- Email: [email protected]
Synchronous machine model parameters estimation by a time-domain identification method
Electrical Power and Energy Systems 32 (2010) 524–529
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Synchronous machine model parameters estimation by a time-domain identification method M. Dehghani a,*, M. Karrari a, W. Rosehart b, O.P. Malik b a b
Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran Department of Electrical and Computer Engineering, University of Calgary, Calgary, Canada
a r t i c l e
i n f o
Article history: Received 14 May 2009 Accepted 27 July 2009
Keywords: Synchronous machines Identification Nonlinear systems Power system modeling H1 identification methods
a b s t r a c t In this paper, the captured data from a real physical machine is used to investigate a nonlinear identification method. Since many practical systems, such as synchronous machines are nonlinear, linear models identified for a particular operating condition may not perform well for other operating conditions. The nonlinear H1 identification is used to overcome this shortcoming and to find a robust model despite noise and system uncertainities. The proposed method is applied on a simulated nonlinear model of a synchronous machine and is also tested on a physical machine. The field voltage is considered as the input, and the active output power and the terminal voltage are considered as the outputs of the synchronous machine. Simulation and experimental results show good accuracy of the identified model. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Adaptive control techniques have been proposed to improve the dynamic performance of power systems [1–4]. Among the published literature on adaptive power system stabilizers (PSSs), self-tuning regulators are the most common. A self-tuning regulator consists of a system model of a known structure and a controller. Control is calculated based on the model parameters. To compute the control appropriate to the operating conditions, parameters of the system model are obtained by an on-line identification technique. Correctness of the identification determines the preciseness of the identified model that tries to track the dynamic behaviour of the plant. For a time varying system, good tracking ability of the identification method is very desirable. The objective of this paper is to identify a nonlinear model of a synchronous machine that can be used for system analysis and controller design, especially the design of a PSS. The model can be used either in a predictive control structure for an on-line design, or as a simulator to test an off-line design. The main problem associated with modeling are the measurement noise and model uncertainities. To overcome these problems for linear autoregressive (ARX) models, the H1 identification method has been proposed [5]. Synchronous generators, however, * Corresponding author. Tel.: +98 711 2303081. E-mail address: [email protected] (M. Dehghani). 0142-0615/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2009.07.010
are highly nonlinear systems. To develop a robust identification method for such a nonlinear system, in the presence of measurement noise and model uncertainities, the H1 identification method is first generalized to cover the nonlinearities of the system. The developed method is then applied to identify the synchronous generator. The identification method and the nonlinear model of the system used in this paper are described in Sections 2 and 3, respectively. The system input–output data used for model identification is presented in Section 4 and the performance of the method is demonstrated on a simulated nonlinear model of a synchronous generator. The application of the proposed method on a physical machine and experimental test results are described in Section 5. Section 6 concludes the paper.
2. Identification method Consider a system in a noisy environment represented by a linear discrete time model:
zk ¼
n X i¼1
ai zki þ
m X
bi uki þ c0 xk
ð1Þ
i¼1
where zk is the unknown real output at time step k; uk1 is the system input at time step (k 1); ai, bi are the system parameters, and xk is the unknown driving disturbance. The system under study in this paper, i.e. a synchronous machine, however, is a nonlinear system with both physical, e.g.
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M. Dehghani et al. / Electrical Power and Energy Systems 32 (2010) 524–529
saturation, and mathematical nonlinearities (as described in Section 3). Therefore, the linear model parameters would change with changes in the operating conditions. To take this into account, the linear regression model of (1) has been generalized to a nonlinear model of (2) using Taylor series:
zk ¼
n X
a1i zki
i¼1
þ
n X
a2i ðzki Þ2
i¼1
m X
n X
aNi ðzki ÞN
i¼1
b1i uki þ þ
m X
i¼1
N
bNi ðuki Þ þ c0 xk
ð2Þ
xk hkþ1 ¼ Ahk þ B vk xk yk ¼ C k ðhk Þhk þ D
ð3Þ
htk ¼ ½a11 a1n aN1 aNn ; b11 b1m bN1 bNm ; v k1 v kn
ð4Þ
and
2
INðnþmÞ 6 A¼6 40
"
nNðnþmÞ
2
3
0NðnþmÞn 01ðn1Þ
011
In1
0ðn1Þ1 3
#7 7; 5
7 5;
1
ð5Þ
0ðn1Þ1 N
uk1 ukm ; ; ðuk1 Þ ðukm Þ ; a1 an
n X i¼1 m X
^1i ^zki a
n X
^2i ð^zki Þ2 a
i¼1
^ u þ þ b 1i ki
n X
^Ni ð^zki ÞN a
i¼1 m X
^ ðu Þ b Ni ki
N
ð6Þ
i¼1
where ^zk is the best estimate of zk. To solve the estimation problem, the following estimation algorithm has been suggested and the stability of the algorithm under certain conditions is proved in [5]:
(
^hkþ1 ¼ A^hk þ Gk ð^hk Þ ðy ^zk Þ; k ^zk ¼ C k ð^hk Þ^hk
^h0 ¼ 0
ð7Þ
The only unknown in the above estimator is Gk. The following formulas are used to estimate the vector:
b k ¼ ðAQ 1 C e t ð^hk Þ þ BDt ÞU1 ð^hk Þ G k k k where
I
0 0
e k ðhk Þ ¼ C e k ðhk Þ þ ^ht 6 C k4 0
3
7 0 05
The next step is to assume that there is a nominal vector of parameters, h, and a given constant M > 0 such that:
kC k ðhk Þ C k ðhÞk 6 M Define:
dk ¼
^ bt AQ 1 k dk G k
b k ^dt Q 1 At G k k
3 t 6 AQ k b b t 7 þ M4 b Gk Gk5 Gk
2 3 b Gk t t 1 þ M 4 t AQ k Q k A 5 AQ k
et e e t ð^hk Þ^dt þ ^dk C e k ð^hk Þ þ M C ~dk ¼ C k k e k ðhÞ C k ðhÞ C k ðhÞ 0 e 1 C k ðhÞ 0 0NðnþmÞn 2@ A NðnþmÞNðnþmÞ þM 1þ M 0nNðnþmÞ In
ð10Þ
ð11Þ
ð12Þ
h i e kþ1 ð^hkþ1 Þ þ ^dkþ1 g~k > 0 e t ð^hkþ1 Þ C I g~tk Q kþ1 þ c2 C kþ1
ð13Þ
1
The predictive form of model (2) is defined as:
i¼1
0 0
where g~k ¼ B Gk D. Using this approach, the identification method is summarized below:
C k ðhk Þ ¼ ½yk1 ykn ; ; ðyk1 ÞN ðykn ÞN ; N
0
e k ð e k ð^ where ^ dtk ¼ C hÞ C hk Þ. It is shown in [5] that the algorithm converges if and only if Q is positive definite and
0NðnþmÞ1
6 B ¼ 4 0½NðnþmÞþn1
þ
2
1 ^ 2 e ^ et ^ ^t 1 ^ t Dk ¼ ^dtk Q 1 k C k ðhk Þ þ C k ðhk ÞQ k dk þ dk Q k dk þ M r Q k et þ 2MQ 1 k C k ðhÞ
where
^zk ¼
et Uk ðhk Þ ¼ DDt þ Ce k ðhk ÞQ 1 k C k ðhk Þ
2
vk
ð9Þ
In (9), c is the disturbance rejection factor. It should be selected as small as possible. On the other hand, it should be large enough to guarantee the convergence of the algorithm. Also:
i¼1
This allows the nonlinearities, such as saturation, to be modeled. In (2), {aki} and {bki} are the model parameters to be identified, yk is the measured output (yk = zk + vk), and vk is the measurement noise. In (2), n is the order of the system and N is the degree of the nonlinearity. A larger N results in better approximation of nonlinearities, but increases model complexity. If N = 1 is selected, as in [5], a linear ARX model is obtained. To identify the parameters, the above system structure can be written as:
D ¼ ½ c0
n o1 t t b bt ^ bt b Q kþ1 ¼ AQ 1 k A þ BB G k Uk ðhk Þ G k þ G k Dk G k þ dk h i e kþ1 ð^hkþ1 Þ þ ~dkþ1 ; Q > 0 e t ðh^kþ1 Þ C c2 C 0 kþ1
ð8Þ
(a) Select a proper input signal to be applied to the system. The input signal should have a wide spectrum to cover all system dynamics. It should also have a proper magnitude. The magnitude should be large enough to cover the nonlinearities and also should be small enough to be safe to perform the test. (b) Select a proper sampling time and final time (the total time for the experiment). (c) Apply the selected input signal (item a), to the system and sample the input–output data by a data acquisition system. (d) Select the order of the model (n) and the number of the terms (N) in (2). Although the real values for n and N could be very high, a lower order is preferred if it can capture the required dynamics. In particular, for real time control applications, since the parameters are identified on-line to track the system, a low order model, generally n = 3, is sufficient. (e) Select proper values for c and M. Smaller value of c means better disturbance rejection. The smallest value of c is selected by trial and error, such that the condition in (13) is met and Q remains positive definite.
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(f) Estimate the linear parts, (a1i, b1i), of vector ht0 in (4) using least squares, as used in this paper, or any other identification algorithm developed for linear systems [6]. (g) Establish A, B, C0(h0) and D matrices. b k using (8) and calculate the other vectors given by (h) Update G (10)–(12) for the next iteration. (i) Update the estimated parameters ^ hkþ1 using (7) and use the estimated value of Gk obtained in the previous step. ^k Þ (j) With the new estimated parameters, form the output ðy ^ k ¼ C k ð^ hk Þ^ hk and compare with the measured values. using y (k) Go to step (h) and update the parameters, unless the parameters have converged.
