Parameter Identification from SSFR Tests and d-q Model Validation of Synchronous Generator S. Rakotovololona, M. Bergeron Student member, IEEE, J. Cros, P. Viarouge
Abstract-- This paper introduces a fast parameter identification technique for a two-order d-q model of a synchronous generator from Standstill Frequency Response (SSFR) tests. This identification method is based on a modified SSFR test and the curves of the noload and sustained short-circuit tests. This allows the lowest SSFR frequency to be 0.01Hz instead of 0.001Hz. The phase and field resistances are measured from fast and precise DC tests during the SSFR test. In this work, the value of the unsaturated synchronous inductance is defined from the no-load air gap linearization and the short-circuit saturation curve. This definition fits better when the saturation is considered and we present how to implement this approach with the Matlab-Simulink SimPowerSystem standard d-q models. Validations are made using sudden short-circuit tests at nominal field current to evaluate d-q model performances with magnetic saturation. Index Terms-- d-q model, Electric Machine, Parameter Extraction, Standstill Frequency Response, Transfer Functions
T
I. INTRODUCTION
HE d-q model is widely used for the simulation of electrical machines. It is a relatively simple and versatile model that sufficiently represents synchronous machine behavior for most simulation needs, such as grid stability studies [1]. For example, for the integration of new generators on a grid, the grid operator will define a range of values for the machine parameters. These values have to be respected by the manufacturer and an experimental parameters identification must be achieved during the commissioning tests. The values can be obtained from the classical tests such as those defined for hydro generators in [2], or from the SSFR tests explained in [3]. __________________________ This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) and Alstom Renewable Power Canada. S. Rakotovololona, M. Bergeron, J. Cros and P. Viarouge are with the LEEPCI laboratory (Laboratoire d'Électrotechnique, Électronique de Puissance et Commande Industrielle) in Laval University, Québec, Canada. (e-mail:
[email protected],
[email protected],
[email protected],
[email protected])
The SSFR test is interesting since it is less expensive than the classical tests and can be carried during down-time. In some cases, for example in a partial refurbishment, it can replace the potentially cumbersome and risky sudden shortcircuit tests. There are some issues with the SSFR. To name a few, the parameters are only evaluated for unsaturated conditions and there are some questions regarding the effect of rotation on the parameters [4]. The precision of the rotor positioning during the SSFR is also a concern [5]. In this paper, we apply the SSFR parameter identification in both axis to a laboratory round rotor generator. We use a different method than the one proposed in the standard [3]. Our method avoid testing the machine at frequencies lower than 0.01Hz. Proceeding to the measurements between 0.001Hz and 0.01Hz takes roughly 9 hours and is subject to a lot of noise. Instead, we proceed to precise DC resistance measurements for the windings and apply the curve fitting on the 0.01-1000Hz frequency range. We used a classical curve fitting method, a few techniques are well described [6], [7]. In the curve fitting process, we consider that the stator resistance is influenced by the setup [8]. In the d-q model, we use the rated resistance value. To speed up the test, it