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Wideband modeling of a 45-MVA generator step-up transformer for network interaction studies
Electric Power Systems Research 142 (2017) 47–57
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Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Wideband modeling of a 45-MVA generator step-up transformer for network interaction studies Bjørn Gustavsen a,∗ , Bjørn Tandstad b a b
SINTEF Energy Research, NO-7465 Trondheim, Norway Statkraft, NO-0216 Oslo, Norway
a r t i c l e
i n f o
Article history: Received 20 June 2016 Received in revised form 3 August 2016 Accepted 30 August 2016 Keywords: Transformer Black-box model Frequency dependency Simulation Transients Transient recovery voltage
a b s t r a c t A stable and passive six-terminal model is developed for a 45-MVA generator step-up transformer based on frequency sweep measurements in the range 5 Hz–10 MHz and curve fitting with rational functions. The modeling employs separate treatment of the zero sequence system in combination with a new variant of the fast residue perturbation method for passivity enforcement. For use in system level simulations, the model can be employed in PSCAD, EMTP-RV and ATP. With ATP, the representation with an equivalent electrical circuit requires some care to prevent the occurrence of near zero circuit elements. Comparison with measured time domain responses at reduced voltage shows that the model can accurately reproduce transient voltage transfer between the windings while taking into account the loading effect of the connected system. The model also reproduces the transient recovery voltage (TRV) when interrupting short-circuit currents. As an application example, the model is demonstrated for simulation of resonant voltage transfer from the high-voltage side to the low-voltage side. In this setting, it is shown that the transformer input impedance can at high frequencies greatly influence the shape of the impinging overvoltages, thereby giving a self-protective effect against incoming overvoltages. © 2016 Elsevier B.V. All rights reserved.
1. Introduction The simulation of high-frequency transformer-network interactions [1] requires transformer models that are sufficiently accurate within the considered frequency band in terms of voltage ratios and impedance characteristics. Relevant simulation studies include voltage transfer between windings, resonant overvoltage buildup, and circuit breaker transient recovery voltage. A number of works have been presented that demonstrate successful application of measurement-based black-box modeling of distribution transformers [2–8] and power transformers [9–14], using low-amplitude measurements. These works assume linearity for the transformer behavior, which is usually considered a valid assumption at higher frequencies. Despite the many positive reports, the successful modeling of a transformer is non-trivial and still requires improvements [15].
∗ Corresponding author at: SINTEF Energy Research, P.O. Box 4761 Sluppen, NO7465 Trondheim, Norway. E-mail addresses: [email protected] (B. Gustavsen), [email protected] (B. Tandstad). URL: http://www.sintef.no (B. Gustavsen). http://dx.doi.org/10.1016/j.epsr.2016.08.035 0378-7796/© 2016 Elsevier B.V. All rights reserved.
This work presents exhaustive results from the measurementbased modeling of a 45-MVA three-phase generator step-up transformer. Its modeling is challenging because of its high voltage ratio (137 kV/8.5 kV) which leads to an undesirable scaling of the four sub-blocks of the terminal admittance matrix, and because its low-voltage winding is connected in delta. The latter results in the terminal admittance matrix having one very small eigenvalue at low frequencies which can lead to large error magnifications in applications if not handled properly. The measurement procedure is first described, emphasizing procedures for retaining the accuracy at low frequencies and noise removal. Following the procedure in Ref. [4], the zero sequence system is measured separately in order to capture the very small eigenvalue at low frequencies that pertains to the delta winding. A passive pole-residue model is extracted for the zero sequence system alone, and for the remaining admittance matrix with the zero sequence system excluded. The two models are finally combined into a single model. Special challenges are addressed and resolved for the passivity correction step which were not considered in Ref. [4]. The accuracy of the model is validated against measured voltage transfer functions in the frequency domain. Using convolution, the model is also validated in the time domain for voltage transfer between windings, with alternative loading conditions. Finally,
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Table 1 Transformer main data. Sn
Upn
Usn
er
ek
45 MVA
8.5 kV
137 kV
0.21%
11.2%
Table 2 List of Instruments. Current sensor Vector network analyzer
Ion Physics, model CM-100-6L Agilent E5061B-3L5
the model is also validated when applied in calculation of breaker transient recovery voltage (TRV). The accuracy at 50 Hz is also discussed. For use with general simulations, EMTP-type circuit simulators should be used. The export to the programs PSCAD, EMTP-RV and ATP is reviewed, and special challenges related to generation of ATP circuit netlist are described and resolved. The model is applied in EMTP-RV for simulation of resonant voltage transfer from the high-voltage side to the low-voltage side.
