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Identification of Synchronous Machine Parameters Using Load Rejection Test Data Edson da Costa Bortoni, Member, IEEE, and José Antônio Jardini, Fellow, IEEE
Abstract—This work shows a computational methodology for the determination of synchronous machines parameters using load rejection test data. By machine modeling one can obtain the quadrature parameters through a load rejection under an arbitrary reference, reducing the present difficulties. The proposed method is applied to a real machine. Index Terms—Parameter identification, synchronous machines.
LIST OF SYMBOLS
0:
Effective terminal voltage (in volts). Effective induced voltage (in volts). Windings voltage (in volts). Windings current (in amperes). Armature phases windings. Direct and quadrature axes. Field and damper windings. Transient and subtransient time constants (in seconds) Synchronous, transient, and subtransient inductances (per unit). Active and reactive power (in watts, var). Resistance (in ohms). Laplace variable. Open circuit, initial conditions and 0 axis. Power angle. Linkage fields. Angular speed (in radians per second). I. INTRODUCTION
T
HE identification of synchronous machines parameters is very important to the analysis of generation systems both in steady state and in transient processes. Within many methods which have been developed in the past decades, two of them gained prominence due to their practice, low risk level imposed to the machine under test and moreover, the quality of the obtained data. One is the frequency response method [1] and the other one is the load rejection test [2]. The frequency response test is carried out by applying currents about 0.5% of the rated current, with frequencies varying in the range from 0.001 Hz to 1000 Hz. The rotor must be prop-
Manuscript received August 27, 1997; revised February 26, 2002. E. C. Bortoni is with the DEE-FEG-UNESP, São Paulo State University, Guaratinguetá 12500-000, Brazil. J. A. Jardini is with the Escola Politécnica da Universidade de São Paulo, São Paulo 05508-900, Brazil (e-mail:
[email protected]). Publisher Item Identifier S 0885-8969(02)05405-0.
erly positioned in order to obtain the direct and quadrature axis parameters. Using the frequency response data one can evaluate the operational reactances and consequently determine the parameters and time constants currently used in power systems studies. A computer based equipment to run these tests with low costs is present in [3]. The second method, not standardized yet, also allows the determination of direct and quadrature axis parameters by executing load rejection in two special operational points, in which the components of current do exists in the axis of interest only. The load point concerning the direct axis can be easily obtained under-excitating the machine as it has been synchronized to the system. The machine must be running at negligible active power and driving a considerable amount of reactive power from the system. This procedure garantees the obtaining of nonsaturated parameters and avoids undesirable over voltages during the tests. The localization of the quadrature axis is not so trivial, since the machine being under-excitated, one must find a loading point in which the absolute value of the power factor angle is equal to the power angle . In practice it can be found after successive load rejections with different powers, aiming at minimize the field current variations. An alternative procedure is the employment of a power angle meter [4]. These difficulties can, sometimes, make this test impracticable. Nevertheless, several papers can be found in the international technical literature presenting field experience of the industry in applying the load rejection test [5]–[7]. It is possible when advanced system identification techniques are applied in such phenomenological procedures [8], [9]. Through the development of the equations which govern the armature voltage behavior during a load rejection, this work will show that the data obtained with a load rejection under an arbitrary reference is sufficient to determine the transient parameters using numerical methods. The special cases in which the components of the current do exist only in the direct axis or only in the quadrature axis will be studied.
II. TRANSIENT PROCESS MODELING The synchronous machine transient process modeling will be made in three steps. The first takes into account only the steady state, before the load rejection; the second analyzes the transient process after the rejection and in the latter, a composite of the previous results will be done in order to obtain the complete behavior of the machine under a local rejection. In these analyses, all the variables are in per unit.
0885-8969/02$17.00 © 2002 IEEE
BORTONI AND JARDINI: IDENTIFICATION OF SYNCHRONOUS MACHINE PARAMETERS
Fig. 1.
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Equivalent circuits of a synchronous machine.
