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A tool for extracting synchronous machines parameters from the dc flux decay test
Computers and Electrical Engineering 31 (2005) 56–68 www.elsevier.com/locate/compeleceng
A tool for extracting synchronous machines parameters from the dc flux decay test F.S. Sellschopp, M.A. Arjona L.
*
Departamento de Ingenierı´a Ele´ctrica, Instituto Tecnolo´gico de la Laguna, Carrara 371, Col. Torreo´n Residencial, 27250 Torreo´n, Coah., Mexico Received 5 September 2003; received in revised form 5 July 2004; accepted 28 October 2004
Abstract A novel computer system tool for the parameter estimation of mathematical synchronous-machine models is presented. It uses state-of-the-art information technology software. The tool allows the electrical power engineers to obtain the electrical parameters of a synchronous machine model. The practical usefulness of the developed system is demonstrated by obtaining the parameter set for conventional and high order d and q axis models from the dc flux decay test. The two-axis model parameters are estimated with the well-known maximum likelihood algorithm. The user can also validate its set of estimated parameters against experimental data. Consequently, a better behavior prediction of synchronous generators within a power system can be achieved. The tool was developed using LabVIEW as a software environment and a low cost data acquisition card, which has a user friendly interface. A solid salient-pole synchronous generator of 7 kVA was used to illustrate its practical impact of the proposed system, where acceptable accuracy in the set of estimated parameters was achieved. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Synchronous machines; Parameter estimation; Flux decay test; Computer; Data acquisition; User interface
*
Corresponding author. Tel.: +52 871 7051331x115; fax: +52 871 7051326. E-mail addresses: [email protected] (F.S. Sellschopp), [email protected] (M.A. Arjona L.).
0045-7906/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compeleceng.2004.10.001
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1. Introduction Synchronous generators are the main devices in the production of electric energy all over the world, and they are operated in parallel, interconnected by transmission lines to supply electric energy to industrial and domestic loads. As a consequence, an understanding of their electromechanical behavior is usually required by electric utilities and manufacturers. A practical and economical way to gain the understanding of the synchronous machine behavior is usually carried out throughout digital simulations of mathematical models. Engineers concerned with the planning of power systems also require machine models, but their requirements differ from those of designers. Power system engineers are usually concerned with the behavior of the system as a whole, and not just with the performance of a single generator. For these reasons, power system analysts have preferred lumped-parameter equivalent circuit models, based on the two-axis theory [1]. There are well-established model structures that are recommended in the literature, however, a power analyst also requires a set of electrical parameters of the synchronous machine to be able to predict its behavior under normal or abnormal conditions. Many research centers have invested a lot of effort in developing test procedures that can lead to a reliable set of machine parameters, some of the research results are found in standards [2,3]. Synchronous machine manufacturers normally provide a set of electrical parameters to their customers, such parameters are usually derived from standard electric tests i.e. sudden short-circuit at low terminal voltage and frequency response tests [3]. Traditionally, the parameters are derived for a model with one damper winding along each axis because most commercial power system simulators use this model order. Nowadays, economic issues have led the electric utilities to drive its generators close to their operating limits. Hence, more accurate mathematical models of generators must be used; when a higher order model, i.e. multiple damper windings, is needed, a different set of parameters are required. The electric utility can buy the new parameters from the machine manufacturer, but this is usually prohibitive because of their high cost. Alternatively, the parameters can be obtained with simple experiments, which can be carried out by the utility engineers. There are several procedures to determine machine quantities; they range from standardized to non-confirmed test methods [2,3]. The tests can be classified as those carried out when the machine is at standstill, i.e. step voltage, frequency response and direct current (dc) flux decay, and when the machine is running at its rated speed, i.e. sudden short-circuit, load rejections, etc. Obviously, the standstill-based tests are easier to carry out, and they provide information for the d and q-axis machine models. However, the standstill tests have been criticized because: (a) they do not take into account the rotational effects, i.e. variation of contact resistance from standstill to full speed, and (b) the machine does not represent its rated saturation conditions [4]. The machine parameters can also be derived from finite element calculations of the electromagnetic machine model, but this approach is more difficult because machine design characteristics are required [5,6]. The dc decay test has the advantage that it can be done with the machine stationary, and only requires a dc supply that can be passed through one or two phases of the stator [7]. While the standstill frequency test requires expensive power signal generator equipment, and it is a time consuming at low frequencies. In addition, higher excitation currents can be applied to the machine in the dc decay test when compared with the standstill frequency response method. Although the procedure is considered an unconfirmed test by [2], there has been some research interest in the analysis and d–q axis parameter extraction from the dc flux decay test [7–11]. However, the main
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focus of those publications was on the parameter estimation algorithms, and little attention was given to the experimental stage of dc decay test. The paper contribution consists in a novel computer system for estimating synchronous machine parameters from the dc flux decay test. The system was developed using state of the art information technology. The developed dc flux decay-testing instrument (DCFDTI) allows the data acquisition, and the parameter estimation of high order synchronous machines models. It is important to emphasize its practical relevance to electric power utilities, because it allows the machine parameter estimation in the power station site, and it yields economical savings. It has a user friendly interface, which was developed in LabVIEW. The maximum likelihood method was employed for the machine parameter estimation. The computer tool was applied to a solid-salient four-pole machine of 7 kVA, 1800 rpm, 220 V. However, it can be adapted to any machine rating. Fundamental parameters were obtained for a third order d-axis model and a first order q-axis model; good results were obtained in both axes.
2. Dc flux decay test description The dc flux decay test does not require sophisticated equipment, it only needs a dc source connected to two machine stator phases and the field winding in short-circuit as it is shown in Fig. 1. In order to carry out the test, the synchronous generator must be at standstill and positioned either in the d or q-axis. Voltages and currents were measured with Hall effect sensors. Initially the switch S1 is closed and the stator windings are excited with a constant dc current, which will set up a magnetic flux density along the machine axis. A small dc excitation current is usually applied, hence, the machine is in the unsaturated condition. Once the stator excitation current is at steady state, the switch S2 is closed to short-circuit the generator stator terminals. The machine voltage and winding currents are acquired at a rate of 10 kHz throughout the transient. The decrement of the stator current during the short-circuit is used to determine reactances, time constants and fundamental parameters of the electric equivalent circuits. In this paper, it is important to highlight that fundamental
Fig. 1. Diagram of the direct-axis dc flux decay test.