The unit is a three-phase two-winding YNd11 transformer of rated power Sn = 45 MV that is used for connecting a hydro power generator to the grid. The primary voltage has two settings: 8.5 kV or 7.5 kV while the secondary voltage is 137 kV. In this work, a model is developed with the secondary winding neutral point solidly grounded and with the 8.5 kV setting on the primary, leading to a component with six external terminals. The main electrical data for the transformer are given in Table 1, referenced to 20 ◦ C. Parameters er and ek denote respectively the real and imaginary part of the short-circuit impedance. 3. Problem statement The transformer behavior is to be characterized (measured) in the frequency domain with respect to its external terminals. Since linearity is assumed, the characterization will be based on the admittance matrix Y (6 × 6) which relates the terminal voltages v (6 × 1) with the terminal currents i (6 × 1), i(ω) = Y meas (ω) v(ω)
(1)
The (measured) admittance matrix Ymeas is to be approximated (fitted) by a pole-residue type rational model (2) which is stable, passive, and symmetrical. The modeling is to be performed in such way that the model can be utilized with different terminal conditions, i.e. without excessive error magnifications. N Rk k=1
jω − ak
(2)
The model is to be utilized in an EMTP-type simulation environment and be validated against measurements. 4. Measurement setup and processing 4.1. Measurement setup The admittance matrix Y was measured in the frequency range 5 Hz–10 MHz using a setup similar to the one described in Ref. [3]. The setup utilizes the gain-phase measurement capability of a Vector Network Analyzer (VNA) in combination with a wide-band current monitor (Table 2) that is integrated within a connection box, see Fig. 1. Manual reconnections on the connection box allows
a
N
A
2. Transformer main data
Ymeas (ω) ∼ = Y(ω) = R0 +
Fig. 1. VNA and connection box with built-in current monitor.
B
b
c
C
Fig. 2. Grounding plane (solid line) on top of transformer.
to measure the elements of Y one-by-one, and to perform voltage transfer measurements using voltage probes. The six transformer terminals were connected to the box using shielded coaxial cables which ranged from about five to six meters in length. The use of such long lengths was necessary as the measurement equipment had to be placed on a platform since the space on top of the transformer was too small to allow a suitable work environment. A braided wire flat was placed on top of the transformer and connected to the neutral point (“N”) and to the tank at several local points, see Fig. 2. The braided wire served as a local earth to which the cable screens were connected. Calibration was performed to account for the non-ideal frequency response and insertion impedance effect of the current monitor [16]. The setup gives the admittance matrix seen from the connection box, i.e. of the transformer with the measurement cables in series. The measurement cables do noticeably influence the measurements and thus the model that is extracted, and their effect should therefore be eliminated. However, in order to validate the accuracy of the measurement and model extraction procedure against additional measurements on the connection box, the results after cable elimination will not be shown. The connection points on the box are therefore referred to as transformer terminals, in both frequency domain and time domain measurements. The effect of cable elimination is presented in detail in Ref. [17] for this 45 MVA transformer where a new cable elimination method based on a transmission line equivalent is applied. 4.2. Series resistors on LV side At low frequencies, the admittance matrix elements are large in magnitude with a small real part. As a result, the insertion impedance from measurements cables and connections may affect the measured admittance elements significantly, thereby corrupting the behavioral information such as voltage ratio and passivity characteristics. Also, since the output impedance of the VNA is 50 , the voltage at the VNA reference input can become too low to be measured with sufficient accuracy. As a precaution, small resistors (0.1 ) were placed in series with the LV terminals. Their effect are discussed in Section 11.
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Fig. 3. Noise removal by fitting a rational model to the data using with L1 -norm minimization. The model uses 12 logarithmically spaced real poles between 1 Hz and 10 kHz.
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Fig. 5. Voltage transfer from HV side to LV side.
Fig. 6. Zero sequence measurement.
side and the low-voltage (LV) side via the partitioned admittance matrix
iH
=
iL
YHH
YHL
YLH
YLL
vH
(3)
vL
which gives the voltage transfers as Fig. 4. Admittance matrix elements.
