200 IEEE Transactions on Energy Conversion, Vol. 12, No. 3, September 1997

On-Line Estimation of Variable Parameters of Synchronous Machines sing a Novel Adaptive Algorithm - Estimation and Experimental Verification

*Longya Xu Senior Member, IEEE

*Department of Electrical Engineering The Ohio State University Columbus, Ohio 43210

**Zhengming Zhao **Jianguo Jiang Member, TEEE

Abstract- On-line estimation of variable parameters of synchronous machines based on a novel adaptive algorithm is presented. The estimation process involves instantaneous parameter tracking and subsequent data processing. The variable parameters are ultimately expressed as nonlinear functions of operating conditions. On-line estimation process is applied to a IOOMVA turbogenerator. While this paper deals with application of the on-line estimation method and experimental verification, principle discussion and development of the novel adaptive algorithm are detailed in a companion paper.

1. Introduction

For the last two decades, parameter estimation in frequency or time domain has improved considerably for calculating nodinear behavior

applications. It is well known that SM parameters are significant& dependent on

saturation levels, internal magnetic field distributions 'and variations, and rotor speed changes. For example, synchrono?s reactance is closely related to the level of airgap flux; leakage dactance can be heavily influenced by armature current causing local saturation and distorted flux distribution; and damper resistance can be substantially altered by rotor speed with respect to the rotating flux due to the induced eddy current in rotor iron. It is for the above keasons the SM parameters should be more accurately termed variable parameters. It is apparent that in a SM circuit model, the paramiters should be continuously adapted to the operating conditions, othkrwise, reliable

Recently, considerable attention has been given to the possibility of estimating SM parameters as functions of operating cbnditions [5,6]. Around this issue, there are three major perplexing droblems: 1) to properly select the model to be identified, that is, td determine the order of the governing equations of the SM modkl that will be consistent with the operating conditions; 2) to find an optimal adaptive algorithm to track the parameter variation, t h h is, to develop an effective algorithm to track the parameters lpromptly and

calculation results can not be guaranteed. t

for presentation at the 1996 IEEEPES August 1, 1996, in Denver, Colorado 1 995, made available for printing May 10, 1996. I

I

**Department of Electrical Engineering Tsinghua University

Beijing 100084, P. R. China

properly in calculating nonlinear SM performance in a practical operating situation.

The f i s t problem listed above was well addressed in [7]. This paper and its companion paper [8] contribute to solving the remaining two problems. In organizing this paper, initially, a set of discrete, eighth order incremental equations are derived as the SM model. Then, a brief description is given of the novel adaptive algorithm and its application for variable parameter estimation of a SM. The step- by-step implementation of the novel adaptive algorithm is as follows: * The actual parameters of a lOOMVA turbogenerator are estimated

on line with a perturbation of the field excitation voltage. A regression method is used to find mathematic expressions for the variable parameters as functions of ,operating conditions (including equivalent airgap EMF, armature current, and rotor slip speed).

In order to demonstrate the effective application of the estimated variable parameters to power systems, the estimated parameters are used to predict behavior of the lOOMVA generator connected to a grid the following steps were taken:

Dynamic simulations were done using the estimated parameters and the designed nominal parameters, respectively, for a new operating condition. These simulated responses are compared with the experimental results.

Finally, the paper ends with a brief summary and conclusions.

2. Identification Model

A systematic approach has been presented in [7] to determine the order of a SM model, consistent with the operation conditions. According to this approach, a dynamic model with one damper circuit in d-axis and two damper circuits in q-axis (in rotor reference frame) is appropriate as the standard SM identification model. This model is adopted. The SM equivalent circuits are shown in Fig. 1.

Xdl xfdl Xfl

Fig. 1 SM equivalent circuits

where Vd and v, are the stator d- and q-axis voltages; x,d and x,, are the d- and q-axis synchronous reactance; xdl, X+ xfl, xldl, X2,1, R,, Rfd, Rid, RI,, and R2, are leakage reactance and resistance; and xfd is the mutual leakage reactance between the field and d-axis damper.

A. Basic SM State and Incremental Equations

0885-8969/97/$10.00 0 1996 IEEE

Electrical transients of a typical (round or salient rotor) SM can be described by the following sixth order electrical equation in a matrix form:

V = Z p I + R I (1)

where, V is a voltage vector as the input, p the derivative operator, I a current vector as the output, and Z and R are inductance and resistance matrices, containing the basic parameters of the SM model. Explicitly,

v [ vq, vd. 0, 0, vfd. 0 1 (2)

I = t iq. 41. iiq, i2q. ifd, i id 1 T (3)

(4)

B1 =

where v f d is the field voltage; id, ifd and ild are the stator d-axis, the field and the rotor d-axis damper currents; iq, ilq and i2q are the stator q-axis, the first q-axis damper and the second q-axis damper currents. The reactance parameters follow the conventional equations as: xd = Xad + xdl; xq = xaq + xq1; xllq = xaq + Xlql; X q = xaq + X%1; xffd = &d xfl; X l l d = %ad + XI&; and Xfld = xad Xfd.

TWO additional equations are used to describe the SM mechanical motion as

(7)

where Tm is the mechanical torque, Te the electromagnetic torque, w the angular velocity of the rotor, WO the synchronous velocity, and 6 power angle; H and D are the inertia constant and mechanical damping coefficient, respectively. Note that o and 00 in the unit of electrical radians per second; 6 is in electrical radians.