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parameters are our goal. Derived parameters can readily be obtained. The test duration is considered to be finished when the stator current has reached its steady state in the d and q axes; in case of testing the direct axis, the field current must also be at steady state. It is important to employ the field current in the estimation process because the model will be able to predict the field winding dynamic behavior with high accuracy. When the q-axis test is carried out, the field winding is in open-circuit. Fig. 1 shows the current (Sc) and voltage (Sv) Hall effect transducers. The limiting resistor Rlim will protect the power source against an overload and will limit the current applied to the machine windings. The value of the resistance must be calculated before the test, and its value must guarantee a negligible current variation before and after the test because the dc source is always connected [2]. The switch S2 internal resistance must have a much lower value than the stator winding resistance; with this consideration, the decrement in the winding currents are not affected. In addition, it is recommended that the current sensor has a low internal resistance when compared to the winding resistance.
3. State-space machine model Since the dc flux decay test is carried out at standstill, the generator mathematical model in the two-axis reference frame can be represented by uncoupled equivalent circuits [12,13]. The above is valid because the d and q-axis rotational voltages of the electrical machine are zero when it is stationary. A standardized model structure for the machine was used as it is described in [14]. In the d-axis model, the field winding is always in short-circuit, i.e. its voltage is zero. When the switch S2 is closed, it was found that the contact resistance of S2 was negligible when it was compared to the machine stator resistance, therefore, measuring zero volts at the terminals of S2. A third order d-axis model for the machine may be represented in terms of fundamental parameters as indicated by (1). 32 3 2 3 2 id Rs 0 0 0 vd 7 6 6v 7 6 0 R 0 0 7 f 76 i f 7 6 f7 6 76 7 6 7¼6 4 0 5 4 0 0 Rkd1 0 54 ikd1 5 0
0 2
0 0 Lls þ Lmd
Rkd2 Lmd
Lmd
6 6 þ6 4
ikd2 Lmd
Lmd
Lfd þ Lmd
Lmd
Lmd
Lmd
Lmd
Llkd1 þ Lmd
Lmd
Lmd
Lmd
Lmd
Llkd2 þ Lmd
3
2
id
3
ð1Þ
7 d 6i 7 7 6 f 7 7 6 7 5 dt 4 ikd1 5 ikd2
where Rs, Rf, Rkd1 and Rkd2 stand for armature, field and d-axis damper resistances respectively. Lls, Lfd, Llkd1 and Llkd2 represent the leakage inductances for the armature, field and d-axis damper windings; Lmd is the mutual inductance along the d-axis. It is important to mention that any possible model structure can be used. The corresponding electric network of (1) is shown in Fig. 2. The derivative of vector currents of (1) may be represented in matrix form as: 1
1
½I_ d ¼ ½Ld ½Rd ½Id þ ½Ld ½V d
ð2Þ
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Fig. 2. D-axis electric equivalent circuit.
where subscript d specifies d-axis, Ld stands for d-axis matrix inductances, Rd is the resistance matrix, Id represents the current vector and Vd is the input voltage vector. Since the contact resistance of S2 is assumed to be zero, then the Vd vector is null in (2) and the resulting set of statespace equations are, ½I_ d ¼ ½Ad ½Id
ð3Þ
where ½Ad ¼ ½Ld 1 ½Rd
ð4Þ
Considering that experiments are usually noisy processes, a stochastic model must be used. Then, a discrete time representation of (3), adding a variable that handles noise is written as Id ðk þ 1Þ ¼ Fd Id ðkÞ þ xd ðkÞ
ð5Þ
The output with its noisy measurements is stated as, yd ðkÞ ¼ Cd Id ðkÞ þ ed ðkÞ
ð6Þ
Fd is a discrete time representation of the Ad continuous time matrix, Cd relates the states with measurements, yd is the vector output, k stands for discrete time, x(k) and e(k) denote the process and measurement noise, respectively. Fig. 3 shows an equivalent circuit for the q-axis with two damper windings and the state space equations can be achieved from the figure, so they are shown in a simplified way by (7). ½I_ q ¼ ½Aq ½Iq
ð7Þ
where, ½Aq ¼ ½Lq 1 ½Rq
ð8Þ
Fig. 3. Q-axis electric equivalent circuit.
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The subscript q indicates q-axis, matrices Lq and Rq stand for q-axis inductances and resistances. The model currents are represented by the vector Iq. Similarly to d-axis, a discrete time representation of (7) can be written as Iq ðk þ 1Þ ¼ Fq Iq ðkÞ þ xq ðkÞ
ð9Þ
The output with its noisy measurements is stated as, yq ðkÞ ¼ Cq Iq ðkÞ þ eq ðkÞ
ð10Þ
4. Equivalent circuit parameter estimation Many estimation algorithms have been reported in literature, the method of maximum likelihood has been considered as a good estimator [15], and it was adopted in this paper. Maximum likelihood estimation works by developing a likelihood function based on the available data, and finding the values of either a parameter or a vector parameter that are able to maximize the likelihood function. In terms of a formal optimization problem, maximization of the likelihood function is accomplished by minimizing the negative logarithm of the likelihood function [15]. The basic idea behind the maximum likelihood estimator is to obtain the most likely parameter values, for a given distribution, that will best describe the data. This can be achieved by using iterative methods to determine the parameter values that minimize the negative log-likelihood function. In the special case when the data has a normal or Gaussian distribution, the likelihood function is defined as: ( !) N Y 1 eTk RðhÞ1 ek pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð11Þ exp LðhÞ ¼ 2 ð2pÞm jRðhÞj k¼1 where h represents the parameter vector, N is the number of data points, j*j is the determinant of *, m is the number of outputs, R(h) denotes the covariance of estimation error, e is the estimation error that is defined as follows: ek ¼ y k y 0k
ð12Þ
where y and y 0 are the measured and estimated outputs respectively, and R(h) is computed as suggested in [15], and it is calculated as, RðhÞ ¼
N 1 X ek eTk N k¼1
ð13Þ
As it was stated above, maximizing the likelihood function is equivalent to minimizing its negative logarithm function, which is defined as follows: N N 1X 1X 1 eTk RðhÞ1 ek þ log ðjRðhÞjÞ þ Nm logð2pÞ ð14Þ V ðhÞ ¼ logðLðhÞÞ ¼ 2 k¼1 2 k¼1 2 The parameter vector may be computed with optimization algorithms by performing a Gauss– Newton or Levenberg–Marquardt constrained non-linear optimization, where (14) is the function to be minimized. The estimated output is obtained using the Kalman prediction–correction
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Fig. 4. Kalman predictor–corrector block diagram.
