An Observer for the Estimation of Synchronous ...

An Observer for the Estimation of Synchronous Generator Damper Currents for Use in Parameter Identification Elias Kyriakides and Gerald T. Heydt Published in: IEEE Transactions on Energy Conversion Publication date: 2003 Document Version: Peer reviewed version Digital Object Identifier (DOI): 10.1109/TEC.2003.808413 Citation for published version (IEEE format): E. Kyriakides and G. T. Heydt, “An observer for the estimation of synchronous generator damper currents for use in parameter identification,” IEEE Transactions on Energy Conversion, vol. 18, no. 1, pp. 175-177, Mar. 2003. General rights: Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.  Personal use of this material is permitted. Permission from the copyright owner of the published version of this document must be obtained (e.g., IEEE) for all other uses, in current or future media, including distribution of the material or use for any profit-making activity or for advertising/promotional purposes, creating new collective works, or reuse of any copyrighted component of this work in other works.  You may freely distribute the URL/DOI identifying the publication in the public portal. Policy for disabling free access: If you believe that this document breaches copyright please contact us at [email protected] providing details, and we will remove access to the work immediately and investigate your claim.

IEEE Transactions on Energy Conversion, Vol. 18, No. 1, pp. 175-177, Mar. 2003.

An Observer for the Estimation of Synchronous Generator Damper Currents for Use in Parameter Identification E. Kyriakides, G. T. Heydt Author Affiliation: Arizona State University, Tempe AZ, USA. Abstract: The main objective of this letter is to design an observer for synchronous machine damper currents. The observer is used in a parameter estimator to identify synchronous machine parameters from on-line measurements. Possible internal machine fault conditions can be detected and remedial action can be undertaken. Keywords: damper currents, observer, on-line measurements, parameter estimation, state estimation, synchronous generator. Introduction: Synchronous generator parameter identification is a problem that has attracted the attention of many researchers. Knowledge of the operational parameters of generators is necessary for performing accurate stability studies and post mortem analysis of power systems. Additionally, anomalous machine operation might be identified through machine parameter estimation. Traditionally, synchronous machine parameters are obtained by off-line tests as described in the IEEE Standards [1]. However, off-line methods may not be practical and parameters obtained by these methods may not be accurate. Decommiting a machine for parameter measurement may not be convenient especially if the machine is a base loaded unit. The parameters of a synchronous machine vary under different loading conditions because of changes of the machine internal temperature, magnetic saturation, and coupling between the machine and external systems. Aging is another factor that causes variation of the operational parameters. References [2-6] present various methods that have been proposed in the literature. In [7], a new approach is presented where the authors use Park’s transformation and synthetic data to estimate synchronous machine parameters employing a least squares minimization technique. The paper also demonstrates a graphical user interface (GUI) that enables fast and user friendly estimation. A significant element in the analysis of synchronous machine is the damper currents. Synchronous generators comprise a number of damper windings which are metal bars placed in slots in the pole faces and connected together at each end. A technology for direct damper current measurement has not yet been fully developed, and therefore it is desired to dynamically estimate the damper currents so that damper currents can be utilized in the parameter estimation process. Modeling of synchronous machines: In order to formulate the state estimation equation for a synchronous generator, it is necessary to employ a mathematical model which represents the synchronous generator in the conditions under study. This model will comprise three stator windings, one field winding and two damper windings as shown in Fig. 1. The generator windings are magnetically coupled. This coupling is a function of the rotor position and therefore, the flux linking each winding is also a function of the rotor position

[8]. The instantaneous terminal voltage of any winding takes the form, (1) v  ri   where r is the winding resistance, i is the current and λ is the flux linkage. It should be noted that in this notation it is assumed that the direction of positive stator currents is out of the terminals, since the synchronous machine under consideration is a generator. ia a

rF vF

iF

ra

LF

Laa Lbb

ib

rb

rD vD=0

iD

va b

LD

Lc c

rn Ln

vn

vb

rc

ic

rQ

vQ=0

iQ

c vc

LQ

n in

Fig. 1. Model of a synchronous machine

The machine equation is transformed into the 0dq reference frame via a Park’s transformation [9,10] to obtain a linear time invariant model. The development of the model in the 0dq reference frame can be seen in [7] and [11]. In its final form the dynamic model becomes, 0 0 0 0   r  3rn 0 v0   v  r Lq 0 0 kM Q   0  d  0  vq     0  Ld r  kM F  kM D   vF  0 0 0 r 0 0  F     0 0 0 0 rD 0  0    0  0 0 0 0 rQ   0

 i0  i  d  iq    iF    iD   iQ 

0 0 0 0   i0   L0  3Ln 0  0 Ld 0 kM F kM D 0   id  .     0 0 Lq 0 0 kM Q   iq     kM F 0 LF M R 0   iF   0  0 kM D 0 M R LD 0   iD     0 kM Q 0 0 LQ   iQ   0