In order to emphasize the parameter variation during transient, a set of incremental equations are derived based on Eqs. (1) through (7). Neglecting the high order terms and viewing the parameters as constants within each incremental interval, a set of incremental equations are derived

In Eq.(9), AT = T, - T, and in Eq. (12) x& = -xd ido + Xad ifdo (id0 and ifdo are the initial values of id and ifd before a transient). Expressing Eq. (8) in a standard state variable matrix form, it follows that

P A X = A2AX + B2 AV (13)

where A2 = -Aiv1 B1 and B2 = A1-l

B. Discrete Difference Equation

To facilitate adaptive parameter estimation with a digital system, it is necessary to transform the continuous state variable model of Eq. (13) into a discrete difference equation form as follows

X (k) = A3 X(k -1) + B 3 V(k) (14)

m0

where A3 = xi (A2 . At)i (15) (mo=1,2, ...) i=l

202

In fiqL(14) and (15), "k" is the instant of concern and At the discrete time-s- that . the following two equations also hold.

A 2 = M C M - l (17)

B 2 = ( A g - I ) - ' A z B j (18)

where M is a matrix consisting of eigenvectors of A3, and C a diagonal matrix consisting of eigenvalues of A3. Eqs. (17) and (18) are used when A2 and B2 are computed from matrices A3 and B3 which are made available from estimation process.

3. Adaptive Estimation Algorithm

A block diagram of the proposed adaptive parameter estimation configuration is shown in Fg. 2. The novel synthesized information factor (SIF) adaptive estimation algorithm, discussed in detail in the companion paper, is used.

Fig. 2 Block diagram of adaptive estimation configuration

The basic concept of the SIF adaptive algorithm is that in processing the data (measured and observed currents) within a sliding window, a sequence of SIFs are used as the weight factors. Since SIFs are obtained by multiplying the correlating factors h(k,i) (characterizing correlation between two data sampled at different instants) by a proper forgetting factor hk-l (characterizing the degree of forgetting), the SIFs efficiently synthesize information contained in previous data with that of the present data. This synthesized information then is used as fhe driving force for the adaptive mechanism. Essentially, SIFs attempt to establish an optimal and balanced relationship between the history and current status of a continuous event. Mathematical derivation, physics explanation, and application of SIFs are to be found in the companion paper [SI.

4. On-Line Estimation

Following the principles and estimation process based on the novel adaptive algorithm, presented in [SI, an on-line estimation was carried out on a lOOMVA turbogenerator. The estimation system diagram is shown in Fig. 3

1 2 3 4

I 1 -field winding of excirer, 2 - exciter, 3 -field winding of generator, 4 - generator, PT - voltage transformer, CT - current tran.$ormer, L - variable inductor, K - switch, AVR - automatic voltage regulator

Fig. 3 System diagram of estimation tests

In order to create a sufficient perturbation so as to excite the inner modes of the SM and, at the same time, not affect the synchronous generator normal operation, a field excitation change shown in Fig. 3 is adopted. Before perturbation, switch K is open and the machine is in steady state. Then, K is closed rapidly to create a sudden change in excitation voltage. This action produces a reactive power perturbation. About 30% step change of Vfd is achieved on the generator by this method. Immediately following the field voltage perturbation, the transient variables, including vab, vbc, i, ib, ic, ifd, vfd and the rotor angle 6 are sampled and recorded by a PC data acquisition system. Then, the adaptive algorithm starts to estimate the parameter matrices A3 and B3. In the process, two steps are involved: 1) a self-leaming is carried out to obtain starting values of the estimated parameters; and 2) the adaptive estimation is executed to track the trajectories of the variable parameters. Prior to the system perturbation, the self-learning algorithm uses the design parameters as the initial conditions to estimate the parameters in steady state. The results of self-leaming algorithm are then used as the starting points for the variable parameter estimation. In the second step, the transient currents, voltages, power angle, and rotor speed obtained from the test and observer are used to form the inpuvoutput data. Then, solving related equations defined in the adaptive algorithm, the trajectories of parameters during the transient are obtained. The measured transient vfd and 6 are shown in FigsA(a) and (b).

180 '4

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 (a) Measured field voltage Vfd

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 (b) Measured power angle 6

Fig. 4 Transient vfd and power angle 6

2.0

q 1.6

,a 1.2 ?

3 H 0.8

," 0.4

0.0

estimated SM variable parameters corresponding to the

- - - _-_.--- ------ "d 'b. - '.

--.------- ---- -. ** - - - l d -

- - Time (sec.) 1 1 1 1 1 1 1 1 1 ~ ~ .

transient are shown in Figs. 5(a) i rough (g). -

2.0

q 1.6

,a 1.2 ?

3 H 0.8

," 0.4

0.0

. . . . . . . . . . . . . - - - _-_.--- ------ "d 'b. - '.

--.------- ---- -. ** - - - l d -

- - Time (sec.) 1 1 1 1 1 1 1 1 1 ~ ~ .

Time (sec.)

0 1 2 3 4 5 6 (b) Estimated variable parameters xq and xaq

0.10 - 0.08 -

?, 0.06

0.02

0.00 0 1 2 3 4 5 6 (c) Estimated variable armature parameter R,

203

0.15 '1

0.00 1 l 1 1 1 1 l 1 1 1 1

0 1 2 3 4 5 6 (d) Estimated variable field parameter R f d

- 1 . . . . . . . . . .

h 208 2*al 7 1 i

0.52

0.00 0 1 2 3 4 5 6 (e) Estimated variable damper parameters RI,

1.60

1.28

-

6

- ? a 6 O.% 8 0.64 -

- Fc

) Time(sec.)

0 1 2 3 4 5 (f) Estimated variable damper parameter R z ~

0.32 - 0.00

. . . . . . . ' 1 7 . .