is also proposed to use a broadband signal (multisinusoidal) or periodic chirp (swept sine) method [9], [10]. To consider the saturation, we introduce a saturation curve that is applied on the no-load air gap linearization. That definition differs from the one used in d-q machine model in the Matlab-Simulink SimPowerSystem toolbox. We believe this approach to be more intuitive [1]. Doing so, the machine parameters have to be evaluated with a SSFR test in unsaturated conditions [11]. To evaluate the validity of the identified parameters, we compare the experimental and simulated waveforms obtained during a phase to phase and three phases shortcircuit. The tests are done with the nominal field current, thus the machine is saturated during the tests. II. D-Q MODELLING The mathematical models used in power system stability studies are based on several simplifications [1], [12]. The standard d-q model uses a transformation in order to obtain a reference frame rotating with the rotor instead of the fixed stator frame. The direct d-axis is aligned with the magnetic
axis of the field winding, and the quadrature q-axis is at 90 electrical degrees from the d-axis. That transformation is known as the d-q transformation or Park transformation. The assumptions associated with the use of d-q transformation are presented below [1]: • Magnetic saturation effects are neglected. By choosing two orthogonal axes, we suppose that currents flowing in one axis do not produce flux in the other axis. • The armature windings are sinusoidally distributed along the air-gap so that each phase winding produces a sinusoidal mmf wave. Hence, space harmonics are negligible and the induced voltage is purely sinusoidal. • The stator slots do not cause appreciable variation of the rotor inductances with rotor position • Magnetic hysteresis is negligible. The d-q transformation has a few advantages compared to the fixed stator frame model: • When the neutral is not connected, zero sequence quantities disappear, since the neutral current is zero. • For steady state operations, time-varying stator quantities in the 𝑎𝑎𝑎 reference appear constant in the dq reference frame. • The inductance matrix is now a matrix of constant terms. Thus, dynamic performance analyses are performed with constant inductances which improves the simulation time. • The current flowing in one axis creates flux in the same axis only. Therefore, the generator can be modeled by two magnetically uncoupled circuits: the d-axis network in which the flux links the field winding and the q-axis network in which the flux does not link the field winding. The direct-axis model includes the d-axis armature winding, the field winding and any additional equivalent damper windings (Fig. 1). The damper bar elements (𝐿𝑛𝑛 , 𝑅𝑛𝑛 ) represent equivalent current paths in the d-axis damper bars or in the rotor steel body. The rotor often has copper rods inserted in the pole face (for salient poles rotor) or underneath the rotor slot wedges (for round rotor). They are connected to each other on each pole to form a grid. When they are connected from pole to pole, they form a full cage. Currents can also be induced in the rotor body of a solid-rotor machine. The number of assumed rotor circuits determines the order of the model. The same approach applies to the q-axis network, except that the field winding is not present (Fig. 2).
Fig. 1. d-axis general equivalent circuit
Fig. 2. q-axis general equivalent circuit