formulation [15]. The block diagram of the Kalman predictor–corrector process is shown in Fig. 4, where x is the vector of process states, u is an exogenous input, k is the time step, and F and G are the state transition and the control gain matrices respectively, C relates states and measurements, and K is the Kalman gain matrix. The computation of the Kalman gain K can be obtained as [15–17]: K ¼ PCT R1
ð15Þ
The error covariance matrix is given by: R ¼ RðhÞ þ CPCT
ð16Þ
and the state estimation covariance matrix P is a solution of discrete Riccati matrix equation, given by (17), [17] P ¼ FPFT FPCT RðhÞ þ CPCT FT þ Q ð17Þ The matrix covariance of the error process Q depends on the process noise sequence x, and it can be computed as a part of the unknown parameters, however, a reasonable value of Q is necessary to initialize the Kalman predictor–corrector algorithm [15]. The entire estimation process is illustrated in Fig. 5.
Fig. 5. Maximum likelihood parameter estimation process.
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5. Computer user interface 5.1. Virtual instrument The user interface was designed on the basis of having a friendly interaction between the user and the DCFDTI system. This program was implemented in the LabVIEW environment since it combines the advantages of graphical programming and high quality, user interface tools [18,19]. The front panel can be constructed and viewed like a physical instrument, where the analyst can visualize the results on the computer screen; because of this, it is called virtual instrument (VI). The front panel is driven by the G-language code or ‘‘block diagram’’, which is the actual code of the program. This part of the VI receives data from the front panel and sends them to the main program. The hardware interface to the computer is a National Instruments PCI-6025E, multifunction, low-cost data acquisition (DAQ) card. The system front panel has a set of buttons that analysts use to control the machine experiment (Fig. 6). The dc signal amplitude is supplied through a 2000 Watts dc power source that the user must setup before performing the test [20]. For the transient period of interest, the system offers the possibility to establish the duration of the data acquisition for the test through a radio knob. In order to acquire the steady state current signal, the data acquisition begins before closing the switch S2 and the acquired measurements are recorded into a file. After data is acquired, the system links another virtual instrument (VI) that shows the front panel of a program that performs the model parameter estimation. The machine parameters are usually expressed in per unit values with respect to the machine power rating. Therefore, the estimation program needs information about the machine voltages
Fig. 6. DCFDTI front panel illustrating the response of a d-axis dc flux decay test.
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and currents bases. The user must also give the desired model order, and a set of initial values in a file for the parameter vector (h); this information is needed by the estimation algorithm to carry out the parameter determination. After all information has been provided to the system through the user interface, the parameter estimation process must be triggered by means of the start switch. The parameter determination algorithm is programmed in Matlab environment, since LabVIEW allows interacting with Matlab [21]. The DCFDTI also shows the validation results, where the prediction of a predetermined model structure is compared with experimental data. 5.2. DCFDTI architecture The DCFDTI front panel has three fields that the user has to fill in. The fields represent input variables that provide the necessary information for the block diagram of the VI to operate according to the test required. The required data includes the test duration time, filename of experimental data and d–q axes option (direct or quadrature axis). The decrement test begins at the moment the relay of switch S2 is activated by the system. Once the data acquisition has finished, a d or q-axis estimation interface appears. The information that the user has to provide is: (a) model order, (b) base quantities for the synchronous machine under test, (c) initial parameter set and (d) filename of estimation results (Fig. 7). After this, the LabVIEW calls the Matlab numerical engine, which performs the parameter estimation. The estimation program allows definition of the convergence criteria as well as the optimization algorithm and returns the set of estimated parameters [21].
Fig. 7. VI front panel of d-axis model validation.
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6. Results and model validation The system developed was tested in a solid-salient pole micro-alternator. A third order model structure for the direct axis, and a first order model for the quadrature axis were assumed. These model structures were selected on the basis of having a better fit to the dc flux decay experimental data. The d-axis dc flux decay test was conducted with the field winding in short-circuit, and consequently both field and stator currents were registered. The q-axis equivalent circuit can be considered as one-port electrical network, hence, the stator current decrement was only acquired. The developed testing system allows the electrical engineer to carry out the flux decay at different excitation levels, which will affect to some extent the machine parameter values. The system allows the comparison of the predicted results, with either d or q-axis models, against the experimental data. The d-axis dc flux decay test was done with an excitation current of 0.13 pu. In order to acquire the whole transient dynamics, a test duration of 5000 ms was chosen. The parameter estimation was done in the background with Matlab. The resulting parameter set allows having an acceptable curve fitting of field and d-axis stator currents. The d-axis model validation is shown in Fig. 7, where a close up has been made to show the accuracy of the estimated parameters. In order to have similar magnetic conditions for both axes, the same excitation current and test duration was considered while testing the q-axis. The estimated parameters allow an adequate q-axis model prediction as it is clearly seen in Fig. 8. Table 1 shows the set of estimated parameters for the d and q axes equivalent circuit models. The stator and field winding resistances, Rs and Rf, can easily be calculated from steady state measurements of voltage and current. Hence they are considered as known parameters in the estimation process (values enclosed within parenthesis means known parameters). Different values for the stator leakage inductance Lls were estimated for each axis, which may have a physical meaning because the estimation process was made for a solid
Fig. 8. VI front panel of q-axis model validation.