0. io F

0.00 1 1 1 1 1 1 1 l ~ ~ ~

0 1 2 3 4 5 6 (g) Estimated variable damper parameter R l d

Fig. 5 Estimated variable parameters

As seen from the above figures, in a negative step change of field voltage, Xd, xq and Xaq show small changes, but %d shows a larger change. Note that since the armature leakage reactance Q = Q -$d, the leakage reactance in d-axis has changed substantially. The large change of the armature leakage reactance can be understood by observing the intemal flux distribution during the transient. That is, the transient currents tend to alter the flux path in such a way that

204

more flux lines go through the air instead of through the iron. This phenomenon is more apparent along the d-axis since in steady state nearly all flux lines go through the iron core. As soon as the transients start, part of flux lines are pushed out of the iron core into the air, resulting in an increased xs.

Another important feature of the estimation results is that the d- axis resistances,Ra, Rfd, and R i d , show large changes, initially increasing and then decreasing (impulse change). The damper resistances in q-axis, RIq and R2* are also in a large change initially and then stay larger (step change). All these changes correspond to the change in power angle.

The large changes (about eight times the initial value) of the winding resistances can be explained as follows. When a step change (30%) of excitation voltage is imposed to the field, substantial currents are induced in all windings coupled to the field. These currents are in such a direction and rate that they are firmly against the attempted change of the magnetic field. Therefore, the di/dt of the induced currents is very high, proportional to that of the perturbed field voltage. A detailed spectrum analysis [7] shows that the equivalent frequency of the rapid perturbation is more than 500 Hz, starting from 60 Hz. Considering the strong skin effects created by the high frequency currents through the windings, it is not surprising to see that resistances of all windings increase dramatically. Simple calculation also shows that resistance, taking skin-effect into account, increases approximately proportional to the increase of operation frequency. This explains well that when the frequency of the magnetic field and the induced currents vary by 8-10 times, the resistance correspondingly vary by 8-10 times. It is interesting to observe that when the transient and sub-transient end, the stator and field resistances return to their steady-state values. However, the damper resistances keep their transient values. In fact, when the transient ends, the damper resistance in steady state is no longer estimated by this algorithm. Fortunately, it is not significant to know the damper resistance since before and after any transients (i.e. in steady state), there is no currents passing through the equivalent damper winding.

Several similar transient testing and parameter estimations have been made at different steady-state operation points and similar results have been obtained. These results all show that the reactances in the d-axis and the resistances (i.e. R,, Rfd, Rid, RI¶, and R q ) are very sensitive to the transients of a SM.

5. Parameters Function and Experimental Verification

The ultimate goal of parameter estimation is to establish an accurate dynamic model based on estimated variable parameters for predicting SM behavior in a power system. To this end, parameter trajectories were estimated in three operating conditions l.OPn, O.7Pn and 0.5Pn (P, is the rated output power), and are formulated into parameter functions corresponding to the operating conditions.

A. Parameter Functions

Without loss of generality, the estimated variable parameters are divided into two parts: the fist part concerns steady state operating condition and the second transient operating condition. Both parts are functions of operating conditions (V, I, cos@, Ifd. 6, 0 and their increments):

Z = Z , + Z t (19) For simplicity, only the airgap voltage AEg, armature current AI, and

slip speed A O are employed as the arguments of the parameter

functions, accounting for saturation, temperature variation, and eddy current effects in transient. The parameter functions are approximated by polynomials:

m l m2 m3

j=l j=l j=l z =Q + a0 + C a l j A E ~ + A i + c a 3 j AJ

m l m2 m3 ~ ...-

z =Q + a0 + C a l j A E ~ + A i + c a 3 j AJ j=l j=l j=l

where m i , m2 and m3 are the maximum order of each operating argument (i.e. AE& AI, and Am), which are automatically determined in regression procedure and depend on the behavior of each argument in transient; and (Q, a l j , a2j, a3j) are the coefficients to be determined by the regression method. Thus the results, a group of parameter functions (include Xd, Xad, $, X q Xffd, x i id , Xllq. X22q, Ra, Rfd, Rid, R I q and Rzq), are found by using the regression method. For example, Xad and Xaq are found as functions of operating arguments are:

Xad = XadO + 0.031 - 2.134 - 41.7 &g2+ 549.3 - 14O1AEs4- 5.97AI + 37.9A12 - 103.7 AI3 + 52.6 Aw (21)

xaq= Xaqo + O.OOO9 - 1.5 AE6 + 5.03 L\E62 + 0.44 AI - 0.27A12 + 1.68 Aw (22)

where XadO and XaqO are the steady-state components. The other parameter expressions are also obtained in the same manner, but not listed to save space. Note that the parameter functions may also be expressed in other formats for convenience, depending on the application needs. The coefficients in Equation (20) are derived from three excitation perturbation transients with different steady-state operating points (1 .OPn, 0.7Pn and OSP,). The perturbation magnitudes are 25%, 30%. and 40% of Vfd, which covers a large range of operating conditions. The selection of a step excitation change as the perturbation for the adaptive estimation is based on the fact that without disturbing the normal operation of the power system, this perturbation can stimulate all modes of the dynamic characteristics of a SM.

B. Experimental Verification

To validate the SIF adaptive algorithm and the parameter functions, another transient test with a step change in field voltage is carried out for the same turbogenerator with operating conditions shown in Table 1.

Table 1: Operating conditions before and after transient

205

0.4 E

~ 0.3 24 - 0.2

a

a3

C O

a 0.1

0.0

The transient outputs of the turbogenerator (id, i ifd, and 6) are 9' compared with those from computer simulation using the parameter functions estimated. The results are shown in Fig. 6.