A. Steady State
B. Transient Process
In a three-phase system with perfectly symmetric voltages yields
The transient process will be studied evaluating the linkage fluxes variations both in direct and quadrature axis. At the end, the initial conditions previously calculated should be added. Representing the direct axis by the equivalent circuit showed in Fig. 1(a), and introducing the Laplace variable, the direct axis linkage flux variations for a given excitation voltage is
(1)
(9)
In a dqo frame, one can obtain
For synchronous machines without damper windings and fully laminated poles (2) The armature phase voltages can be recalculated as (10)
(3) In case of balanced load and neglecting the armature resistance, the currents in dqo frame are obtained as follows:
is the armature leakage inductance and the direct Where . axis mutual inductance Considering that in the load rejection there is a negative current variation (11)
(4) The linkage flux equations are
Applying the inverse Laplace transformation, one can obtain the linkage flux variation in a transient process (12)
(5)
Taking into account the damper windings effects, yields
It is well known that (13) (6) In steady state, with no damper winding currents, balanced load, neglecting the armature resistance and taking the angular equal to 1 p.u. speed
In case of quadrature axis, the equivalent circuit of Fig. 1(b) can be adopted. Applying the same procedure used for the direct axis, results (14) When there is no damper winding in the quadrature axis
(7) (15) Using (2), one can obtain the initial conditions of the direct and quadrature linkage fluxes
(8)
C. Description of the Load Rejection Phenomena The machine behavior in a load rejection transient process can be obtained by composing the phenomena which occur before
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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 17, NO. 2, JUNE 2002
TABLE I RATED CHARACTERISTICS
and after to the load rejection. Thus, the linkage flux in the direct axis will be
(16) In the same way, for the quadrature axis
(17) With these values in the (6), taking armature resistance, one can obtain
and neglecting the
(18) and parcels represent the transformer efThe fect which occurs in the machine when the load is suddenly changed, producing nonperiodic and second harmonic components in the armature quantities. This phenomena can be neglected without including large errors [11]
These are the basic expressions to the load rejection description and they could be used to identify the synchronous machines parameters. The expressions (21) and (22) describe particular load rejections cases. The expression (20) describes the load rejection under an arbitrary condition. Thus, one can propose two procedures. • Determine all the synchronous machines parameter with a load rejection under an arbitrary condition using (20); or, • Make a load rejection test in the direct axis, using (21) to determine the direct axis parameters, and make another load rejection, under an arbitrary axis, to calculate the quadrature axis parameters with (20), using the previous calculated direct axis parameters as constants. These procedures can simplify the identification process, and can spread the use of load rejection tests. It is important to notice that the field current behavior can also be modeled and used to improve the parameter identification task. III. PARAMETERS IDENTIFICATION The synchronous machines parameters will be identified in order to minimize the error between the theoretical model (20), (21) or (22) and the experimental data obtained by test. The Levenberg–Marquardt method [10] will be employed to minimize the error function. This technique was selected because it combines the major characteristics of two other methods—the steep descent and the conjugate directions, or in other words, for large errors the stepest descent is used, but as the error decreases, the conjugate directions begin to be used. The transition with the methods is made intrinsically by the Levenberg–Marquardt technique. Another advantage of this method is that it avoids the destruction of a convergence work when a parameter runs away from its real value. The goodness function is defined as
(19) (23) Substituting (3) for (19), yields are the measured in which is the parameter vector, and and the calculated values at the instant . The parameter vector will be iterativelly calculated in order to minimize the goodness function, as follows: (20) When there is no current component in the quadrature axis and ) (
(21) On the other hand, when there is no direct axis current component, results
(22)
(24) where is a calculated vector responsible for the transition with the previous methods [10], and is the iteration counter. Naturally, it is important to notice that any identification method has its performance improved if the iterative process begins with adequate initial approximations for the unknown parameters. In this case, considering the typical voltage behavior in face of a load rejection in the -axis and in the -axis, the initial parameters could be obtained by using the basic relationships presented in [2]. On the other hand, one can observe in several textbooks that the synchronous machines parameters are within a well known range as a function of the machine characteristics.
BORTONI AND JARDINI: IDENTIFICATION OF SYNCHRONOUS MACHINE PARAMETERS
Fig. 2.
Fig. 3.
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(a)
(b)
(a)
(b)
Load rejection—direct axis.
Load rejection—arbitrary axis.
Thus, in the absence of more detailed data, one can use these information as initial conditions. This technique will be employed for the direct and quadrature axis time constants, avoiding the laborious graphical approximation. IV. APPLICATION The proposed method was applied to one rounded rotor synchronous machine, the rated characteristics are presented in Table I. It was not possible to determine the quadrature axis during the tests, thus the test in this axis could not be done. However, load rejection tests under an arbitrary axis and under the direct axis was made. Fig. 2 shows (a) the transient and (b) subtransient response for the direct axis. Fig. 3 shows the (a) transient and (b) subtransient response for the arbitrary axis.
A. Direct Axis Parameters The quantities values in the instant immediately before the rejection are MW
p.u. Mvar V p.u. A p.u.
p.u.
After the transient process, the terminal voltage became equal to the induced voltage, with its value stabilized in V
p.u.
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The identification process starts obtaining the initial guesses. Using the expressions given in [2], yields
TABLE II DIRECT AXIS PARAMETERS
p.u. p.u. p.u.
TABLE III QUADRATURE AXIS PARAMETERS
The following typical values were adopted as initial guesses for the time constants s s Applying the Levenberg–Marquardt algorithm to the expression (21), one can obtain the following refined parameters, with relative error smaller than 1%, after six iterations. It is important to notice that in this test the field voltage was kept constant, since the synchronous machine was independently excitated using a dc generator (see Table II). B. Quadrature Axis Parameters The electrical quantities values at the instant immediately before the load rejection are MW Mvar V A
p.u. p.u. p.u. p.u.
In the steady state, after the rejection, the terminal voltage was V
p.u.