Since they are magnetically uncoupled, d- and q- axis networks are considered independently. The d-axis network is a quadrupole. It consists of four variables transformed in the Laplace domain: the field current 𝑖𝑓𝑓 (𝑠), the field voltage 𝑣𝑓𝑓 (𝑠), the d-axis armature current 𝑖𝑑 (𝑠) and the daxis armature voltage 𝑣𝑑 (𝑠). When two of the four variables are specified, the remaining two variables can be determined through the network parameters. For example, if 𝑖𝑑 (𝑠) and 𝑖𝑓𝑓 (𝑠) are given, the set of network parameters is defined as follow (1). 𝑍𝑑𝑑 is the armature input impedance with an opened field winding. 𝑍𝑓𝑓 is the field input impedance with an opened stator winding. 𝑍𝑎𝑎𝑎 is the open-circuit transfer impedance. 𝑣𝑑 𝑍𝑑𝑑 (𝑠) 𝑍𝑎𝑎𝑎 (𝑠) −𝑖𝑑 �𝑣 ′� = � �� � 𝑍𝑎𝑎𝑎 (𝑠) 𝑍𝑓𝑓 (𝑠) 𝑖𝑓𝑓 ′ 𝑓𝑓
(1)
With a choice of 𝑖𝑑 (𝑠) and 𝑣𝑓𝑓 (𝑠) as independent variables, (2) shows the terminal relations for the d-axis. 𝑣𝑑 𝑍𝑑 (𝑠) 𝑠𝑠(𝑠) −𝑖𝑑 �𝑖 ′� = � �� � 𝑠𝑠(𝑠) 𝑌𝑓𝑓 (𝑠) 𝑣𝑓𝑓 ′ 𝑓𝑓
(2)
We obtain network parameters that are easily measurable with the field winding shorted or opened during the test. 𝑍𝑑 (𝑠) is the d-axis operational impedance. 𝑠𝑠(𝑠) is the armature to field transfer function. 𝑍𝑎𝑎𝑎 (𝑠) is the opencircuit transfer impedance. 𝑍𝑑 (𝑠) = − 𝑠𝑠(𝑠) =
𝑍𝑎𝑎𝑎 (𝑠) =
𝑣𝑑 � 𝑖𝑑 𝑣
𝑖𝑓𝑓 ′ � 𝑖𝑑 𝑣
(3)
𝑓𝑓 ′=0
(4)
𝑓𝑓 ′=0
𝑠𝑠(𝑠) 𝑣𝑓𝑓 ′ = � 𝑌𝑓𝑓 (𝑠) 𝑖𝑑 𝑖
(5) 𝑓𝑓 =0
Any choice of two independent variables is valid in determining the network parameters. However, three frequency-domain transfer functions are required to completely define the d-axis network [13]. For q-axis network, only one transfer function is necessary since it is a dipole. The relationship between
voltage and armature current is shown in (6). Shortcircuiting the field winding is not necessary for this identification. However, a quick test can be done to validate that the current flowing in the field winding is negligible and the rotor correctly aligned. 𝑍𝑞 (𝑠) = −
𝑣𝑞 � 𝑖𝑞 𝑣
𝑓𝑓 =0
(6)
III. GENERATOR USED FOR THE STUDY The machine used for the experiments is a three phases 5.4 kVA round rotor synchronous generator described in [14]. The stator has 54 slots. The rotor has 3 concentric coils per pole and a total of 24 copper damper bars connected by short-circuit rings. The main parameters are presented in Table . Fig. 3 present the characteristic curves. Table I: Generator main parameters
resistance of the field winding referred to the stator. All quantities are referred to the stator. In practice, measurements of field quantities are made at the field terminals. To use a common base, a stator/field transformation ratio (𝑁𝑎𝑎𝑎 ) is defined for the d axis: 𝑁𝑎𝑎𝑎
�2 𝑉𝑥 3 1 3 = × × 2 𝜔𝑛 𝐿𝑎𝑎𝑎 × 𝐼𝑓𝑓
(7)
The value of the transformation ratio is constant. It does not vary with the operating point. This condition is satisfied when the ratio (Vx ⁄Ifx × Ladx ) is constant; the open-circuit saturation curve is approximated as a straight line. Different methods can be used to obtain (𝑁𝑎𝑎𝑎 ). A first one is to use the straight line through the nominal point of operation (𝐼𝑓𝑓 , 𝑈𝐿𝐿 ). This method is based on the same approximation as the Behn-Eschenburg model. The set {𝑉𝑥 , 𝐼𝑓𝑓 , 𝐿𝑎𝑎𝑎 } becomes {𝑈𝑙𝑙 , 𝐼𝑓𝑓 , 𝐿𝑎𝑎_𝐵𝐵 } and we obtain (8). The method used in this work uses the air gap line to approximate the open-circuit saturation curve. The set {𝑉𝑥 , 𝐼𝑓𝑓 , 𝐿𝑎𝑎𝑎 } becomes {𝑈𝑙𝑙 , 𝐼𝑓𝑓 , 𝐿𝑎𝑎_𝑎𝑎 } and we obtain (9). We believe that the method based on the air gap line is more intuitive when the saturation is considered in order to obtain a valid model for all operating points. Lad_Behn_Eschenburg = Lad_air_gap =
Ull
√3 × ωn × Iccn Ull
Ifg √3 × ωn × Iccn × I
fn
− 𝐿𝑎
− La
(8)
(9)