Other variables and constants are defined in Appendix. To show the strength of the identification method in a more challenging problem, the effect of saturation is also considered in the model. The first step towards the representation of saturation is to introduce the saturation factors (Ksd, Ksq) as [7]:
xad ¼ K sd xadu ;
xaq ¼ K sq xaqu
where xadu, xaqu are unsaturated values of xad, xaq and Ksd, Ksq are parameters which depend on the operating conditions. They are calculated using open circuit saturation curve (OCC). For K sd ¼ V atVþatDv , Vat is the air-gap voltage and can be calculated by:
V at ¼ jv t þ jxl ij 3. Study system A synchronous machine connected to an infinite bus through a transmission line, Fig. 1, is considered here as the study system. Parameters of the synchronous machine used in the simulation studies are given in Appendix. The nonlinear structure of the synchronous generator, derived in [7,8], is used to model the system. The model is described by (14)–(16):
v d ¼ ra id þ w_ d xwq v q ¼ ra iq þ w_ q þ xwd v f ¼ rf :if þ w_ f 0 ¼ v D ¼ r D :iD þ w_ D 0 ¼ v Q ¼ rQ :iQ þ w_ Q
ð14Þ
ð18Þ
Dv depends on the operating conditions. To model such a dependence, the commonly used formula is:
Dv ¼ AeBðv t VT1Þ
ð19Þ
where A, B and VT1 are proper constants which are obtained given the OCC curve. For salient pole machines, xaq does not vary significantly with saturation. Therefore, Ksq can be assumed to be equal to one for all loading conditions. For cylindrical rotor machines, however, Ksq = Ksd is assumed. For further treatment of saturation effects refer to [7]. This model is used in the simulation studies described in this paper for the identification of the synchronous machine model. 4. Simulation results
and
2
x0 wd 3 2 xd 4 x0 wf 5 ¼ 4 xmd
ð17Þ
x0 wD x md x0 wq xq ¼ x0 wQ xmq
32 3 xmd xmd id 5 4 xf xmd : if 5 xmd xD iD xmq iq : xQ iQ
To illustrate the proposed identification method, a Psuedo Random Binary Sequence (PRBS) signal is applied to the field voltage, and the electric power, terminal voltage and field voltage are sampled. The sampling time was selected to be 1 ms. The input–output data collected from the system model, shown in Fig. 2, is used for
The rotor dynamics is described by:
d_ ¼ x x_ ¼ 1J ðT m T e D xÞ
ð15Þ
where Te, electrical torque, is usually approximated by electrical power (Pe) when the system is connected to an infinite bus (which means x ffi x0) [7,8], i.e.:
T e ffi P e ¼ v d id þ v q iq i¼
P2 þ Q 2
; u ¼ tan1
Q
P vt v d ¼ v t sin d; v q ¼ v t cos d id ¼ i sinðd þ uÞ; iq ¼ i cosðd þ uÞ v bd ¼qv dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r e id þ xe iq ; v bq ¼ v q re iq xe id ffi v B ¼ v 2bd þ v 2bq
ð16Þ
xd ¼ xad þ xl ; xq ¼ xaq þ xl
Fig. 1. Structure of the study system.
Fig. 2. Data collected from the seventh order nonlinear synchronous generator model.
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M. Dehghani et al. / Electrical Power and Energy Systems 32 (2010) 524–529
Fig. 3. Identification results with the identified model and the measured variables, Fig. 2, at P = 0.9; Q = 0; vt = 1.05 p.u.
the identification procedure. This data is for the operating conditions:
P ¼ 0:9 p:u:; Q ¼ 0:0 p:u: and
Fig. 5. Identification results with the identified model and the nonlinear synchronous model at P = 1.1 p.u.; Q = 0.1 p.u.; vt = 1.05 p.u.
Table 1 Some parameters of the identified model in (2).
v t ¼ 1:05 p:u:
The identification method described in Section 2 is used to identify a synchronous machine simulated using the seventh order nonlinear model with saturation described in Section 3. Using the first 300 samples and least squares identification method [6], the parameters of the equivalent linear model were estimated and used for ht0 . The parameters in the identification method were set to c = 7 and M = 0.1. These values were selected by trial and error to give a robust identification. In this study, the input is the sampled field voltage (vf) and the outputs are the sampled active power (P) or terminal voltage (vt). Identification results with the identified model and the measured variables, Fig. 2, are shown in Fig. 3. It can be seen that the proposed method is very successful in identifying the system dynamics. Since the system output and the model output are not distinguishable in Fig. 3, the error signals are shown in Fig. 4. To show that the identified model has successfully covered the main nonlinearities of the system, more studies were carried out. A
b11
b12
v1
Modeling electrical power 0.6568 0.1165 First OPa Second OP 0.6481 0.1154
10.116 10.606
17.713 17.577
0.313 104 0.299 104
Modeling terminal voltage First OP 0.6407 0.1146 Second OP 0.6416 0.1143
11.44 11.55
17.218 17.218
0.399 104 .371 104
a11
a
a12
OP stands for the operating point.
comparison of the performance of the identified model and the system at P = 1.1 p.u.; Q = 0.1 p.u.; vt = 1.05 p.u. is shown in Fig. 5. It can be seen that the identified model has modeled the system correctly. Additional studies showed similar performance when the identification algorithm is left on and updates the parameters as the operating conditions change. Values of some of the parameters obtained for these two operating points are given in Table 1. In the upper part of the table the parameters are for field input and power output model and in the lower part of the table the parameters are for field input and terminal voltage output model. With the identifier working continuously, the performance of the identified model in response to changes in terminal voltage and power output is shown in Fig 6. In this study, the system is subjected to a relatively large change of operating conditions and the estimator updates the parameters as the operating conditions of the system change.
5. Implementation of the proposed method on a micromachine
Fig. 4. Error signals of Fig. 3.
The system under consideration is a 3 kVA, 208 V, 60 Hz, threephase micro-machine, driven by a DC motor. The micro-machine can represent dynamic response of much larger synchronous machines when the parameters and variables are considered in a normalized per unit version [8]. The main problem with a micromachine can be the field time constant, which is much lower than that of the larger machines. This problem has been overcomed using a time constant regulator, which is used to increase the effective field time constant to match that of the larger units.
