
4
8
12
20
10 t
s
----+
Fig. 4 Tut on IfIIlog model Oscillations without state-sp8Ce regulator
Fig. 5 Damped oscillations after action of regulltor
Adaptive Regulator for Excitation of Synchronous Generators Td' TclO' T.
Idual field lime conslanl no-load fi"ld lime cons;lanl inerlia lime constant of generator-rotor a load-angle Index 0 stationary state s laplace-operator Except for the time constants and the angles, all quantities per-unit values as given by IEC 34-10
49
~3=----
T. The parameters of matrix 8 are : b ll
er.
=
1/ Td'
b 12 = 0 b21 = 0
APPENDIX 1 The Demello-Concorclla-moclel In the following, a short description of lhe basic formulas of the model of Demello and Concordia is given . For a full description, see Demallo and Concordia (1969), Anderson and Fouad (1977) or Schreurs (1986).
b = 22
IIln/
Ta
-----5sin &0 d xd + xe
T '
For small changes of voltage behind synchronous reactance u q and load angle AoS , the change of electrical torque is given by: eqOeeO Am e
:
eqO
~os
xd + xe
30 AoS+
--51n
xd + xe
30 u q +sDeAoS
The change of voltllge behind synchronous rellctllnce is given by : Xd - xd' Ae q :~Aef+ sTd' ~eO sin I + sTd' xe + xd'
3o AoS )
And lhe motion of lhe geneI'll tor-rotor Is described by :
Am a
:
Am
2 Ta e+ sDmAoS + 5 --AoS
The actual field time constant Td' is :
From this, the state-space equation 1 can be derived with the following parllmeters: 1
/ Td'
xd - xd'
a12 :
eeO sin
30
xe + xd' a13: 0 a21 : 0 a22 = - OmiT
a
8 eO 131 = - -----5sin 80 Td' xd + xe
Xd - xd'
'qO'.o xd + x,
Xd' : 0.36 pu Un = 21 ,000 V TdO': 7.37 s
Operating point: P = 495 MW U : 21.000 V
Q
,,207 MVA R
xe" 0.21 pu
Anderson, P. M. and A. A. Foulld (1977). Power system control and stability . Iowa State University Press, Ames, lo.wa, . USA. Dandeno, P. L. (1982) . General overview of steady-state (small signal) stability in bulk electricity systems : a Norlh American perspective . 1nl. J. of Electrical Power ~ Eneray Systems , ~ , 253-264. Demello, F. P and Ch . Concordia, (1969) . Concepts of synchronous mllchine stllbility 115 IIffected by excitation control . IEEE Transactions on Power Apparatys and Syst.ms .. PAS-88 , 316-329. Eykhoff, P (1974). System Identification, Parameter and State Estimation . Wiley, London. Fork, K., W. Juling and W. Kaufhold (1981). Dimpfung van Leistungspendelungen durch Beeinfliissung der G.neralorerregung . Si.mens En.rai.l.chnik ,~, 9-12. IEEE (1980). Power system stabilizlltion by .xcitation control. IEEE Tulorial Course No . 81 EHO 175-0 PINR. Kuo, B. C. (1980). Digital Control SystelTl5. Holt, Reinehrt 8. INinston, New York. Park, R. H. (1929). Two reaction theory of synchronous machines I. AIEE Transaclions .. ~, 716-730 . Perk, R. H. (1933). Two reaction theory of synchrOllOU! machines 11. AIEE Transactions • ~ .. 352-355. Prager, D. l. IInd P. E. Wellstead (1980). Mullivarilble poleIssignment s.lf-tuning regUlators. ~. 9-182 . Schreurs. C. end J . Simons (1985). Messungen in der "CllUScentrll.". Elektrotachniek . M. 291-300 . Schreurs. C. (1986). A.tive Regelverflhren fUr die EIT!C!!!!Q yoo SyocbrOO!!lll!!Cltoreo. Eiodhoven University of Technology. Dissertation. Eindhoven. Netherlands. Wlldmlllll. H.• M. Weibelzihl IOd J . Wolf (1972). Ein elektroni!IChH I10deIl der Synchronml!lChine . ~ forsclMlgs UOd Enlwicklyngsberjchle • .1. 157-166.
m.
-----5sln &0 --~e,O sin xd + x, x, + xd'
+
Xd = 2.25 pu . Sn z 770 MVA Pn : 600.6 MW T. =7 .175
REFERENCES
~n
1111 : -
APPENDIX 2 Paramelers of ·Claus· -unit
D,Dm~n
cos·OTa
30