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Table 1 Parameter estimation for the d and q-axis equivalent circuits d-axis
q-axis
Third order
First order
Rs (pu) Rf (pu) Rkd1 (pu) Rkd2 (pu) Lls (pu) Lmd (pu) Lfd (pu) Llkd1 (pu) Llkd2 (pu) V(h)
(0.060839) (0.009523) 0.111232 0.038962 0.205124 0.962655 0.002356 0.000103 0.128339 89,853.4025
Rs (pu) Rkq1 (pu) Lls (pu) Lmq (pu) Llkq1 (pu)
(0.060839) 0.181496 0.295154 0.442728 0.327947
V(h)
29,846.5909
Fig. 9. Experimental setup for testing a synchronous generator.
salient-pole synchronous machine. It is well-known that a salient-pole machine exhibits different equivalent reluctances along each magnetic axis because they have different air-gap lengths. The estimated mutual inductances Lmd and Lmq clearly indicate the saliency effect of the alternator and a saliency ratio (Lmq/Lmd) of 0.43 was estimated. This estimated ratio has a physical relationship with the nature of salient pole synchronous machine i.e. the q-axis magnetic circuit has higher reluctance than the d-axis one. Consequently, the q-axis magnetizing inductance is smaller than the d-axis mutual inductance. Fig. 9 shows a photograph of the dc decay test setup in an electrical machines laboratory.
7. Conclusions A new computer system to extract synchronous machine parameters from the dc flux decay procedure was presented. The system allows electrical power engineers to perform the parameter
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estimation of synchronous machine models in the two-axis reference frame. It can estimate the parameters of a generator with any number of fictitious damper windings along each axis, including differential inductances on d-axis. The experimental data required to estimate the machine parameters was obtained by applying the dc flux decay test, which is carried out when the machine is stationary along any machine orthogonal axis. One important feature of the developed system is that it allows the validation of synchronous machine models against the experimental data. A synchronous generator of 7 kVA was tested to illustrate the performance and practical relevance of the developed tool. The maximum likelihood method was successfully used to estimate the machine parameters. Two damper windings were considered in the d-axis model and acceptable fitting were obtained. The q-axis model was represented properly with one fictitious damper winding. The set of estimated parameters from the dc flux decay test may give acceptable prediction results when the generator is at rated load condition. It was also observed that leakage and mutual inductances are representative of the machine saliency. Although the tool has been tested in a small generator, it can be readily used in any machine rating.
Acknowledgments The authors are grateful to Consejo Nacional de Ciencia y Tecnologı´a (CONACYT) and to Consejo del Sistema Nacional de Education Tecnolo´gica (CoSNET) for financial support. The authors thank to Susan Diekvoss for checking the English writing.
References [1] Adkins B, Harley RG. The general theory of alternating current machines. London: Chapman and Hall; 1975. [2] IEC 34-4. British standard, General requirements for rotating electrical machines, Part 104. Methods of test for determining synchronous machines quantities, 1985. [3] IEEE Std 115-1995 IEEE Guide: Test Procedures for Synchronous Machines, 1995. [4] Jack AG, Bedford TJ. An analysis of the results from computation of transients in synchronous generators using frequency domain methods. IEEE Trans Energy Convers 1988;3(2):375–83. [5] Tang R, Gu H, Geng L. The calculation of transient field and parameters of REPM synchronous generator. IEEE Trans Magn 1985;21(6):2336–9. [6] Arjona LMA, Macdonald DC. Characterising the d-axis machine model of a turbogenerator using finite elements. IEEE Trans Energy Convers 1999;14(3):340–6. [7] Turner PJ, Reece ABJ, Macdonald DC. The DC decay test for determining synchronous machine parameters: measurement and simulation. IEEE Trans Energy Convers 1989;4(4):616–23. [8] Tumageanian A, Keyhani A, Moon SI, Leaksan TI. Maximum likelihood estimation of synchronous machine parameters from flux decay data. IEEE Trans Ind Appl 1994;30(2):433–9. [9] Keyhani A, Moon SI. Maximum likelihood estimation of synchronous machine parameters and study of noise effect from DC flux decay data. Gener, Trans Distrib IEE Proc C 1992;139(1):76–80. [10] Ara T, Yamamoto S, Oda S, Matsuse K. Prediction of starting performance of synchronous motor by DC decay testing method with rotor in any arbitrary position. IEEE Trans Ind Appl 1998;34(4):752–7. [11] Vas P. Parameter estimation, condition monitoring, and diagnosis of electrical machines. New York: Oxford Science Publications; 1993. [12] Park RH. Two reaction theory of synchronous machines, generalized method of analysis. Part I Trans AIEE 1929;48(3):716–30.
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[13] Park RH. Two reaction theory of synchronous machines. Part II Trans AIEE 1933;52(2):352–5. [14] IEEE guide for synchronous generator modeling practices and applications in power system stability analyses, IEEE Standard 1110-2002, Sponsor: Electric Machinery Committee of the IEEE PES, November 11, 2003. [15] Wamkeue R, Kamwa I, Dai-Do X, Keyhani A. Iteratively reweighted least squares for maximum likelihood identification of synchronous machine parameters from on-line tests. IEEE Trans Energy Convers 1999;14(2):159–66. [16] Brown RG, Hwang PYC. Introduction to random signals and applied Kalman filtering with Matlab: exercises and solutions. 3rd ed. New York: John Wiley and Sons; 1997. [17] Chui CK, Chen G. Kalman filtering with real-time applications. 2nd ed. Germany: Springer-Verlag; 1991. [18] LabVIEW user manual, National Instruments Corporation, 2000. [19] LabVIEW measurements manual, National Instruments Corporation, 2000. [20] Hewlett Packard, Operating manual, analog programmable dc power supplies, series 654xA, 655xA, 657xA, HP part no 5959-3374, USA, 1999. [21] UserÕs guide, Optimization toolbox for use with Matlab, The Mathworks, 2001. F.S. Sellschopp, Electrical Engineer, graduated in 1994, from the Instituto Tecnolo´gico de Tepic, Me´xico. From 1996 to 1999, he studied a masterÕs degree at the Instituto Tecnolo´gico de la Laguna, Me´xico, where he received the M.Sc. in Electrical Engineering. Presently, he is studying for a Science Doctorate at the Instituto Tecnolo´gico de la Laguna. His research interests include parameter estimation, electrical machine modeling, and control.
M.A. Arjona L. is a Professor of Electrical Machines at the Instituto Tecnolo´gico de la Laguna, Me´xico where he obtained his M.Sc. degree in Electrical Engineering in 1990. He received the B.Sc. degree in Electrical Engineering from the Instituto Tecnolo´gico de Durango in 1988. He was with the Simulation Department of the Instituto de Investigaciones Ele´ctricas from 1991–1999. He obtained his Ph.D. in Electrical Engineering at Imperial College, London, in 1996. His interests include modeling and control applications of electrical machines.