- - - - - - - -

Time (sec.) 1 1 1 1 1 1 1 1 1 1 ,

. . . . . . . . . . .

o . 0 5 1 7 0.00 1 1

'?1 -0.10

-0.15

-0.20 0 1 2 3 4 5 6

.4

d

?

a ?

z zl v

- Simulated

0.20 - c' 0.15 - ? 2 0.10 -

0.05 -

Time (sec.)

0 1 2 3 4 5 6 -0.05 1 1 1 1 1 1 1 1 1 1 1

0.20;

0 1 2 3 4 5 6

0.5 I- 4

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0.20 1

0 1 2 3 4 5 6

. . . . . . . . . . 0.5 1 ' 1

0 1 2 3 4 5 6

Fig. 7 Dynamic responses from experimental testing and from computer simulation with designed parameters Fig. 6 Dynamic responses from experimental testing and from

computer simulation with estimated parameters

It is very encouraging to observe that once the parameter functions of a SM is obtained, the dynamic behavior of a SM can be predicted

6. ~~~~~~~i~~~

quite accurately by computer simulations. As a c o m P ~ ~ s o n ~ the results from simulations using designed values of the parameters are presented against the actual responses from the tests in Fig. 7. The ~ S ~ U r i ~ in using the behavior of a SM by computer simulation is obvious from these results.

The trajectories of the variable parameters of a SM under step excitation change are first shown. T~~ of the three problems concerning variable identification of S M ~ , namely, tracking the trajectories of the variable parameters according to operating conditions and Fmding the proper parameter functions for applications, are the target of this paper. Based on the concept of SIF

Parmeters to Predict

206

(optimally combining forgetting with memorizing effects), a new adaptive estimation algorithm is developed to obtain the trajectories of variable parameters. The parameter functions can be found in terms of the trajectories by a conventional regression method. The parameters obtained by this new method have been successfully applied to predict transient behavior of an actual lOOMW turbogenerator. The output responses of simulation employing parameter functions are in a very good agreement with the experimental testing of the actual SM. At present, the on-line adaptive estimation method is favorably accepted by utility companies for power system stability calculations and analysis in PR China. Since the estimated parameters are based on the general SM model of eighth order, other dynamic behavior of the SM, such as the transient after a sudden load change, can also be predicted satisfactorily. It is recommended that the estimated SM parameters be used as an unseparated group in computer simulation in a properly selected SM model structure (an eighth order model in this paper) for accurate prediction. This is because the SIF adaptive estimation algorithm is conducted at the system level and generates a set of estimated parameters for a completed description of the SM. Therefore, a mixture of any individual estimated parameters obtained by SIF method with those obtained by other methods may not produce meaningful results if it is used in a computer simulation. The authors believe that the application of the adaptive estimation algorithm described in the paper is just a beginning, and more applications such as real time control of a SM can be processed. The results of combining the on-line adaptive estimation with real-time adaptive control will be presented in future papers.

References P. L. Dandeno, D. H. Baker, et al., "Current usage and suggested practices in power system stability simulations for synchronous machines", LEEE Trans. on EC-1, No. 1, pp. 77- 93,1986 E. Eifelberg and R. G. Harley, "Estimating synchronous machine electrical parameters from frequency response tests", IEEE Trans. on EC-2, No. 1, pp. 125-132,1987

L. X. Lee and W. J. Wilson, "Synchronous machine parameter identification: a time domain approach", IEEE Trans. on EC-3,

Z. M. Zhao, F. S. Zheng, J. D. Gao and L. Y. Xu, "A dynamic on-line parameter identification and full-scale system experimental verification for large synchronous machines", IEEE Trans. on EC-10, No.3, pp. 392-398, Sept. 1995 C. X. Mao, J. Tan, 0. P. Malik and 6. S . Hope, "Studies of real-time adaptive optimal excitation control of synchronous generator and power system stabilizer", LEEE Trans. on EC-7, No. 3. pp. 598-605, Sept. 1992

A. M. Farhoud, et al., "Adaptive enhancement of synchronous generator stabilizer performance using parameter optimization technique", IEEE IAS, Vol. 1 1993, 147-154 Z. M. Zhao and F. S . Zheng, "Model structure identification with SVD for synchronous machines", Proceeding of International conference on EMASM, Zurich, Switzerland, pp. 86-89, August 1991 Z. M. Zhao, L. Y. Xu, and J. G. Jiang, "On-.line estimating variable parameters of a synchronous machine by a novel adaptive algorithm -- principles and procedures". IEEE PES Summer Meeting, July 1996

N0.2, pp. 241-248, 1988

Longya Xu was bom in Hunan, China. He graduated from Shangtan Institute of Electrical Engineering in 1970. He received the B.E.E. from Hunan University, China, in 1982, and M.S. and Ph.D. from the University of Wisconsin-Madison, in 1986 and 1990 both in Electrical Engineering.

From 1982-1984, he worked as a researcher for linear electric machines in the Institute of Electrical Engineering, Sinica Academia of China. Since he came to the U.S., he has served as a consultant to several industry companies including Raytheon Co., US Wind Power Co., Pacific Scientific Co., and Unique Mobility Inc. for various industrial concerns. He joined the Department of Electrical Engineering at the Ohio State University in 1990, where he is presently an Associate Professor. Dr. Xu received the 1990 First Prize Paper Award in the Industry Drive Committee, IEEEDAS. In 1991, Dr. Xu won a Research Initiation Award from National Science Foundation for his research project "A High-efficiency, Low-cost Flexible Variable Speed Wind Power Generating System." His research and teaching interests include dynamic modeling and converter optimized design of electrical machines and power converters for variable speed generating and drive system. He is an active IEEE member, currently serving as the Secretary of Electric Machine Committee of IEEE/IAS and an Associate Editor of IEEE Transaction on Power Electronics.