The equivalent circuit parameters are related to the wellknown standard parameters (Xd, X’d, X’’d, T’d, T’’d, T’do, T’’do, Xq, X’q, X’’q, T’q, T’’q, T’qo, T’’qo) through the poles and zeros of 𝐿𝑑 (𝑠) transfer function [1]. V. EXPERIMENTAL SSFR
Fig. 3. Generator characteristic curves
IV. GENERATOR MODEL A second order d-q model was chosen for the synchronous generator at hand. Thus, we consider one equivalent damper winding in d-axis and two damper windings in q-axis. The d-axis second order d-q model has seven parameters in both axes. The set of circuit parameters is {𝑅𝑎 , 𝐿𝑎 , 𝐿𝑎𝑎 , 𝐿1𝑑 , 𝑅1𝑑 , 𝐿𝑓𝑓 ’, 𝑅𝑓𝑓 ’, 𝐿𝑎𝑎 , 𝐿1𝑞 , 𝑅1𝑞 , 𝐿2𝑞 , 𝑅2𝑞 , 𝑁𝑎𝑎𝑎 } where: 𝑅𝑎 is the rated phase resistance, 𝐿𝑎 is the stator leakage inductance, 𝐿𝑎𝑎 and 𝐿𝑎𝑎 are the d- and q-axis magnetizing inductance, {𝐿1𝑑 , 𝑅1𝑑 } and {𝐿1𝑞 , 𝑅1𝑞 , 𝐿2𝑞 , 𝑅2𝑞 } are d- and q-axis leakage inductance and resistance of the equivalent damper windings, and {𝐿𝑓𝑓 ’, 𝑅𝑓𝑓 ’} are the leakage inductance and
Test procedures and recommendations when performing SSFR tests are well documented in [3]. Tests in d- and qaxis are carried separately in specific rotor positions. For 𝐿𝑑 (𝑠) and 𝑠𝑠(𝑠), the field-winding is shorted and the rotor is aligned along the direct axis. For 𝑍𝑎𝑎𝑎 (𝑠), the rotor is in d-axis, but the field winding is left open. Finally, for 𝐿𝑞 (𝑠), the rotor is short-circuited and aligned along the quadrature axis. Fig. 4 shows the d-axis experimental SSFR setup. A sinusoidal voltage with varying frequency is applied to two armature phases in series.
Fig. 4. Experimental SSFR setup
The IEEE Standard suggests a frequency range of 0.001 Hz to 1000 Hz [3]. However, we chose to consider only frequencies above 0.01 Hz. Not only very low frequency measurements have very bad signal-to-noise ratio, but they also takes many hours to perform. When the machine characteristic curves and rated stator and rotor windings’ resistance are available, the 0.01-1000 Hz range is sufficient to complete the identification process. If the generator is unavailable for experimental testing, the SSFR results can be obtained by simulation [15], [16]. Since we limit the lowest frequency at 0.01Hz, we need to measure precisely the DC armature resistance (𝑅𝑎_𝑚𝑚𝑚 ) of the SSFR setup. The power amplifier cables are usually connected to the bus bars exiting the generator. (𝑅𝑎_𝑚𝑚𝑚 ) includes the generator leads resistance and should be used in the curve fitting process. In the d-q model (𝑅𝑎 ) is used. This is especially important for units with small phase resistances. VI. PARAMETER EXTRACTION Parameter determination using SSFR method is based on curve-fitting techniques applied to nonlinear functions. We fit simultaneously the three transfer functions discussed before: 𝑍𝑑 (𝑠) or 𝐿𝑑 (𝑠), 𝑠𝑠(𝑠) and 𝑍𝑎𝑎𝑎 (𝑠) [3]. Relationships between these transfer functions and the circuit elements {𝑅𝑎 , 𝐿𝑎 , 𝐿𝑎𝑎 , 𝐿1𝑑 , 𝑅1𝑑 , 𝐿𝑓𝑓 ’, 𝑅𝑓𝑓 ’, 𝐿𝑎𝑎 , 𝐿1𝑞 , 𝑅1𝑞 , 𝐿2𝑞 , 𝑅2𝑞 } of the equivalent circuit model are established using simple network theory. The steps followed to determine the model parameters from SSFR tests are summarized in the Fig. 5. Some parameters are defined before the curve-fitting process. The value of 𝑅𝑎_𝑚𝑚𝑚 and 𝑅𝑓𝑓 are obtained by DC measurements and are fixed. Two variables {𝑁𝑎𝑎𝑎 , 𝐿𝑎𝑎 } are defined by their respective expressions (7) and (9). Their value changes when the leakage inductance 𝐿𝑎 is modified during the curve-fitting process. The field winding resistance 𝑅𝑓𝑓 ′ viewed from stator side is also a dependent variable given its relation with the transformation ratio (10). 𝑅𝑓𝑓 ′ = 𝑅𝑓𝑓 × 3⁄2 × 1⁄𝑁𝑎𝑎𝑎 2
(10)
The remaining parameters {𝐿𝑎 , 𝐿1𝑑 , 𝑅1𝑑 , 𝐿𝑓𝑓 ′, 𝐿𝑎𝑎 , 𝐿1𝑞 , 𝑅1𝑞 , 𝐿2𝑞 , 𝑅2𝑞 } are variables during the optimisation.