A tool for extracting synchronous machines parameters from the dc flux decay test F.S. Sellschopp, M.A. Arjona L.
*
Departamento de Ingenierı´a Ele´ctrica, Instituto Tecnolo´gico de la Laguna, Carrara 371, Col. Torreo´n Residencial, 27250 Torreo´n, Coah., Mexico Received 5 September 2003; received in revised form 5 July 2004; accepted 28 October 2004
Abstract A novel computer system tool for the parameter estimation of mathematical synchronous-machine models is presented. It uses state-of-the-art information technology software. The tool allows the electrical power engineers to obtain the electrical parameters of a synchronous machine model. The practical usefulness of the developed system is demonstrated by obtaining the parameter set for conventional and high order d and q axis models from the dc flux decay test. The two-axis model parameters are estimated with the well-known maximum likelihood algorithm. The user can also validate its set of estimated parameters against experimental data. Consequently, a better behavior prediction of synchronous generators within a power system can be achieved. The tool was developed using LabVIEW as a software environment and a low cost data acquisition card, which has a user friendly interface. A solid salient-pole synchronous generator of 7 kVA was used to illustrate its practical impact of the proposed system, where acceptable accuracy in the set of estimated parameters was achieved. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Synchronous machines; Parameter estimation; Flux decay test; Computer; Data acquisition; User interface
*
Corresponding author. Tel.: +52 871 7051331x115; fax: +52 871 7051326. E-mail addresses: [email protected] (F.S. Sellschopp), [email protected] (M.A. Arjona L.).
0045-7906/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compeleceng.2004.10.001
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1. Introduction Synchronous generators are the main devices in the production of electric energy all over the world, and they are operated in parallel, interconnected by transmission lines to supply electric energy to industrial and domestic loads. As a consequence, an understanding of their electromechanical behavior is usually required by electric utilities and manufacturers. A practical and economical way to gain the understanding of the synchronous machine behavior is usually carried out throughout digital simulations of mathematical models. Engineers concerned with the planning of power systems also require machine models, but their requirements differ from those of designers. Power system engineers are usually concerned with the behavior of the system as a whole, and not just with the performance of a single generator. For these reasons, power system analysts have preferred lumped-parameter equivalent circuit models, based on the two-axis theory [1]. There are well-established model structures that are recommended in the literature, however, a power analyst also requires a set of electrical parameters of the synchronous machine to be able to predict its behavior under normal or abnormal conditions. Many research centers have invested a lot of effort in developing test procedures that can lead to a reliable set of machine parameters, some of the research results are found in standards [2,3]. Synchronous machine manufacturers normally provide a set of electrical parameters to their customers, such parameters are usually derived from standard electric tests i.e. sudden short-circuit at low terminal voltage and frequency response tests [3]. Traditionally, the parameters are derived for a model with one damper winding along each axis because most commercial power system simulators use this model order. Nowadays, economic issues have led the electric utilities to drive its generators close to their operating limits. Hence, more accurate mathematical models of generators must be used; when a higher order model, i.e. multiple damper windings, is needed, a different set of parameters are required. The electric utility can buy the new parameters from the machine manufacturer, but this is usually prohibitive because of their high cost. Alternatively, the parameters can be obtained with simple experiments, which can be carried out by the utility engineers. There are several procedures to determine machine quantities; they range from standardized to non-confirmed test methods [2,3]. The tests can be classified as those carried out when the machine is at standstill, i.e. step voltage, frequency response and direct current (dc) flux decay, and when the machine is running at its rated speed, i.e. sudden short-circuit, load rejections, etc. Obviously, the standstill-based tests are easier to carry out, and they provide information for the d and q-axis machine models. However, the standstill tests have been criticized because: (a) they do not take into account the rotational effects, i.e. variation of contact resistance from standstill to full speed, and (b) the machine does not represent its rated saturation conditions [4]. The machine parameters can also be derived from finite element calculations of the electromagnetic machine model, but this approach is more difficult because machine design characteristics are required [5,6]. The dc decay test has the advantage that it can be done with the machine stationary, and only requires a dc supply that can be passed through one or two phases of the stator [7]. While the standstill frequency test requires expensive power signal generator equipment, and it is a time consuming at low frequencies. In addition, higher excitation currents can be applied to the machine in the dc decay test when compared with the standstill frequency response method. Although the procedure is considered an unconfirmed test by [2], there has been some research interest in the analysis and d–q axis parameter extraction from the dc flux decay test [7–11]. However, the main
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focus of those publications was on the parameter estimation algorithms, and little attention was given to the experimental stage of dc decay test. The paper contribution consists in a novel computer system for estimating synchronous machine parameters from the dc flux decay test. The system was developed using state of the art information technology. The developed dc flux decay-testing instrument (DCFDTI) allows the data acquisition, and the parameter estimation of high order synchronous machines models. It is important to emphasize its practical relevance to electric power utilities, because it allows the machine parameter estimation in the power station site, and it yields economical savings. It has a user friendly interface, which was developed in LabVIEW. The maximum likelihood method was employed for the machine parameter estimation. The computer tool was applied to a solid-salient four-pole machine of 7 kVA, 1800 rpm, 220 V. However, it can be adapted to any machine rating. Fundamental parameters were obtained for a third order d-axis model and a first order q-axis model; good results were obtained in both axes.
2. Dc flux decay test description The dc flux decay test does not require sophisticated equipment, it only needs a dc source connected to two machine stator phases and the field winding in short-circuit as it is shown in Fig. 1. In order to carry out the test, the synchronous generator must be at standstill and positioned either in the d or q-axis. Voltages and currents were measured with Hall effect sensors. Initially the switch S1 is closed and the stator windings are excited with a constant dc current, which will set up a magnetic flux density along the machine axis. A small dc excitation current is usually applied, hence, the machine is in the unsaturated condition. Once the stator excitation current is at steady state, the switch S2 is closed to short-circuit the generator stator terminals. The machine voltage and winding currents are acquired at a rate of 10 kHz throughout the transient. The decrement of the stator current during the short-circuit is used to determine reactances, time constants and fundamental parameters of the electric equivalent circuits. In this paper, it is important to highlight that fundamental
Fig. 1. Diagram of the direct-axis dc flux decay test.