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M. Dehghani et al. / Electrical Power and Energy Systems 32 (2010) 524–529
Fig. 6. Identification results with the identified model and the nonlinear synchronous model while the identification algorithm is on.
Fig. 8. Identification results with the identified model and the micro-machine outputs at P = 0.6, Q = 0.53, Vt = 1.22 p.u.
The setup used for the experiment is shown in Fig. 7. The synchronous generator is driven by a DC motor. The excitation input signal is applied to the synchronous machine field through a D/A converter. The field voltage, terminal voltage and the electrical power are measured and sampled by the data acquisition system. The machine is connected to a constant voltage bus by a double circuit transmission line modeled by lumped elements. Each circuit consists of six p sections and simulates the performance of a 300 km long 500 kV transmission line. The sampling time was selected to be 50 ms. This sampling time proved to be fast enough to capture the required dynamics. To apply the identification method described in Section 2 to the micro-machine described here, a number of experiments were carried out on the system. In the first experiment, a PRBS signal was applied. The initial operating condition was selected to be P = 0.6 p.u., Q = 0.53 p.u. and vt = 1.22 p.u.
Using the first 300 samples and the least squares method [6], the parameters of the equivalent linear model were estimated and used in ht0 . For the experimental results c = 5 and M = 0.1 were selected by trial and error to give a robust identification. Comparison of identification obtained by the identification method with the measured variables is shown in Fig. 8. It can be seen from the figure that the proposed method is very successful in identifying the system dynamics. In the second experiment, the initial operating condition was selected to be P = 1.2 p.u., Q = 0.72 p.u. and vt = 1.31 p.u. A comparison of the performance of the identified model and the system for the second experiment is shown in Fig. 9. The results show that the identified model has modeled the system quite accurately. Values of some of the parameters obtained for the two operating points are given in Table 2.
Fig. 7. Experimental set-up for the micro-machine.
M. Dehghani et al. / Electrical Power and Energy Systems 32 (2010) 524–529
529
lated seventh order nonlinear model of the synchronous machine and then by experiments on a physical machine. Simulation and experimental results show that the proposed method can be used successfully for the identification of the parameters of a synchronous machine model. All signals required for the proposed method are easily measurable. Based on the results, the proposed method seems feasible of application in an adaptive control scheme being considered for a synchronous machine. Appendix A The main variables and constants of (14) are:
i¼
P2 þ Q 2
;
u ¼ tan1
Q
P vt v d ¼ v t sin d; v q ¼ v t cos d id ¼ i sinðd þ uÞ; Fig. 9. Identification results with the identified model and the micro-machine outputs at P = 1.2, Q = 0.72, Vt = 1.31 p.u.
xd ¼ xad þ xl ; Table 2 Some parameters of the identified model in (2) in the experimental studies. a12
b11
b13
v1
Modeling electrical power 0.5529 First OPa Second OP 0.7818
0.4141 0.4901
0.018 0.015
0.0135 0.0241
0.163 105 0.183 105
Modeling terminal voltage First OP 0.4583 Second OP 0.3200
0.3082 0.4785
0.055 0.007
0.038 0.0066
0.696 106 0.135 105
a11
a
OP stands for the operating point.
-7.61
x 10
-7.62
b 11
J, D T 0do xl xad, xaq x0d xe, re d
x Tm EFD e0q P, Q
vt vB
-3
iq ¼ i cosðd þ uÞ
v bd ¼ v d re id þ xe iq ; v bq ¼ v q re iq xe id qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v B ¼ v 2bd þ v 2bq xq ¼ xaq þ xl
rotor inertia and damping factor direct-axis transient time constant stator leakage reactance direct and quadratic axis mutual reactance direct transient reactance line and transformer reactance and resistance rotor angle with respect to the machine terminals rotor speed mechanical input torque steady state internal voltage of armature transient internal voltage of armature terminal active and reactive power per phase terminal voltage infinite bus voltage
-7.63
The parameters of machine, considered in the simulation, are as follows:
-7.64
xmd ¼ 1:236
xmq ¼ 1:196
-7.65
xd ¼ 1:386
xq ¼ 1:344
r a ¼ 0:003
-7.66
r f ¼ 0:006
xf ¼ 0:165
-7.67
RD ¼ 0:0284
xD ¼ 0:1713
-7.68
RQ ¼ 0:02368 xQ ¼ 0:125
-7.69
D ¼ 0:0025 A ¼ 0:031
-7.7
0
20
40
60
80
J ¼ 0:0252 B ¼ 6:93
100 time(s)
References Fig. 10. Variation of parameter b11.
As an illustration, one parameter, b11, computed by the identification algorithm is shown in Fig. 10 for the two operating points. This parameter is used in modeling the terminal voltage. 6. Conclusions Nonlinear identification of a synchronous generator using H1 identification method is described in this paper. The normal linear H1 identification method has been modified to cover the nonlinearities of the system, such as saturation effects in synchronous generators. The proposed method has been tested first on a simu-
[1] Pierre DA. A prospective on adaptive control of power systems. IEEE Transactions on Power Systems 1987;PWRS-2(5):387–96. [2] Chen GP, Malik OP, Hope GS, Qin YH, Xu GY. An adaptive power system stabilizer based on the self-optimizing pole-shifting control strategy. IEEE Transactions on Energy Conversion 1993;8(4):638–45. [3] Lee Sang-Seung, Park Jong-Keun. Sliding mode power system stabilizer based on LQR: part I. Journal of Electrical Engineering and Information Science 1996;1(13):39–45. [4] He J, Malik OP. An adaptive power system stabilizer based on recurrent neural networks. IEEE Transactions on Energy Conversion 1997;12(4):413–8. [5] Theodor Y, Shaked U, Berman N. Time-domain H1 identification. IEEE Transactions on Automatic Control 1996;41(July):1019–23. [6] Nelles O. Non-linear System Identification: From Classical Approaches to Neural Networks and Fuzzy Models. Berlin, New York: Springer; 2001. [7] Yu Yau-Nan. Electric Power System Dynamics. Academic Press; 1983. [8] Kundur P. Power System Stability and Control. McGraw-Hill Inc.; 1994.