(2)

Development of an observer for the damper winding currents: Usually, available data for synchronous generators are the stator phase currents and voltages at the terminals of the machine, and the field voltage and current. Often it is possible to measure the rotor torque angle δ, using commercially available instruments. The torque angle enables the transformation of abc quantities to 0dq quantities as they appear in (2). In order to set up the parameter estimation problem, it is necessary to have measurements for the damper currents iD and iQ. An alternative approach would have been the transformation of (2) in the form x  Ax  Bu and the use of adaptive observers to estimate both the unavailable states and the unknown parameters [12]. Observation of the synchronous generator model in (2), shows that it is possible to use the last two rows, rearranging them to obtain expressions for the damper winding currents. The parameters that are involved in this process are not operational parameters of the machine, but parameters of the damper windings. These are constant and well known from manufacturers’ data. Further, there is low

interest in estimating those parameters, and hence one can separate the two equations from the rest of the model. Rewriting the last two equations of (2), 0  rD i D  kM D i d  M R i F  LD i D (3) 0  rQ i Q  kM Q i q  LQ i Q . The current derivatives can be approximated by the forward difference formula, i (t  t )  i (t ) . (4) i(t )  t Equation (3) is rearranged in discrete form as, rD kM D M t ]i D (n) t id (n) R t iF (n) LD LD LD rQ kM Q i Q (n  1)  [1  t ]i Q (n)  t  i q (n) LQ LQ

The MSE is calculated to be 0.705 p.u., but still the observed current is in phase with the EMTP simulated current as in the steady state case.

i D (n1)[1

(5)

Equation (5) enables the calculation of the damper currents. All parameters can be accurately calculated using manufacturer’s data, while the time varying quantities are available measurements. The only ambiguity in (5) is the value of iD(0) and iQ(0). These are needed to initiate the observation process. The initial conditions can be assumed to be zero without loss of accuracy as will be shown in the two case studies in the next section. Case studies for damper winding current observer: In order to ascertain the validity of the proposed method, it is desired to perform a number of case studies comparing the estimated damper currents to damper currents generated using the Electromagnetic Transients Program (EMTP). Two case studies are presented. A synchronous generator was simulated in EMTP both in steady state and in transient mode, using parameters calculated from the manufacturer’s data for the machine. The machine under consideration is a crosscompound generator that is located at the U. S. southwest. It comprises two generators; a high pressure generator rated at 483 MVA and a low pressure generator rated at 426 MVA. In the first case study, the machine is operating nearly in steady state. The damper currents are observed according to (5) with the initial conditions assumed to be zero. Fig. 2 shows the EMTP simulated and estimated damper currents in the direct axis and quadrature axis damper windings. The only reason for nonzero damper currents is the use a starting point in the simulation that is not precisely at a steady state operating condition. The estimated current for the d-axis damper is in good agreement with the EMTP simulated current. The mean square error (MSE) of the two signals is nearly at the computer accuracy as zero. For the q-axis damper the observed state is in phase with the EMTP simulated value. There is some difference between the two states. As it will be shown later, this difference does not affect the accuracy of the estimated parameters. In the second case study, transient data were considered. A permanent line to line fault was applied at 0.25 seconds between phases b and c. The observed damper currents as compared to the EMTP simulated damper currents for each axis can be seen in Fig. 3. The direct-axis damper model seems to offer an exact observed state. The MSE is calculated to be 0.0147 p.u. on the machine base. On the other hand, the quadrature-axis damper current has a more significant error.

Fig. 2. Comparison between EMTP simulated and estimated damper currents using steady state data

Fig. 3. Comparison between EMTP simulated and estimated damper currents using transient data

Estimation of machine parameters and testing of the algorithm: Figure 4 depicts the block diagram for the

damper current observer and machine parameter identification as used in the estimation algorithm. The system model is rearranged in the form H  x  z , where H is a known coefficient matrix of dimension m  n , x is a vector of dimension n containing the unknown parameters, and z is a vector of dimension m containing known values. In this notation m is the number of measurements and n is the number of parameters to be estimated. The estimated parameters are estimated in a least squares sense. The machine parameter estimator algorithm was tested using the available steady state EMTP data. There are six parameters that are desired to be estimated in the two matrices of (2). Table 1 depicts the ‘actual’ (i.e., values used in the EMTP simulation) and estimated parameters and the percent error for each parameter. Unknown plant

V

 I 4 x1   I 4 x1  V4 x1       0    R6 x 6  i D   L6 x 6  i D   2 x1   iQ   iQ     

I iˆD , iˆQ

+ e

Observer -

Iˆ

a priori system knowledge

Parameter identification algorithm

Estimated parameters

Fig. 4. Block diagram for observer implementation and parameter identification algorithm

Parameter

Table 1. Estimated parameters using EMTP data Actual value (p.u.) Estimated value (p.u.)