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parameters are our goal. Derived parameters can readily be obtained. The test duration is considered to be finished when the stator current has reached its steady state in the d and q axes; in case of testing the direct axis, the field current must also be at steady state. It is important to employ the field current in the estimation process because the model will be able to predict the field winding dynamic behavior with high accuracy. When the q-axis test is carried out, the field winding is in open-circuit. Fig. 1 shows the current (Sc) and voltage (Sv) Hall effect transducers. The limiting resistor Rlim will protect the power source against an overload and will limit the current applied to the machine windings. The value of the resistance must be calculated before the test, and its value must guarantee a negligible current variation before and after the test because the dc source is always connected [2]. The switch S2 internal resistance must have a much lower value than the stator winding resistance; with this consideration, the decrement in the winding currents are not affected. In addition, it is recommended that the current sensor has a low internal resistance when compared to the winding resistance.
3. State-space machine model Since the dc flux decay test is carried out at standstill, the generator mathematical model in the two-axis reference frame can be represented by uncoupled equivalent circuits [12,13]. The above is valid because the d and q-axis rotational voltages of the electrical machine are zero when it is stationary. A standardized model structure for the machine was used as it is described in [14]. In the d-axis model, the field winding is always in short-circuit, i.e. its voltage is zero. When the switch S2 is closed, it was found that the contact resistance of S2 was negligible when it was compared to the machine stator resistance, therefore, measuring zero volts at the terminals of S2. A third order d-axis model for the machine may be represented in terms of fundamental parameters as indicated by (1). 32 3 2 3 2 id Rs 0 0 0 vd 7 6 6v 7 6 0 R 0 0 7 f 76 i f 7 6 f7 6 76 7 6 7¼6 4 0 5 4 0 0 Rkd1 0 54 ikd1 5 0
0 2
0 0 Lls þ Lmd
Rkd2 Lmd
Lmd
6 6 þ6 4
ikd2 Lmd
Lmd
Lfd þ Lmd
Lmd
Lmd
Lmd
Lmd
Llkd1 þ Lmd
Lmd
Lmd
Lmd
Lmd
Llkd2 þ Lmd
3
2
id
3
ð1Þ
7 d 6i 7 7 6 f 7 7 6 7 5 dt 4 ikd1 5 ikd2
where Rs, Rf, Rkd1 and Rkd2 stand for armature, field and d-axis damper resistances respectively. Lls, Lfd, Llkd1 and Llkd2 represent the leakage inductances for the armature, field and d-axis damper windings; Lmd is the mutual inductance along the d-axis. It is important to mention that any possible model structure can be used. The corresponding electric network of (1) is shown in Fig. 2. The derivative of vector currents of (1) may be represented in matrix form as: 1
1
½I_ d ¼ ½Ld ½Rd ½Id þ ½Ld ½V d
ð2Þ
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Fig. 2. D-axis electric equivalent circuit.
where subscript d specifies d-axis, Ld stands for d-axis matrix inductances, Rd is the resistance matrix, Id represents the current vector and Vd is the input voltage vector. Since the contact resistance of S2 is assumed to be zero, then the Vd vector is null in (2) and the resulting set of statespace equations are, ½I_ d ¼ ½Ad ½Id
ð3Þ
where ½Ad ¼ ½Ld 1 ½Rd
ð4Þ
Considering that experiments are usually noisy processes, a stochastic model must be used. Then, a discrete time representation of (3), adding a variable that handles noise is written as Id ðk þ 1Þ ¼ Fd Id ðkÞ þ xd ðkÞ
ð5Þ
The output with its noisy measurements is stated as, yd ðkÞ ¼ Cd Id ðkÞ þ ed ðkÞ
ð6Þ
Fd is a discrete time representation of the Ad continuous time matrix, Cd relates the states with measurements, yd is the vector output, k stands for discrete time, x(k) and e(k) denote the process and measurement noise, respectively. Fig. 3 shows an equivalent circuit for the q-axis with two damper windings and the state space equations can be achieved from the figure, so they are shown in a simplified way by (7). ½I_ q ¼ ½Aq ½Iq
ð7Þ
where, ½Aq ¼ ½Lq 1 ½Rq
ð8Þ
Fig. 3. Q-axis electric equivalent circuit.
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The subscript q indicates q-axis, matrices Lq and Rq stand for q-axis inductances and resistances. The model currents are represented by the vector Iq. Similarly to d-axis, a discrete time representation of (7) can be written as Iq ðk þ 1Þ ¼ Fq Iq ðkÞ þ xq ðkÞ
ð9Þ
The output with its noisy measurements is stated as, yq ðkÞ ¼ Cq Iq ðkÞ þ eq ðkÞ
ð10Þ
4. Equivalent circuit parameter estimation Many estimation algorithms have been reported in literature, the method of maximum likelihood has been considered as a good estimator [15], and it was adopted in this paper. Maximum likelihood estimation works by developing a likelihood function based on the available data, and finding the values of either a parameter or a vector parameter that are able to maximize the likelihood function. In terms of a formal optimization problem, maximization of the likelihood function is accomplished by minimizing the negative logarithm of the likelihood function [15]. The basic idea behind the maximum likelihood estimator is to obtain the most likely parameter values, for a given distribution, that will best describe the data. This can be achieved by using iterative methods to determine the parameter values that minimize the negative log-likelihood function. In the special case when the data has a normal or Gaussian distribution, the likelihood function is defined as: ( !) N Y 1 eTk RðhÞ1 ek pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð11Þ exp LðhÞ ¼ 2 ð2pÞm jRðhÞj k¼1 where h represents the parameter vector, N is the number of data points, j*j is the determinant of *, m is the number of outputs, R(h) denotes the covariance of estimation error, e is the estimation error that is defined as follows: ek ¼ y k y 0k
ð12Þ
where y and y 0 are the measured and estimated outputs respectively, and R(h) is computed as suggested in [15], and it is calculated as, RðhÞ ¼
N 1 X ek eTk N k¼1
ð13Þ
As it was stated above, maximizing the likelihood function is equivalent to minimizing its negative logarithm function, which is defined as follows: N N 1X 1X 1 eTk RðhÞ1 ek þ log ðjRðhÞjÞ þ Nm logð2pÞ ð14Þ V ðhÞ ¼ logðLðhÞÞ ¼ 2 k¼1 2 k¼1 2 The parameter vector may be computed with optimization algorithms by performing a Gauss– Newton or Levenberg–Marquardt constrained non-linear optimization, where (14) is the function to be minimized. The estimated output is obtained using the Kalman prediction–correction
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Fig. 4. Kalman predictor–corrector block diagram.