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Synchronous machine model parameters estimation by a time-domain identification method M. Dehghani a,*, M. Karrari a, W. Rosehart b, O.P. Malik b a b
Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran Department of Electrical and Computer Engineering, University of Calgary, Calgary, Canada
a r t i c l e
i n f o
Article history: Received 14 May 2009 Accepted 27 July 2009
Keywords: Synchronous machines Identification Nonlinear systems Power system modeling H1 identification methods
a b s t r a c t In this paper, the captured data from a real physical machine is used to investigate a nonlinear identification method. Since many practical systems, such as synchronous machines are nonlinear, linear models identified for a particular operating condition may not perform well for other operating conditions. The nonlinear H1 identification is used to overcome this shortcoming and to find a robust model despite noise and system uncertainities. The proposed method is applied on a simulated nonlinear model of a synchronous machine and is also tested on a physical machine. The field voltage is considered as the input, and the active output power and the terminal voltage are considered as the outputs of the synchronous machine. Simulation and experimental results show good accuracy of the identified model. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Adaptive control techniques have been proposed to improve the dynamic performance of power systems [1–4]. Among the published literature on adaptive power system stabilizers (PSSs), self-tuning regulators are the most common. A self-tuning regulator consists of a system model of a known structure and a controller. Control is calculated based on the model parameters. To compute the control appropriate to the operating conditions, parameters of the system model are obtained by an on-line identification technique. Correctness of the identification determines the preciseness of the identified model that tries to track the dynamic behaviour of the plant. For a time varying system, good tracking ability of the identification method is very desirable. The objective of this paper is to identify a nonlinear model of a synchronous machine that can be used for system analysis and controller design, especially the design of a PSS. The model can be used either in a predictive control structure for an on-line design, or as a simulator to test an off-line design. The main problem associated with modeling are the measurement noise and model uncertainities. To overcome these problems for linear autoregressive (ARX) models, the H1 identification method has been proposed [5]. Synchronous generators, however, * Corresponding author. Tel.: +98 711 2303081. E-mail address: [email protected] (M. Dehghani). 0142-0615/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2009.07.010
are highly nonlinear systems. To develop a robust identification method for such a nonlinear system, in the presence of measurement noise and model uncertainities, the H1 identification method is first generalized to cover the nonlinearities of the system. The developed method is then applied to identify the synchronous generator. The identification method and the nonlinear model of the system used in this paper are described in Sections 2 and 3, respectively. The system input–output data used for model identification is presented in Section 4 and the performance of the method is demonstrated on a simulated nonlinear model of a synchronous generator. The application of the proposed method on a physical machine and experimental test results are described in Section 5. Section 6 concludes the paper.
2. Identification method Consider a system in a noisy environment represented by a linear discrete time model:
zk ¼
n X i¼1
ai zki þ
m X
bi uki þ c0 xk
ð1Þ
i¼1
where zk is the unknown real output at time step k; uk1 is the system input at time step (k 1); ai, bi are the system parameters, and xk is the unknown driving disturbance. The system under study in this paper, i.e. a synchronous machine, however, is a nonlinear system with both physical, e.g.