Fig. 5. Parameter identification flowchart
A. Determination of measured transfer functions To obtain the transfer functions, the measured values (𝑖𝑎𝑎𝑎 , 𝑣𝑎𝑎𝑎 , 𝑣𝑓𝑓 , 𝑖𝑓𝑓 ) have to be transformed to d-q quantities. 𝑖𝑑 =
2
√3
𝑣𝑑 = −
𝑖𝑎𝑎𝑎
1
√3
vfd ′ =
ifd ′ = ifd ×
𝑣𝑎𝑎𝑎
vfd Nafd
2 × Nafd 3
(11) (12)
(13)
(14)
The expression of the transfer functions in terms of measured values are obtained by replacing (𝑣𝑑 , 𝑖𝑑 , 𝑣𝑓𝑓 ′, 𝑖𝑓𝑓 ′) in expressions (3) to (6). Zd (s) =
sG(s) = Zafo (s) =
1 varm 2 iarm
ifd ′ Nafd ifd = id √3 iarm
vfd ′ vfd √3 = id iarm 2 × Nafd
Zq (s) =
1 varm 2 iarm
(15) (16)
(17)
(18)
To obtain the operational inductance 𝐿𝑑 (𝑠), IEEE Std 115-2009 suggests plotting the real component of 𝑍𝑑 (𝑠) as a function of frequency and extrapolate it to zero frequency to get the armature dc resistance of the SSFR setup (𝑅𝑎_𝑚𝑚𝑚 ). Then, obtain 𝐿𝑑 (𝑠) as follow:
Ld (s) =
Zd (s) − R a_mes s
(19)
However, we prefer to make a DC measurement of the armature winding with the same SSFR setup in order not to perform the tests at low frequencies (< 0.01 Hz). The same method is applied for 𝐿𝑞 (𝑠). This reduce the duration of the SSFR from roughly 10 hours to 30 minutes per test. Thus, we save 28.5 testing hours for the three tests. B. Expression of the transfer functions with equivalent circuit elements In order to proceed to the curve fitting, we have to express the transfer functions in terms of the R-L elements in the second order circuit model. The d-axis operational impedance is given by (3). When the field winding is in short-circuit, we have three branches in parallel (Fig. 6) having an equivalent parallel impedance 𝑍𝑝 (20) and one in series 𝑍𝑠 (21). We can then express Zd (22) as the sum of 𝑍𝑝 and 𝑍𝑠 .