Zhengming Zhao received the B.S. and M.S.degrees from Hunan University,PR China in 1982 and 1985 respectively, the Ph.D. degree from Tsinghua University, PR China in 1991, all in electrical engineering.

He worked in the Dept. of Electrical Engineering, Hunan University during 1985-1987 as a lecturer. He is an Associate Professor in Tsinghua University since 1993 and currently on leave in the Department of Electrical Engineering, The Ohio State University as visiting scholar. His areas of interest include parameter identification and signal processing, transient analysis of power system, and design, analysis and control of electric machines.

Jianguo Jiang received B.S. and M.S. degrees from Tsinghua University in 1961 and 1965, respectively, all in Electrical Engineering.

From 1965-1988, he worked in the Dept. of Electrical Engineering of Tsinghua University as a lecturer and Associate Professor. In 1989, he was a visiting scholar at the Toronto University in Canada, working on fault diagnosis of electric machines and digital signal processing. He is currently a Professor in the Dept. of Electrical Engineering of Tsinghua University. His areas of interest include machine fault diagnosis and dynamic analysis, signal processing, design and control of electric machines.

207

estimation scheme, if they seriously believe that their scheme is statistically consistent and unbiased, wen when assuming a perfectly white measurement noise.

Discussion

[This is a combined discussion of the companion papers, “On-Line Estimation of Variable Parameters of Synchronous Machines Using a Novel Adaptive Algorithm - Principles and Procedures” (96 SM 356-6) by Z. Zhao, L. Xu, and J. Jiang and “On-Line Estimation of Variable Parameters of Synchronous Machines Using a Novel Adaptive Algorithm - Estimation and Experimental Verification” (96 SM 355-8 EC) by L. Xu, Z. Zhao, and J. Jiang.]

I. Kamwa (Hqdo-Qudbec, IREQ, Varennes, Qudbec, Canah J3X ISI). and A. Keyhani (The Ohio State University, Lkpartment of Electrical Engineering, Columbus, OH 43210). The authors have used test data for confirming their proposed parameter estimation approach. Nevertheless, their two companion papers rise a number of questions both from the theoretical view point and practical use of their scheme. We will discuss these issues in the following:

I- Development of the Novel Adaptive Algorithm (SM 96 356-6 EC) El) What the authors have rederived is just yet another form of the recursive weighted least-squares previously published in well known scholar books [A-C]. Hence, using a more conventional formulation, let denote by S, the j-th parameter vector, fi the corresponding measurement vector, A, the observation matrix and W, a suitable diagonal weighting matrix. The fundamental equation (9) of SM96 356-6 EC then takes the form:

This is exactly the equation (4.10) on p. 52 in [A]. Replacing W, in (A) with the correlation matrix r,given in eq. 13 of SM96 356-6 EC obviously yields a specific version of the batch weighted least-squares algorithm, which can be made recursive in a straightforward manner as outlined in [A] (p. 52-58). In the same reference, the resulting algorithm was called sequential weighted least-squares (WLS), and the proof for its convergence was rather complete. The authors should clearly state what is novel in their algorithm, comparatively to the conventional sequential WLS with a forgetting factor. If the time-varying weighting matrix TJ(k) is the only significant difference, then the rather clumsy demonstration extending from eq. (16) to (30), and including theorems 1 and 2 in appendix was irrelevant for a Transactions paper, just because more readable proofs are easy to find [A-C].

1-2) The Algorithm development fails to show that the estimated parameters will always be biased if measurement noise is present, as is normally the case:

where Qp is the true value of the j-th parameter and n, the noise penetrating in the j-th estimation channel. Refereeing to theorem 1 in appendix of SM96 356-6 EC, the two criteria ensuring uniform convergence of the algorithm to the true value Qp are: model noise n, is (1) a zero mean-value sequence and (2) independent of the data matrix Hi(k). While the former condition is already difficult to ensure practically, due to unavoidable biases in transducers for instance, the latter condition is simply never verified by the proposed method: since HF) always involves some terms already present in H,&-I) and hence, dependent upon nf i - l ) , it follows that the two sequences HF) and nfi) are necessarily correlated with each other. This result is well known [A,F] for ARMA models, and has motivated in the past, advanced schemes such as Generalized Least Squares [A, p.58-611, Instrumental Variables [D] and Recursive M a x i ” Likelihood @?I, which are statistically much more efficient than the simplistic sequential weighted least-squares [A] used in the papers under discussion. These discussers will appreciate if the authors could outline a rigorous proof of the unbiaseness of their

ZJ(k)= H J ( k @ , O +nJ(k)

11- Experimental Verification of the Algorithm (SM96 355-8 EC) 11-1) According to our understanding, the authors first estimate the discrete matrices A3 and B,, then equations (17-18) are used to find the continuous system matrices AI and BI which contain explicitly the desired synchronous machine parameters. Equation (17) is certainly income4 since: A3=erp(A2*T), where T is the sampling period [A]. It seems by this reasoning that the matrix C should involved the logarithm of the eigenvalues of A3, not these eigenvalues themselves. The authors should check our claim and provide in their closure, suitable modifications to ~uat ions (17-18).