In practice some care must be taken regarding the influence of the speed variation over the voltage values registered in the test, since the rotor will always be accelerated in a active load rejection. This influence can be neutralized simply dividing the voltage record by the speed value in per unit for the same instant of time. This problem practically inexist in the direct axis rejection since there is no or few active power. The saturation level is another important point to the success of the method application. It is important to maintain the machine under excitated in order to obtain unsaturated quantities. Since the studied machine was a rounded rotor type, the same initial guesses for the direct axis was adopted. Furthermore, the load angle will also be identified, its initial guess is obtained through the following approximation:
It is important to notice that this value will be grater than the true for salient pole machines, and will be very approximated for the rounded rotor ones.
Applying the proposed technique to expression (20), one can obtain the following refined parameters, with relative error smaller then 1%, after nine iterations (see Table III). V. CONCLUSION The work showed a mathematical modeling of the load rejection phenomena, allowing the parameter identification with numerical methods. This technique avoids the employment of a load angle meter, making possible its generalized use. The parameters identification explores all the transient, subtransient and steady state information, and not only specified points, reducing the errors and conducting to more representative parameters. With the use of data acquisition systems and monitoring other quantities of interest, one can obtain other useful parameters through the load rejection test and speed decay, like moment , efficiency and others. of inertia REFERENCES [1] IEEE Guide: Test Procedures for Synchronous Machines, IEEE Std. 115 A, 1983. [2] F. P. de Mello and J. R. Ribeiro, “Derivation of synchronous machine parameters from tests,” IEEE Trans. Power Apparat. Syst., vol. 96, pp. 1211–1218, July/Aug. 1977. [3] E. C. Bortoni, “Parameter Identification and Modeling of Salient Pole Synchronous Machine Using Frequency Response Methods,” Ph.D. thesis, Univ. São Paulo, São Paulo, Brasil, 1998. in Portuguese. [4] F. P. de Mello, “Measurement of synchronous machine rotor angle from analysis of zero sequence harmonic components of machine terminal voltage,” IEEE Trans. Power Delivery, vol. 9, pp. 1770–1777, Oct. 1994. [5] E. Bortoni, E. Beltrame, A. H. M. Santos, and E. K. Diederich, “Identification of characteristics and parameters replan generators groups,” GENERCO Rep., Petrobras, 1994/1995. in Portuguese. [6] P. A. E. Rusche, G. J. Brock, L. N. Hannett, and J. R. Willis, “Test and simulation of network dynamic response using SSFR and RTDR derived synchronous machine model,” IEEE Trans. Energy Conv., vol. 5, pp. 145–155, Mar. 1990. [7] T. H. Ling, T. N. Wen, T. D. Sheng, and L. J. Yu, “Mathematical model of synchronous machine in transient process,” in Proc. ICEM’88—Int. Conf. Elect. Machines, 1988. [8] I. Kamwa, P. Viarouge, and E. J. Dickinson, “Optimal estimation of the generalized operational impedances of synchronous machines from short-circuit tests,” IEEE Trans. Energy Conv., vol. 5, pp. 401–407, June 1990. [9] , “Identification of generalized models of synchronous machines from time-domain tests,” Proc. Inst. Elect. Eng. C, vol. 137, no. 6, pp. 445–452, 1991. [10] W. H. Press, B. P. Flannery, S. A. Tenkolsky, and W. T. Vetterling, Numerical Recipes in C. Cambridge, U.K.: Cambridge Univ. Press, 1988. [11] P. C. Krause, F. Nozari, T. L. Skvarenina, and D. W. Olive, “The theory of neglecting stator transients,” IEEE Trans. Power Apparat. Syst., vol. 98, pp. 141–148, Jan.–Feb. 1979.
BORTONI AND JARDINI: IDENTIFICATION OF SYNCHRONOUS MACHINE PARAMETERS
Edson da Costa Bortoni (M’94) was born in Maringá, Brazil on December 1, 1966. He received the B.S. degree from the Itajubá Federal Engineering School (EFEI), Brazil, in 1990, the M.Sc. degree in energy systems planning from UNICAMP, Brazil, in 1993, and the Ph.D. degree in synchronous machines parameters identification and modeling from the University of São Paulo, São Paulo, Brazil, 1998. He was a Professor at EFEI and Amazon Federal University. Presently, he is an Assistant Professor at São Paulo State University, Guaratinguetá. His areas of interest are power generation systems design and control, synchronous machines, and energy conservation.
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José Antônio Jardini (M’66–SM’78–F’90) was born in Espírito Santo do Pinhal, Brazil, on March 27, 1941. He received the B.S., M.Sc., and Ph.D. degrees from Escola Politécnica da Universidade de São Paulo (EPUSP), São Paulo, Brazil, in 1963, 1971, and 1973, respectively. His professional experience included the Themag Engineering, Ltd. (an engineering company), Brazil, for 25 years, where he conducted many power systems studies and participated in the Itaipu transmission system project. He is a Professor at EPUSP, teaching power system analysis and digital automation. He is also Coordinator at the Distribution Excellence Center (CED), where he is responsible for projects like load profile evaluation.