Fig. 6. Equivalent circuit with shorted field winding
𝑍𝑝 =
1 1 1 1 + + sLad R1d + sL1d R fd ′ + sLfd ′ 𝑍𝑠 = 𝑅𝑎 + 𝑠𝐿𝑎
Zd (s) = R a + sLa +
1 1 1 1 + + sLad R1d + sL1d R fd ′ + sLfd ′
Zd (s) = R a + sLd (s)
(20)
sG(s) =
1 1 1 + + sLad R1d + sL1d R fd ′ + sLfd ′
1 × R fd ′ + sLfd ′
Fig. 7. Equivalent circuit with opened field winding
Zpo =
1 1 1 + sLad R1d + sL1d
Zafo (s) =
1 1 1 + sLad R1d + sL1d
(25)
(26)
Determining the q-axis circuit parameters follows the same approach as for the d-axis operational impedance. C. Fitting of the transfer functions This is the final step to obtain the model parameters in the form of equivalent circuit. The circuit parameters {𝐿𝑎 , 𝐿1𝑑 , 𝑅1𝑑 , 𝐿𝑓𝑓 ′, 𝐿𝑎𝑎 , 𝐿1𝑞 , 𝑅1𝑞 , 𝐿2𝑞 , 𝑅2𝑞 } are set as variable during the optimisation process. Their values are iteratively modified until the sum of errors between measured and R-L transfer functions is minimized [6]. Remember that 𝑅𝑎 and 𝑅𝑓𝑓 ′ are constant, and {𝐿𝑎𝑑 , 𝑅𝑓𝑓 ’, 𝑁𝑎𝑎𝑎 } are dependant variables. The process ends when the best fit is obtained for all transfer functions. The values obtained are presented in Table . Table II: d-q model parameters
(21)
(22)
(23)
Referring back to Fig. 6, we see that the current flowing through (𝐿𝑓𝑓 ′, 𝑅𝑓𝑓 ′) is 𝑖𝑓𝑓 . The voltage across this impedance is 𝑉𝑎𝑎 , which is the same voltage across the equivalent impedance 𝑍𝑝 . Thus, Vab = id × Zp = ifd × (R fd ′ + sLfd ′) which can be rearranged to give the expression of 𝑠𝑠(𝑠) in terms of the equivalent circuit parameters: 1
(25). By extension, Zafo (s) is obtained (26).
(24)
For the open-circuit transfer impedance 𝑍𝑎𝑎𝑎 (𝑠), the equivalent circuit is given by Fig. 7. The voltage 𝑣𝑓𝑓 across the field winding is equal to the voltage across the equivalent impedance. The impedance of Zpo is given by
VII. CONSIDERING SATURATION In synchronous machines, the magnetic saturation has a significant impact on the no-load characteristic. Approximating the open circuit saturation curve with a linear line (air gap or Behn-Eschenburg) provides a model that is only correct for a specific region of operation in steady state. For large transients, it is necessary to consider the saturation to obtain correct waveforms.
It is generally assumed that the leakage inductance does not vary regarding the level of saturation. Only the magnetizing inductances (𝐿𝑎𝑎 , 𝐿𝑎𝑎 ) are affected. It is also assumed that the stator armature reaction has no influence on the magnetic saturation. Consequently, the magnetic saturation can be deduced from the machine no-load operation [1], [17]. The simplest method uses two saturation factors (𝑘𝑠𝑠 , 𝑘𝑠𝑠 ) to modify the value of the magnetizing inductances (𝐿𝑎𝑎 , 𝐿𝑎𝑎 ) (27). The factor (𝑘𝑠𝑠 ) is function of the excitation current (𝑖𝑓 ). It is computed from the no-load characteristic and the air gap line. For a given field current, we compute the ratio between the no-load voltage (𝐸𝑣 ), and the equivalent voltage on the air gap line: 𝐸𝑙𝑙𝑙𝑙𝑙𝑙 (28). 𝐿𝑎𝑎_𝑠𝑠𝑠 = 𝑘𝑠𝑠 (𝑖𝑖) ∗ 𝐿𝑎𝑎 � 𝐿𝑎𝑎_𝑠𝑠𝑠 = 𝑘𝑠𝑠 (𝑖𝑖) ∗ 𝐿𝑎𝑎 𝐾𝑠𝑠 (𝑖𝑖) =
𝐸𝑣 (𝑖𝑖) 𝐸𝑙𝑙𝑙𝑙𝑙𝑙 (𝑖𝑖)
(27) (28)
For salient pole machines, (𝑘𝑠𝑠 ) is usually set to 1 for all loading conditions since the path of q-axis flux is mostly in the air [1]. For round rotor alternator, (𝑘𝑠𝑠 ) is often put equal to (𝑘𝑠𝑠 ) since the q-axis saturation data is usually not available. In reality, magnetic saturation creates a crosscoupling between d- and q-axes [17]. For the generator under study, we evaluated (𝑘𝑠𝑠 ) using the no-load characteristic at up to twice the nominal field current. When the field current is greater than this value, the saturation coefficient is kept constant at its smallest value. The Fig. 8 illustrates the effect of not considering the saturation. It compares the experimental and d-q model field current response during a three-phase short circuit at the nominal field current (0.5A). As stated before, the transformation ratio is based on the air gap line approximation. The current waveforms obtained are less damped than the experimental result and there is an overshoot on the first cycle. This leads to incorrect prediction of losses and transient torques. The Fig. 9 presents the field current response with the saturation considered. The response is better. As it can be seen, the first peak is at 12 pu, the machine is very saturated at that current level.