formulation [15]. The block diagram of the Kalman predictor–corrector process is shown in Fig. 4, where x is the vector of process states, u is an exogenous input, k is the time step, and F and G are the state transition and the control gain matrices respectively, C relates states and measurements, and K is the Kalman gain matrix. The computation of the Kalman gain K can be obtained as [15–17]: K ¼ PCT R1
ð15Þ
The error covariance matrix is given by: R ¼ RðhÞ þ CPCT
ð16Þ
and the state estimation covariance matrix P is a solution of discrete Riccati matrix equation, given by (17), [17] P ¼ FPFT FPCT RðhÞ þ CPCT FT þ Q ð17Þ The matrix covariance of the error process Q depends on the process noise sequence x, and it can be computed as a part of the unknown parameters, however, a reasonable value of Q is necessary to initialize the Kalman predictor–corrector algorithm [15]. The entire estimation process is illustrated in Fig. 5.
Fig. 5. Maximum likelihood parameter estimation process.
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5. Computer user interface 5.1. Virtual instrument The user interface was designed on the basis of having a friendly interaction between the user and the DCFDTI system. This program was implemented in the LabVIEW environment since it combines the advantages of graphical programming and high quality, user interface tools [18,19]. The front panel can be constructed and viewed like a physical instrument, where the analyst can visualize the results on the computer screen; because of this, it is called virtual instrument (VI). The front panel is driven by the G-language code or ‘‘block diagram’’, which is the actual code of the program. This part of the VI receives data from the front panel and sends them to the main program. The hardware interface to the computer is a National Instruments PCI-6025E, multifunction, low-cost data acquisition (DAQ) card. The system front panel has a set of buttons that analysts use to control the machine experiment (Fig. 6). The dc signal amplitude is supplied through a 2000 Watts dc power source that the user must setup before performing the test [20]. For the transient period of interest, the system offers the possibility to establish the duration of the data acquisition for the test through a radio knob. In order to acquire the steady state current signal, the data acquisition begins before closing the switch S2 and the acquired measurements are recorded into a file. After data is acquired, the system links another virtual instrument (VI) that shows the front panel of a program that performs the model parameter estimation. The machine parameters are usually expressed in per unit values with respect to the machine power rating. Therefore, the estimation program needs information about the machine voltages
Fig. 6. DCFDTI front panel illustrating the response of a d-axis dc flux decay test.
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and currents bases. The user must also give the desired model order, and a set of initial values in a file for the parameter vector (h); this information is needed by the estimation algorithm to carry out the parameter determination. After all information has been provided to the system through the user interface, the parameter estimation process must be triggered by means of the start switch. The parameter determination algorithm is programmed in Matlab environment, since LabVIEW allows interacting with Matlab [21]. The DCFDTI also shows the validation results, where the prediction of a predetermined model structure is compared with experimental data. 5.2. DCFDTI architecture The DCFDTI front panel has three fields that the user has to fill in. The fields represent input variables that provide the necessary information for the block diagram of the VI to operate according to the test required. The required data includes the test duration time, filename of experimental data and d–q axes option (direct or quadrature axis). The decrement test begins at the moment the relay of switch S2 is activated by the system. Once the data acquisition has finished, a d or q-axis estimation interface appears. The information that the user has to provide is: (a) model order, (b) base quantities for the synchronous machine under test, (c) initial parameter set and (d) filename of estimation results (Fig. 7). After this, the LabVIEW calls the Matlab numerical engine, which performs the parameter estimation. The estimation program allows definition of the convergence criteria as well as the optimization algorithm and returns the set of estimated parameters [21].
Fig. 7. VI front panel of d-axis model validation.
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6. Results and model validation The system developed was tested in a solid-salient pole micro-alternator. A third order model structure for the direct axis, and a first order model for the quadrature axis were assumed. These model structures were selected on the basis of having a better fit to the dc flux decay experimental data. The d-axis dc flux decay test was conducted with the field winding in short-circuit, and consequently both field and stator currents were registered. The q-axis equivalent circuit can be considered as one-port electrical network, hence, the stator current decrement was only acquired. The developed testing system allows the electrical engineer to carry out the flux decay at different excitation levels, which will affect to some extent the machine parameter values. The system allows the comparison of the predicted results, with either d or q-axis models, against the experimental data. The d-axis dc flux decay test was done with an excitation current of 0.13 pu. In order to acquire the whole transient dynamics, a test duration of 5000 ms was chosen. The parameter estimation was done in the background with Matlab. The resulting parameter set allows having an acceptable curve fitting of field and d-axis stator currents. The d-axis model validation is shown in Fig. 7, where a close up has been made to show the accuracy of the estimated parameters. In order to have similar magnetic conditions for both axes, the same excitation current and test duration was considered while testing the q-axis. The estimated parameters allow an adequate q-axis model prediction as it is clearly seen in Fig. 8. Table 1 shows the set of estimated parameters for the d and q axes equivalent circuit models. The stator and field winding resistances, Rs and Rf, can easily be calculated from steady state measurements of voltage and current. Hence they are considered as known parameters in the estimation process (values enclosed within parenthesis means known parameters). Different values for the stator leakage inductance Lls were estimated for each axis, which may have a physical meaning because the estimation process was made for a solid
Fig. 8. VI front panel of q-axis model validation.