4.3. Noise suppression It was found that some of the (small) measured admittance matrix elements had a significant amount of noise at around 50 Hz and at lower harmonics. This noise component is undesirable as it leads to spurious poles in the final model. The noise is suppressed by a signal processing step via rational approximation as follows. First, the number of samples is increased from 401 to 1601 by linear interpolation between the sample points, thereby reducing the probability that the least-squares rational approximation to be calculated will interpolate the sample values. Next, each admittance element is at low frequencies (below 1 kHz) replaced with the response of a low-order rational model with prescribed logarithmically spaced real poles. The residues are calculated using L1 -norm minimization [18] when solving the least-squares problem instead of the standard L2 -norm, further reducing the contribution from the outliers in the data. Fig. 3 shows the effect of the noise removal on the three small elements associated with the two off-diagonal blocks of Y. 5. Frequency domain measurements
v H = H HL v L ,
H HL = −Y −1 HH Y HL
(4a)
v L = H LH v H ,
H LH = −Y −1 LL Y LH
(4b)
The voltage transfers are compared with a direct measurement of these quantities using voltage probes. The agreement is found to be excellent for HHL , but for HLH the accuracy is poor at frequencies below 10 kHz as shown in Fig. 5. 5.2. Zero sequence admittance matrix The reason for the large errors in Fig. 5 is the presence of the ungrounded delta winding on the LV side which has only a weak capacitive coupling to ground at low frequencies. Following the procedure in Ref. [4], the accuracy problem is alleviated by making a separate measurement of the zero sequence system (on the connection box) as shown in Fig. 6, leading to a 2 × 2 matrix Y0 ,
iH iL
= Y0
vL
, Y0 =
yHH
yHL
yLH
yLL
(5)
The accuracy of this measurements is further improved in the lower frequency range by using the capacitive perturbation method described in Section 3 in Ref. [4]. With that method, one measures in addition the voltage transfer from terminal H to L, before (vLH ) and after (ˆvLH ) adding a suitably sized capacitor to the open terminal L. One next replaces elements yLH , yHL and yLL with the following,
5.1. Full admittance matrix yLH = yHL = Fig. 4 shows the measured elements of the admittance matrix Y, after noise removal. The quality of the measurement is assessed by calculating the voltage transfer between the high-voltage (HV)
vH
yLL = −
yLH
vLH
−jωC · vˆ LH 3(1 − vˆ LH /vLH )
(6) (7)
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The resulting model is subjected to iterative passivity enforcement using fast residue perturbation (FRP) [21] in combination with passivity checking. Some remarks are in order regarding the implementation. 6.3. Passivity enforcement of Ya Direct application of FRP to Ya results in a model where property (8) is lost for the sub-blocks of the residue matrices. Therefore, one must repeatedly reapply steps 1 and 2 to the residue matrices and perform the perturbation, leading to a time consuming and unreliable process. This difficulty is avoided by using a modified version of the FRP approach. It follows from (8) that the residue matrices Ri have two eigenvalues that are zero with associated eigenvectors that can be selected as
Fig. 7. Zero sequence admittance matrix.
t 0,a =
1(3×1)
,
t 0,b =
1(3×1)
(9)
Fig. 7 compares the elements of the zero sequence component of the original (6 × 6) Y with those of the directly measured Y0 , and those of Y0 obtained by the perturbation method. The zero sequence component of the original Y is obtained by replacing each of the four sub-blocks in (3) with a single element that equals the sum of the block-elements divided by three. The directly measured Y0 is replaced with the result by the perturbation method at frequencies below 8 kHz. The perturbation used a capacitance of 16.9 nF.
With the approach of FRP [21], only the eigenvalues of the residue matrices are perturbed. The procedure is now modified by excluding from the vector of free variables those two eigenvalues that are zero. That way, the two zero eigenvalues and associated eigenvectors (9) remain unchanged by the passivity enforcement. Also, property (8) remains unchanged.