525
M. Dehghani et al. / Electrical Power and Energy Systems 32 (2010) 524–529
saturation, and mathematical nonlinearities (as described in Section 3). Therefore, the linear model parameters would change with changes in the operating conditions. To take this into account, the linear regression model of (1) has been generalized to a nonlinear model of (2) using Taylor series:
zk ¼
n X
a1i zki
i¼1
þ
n X
a2i ðzki Þ2
i¼1
m X
n X
aNi ðzki ÞN
i¼1
b1i uki þ þ
m X
i¼1
N
bNi ðuki Þ þ c0 xk
ð2Þ
xk hkþ1 ¼ Ahk þ B vk xk yk ¼ C k ðhk Þhk þ D
ð3Þ
htk ¼ ½a11 a1n aN1 aNn ; b11 b1m bN1 bNm ; v k1 v kn
ð4Þ
and
2
INðnþmÞ 6 A¼6 40
"
nNðnþmÞ
2
3
0NðnþmÞn 01ðn1Þ
011
In1
0ðn1Þ1 3
#7 7; 5
7 5;
1
ð5Þ
0ðn1Þ1 N
uk1 ukm ; ; ðuk1 Þ ðukm Þ ; a1 an
n X i¼1 m X
^1i ^zki a
n X
^2i ð^zki Þ2 a
i¼1
^ u þ þ b 1i ki
n X
^Ni ð^zki ÞN a
i¼1 m X
^ ðu Þ b Ni ki
N
ð6Þ
i¼1
where ^zk is the best estimate of zk. To solve the estimation problem, the following estimation algorithm has been suggested and the stability of the algorithm under certain conditions is proved in [5]:
(
^hkþ1 ¼ A^hk þ Gk ð^hk Þ ðy ^zk Þ; k ^zk ¼ C k ð^hk Þ^hk
^h0 ¼ 0
ð7Þ
The only unknown in the above estimator is Gk. The following formulas are used to estimate the vector:
b k ¼ ðAQ 1 C e t ð^hk Þ þ BDt ÞU1 ð^hk Þ G k k k where
I
0 0
e k ðhk Þ ¼ C e k ðhk Þ þ ^ht 6 C k4 0
3
7 0 05
The next step is to assume that there is a nominal vector of parameters, h, and a given constant M > 0 such that:
kC k ðhk Þ C k ðhÞk 6 M Define:
dk ¼
^ bt AQ 1 k dk G k
b k ^dt Q 1 At G k k
3 t 6 AQ k b b t 7 þ M4 b Gk Gk5 Gk
2 3 b Gk t t 1 þ M 4 t AQ k Q k A 5 AQ k
et e e t ð^hk Þ^dt þ ^dk C e k ð^hk Þ þ M C ~dk ¼ C k k e k ðhÞ C k ðhÞ C k ðhÞ 0 e 1 C k ðhÞ 0 0NðnþmÞn 2@ A NðnþmÞNðnþmÞ þM 1þ M 0nNðnþmÞ In
ð10Þ
ð11Þ
ð12Þ
h i e kþ1 ð^hkþ1 Þ þ ^dkþ1 g~k > 0 e t ð^hkþ1 Þ C I g~tk Q kþ1 þ c2 C kþ1
ð13Þ
1
The predictive form of model (2) is defined as:
i¼1
0 0
where g~k ¼ B Gk D. Using this approach, the identification method is summarized below:
C k ðhk Þ ¼ ½yk1 ykn ; ; ðyk1 ÞN ðykn ÞN ; N
0
e k ð e k ð^ where ^ dtk ¼ C hÞ C hk Þ. It is shown in [5] that the algorithm converges if and only if Q is positive definite and
0NðnþmÞ1
6 B ¼ 4 0½NðnþmÞþn1
þ
2
1 ^ 2 e ^ et ^ ^t 1 ^ t Dk ¼ ^dtk Q 1 k C k ðhk Þ þ C k ðhk ÞQ k dk þ dk Q k dk þ M r Q k et þ 2MQ 1 k C k ðhÞ
where
^zk ¼
et Uk ðhk Þ ¼ DDt þ Ce k ðhk ÞQ 1 k C k ðhk Þ
2
vk
ð9Þ
In (9), c is the disturbance rejection factor. It should be selected as small as possible. On the other hand, it should be large enough to guarantee the convergence of the algorithm. Also:
i¼1
This allows the nonlinearities, such as saturation, to be modeled. In (2), {aki} and {bki} are the model parameters to be identified, yk is the measured output (yk = zk + vk), and vk is the measurement noise. In (2), n is the order of the system and N is the degree of the nonlinearity. A larger N results in better approximation of nonlinearities, but increases model complexity. If N = 1 is selected, as in [5], a linear ARX model is obtained. To identify the parameters, the above system structure can be written as:
D ¼ ½ c0
n o1 t t b bt ^ bt b Q kþ1 ¼ AQ 1 k A þ BB G k Uk ðhk Þ G k þ G k Dk G k þ dk h i e kþ1 ð^hkþ1 Þ þ ~dkþ1 ; Q > 0 e t ðh^kþ1 Þ C c2 C 0 kþ1
ð8Þ
(a) Select a proper input signal to be applied to the system. The input signal should have a wide spectrum to cover all system dynamics. It should also have a proper magnitude. The magnitude should be large enough to cover the nonlinearities and also should be small enough to be safe to perform the test. (b) Select a proper sampling time and final time (the total time for the experiment). (c) Apply the selected input signal (item a), to the system and sample the input–output data by a data acquisition system. (d) Select the order of the model (n) and the number of the terms (N) in (2). Although the real values for n and N could be very high, a lower order is preferred if it can capture the required dynamics. In particular, for real time control applications, since the parameters are identified on-line to track the system, a low order model, generally n = 3, is sufficient. (e) Select proper values for c and M. Smaller value of c means better disturbance rejection. The smallest value of c is selected by trial and error, such that the condition in (13) is met and Q remains positive definite.
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(f) Estimate the linear parts, (a1i, b1i), of vector ht0 in (4) using least squares, as used in this paper, or any other identification algorithm developed for linear systems [6]. (g) Establish A, B, C0(h0) and D matrices. b k using (8) and calculate the other vectors given by (h) Update G (10)–(12) for the next iteration. (i) Update the estimated parameters ^ hkþ1 using (7) and use the estimated value of Gk obtained in the previous step. ^k Þ (j) With the new estimated parameters, form the output ðy ^ k ¼ C k ð^ hk Þ^ hk and compare with the measured values. using y (k) Go to step (h) and update the parameters, unless the parameters have converged.