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Table 1 Parameter estimation for the d and q-axis equivalent circuits d-axis
q-axis
Third order
First order
Rs (pu) Rf (pu) Rkd1 (pu) Rkd2 (pu) Lls (pu) Lmd (pu) Lfd (pu) Llkd1 (pu) Llkd2 (pu) V(h)
(0.060839) (0.009523) 0.111232 0.038962 0.205124 0.962655 0.002356 0.000103 0.128339 89,853.4025
Rs (pu) Rkq1 (pu) Lls (pu) Lmq (pu) Llkq1 (pu)
(0.060839) 0.181496 0.295154 0.442728 0.327947
V(h)
29,846.5909
Fig. 9. Experimental setup for testing a synchronous generator.
salient-pole synchronous machine. It is well-known that a salient-pole machine exhibits different equivalent reluctances along each magnetic axis because they have different air-gap lengths. The estimated mutual inductances Lmd and Lmq clearly indicate the saliency effect of the alternator and a saliency ratio (Lmq/Lmd) of 0.43 was estimated. This estimated ratio has a physical relationship with the nature of salient pole synchronous machine i.e. the q-axis magnetic circuit has higher reluctance than the d-axis one. Consequently, the q-axis magnetizing inductance is smaller than the d-axis mutual inductance. Fig. 9 shows a photograph of the dc decay test setup in an electrical machines laboratory.
7. Conclusions A new computer system to extract synchronous machine parameters from the dc flux decay procedure was presented. The system allows electrical power engineers to perform the parameter
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estimation of synchronous machine models in the two-axis reference frame. It can estimate the parameters of a generator with any number of fictitious damper windings along each axis, including differential inductances on d-axis. The experimental data required to estimate the machine parameters was obtained by applying the dc flux decay test, which is carried out when the machine is stationary along any machine orthogonal axis. One important feature of the developed system is that it allows the validation of synchronous machine models against the experimental data. A synchronous generator of 7 kVA was tested to illustrate the performance and practical relevance of the developed tool. The maximum likelihood method was successfully used to estimate the machine parameters. Two damper windings were considered in the d-axis model and acceptable fitting were obtained. The q-axis model was represented properly with one fictitious damper winding. The set of estimated parameters from the dc flux decay test may give acceptable prediction results when the generator is at rated load condition. It was also observed that leakage and mutual inductances are representative of the machine saliency. Although the tool has been tested in a small generator, it can be readily used in any machine rating.
Acknowledgments The authors are grateful to Consejo Nacional de Ciencia y Tecnologı´a (CONACYT) and to Consejo del Sistema Nacional de Education Tecnolo´gica (CoSNET) for financial support. The authors thank to Susan Diekvoss for checking the English writing.
References [1] Adkins B, Harley RG. The general theory of alternating current machines. London: Chapman and Hall; 1975. [2] IEC 34-4. British standard, General requirements for rotating electrical machines, Part 104. Methods of test for determining synchronous machines quantities, 1985. [3] IEEE Std 115-1995 IEEE Guide: Test Procedures for Synchronous Machines, 1995. [4] Jack AG, Bedford TJ. An analysis of the results from computation of transients in synchronous generators using frequency domain methods. IEEE Trans Energy Convers 1988;3(2):375–83. [5] Tang R, Gu H, Geng L. The calculation of transient field and parameters of REPM synchronous generator. IEEE Trans Magn 1985;21(6):2336–9. [6] Arjona LMA, Macdonald DC. Characterising the d-axis machine model of a turbogenerator using finite elements. IEEE Trans Energy Convers 1999;14(3):340–6. [7] Turner PJ, Reece ABJ, Macdonald DC. The DC decay test for determining synchronous machine parameters: measurement and simulation. IEEE Trans Energy Convers 1989;4(4):616–23. [8] Tumageanian A, Keyhani A, Moon SI, Leaksan TI. Maximum likelihood estimation of synchronous machine parameters from flux decay data. IEEE Trans Ind Appl 1994;30(2):433–9. [9] Keyhani A, Moon SI. Maximum likelihood estimation of synchronous machine parameters and study of noise effect from DC flux decay data. Gener, Trans Distrib IEE Proc C 1992;139(1):76–80. [10] Ara T, Yamamoto S, Oda S, Matsuse K. Prediction of starting performance of synchronous motor by DC decay testing method with rotor in any arbitrary position. IEEE Trans Ind Appl 1998;34(4):752–7. [11] Vas P. Parameter estimation, condition monitoring, and diagnosis of electrical machines. New York: Oxford Science Publications; 1993. [12] Park RH. Two reaction theory of synchronous machines, generalized method of analysis. Part I Trans AIEE 1929;48(3):716–30.
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[13] Park RH. Two reaction theory of synchronous machines. Part II Trans AIEE 1933;52(2):352–5. [14] IEEE guide for synchronous generator modeling practices and applications in power system stability analyses, IEEE Standard 1110-2002, Sponsor: Electric Machinery Committee of the IEEE PES, November 11, 2003. [15] Wamkeue R, Kamwa I, Dai-Do X, Keyhani A. Iteratively reweighted least squares for maximum likelihood identification of synchronous machine parameters from on-line tests. IEEE Trans Energy Convers 1999;14(2):159–66. [16] Brown RG, Hwang PYC. Introduction to random signals and applied Kalman filtering with Matlab: exercises and solutions. 3rd ed. New York: John Wiley and Sons; 1997. [17] Chui CK, Chen G. Kalman filtering with real-time applications. 2nd ed. Germany: Springer-Verlag; 1991. [18] LabVIEW user manual, National Instruments Corporation, 2000. [19] LabVIEW measurements manual, National Instruments Corporation, 2000. [20] Hewlett Packard, Operating manual, analog programmable dc power supplies, series 654xA, 655xA, 657xA, HP part no 5959-3374, USA, 1999. [21] UserÕs guide, Optimization toolbox for use with Matlab, The Mathworks, 2001. F.S. Sellschopp, Electrical Engineer, graduated in 1994, from the Instituto Tecnolo´gico de Tepic, Me´xico. From 1996 to 1999, he studied a masterÕs degree at the Instituto Tecnolo´gico de la Laguna, Me´xico, where he received the M.Sc. in Electrical Engineering. Presently, he is studying for a Science Doctorate at the Instituto Tecnolo´gico de la Laguna. His research interests include parameter estimation, electrical machine modeling, and control.
M.A. Arjona L. is a Professor of Electrical Machines at the Instituto Tecnolo´gico de la Laguna, Me´xico where he obtained his M.Sc. degree in Electrical Engineering in 1990. He received the B.Sc. degree in Electrical Engineering from the Instituto Tecnolo´gico de Durango in 1988. He was with the Simulation Department of the Instituto de Investigaciones Ele´ctricas from 1991–1999. He obtained his Ph.D. in Electrical Engineering at Imperial College, London, in 1996. His interests include modeling and control applications of electrical machines.