6. Modeling with rational functions
6.4. Passivity assessment of Ya
The separately measured zero sequence system is utilized in the modeling procedure using the approach described in Ref. [4], followed by a new approach for passivity assessment and enforcement. Here, two rational models are extracted that are stable and passive. The first one (Ya ) is for the admittance matrix with the zero sequence system removed, and the second one (Yb ) is for the zero sequence system alone. Finally, the two models are concatenated into a single model, Y = Ya + Yb . This approach is motivated by the fact that Y has a large spread in its eigenvalues at low frequencies which makes it hard to create a model from Y which retains the accuracy of the small eigenvalues.
The passivity enforcement scheme is employed within an iterative loop where passivity violations are identified in each run. In order to safely determine all violations, the eigenvalues of the Hamiltonian test matrix (10) [22] or the half-size test matrix (11) [23] can be assessed for determining cross-over frequencies between passive and non-passive frequency ranges. Matrices A, B, C, and D represent the state-space model associated with the pole-residue model and are readily obtained as shown in Ref. [25].
6.1. Removal of zero-sequence system
M=
1(3×1)
A − B(D + DT ) −CT (D + DT )
−1
−1
−1(3×1)
B(D + DT )
C
C
The zero sequence system is removed from Y in (3) by applying to each sub-block the following procedure. 1. Subtract from each row the average value of the row elements; 2. Subtract from each column the average value of the column elements.
B
−AT + CT (D + DT )
S = A(BD−1 C − A)
−1 T −1 T
(10)
B
(11)
Both approaches fail as D is singular due to the removed zero sequence system. The singularity is removed by adding to D a contribution (12) associated with the zero sequence system, before performing the passivity assessment. This modification does not change the cross-over frequencies that are to be determined.
2 · 1(3×3)
−1(3×3)
−1(3×3)
2 · 1(3×3)
As a result, each block of the modified admittance matrix Ya gets one eigenvector with unity entries and associated eigenvalue equal to zero,
D→D+
(0 , t0 ) = (0, 1(3×1) )
Fig. 8 shows the resulting model for Ya using N = 100 poleresidue terms, after passivity enforcement. It is observed that the matrix elements are in general fitted with a high degree of accuracy.
(8)
6.2. Pole-residue modeling of Ya Ya is subjected to pole-residue modeling (2) using vector fitting (VF) [19,20] with inverse magnitude weighting. Each residue matrix Ri of the model is subjected to steps 1 and 2 described in Section 6.1 to remove the zero sequence component from the subblocks of Ri . As a result, the model’s admittance matrix Y inherits the same properties (8) as Ya in terms of the zero sequence eigenvalues and eigenvectors.
(12)
6.5. Zero sequence modeling The 2 × 2 zero sequence admittance matrix Y0 (5) is fitted with a pole-residue model using VF and subjected to passivity enforcement using iterative FRP. Fig. 9 shows the resulting model using N = 40 pole-residue terms, demonstrating an accurate result for all elements.
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Fig. 8. Passive model of Ya .
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Fig. 10. Elements of Y, for combined model.
Fig. 11. Eigenvalues of Y, for combined model. Fig. 9. Passive model of Y0 .
Each element of Y0 is expanded into a 3 × 3 block through (13).
i,j Y0
1 = 1(3×3) · 3
i,j r0
+
N i,j rk k=1
=
s − ak
i,j R0
+
N0 i,j Rk k=1
s − ak
(13)
The four sub-models are combined into a single model by stacking the residue matrices,
Yb =
R1,1 0
R1,2 0
R2,1 0
R2,2 0
+
N0 1 i=1
s − ak
R1,1 k
R1,2 k
R2,1 k
R2,2 k
(14)
6.6. Model combination Finally, the model of Ya and of Yb are combined into a single model by appending the two sums of pole-residue terms into a single sum Y = Ya + Yb , giving a model with N + N0 pole-residue terms. Figs. 10 and 11 show respectively the elements of the final Y and its eigenvalues, demonstrating excellent agreement. An exception is for two of the small eigenvalues in Fig. 11 at frequencies below 1 kHz. These two eigenvalues are found to be associated with the transformer no-load current which are unimportant in most transient simulation studies. Fig. 12 verifies that the model is passive within the 5 Hz–10 MHz frequency range as the eigenvalues of (Y + YH ) are positive. The procedure ensures passivity at all frequencies, ω ∈ [0, ∞).
Fig. 12. Eigenvalues of (Y + YH )/2, for combined model.