Other variables and constants are defined in Appendix. To show the strength of the identification method in a more challenging problem, the effect of saturation is also considered in the model. The first step towards the representation of saturation is to introduce the saturation factors (Ksd, Ksq) as [7]:
xad ¼ K sd xadu ;
xaq ¼ K sq xaqu
where xadu, xaqu are unsaturated values of xad, xaq and Ksd, Ksq are parameters which depend on the operating conditions. They are calculated using open circuit saturation curve (OCC). For K sd ¼ V atVþatDv , Vat is the air-gap voltage and can be calculated by:
V at ¼ jv t þ jxl ij 3. Study system A synchronous machine connected to an infinite bus through a transmission line, Fig. 1, is considered here as the study system. Parameters of the synchronous machine used in the simulation studies are given in Appendix. The nonlinear structure of the synchronous generator, derived in [7,8], is used to model the system. The model is described by (14)–(16):
v d ¼ ra id þ w_ d xwq v q ¼ ra iq þ w_ q þ xwd v f ¼ rf :if þ w_ f 0 ¼ v D ¼ r D :iD þ w_ D 0 ¼ v Q ¼ rQ :iQ þ w_ Q
ð14Þ
ð18Þ
Dv depends on the operating conditions. To model such a dependence, the commonly used formula is:
Dv ¼ AeBðv t VT1Þ
ð19Þ
where A, B and VT1 are proper constants which are obtained given the OCC curve. For salient pole machines, xaq does not vary significantly with saturation. Therefore, Ksq can be assumed to be equal to one for all loading conditions. For cylindrical rotor machines, however, Ksq = Ksd is assumed. For further treatment of saturation effects refer to [7]. This model is used in the simulation studies described in this paper for the identification of the synchronous machine model. 4. Simulation results
and
2
x0 wd 3 2 xd 4 x0 wf 5 ¼ 4 xmd
ð17Þ
x0 wD x md x0 wq xq ¼ x0 wQ xmq
32 3 xmd xmd id 5 4 xf xmd : if 5 xmd xD iD xmq iq : xQ iQ
To illustrate the proposed identification method, a Psuedo Random Binary Sequence (PRBS) signal is applied to the field voltage, and the electric power, terminal voltage and field voltage are sampled. The sampling time was selected to be 1 ms. The input–output data collected from the system model, shown in Fig. 2, is used for
The rotor dynamics is described by:
d_ ¼ x x_ ¼ 1J ðT m T e D xÞ
ð15Þ
where Te, electrical torque, is usually approximated by electrical power (Pe) when the system is connected to an infinite bus (which means x ffi x0) [7,8], i.e.:
T e ffi P e ¼ v d id þ v q iq i¼
P2 þ Q 2
; u ¼ tan1
Q
P vt v d ¼ v t sin d; v q ¼ v t cos d id ¼ i sinðd þ uÞ; iq ¼ i cosðd þ uÞ v bd ¼qv dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r e id þ xe iq ; v bq ¼ v q re iq xe id ffi v B ¼ v 2bd þ v 2bq
ð16Þ
xd ¼ xad þ xl ; xq ¼ xaq þ xl
Fig. 1. Structure of the study system.
Fig. 2. Data collected from the seventh order nonlinear synchronous generator model.
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Fig. 3. Identification results with the identified model and the measured variables, Fig. 2, at P = 0.9; Q = 0; vt = 1.05 p.u.
the identification procedure. This data is for the operating conditions:
P ¼ 0:9 p:u:; Q ¼ 0:0 p:u: and
Fig. 5. Identification results with the identified model and the nonlinear synchronous model at P = 1.1 p.u.; Q = 0.1 p.u.; vt = 1.05 p.u.
Table 1 Some parameters of the identified model in (2).
v t ¼ 1:05 p:u:
The identification method described in Section 2 is used to identify a synchronous machine simulated using the seventh order nonlinear model with saturation described in Section 3. Using the first 300 samples and least squares identification method [6], the parameters of the equivalent linear model were estimated and used for ht0 . The parameters in the identification method were set to c = 7 and M = 0.1. These values were selected by trial and error to give a robust identification. In this study, the input is the sampled field voltage (vf) and the outputs are the sampled active power (P) or terminal voltage (vt). Identification results with the identified model and the measured variables, Fig. 2, are shown in Fig. 3. It can be seen that the proposed method is very successful in identifying the system dynamics. Since the system output and the model output are not distinguishable in Fig. 3, the error signals are shown in Fig. 4. To show that the identified model has successfully covered the main nonlinearities of the system, more studies were carried out. A
b11
b12
v1
Modeling electrical power 0.6568 0.1165 First OPa Second OP 0.6481 0.1154
10.116 10.606
17.713 17.577
0.313 104 0.299 104
Modeling terminal voltage First OP 0.6407 0.1146 Second OP 0.6416 0.1143
11.44 11.55
17.218 17.218
0.399 104 .371 104
a11
a
a12
OP stands for the operating point.
comparison of the performance of the identified model and the system at P = 1.1 p.u.; Q = 0.1 p.u.; vt = 1.05 p.u. is shown in Fig. 5. It can be seen that the identified model has modeled the system correctly. Additional studies showed similar performance when the identification algorithm is left on and updates the parameters as the operating conditions change. Values of some of the parameters obtained for these two operating points are given in Table 1. In the upper part of the table the parameters are for field input and power output model and in the lower part of the table the parameters are for field input and terminal voltage output model. With the identifier working continuously, the performance of the identified model in response to changes in terminal voltage and power output is shown in Fig 6. In this study, the system is subjected to a relatively large change of operating conditions and the estimator updates the parameters as the operating conditions of the system change.