% Error

r

0.0027

0.00261

3.3

Ld

1.80

1.7999

5.6x10-3

Lq

1.72

1.72009

5.2x10-3

rF

9.722x10-4

9.7994x10-4

0.8

LF

1.75791

1.746998

0.62

MF

1.33905

1.33908

2.2x10-3

It is also useful to study the effect of estimating more than one parameter at a time. This will indicate whether multiple parameter estimation is feasible and it will enable the user to avoid multiple program executions. For illustration purposes the three parameters to be estimated simultaneously are Ld, Lq and rF. Table 2 shows the estimated quantities and the percent error for each of the parameters. It can be seen that the estimated parameters and the percent error are identical to the previous case study (Table 1), where these parameters were estimated individually. This shows that more than one generator parameters can be estimated at the same time, and it will be particularly useful in case that there is uncertainty about two or more parameters.

Table 2. Multiple simultaneous parameter estimation using EMTP data Parameter Actual value (p.u.) Estimated value (p.u.) % Error Ld

1.80

Lq

1.72

rF

9.722x10

5.6x10-3

1.7999

5.2x10-3

1.72009 -4

-4

9.7994x10

0.8

Conclusions: In this letter, a method to identify synchronous machine parameters from on-line measurements is presented. The method is based on least squares estimation. An observer for identification of the unmeasurable damper currents is also developed to enable the use of the damper currents in the parameter estimation algorithm. Two case studies for both steady state and transient operation were presented. For both cases, the damper currents in the d-axis winding agreed to the simulated data. In the case of the q-axis winding, it was noted that there was a small difference between the estimated and the simulated currents, but the two currents were in phase. The generator parameters were estimated using steady state data obtained from EMTP simulations, and utilizing the observed damper currents. The percent error in the estimated parameters shows that this method is reliable and can be extended to the real data case provided that a noise filter and bad data rejection technique are utilized. Acknowledgments: The authors acknowledge the support of Arizona Public Service and of the Power Systems Engineering Research Center (PSerc), as well as the technical support of Drs. B. Agrawal, A. Keyhani, and J. Rico. References: [1] “Test Procedures for Synchronous Machines,” IEEE Standard 115, March 1965. [2] A. Keyhani, S. Hao, G. Dayal, “Maximum Likelihood Estimation of SolidRotor Synchronous Machine Parameters from SSFR Test Data,” IEEE Transactions on Energy Conversion, Vol. 4, No. 3, September 1989, pp. 551-558. [3] A. Tumageanian, A. Keyhani, “Identification of Synchronous Machine Linear Parameters from Standstill Step Voltage Input Data,” IEEE Transactions on Energy Conversion, Vol. 10, No. 2, June 1995, pp. 232-240. [4] H. B. Karayaka, A. Keyhani, B. Agrawal, D. Selin, G. T. Heydt, “Methodology Development for Estimation of Armature Circuit and Field Winding Parameters of Large Utility Generators,” IEEE Transactions on Energy Conversion, Vol. 14, No. 4, December 1999, pp. 901-908. [5] H. B. Karayaka, A. Keyhani, B. Agrawal, D. Selin, G. T. Heydt, “Identification of Armature Circuit and Field Winding Parameters of Large Utility Generators,” IEEE Power Engineering Society Winter Meeting, Vol. 1, 1999, pp. 29-34. [6] H. B. Karayaka, A. Keyhani, B. Agrawal, D. Selin, G. T. Heydt, “Identification of Armature, Field, and Saturated Parameters of a Large Steam Turbine-Generator from Operating Data,” IEEE Transactions On Energy Conversion, Vol. 15, No. 2, June 2000, pp. 181-187. [7] E. Kyriakides, G. T. Heydt, “A Graphical User Interface for Synchronous Machine Parameter Identification,” North American Power Symposium (NAPS) Proceedings, College Station, TX, October 2001, pp. 112-119. [8] P. M. Anderson, B. L. Agrawal, J. E. Van Ness, “Subsynchronous Resonance in Power Systems,” IEEE Press, New York, 1990. [9] R. H. Park, “Two-Reaction Theory of Synchronous Machines – Generalized Methods of Analysis – Part I,” AIEE Transactions, Vol. 48, July 1929, pp. 716-727. [10] R. H. Park, “Two-Reaction Theory of Synchronous Machines, Part II,” AIEE Transactions, Vol. 52, June 1933, pp. 352-355. [11] P.M. Anderson, A. A. Fouad, “Power System Control and Stability,” The Iowa State University Press, Iowa, 1977. [12] G. Lüders, K. S. Narendra, “An Adaptive Observer and Identifier for a Linear System,” IEEE Transactions on Automatic Control, Vol. AC-18, No. 5, October 1973, pp. 496-499.