11-2) One significant shortcoming of the proposed identification scheme comes from the necessity to observe indirectly the 1 1 1 state of the system. If for instance, 6, W , id, iq and ifd are the only state variables measured in the field, we need to recollstruct ild, ilq and i2q indirectly. This is a rather complicated task, which may requires a sophisticated algorithm on its own [GI. In this context, Figure 2 in SM96 356-6 EC shows a black box called Observer Algorithm which is not described elsewhere. More specifically, there is no explanation of the two symbols XI and X2 used to reconstruct the actual state vector X, which is compared to the predicted value 2 in order to generate the error feedback in the adaptive identifier.

We feel that there are many findamental problems with the proposed scheme since the method is based on designing an adaptive observer to estimate the unmeasurable damper currents that are used to estimate the damper parameters and the rest of the machine parameters recursively. We would appreciate the authors comments on the design steps for construction of their observer which is the key to their parameter estimation methcd.

11-3) In SM96 355-8 EC, attempt is made to estimate Xd and Xd, thus trying to obtain the leakage and magnetbng reactances simultaneously: this is known, since Canay in 1969 m, to be an ill-posed identification problem, which violates the parsimony principle and thus giving rise to an infinite number of admissible solutions. Hence, Xad is reduced significantly on Fig. 5, while the machine is evolving towards a much less saturated state. By contrast, Xd tends to increase (the correct thing), which means that Xd-Xad is shaped like Ra (Fig. 5). If this is true (as explained by the authors), therefore, any currently used saturation model, generally based on the so-called saturation “factors”, is deemed wrong I We would like the authors to comment these observations.

11-4) It is an established fact that the states of a dynamic system is function of time and if the system is nonlinear then the Parameters of the system are also function of states of the system. For a synchronous machine, the machine parameters are function of excitation, operating temperature and saturation. Based on these observations, the authors arc presenting the following time varying parameters from their test data due to a transient condition given in Table 1: 1. The armature resistance Ra, is changing as a function of time from .O 1 pu at time step 1.5 sec to 0.08 pu at time step 3.0 sec. and back to .01 pu at time step 5.0 sec. for total transient duration of 3. sec. as shown in Fig.5 (c). 2. The trajectory of the damper parameters Rlq, R2q, and Rld are changing in time frame of 3.0 sec. as shown in Fig. 5 (e)-&). 3. The trajectory of Xq and Xd parameters are constant for the same operating condition. For the case presented, the authors indicate that the machine k not saturated and thus the parameters Xd and Xq are essentially constant and thus the machine core losses are also constant and it will not be a factor in operating temperature. The authors are suggesting that the change in machine parameters are due to harmonic frequencies that are present in the disturbance signal. One would expect that the dynamic change in system losses (i.e. resistances) will result in corresponding change in

208

operating temperature. It is hard to believe that the machine operating temperature would change and retum to the same value in 3.0 sec?. We believe this is not physically possible. The harmonics present in the perturbation signal are in quasi-steady-state (rectifier operation) and they should not contribute to large time varying losses as presented

III- General comments (SM 96 356-6 EC & SM96 355-8 EC) 111-1) We wonder if a synchronous machine is a system so stochastic or uncertain that it requires an adaptive identification algorithm providing sequences of time-varying inductances, resistances and so forth. To us, the answer is no: It is just a huge static amount of iron, cupper and other stuff with an electrical behavior which evolves very slowly through time. The bottom line here is non-linearity, not time-dependency. Therefore, the suitable solution is not adaptation through time, but good functional models of the non linearity. It is quite interesting that after all, the authors used at various places the concept of functional non linear models to represent the relationships between the machine parameters and its state-variables. If equations of the type (35) and (36) in SM96 356-6 EC or (20-22) in SM96 355-8 EC can describe the machine properly, why not attempt to determine their static parameters directly for various operating points, without first using an adaptive estimator to build transient trajectories and then performing a multivariate polynomial regression? In this context, we direct the authors' attention to a recent related work m. Several questions from the discussion of R.P. Shuh et al. in m applied as well to the papers under discussion, and need therefore, further consideration by the authors.

111-2) Another problem with the SIF is its poor performance when the excitation is not persistent. In such situations, the prediction residuals (unforhmately not shown in the paper) cannot be white, which violates one of the basis hypotheses require by the SIF to function properly. In fact, the damper parameters block at arbitrarily values once the transient Vanished (section 4, SM96 355-8 EC). The given explanation is that the steady-state parameters of the dampers are irrelevant. Nevertheless, the fact that they are undetermined at the end of the estimation process brings serious concerns about the uniqueness of all the other parameters. To sum up, the proposed method seems to provide at convergence a black-box, so- called fitting model only; not our favorite model, which uniquely descr ik the machine behavior and must be invariant whatever the test signal is, as long as the operating domain covered remains unchanged. It will be helpful if the authors can provide a table showing in terms of standard dynamic constants (T'd, T"4 ...), the design parameters for the lOOMva turbogenerator, along with initial and final parameters for one typical run of their algorithm.

111-3) To finish, these discussers want to acknowledge the important work invested by the authors of in order to perform successfuly the extensive set of field tests reported in the two papers. Especially, their test bed which generates an almost purely reactive transient is quite onginal.

However, we feel that a fundamental issue in modeling of dynamical system has been ignored by the authors: Model Structural Identification, i.e., both the linear and saturated model structure. Since the authors model structure is fixed and the saturation is ignored, and the authors have required match between the test data and the model, they have obtained a match by allowing the parameters to change as a h c t i o n of time without regard to physical requirements of the machine. Naturally, with a such large degree of freedom, match can be obtained between the test data and t h e -varying model parameters; however, the results are meaningless. The confirmation of their time-varying model with the test data does not indicate the correctness of the proposed procedure.