5. Implementation of the proposed method on a micromachine
Fig. 4. Error signals of Fig. 3.
The system under consideration is a 3 kVA, 208 V, 60 Hz, threephase micro-machine, driven by a DC motor. The micro-machine can represent dynamic response of much larger synchronous machines when the parameters and variables are considered in a normalized per unit version [8]. The main problem with a micromachine can be the field time constant, which is much lower than that of the larger machines. This problem has been overcomed using a time constant regulator, which is used to increase the effective field time constant to match that of the larger units.
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Fig. 6. Identification results with the identified model and the nonlinear synchronous model while the identification algorithm is on.
Fig. 8. Identification results with the identified model and the micro-machine outputs at P = 0.6, Q = 0.53, Vt = 1.22 p.u.
The setup used for the experiment is shown in Fig. 7. The synchronous generator is driven by a DC motor. The excitation input signal is applied to the synchronous machine field through a D/A converter. The field voltage, terminal voltage and the electrical power are measured and sampled by the data acquisition system. The machine is connected to a constant voltage bus by a double circuit transmission line modeled by lumped elements. Each circuit consists of six p sections and simulates the performance of a 300 km long 500 kV transmission line. The sampling time was selected to be 50 ms. This sampling time proved to be fast enough to capture the required dynamics. To apply the identification method described in Section 2 to the micro-machine described here, a number of experiments were carried out on the system. In the first experiment, a PRBS signal was applied. The initial operating condition was selected to be P = 0.6 p.u., Q = 0.53 p.u. and vt = 1.22 p.u.
Using the first 300 samples and the least squares method [6], the parameters of the equivalent linear model were estimated and used in ht0 . For the experimental results c = 5 and M = 0.1 were selected by trial and error to give a robust identification. Comparison of identification obtained by the identification method with the measured variables is shown in Fig. 8. It can be seen from the figure that the proposed method is very successful in identifying the system dynamics. In the second experiment, the initial operating condition was selected to be P = 1.2 p.u., Q = 0.72 p.u. and vt = 1.31 p.u. A comparison of the performance of the identified model and the system for the second experiment is shown in Fig. 9. The results show that the identified model has modeled the system quite accurately. Values of some of the parameters obtained for the two operating points are given in Table 2.
Fig. 7. Experimental set-up for the micro-machine.
M. Dehghani et al. / Electrical Power and Energy Systems 32 (2010) 524–529
529
lated seventh order nonlinear model of the synchronous machine and then by experiments on a physical machine. Simulation and experimental results show that the proposed method can be used successfully for the identification of the parameters of a synchronous machine model. All signals required for the proposed method are easily measurable. Based on the results, the proposed method seems feasible of application in an adaptive control scheme being considered for a synchronous machine. Appendix A The main variables and constants of (14) are:
i¼
P2 þ Q 2
;
u ¼ tan1
Q
P vt v d ¼ v t sin d; v q ¼ v t cos d id ¼ i sinðd þ uÞ; Fig. 9. Identification results with the identified model and the micro-machine outputs at P = 1.2, Q = 0.72, Vt = 1.31 p.u.
xd ¼ xad þ xl ; Table 2 Some parameters of the identified model in (2) in the experimental studies. a12
b11
b13
v1
Modeling electrical power 0.5529 First OPa Second OP 0.7818
0.4141 0.4901
0.018 0.015
0.0135 0.0241
0.163 105 0.183 105
Modeling terminal voltage First OP 0.4583 Second OP 0.3200
0.3082 0.4785
0.055 0.007
0.038 0.0066
0.696 106 0.135 105
a11
a
OP stands for the operating point.
-7.61
x 10
-7.62
b 11
J, D T 0do xl xad, xaq x0d xe, re d
x Tm EFD e0q P, Q
vt vB
-3
iq ¼ i cosðd þ uÞ
v bd ¼ v d re id þ xe iq ; v bq ¼ v q re iq xe id qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v B ¼ v 2bd þ v 2bq xq ¼ xaq þ xl
rotor inertia and damping factor direct-axis transient time constant stator leakage reactance direct and quadratic axis mutual reactance direct transient reactance line and transformer reactance and resistance rotor angle with respect to the machine terminals rotor speed mechanical input torque steady state internal voltage of armature transient internal voltage of armature terminal active and reactive power per phase terminal voltage infinite bus voltage
-7.63
The parameters of machine, considered in the simulation, are as follows:
-7.64
xmd ¼ 1:236
xmq ¼ 1:196
-7.65
xd ¼ 1:386
xq ¼ 1:344
r a ¼ 0:003
-7.66
r f ¼ 0:006
xf ¼ 0:165
-7.67
RD ¼ 0:0284
xD ¼ 0:1713
-7.68
RQ ¼ 0:02368 xQ ¼ 0:125
-7.69
D ¼ 0:0025 A ¼ 0:031
-7.7
0
20
40
60
80
J ¼ 0:0252 B ¼ 6:93
100 time(s)
References Fig. 10. Variation of parameter b11.
As an illustration, one parameter, b11, computed by the identification algorithm is shown in Fig. 10 for the two operating points. This parameter is used in modeling the terminal voltage. 6. Conclusions Nonlinear identification of a synchronous generator using H1 identification method is described in this paper. The normal linear H1 identification method has been modified to cover the nonlinearities of the system, such as saturation effects in synchronous generators. The proposed method has been tested first on a simu-
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