Fig. 8. Field current during a sudden three phases short-circuit at full voltage without considering the magnetic saturation
Fig. 9. Field current during a sudden three phases short-circuit at full voltage with magnetic saturation considered
VIII. VALIDATION OF THE D-Q MODEL To evaluate the transient responses of the d-q model, we used the Matlab/Simulink block “SI Fundamental Synchronous Machine” with the parameter values identified using the SSFR identification. Using the air gap line to define the transformation ratio imposes using the 𝑖𝑓𝑓 value in place of the 𝑖𝑓𝑓 value. This is necessary in order to consider the saturation as presented in the paper. The difference between 𝑖𝑓𝑓 and 𝑖𝑓𝑓 is readily seen on Fig. 3. The Fig. 10 compares the phase current during the three phases short-circuit. The field current for the same test is given by Fig. 9. The general behavior is well represented. All the tests are done at 900 rpm. Note that the peak phase current is of 7pu, it is necessary to consider the saturation.
Fig. 10. Phase current during a three phases short-circuit at full voltage
The Fig. 11 and Fig. 12 compare the field current and the phase current during a phase to phase short-circuit. The waveforms are quite well represented. The perturbation at two times the stator frequency is well represented on the field winding.
Fig. 11. Phase current during a two phases short-circuit at full voltage
effects during the first period of the sudden short-circuit. X. REFERENCES [1] [2] [3] [4] Fig. 12. Field current during a two phases short-circuit at full voltage
To compare the torque and losses in the damper winding, we carried finite element simulations. The finite element simulation considers the real geometry, thus all space and time harmonics are considered. As it can be seen on Fig. 13, the slow varying component of the torque during the three phases short-circuit is well followed, but the fast varying component is completely absent. On Fig. 14, it can be seen that the damper losses are underestimated by the d-q model. The FE simulation considers the skin effect in the damper bars.
[5] [6]
[7]
[8]
[9]
[10]
[11] Fig. 13. Torque during the three phases short-circuit [12] [13]
[14] [15] Fig. 14. Damper losses during the three phases short-circuit
IX. CONCLUSION This paper presented a fast identification technique using a modified SSFR tests. It was shown how to define the transformation ratio with the air gap line and introduce a comprehensive saturation curve within the Matlab-Simulink d-q model block. The waveforms are compared in saturated conditions and it is shown that the d-q model provide a good fit with the field and phase currents. It is also shown that the d-q model underestimates the extra damper losses due to skin
[16]
[17]
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XI. BIOGRAPHIES Stéphanie Rakotovololona was born in Madagascar on January 10th, 1989. She received the B.Eng in electrical engineering from Moncton University in 2012. She is now pursuing a M.sc. degree in Laval University with the LEEPCI laboratory
Maxim Bergeron (S’2004) was born in the province of Québec, Canada on the January 4th, 1985. He received the B.Eng. and M.Sc. degree in electrical engineering from the Laval University in 2008 and 2011. He is now pursuing a Ph.D. degree in the same University in collaboration with LEEPCI and Alstom Hydro Power. Jérôme Cros received a Doctor degree from the "Institut National Polytechnique" of Toulouse, France in 1992. Since 1995, he is a professor in the Electrical Engineering Department of Laval University, Québec, Canada. His fields of interest include electrical machine design, power electronics and magnetic field calculation. Philippe Viarouge was born in Périgueux, France in 1954. He received the Engineer and Doctor of Engineering degrees, from the "Institut Polytechnique", Toulouse, France in 1976 and 1979, respectively. Since 1979, he has been Professor in the Department of Electrical and Computer Engineering at Laval University, Québec, PQ, Canada. His research interests include power electronics, AC drives, modeling and design of electrical machines.