R~fer~nces [A] N.K. Sinha, B. Kmzta, Modeling and Identification of Dynamic Systems.

p] L. Ljung, T . Soderstrom, Theory and Practice of Recursive Identification.

[C] T.C. Hsia, System Identification - Least-Squares Methods, Lexington

Van Nostrand and Reinhold, New York, 1983.

The MIT Press, 1983

Books, 1977.

p] P.C. Young, "An instrumental variable method for ml-time identification of a noisy process", Automatica, 6, 1970, pp.271-287.

[E] R Hastings-James, M.W. Sage, "Recursive Generalized-Least-Squares Procedures for On-Line Identilication of Process Parameters," Proc. IEE, 116, December 1969, pp. 2057-2062 J. Gertler, C. Banyasz, "A Recursive (on-line) Maximum Likelihood

Identification Method,'' IEEE Trans. on Automatic Control, AC-19, 1974, pp.825-830.

[GI H. Tsai, A. Keyhani, J. Demcko, RG. farmer, "On-Line Synchronous Machine Parameter Estimation from Small Disturbance Operating Data," IEEE Trans. on Energy Conv., EC-IO(I), March 1995, pp.25-36.

N I.M. Canay, "Causes of Discreapancies in calculation of Rotor Quantities and Exact Equivalent Circuits of the Synchronous Machine," IEEE Trans. on Power App. and Syst., PAS-88(7) July 1969, pp. 11 14-1 120.

m A. Keyhani and S. Miri, "Observers for Tracking of Synchronous Machine Parameters and Detection of Inciepient Faults, " IEEE Trans. on Energy Conv., EC-1(2), June 1986, pp. 184-192.

Manuscript received September 3, 1996

Z. Zhao, J. Jiang (Dept. of Elec. Eng, Tsinghua University, Beijing, 100084, China) and L. Xu (Dept. of Elec. Eng., The Ohio State University, Columbus, OH 43220) :

The authors wish to thank the discussers for their interests in the papers since it gives us an opportunity to highlight our contributions to the subject. To clarify the discusser's confusion and misunderstanding, the closure follows the sequence of the discussion.

1-1: With regard to the comments of "rather clumsy demonstration...", we would like to emphasize that the form of Eq. (9) is superficially similar to a conventional recursive weighted least-square (RLS, called WLS by the discussers) equation, but the concept and behavior of the weighting factor of Eq. (9) is totally different from those used in any conventional RLS equations. Upon reviewing the conventional RLS, we pointed out its serious drawbacks, including overshoot, one-sided forgetting factor, poor tracking ability, etc. It is for overcoming the disadvantages of the conventional RLS method that we developed the novel synthesized information factor (SIF) based adaptive algorithm. In effect, the SIF based algorithm includes many new items, such as the concept of SIF which establishes an optimal and balanced relationship between history and current status for a continuous event, the derivation of the expression of SIF (Eqs. 2 and 3), the appropriate selection of an sliding time window, the application of correlation factor in the §IF algorithm, the proper usage of a forecasting gain matrix K'(k), ... To our best knowledge, all the aforementioned items have not appeared on any SM parameter estimation literature. Could discussers compare our proposed algorithm with the mentioned books in more detail to justify their comments?

1-2: In Theorem 1 of the appendix, the discussers can find, if they will, that the uniform convergence proof is provided with the assumption that the model is subject to a zero mean-value noise sequence and independent of data Hk, If other types of noise, not white noise, are present, the SIF will reject them adaptively because the SIF only relates the correlated signals to each other, forgetting the others. Actually, this is one of the salient features exclusively associated with the SIF based algorithm, not a simplistic sequential WLS algorithm that the discussers referred to.

209

short), and different study objectives (stability analysis or routine monitor), a SM should be expressed by different model structures. The model structure selection criterion is also presented in terms of measured state variables. The model chosen in the papers is for purpose of illustrating the principle and effectiveness of the SIF based adaptive algorithm.

As for saturation effects, at least two methods are equally well suited: one is from the model structure point of view, and the other from parameter adaptation point of view. Parameter estimation belongs to the latter. Since the estimated parameters are based on the measured inputs and output variables, saturation and other nonlinearities are automatically included. While questioning the experimental verification for their "meaningful" and "correct" results [5-161, we feel particularly confident about the approach of our experimental verification presented in the paper: a SM at lOOMVA operated in a full-scale large power system is used and consistent results are obtained.

11-1: In Eq. (17), we stated in the paper that "C" is a diagonal matrix consisting of the eigenvalues of "A3". This statement means that "C" is related to the eigenvalues of "A3". The form of the eigenvalues appear in "C", of couse, can be either logarithm or exponential, depending on the match of the eigenvalues with respect to the corresponding eigenvectors in matrix "M". These are fundamental concepts of linear algebra which, we hope, are familar to the discussers.

11-2: In a previous work, published in 1992, we have developed an effective adaptive observer algorithm for the state variables which are not measurable. As we presented during the oral discussion, the discussers are referred to [4] for details.

11-3: For a step change of excitation voltage as described in the paper, the relative positions between the stator MMF and the rotor field will change (the power angle change verifies this). Therefore, the flux distribution inside the machine will change correspondingly. The variations of Xd-Xad during transients are the terminal reflection of the flux distribution change inside the machine. It should be noted that the SIF adaptive parameter estimation is conducted at system level and generates a set of parameters for a completed description of the SM machine. We need to treat the estimated parameters as a group in applications.

11-4: The discussers are obviously confused that the changes of temperature and resistance are two different concepts and could be relatively independent. As well- known, temperature-rise corresponds to energy accumulation, generally with a large time constant, while the resistance change could be purely a result of current distribution change which occurs almost instantaneously in transient. Note that due to skin effect, the transient currents flow through a cross-section which is much smaller than that when the transient is over. Bearing these fundamentals in mind, the discussers will not have difficulty in understanding that the resistance could change very rapidly, de coupled from a slow change of temperature.

111-1: The work of our two papers is an attempt to obtain a more accurate model accounting for nonlinearities of a SM. The variable parameters are not, as the discussers claimed, functions of time, but functions of operating conditions although we have used the time-domain disturbance and responses in the process of estimation.

111-2: As discussed in 11-2, we have developed an effective method to estimate the steady state parameters. The SIF based algorithm proposed in the two papers is used to obtain the parameter trajectories in transient. Combining the transient trajectories with the steady state values, we are able to obtain satisfactory parameter functions for a wide range of operating conditions. As to the comparison of estimated and designed values, we have provided a number of examples in [3].

111-3: While using a model structure with one damper circuit in d-axis and two dampers in q-axis, we did not lower the importance of model structure selection. In fact, in a related work [4], we have specifically developed a method for model structure determination and concluded that for different machines (with salient or round rotor), different operating modes (three-phase or single-phase

Closing Remarks

After clearing up the above confusion and misunderstandings raised by the discussers, we feel that the rest of the discussion is full of biased and unsubstantiated comments, much reflecting the discussers' resent upon the work not fitting to their tastes. Regretfully, we are not impressed. At this point, we would like to share a point of view in general with the discussers: any SM models thus far with lumped circuit parameters is but a subjective understanding of the physics of an objective SM. Neither the discussers nor the authors have and will exhaust abundant research topics and stop enthusiastic exploration for the SM, such a wonderful piece of art. Comparing with the work in [5-161, we feel encouraged by the accomplished results which are not a trivial repetition, slight variation of one from another, or supported by testing on toy-size machines. We welcome any serious, professional discussion. Nevertheless, we will not give up our efforts to pursue the truth in the area of our interests because of unwarranted claims.

References:

[I]. C. Fan and D. Xiao, "System identification", Science and Technology Press, Beijing, 1987

[2]. J. S. Bendat and A. G. Pierco, "Engineering application of correlation and spectral analysis", Wiley, New York, 1980

[3]. Z. Zhao, F. Zheng, J. Gao, and L. Xu, "A dynamic on- line parameter identification and full-scale system experimental verification for large synchronous machines", IEEE Trans. on EC, Vol.10, No.3, Sept. 1995, pp. 392-398

[4] Z. Zhao and F. Zheng, "Model structure identification with SVD for synchronous machines", Proc. of Intern. Conf. on EMASM, Zurich, Switzerland, August 1991,

[S]."Identification of synchronous machine linear parameters from standstill step voltage input data," IEEE Transactions on Energy Conversion,Vol.9, April 1995 (A. Kumageanian, A. Keyhani)

pp.86-89

210

[ 61 ."Estimation of induction machine parameters from standstill time domain data," IEEE Transactions on Industry Applications," NovDec 1994. (S.I. Moon, A. Keyhani)

[7]."Identification of high order synchronous generator models from SSFR test data," IEEE Transactions on Energy Conversion, Vol. 9, March 1995 (A. Keyhani, J. Tsai)

[8j."Estimation of induction machine parameters from standstill time domain data," IEEE Transactions on Industry Applications," Vol. 30, No. 6, Nov/Dec 1994, pp.1609-1615. (S.I. Moon, A. Keyhani)

[9]."Maximum likelihood estimation of synchronous machine parameters from stand-still time response data," IEEE Transactions on Energy Conversion , Vol. 9, pp. 98- 114, March 1994. (A. Keyhani, H. Tsai, T. Leksan)

[ lO]."Maximum likelihood estimation of synchronous machine parameters from flux decay data," IEEE Transactions on Industry Applications , Vol. 30, No. 2, 433-439, March/April 1994. (A. Tumageanian, A. Keyhani, S.I. Moon, T. Leksan, L. Xu)

[ 1 l]."Maximum likelihood estimation of synchronous machine parameters and study of noise effect from flux decay data," IEE Proceedings Part C , Vol. 139, No. 1, pp. 76-79, January 1992. (A. Keyhani, S. I. Moon)

[12]."Synchronous machine parameter identification," Electric Machine and Power Systems , Vol. 20, pp. 45-69, 1992. (A. Keyhani)

[ 13]."Maximum likelihood estimation of generator stability constants using SSFR test data," IEEE Transactions on Energy Conversion , Vol. 6, No. 1, pp. 140-154, March 1991. (A. Keyhani, S. Hao, R. P. Schulz)

[ 14]."Maximum likelihood estimation of transformer high frequency parameters from test data," IEEE Transactions on Power Delivery, Vol. 6, No. 2, pp. 858-865, April 1991. (A. Keyhani, S. Chua, S. Sebo, H. Tsai)

[ 15]."Maximum likelihood estimation of high frequency machine and transformer winding parameters," IEEE Transactions on Power Delivery , Vol. 5 , No. 1, pp. 212- 219, January 1990. (A. Keyhani, H. Tsai, A. Abur)

[ 16]."Maximum likelihood estimation of solid-rotor synchronous machine parameters from SSFR test data," IEEE Transactions on Energy Conversion , Vol. 4, No. 3, pp. 551-558, September 1989. (A. Keyhani, S. Hao, G. DaYd)

Manuscript received November 5, 1996.