7. Frequency domain validation The model accuracy is validated in the frequency domain by comparing the voltage transfers (4a) and (4b) as calculated by the model’s Y with those of a direct measurement. Figs. 13 and 14 show that the voltage transfers are highly accurate in both directions, within the full frequency range. Comparison between Figs. 13 and 5 shows that the previous inaccuracies below 10 kHz are not present in the final model, which is due to the separate measurement and modeling of the zero sequence system. It
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Fig. 13. Voltage transfer from high to low.
Fig. 14. Voltage transfer from low to high.
is noted that in the case of the three small elements in Fig. 14, the relative errors are large below 10 kHz. However, these errors are inconsequential for a simulation since the small magnitudes imply low voltages. 8. Time domain validation: voltage excitation on HV side As a final validation, a set of time domain measurements are performed at reduced voltage and compared against time domain simulations using the pole-residue model. Again, all measurements are performed on the connection box. The setup consists of a function generator with 50 output impedance (Tektronix AFG3052C), a storage oscilloscope (Tektronix DPO 4054B), and a 0.5 m coaxial cable for connecting the function generator to the connection box. When applying the voltage to the LV side, a wide-band amplifier with low output impedance (Toellner 7608) was placed between the function generator and the transformer. The amplifier was necessary as the function generator output voltage would otherwise collapse. The voltage measurements were made using passive voltage probes with 10 M input impedance. 8.1. Unloaded transformer A step-like voltage is applied to terminal 1 with terminals 2 and 3 grounded (on the connection box) and the remaining terminals open, see Fig. 15a. The voltage response on the terminals are measured and the measured voltage on terminal 1 is applied to the model in an EMTP-like time domain simulation with model interfacing based on a Norton equivalent and recursive convolution [24].
Fig. 15. (a) Voltage application to HV side of unloaded transformer. (b) Open circuit voltages on LV side. (c) Open circuit voltages on LV side. Initial response.
Fig. 15b and c (zoomed view) shows the measured and simulated responses. It is observed that highly accurate results are obtained, both for the 60 kHz oscillation in Fig. 15b and the 2 MHz oscillation in Fig. 15c. These oscillations are also found as dominant peaks in the ditto frequency domain responses in Fig. 13. 8.2. Loaded transformer The transformer is loaded with resistors on the connection box as shown in Fig. 16a. The selected values correspond approximately to the characteristic impedance of cables (30 ) and overhead lines (400 ). Fig. 16b and c compares measured and simulated voltage responses at the terminals. The resistors are in the simulation represented by circuit elements and the measured voltage on node 0 is represented by an ideal voltage source. Again, highly accurate results are obtained. It is further observed that the oscillations are strongly damped compared to the previous unloaded transformer case, in particular the 60 kHz oscillation in Fig. 15b. This result implies that the model interacts correctly with the connected circuit.
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Fig. 17. (a) Voltage application to LV side of unloaded transformer. (b) Open circuit voltages on HV side.
Fig. 16. (a) Voltage application to HV side of loaded transformer. (b) Voltage response on LV side of loaded transformer. (c) Voltage response on LV side of loaded transformer. Initial transient.
9. Time domain validation: voltage excitation on LV side A similar investigation is performed when placing the excitation on the transformer low-voltage side. 9.1. Unloaded transformer In the unloaded transformer case (Fig. 17a), the model accurately reproduces the transient response on the high-voltage side as can be seen in Fig. 17b. The oscillation frequency is about 8.5 kHz and is clearly observable by a dominant peak in the corresponding frequency responses in Fig. 14.
Fig. 18. (a) Voltage application to LV side of loaded transformer. (b) Voltage response on HV side of loaded transformer.
10. Time domain validation: TRV applications 9.2. Loaded transformer 10.1. Opening switch on HV side With 400 resistors connected to the HV side (Fig. 18a), the oscillations in Fig. 17b become completely damped out in both the measurement and the simulation, see Fig. 18b.
As a final validation, the accuracy in transient recovery voltage simulation is investigated. In this test a mechanical switch starts
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Fig. 19. (a) Current interruption on HV side. (b) Voltage response on terminal 1. Table 3 Transformer voltage ratio (positive sequence).
Nominal Model
Fig. 20. (a) Current interruption on LV side. (b) Voltage response on terminal 4.
Table 4 Transformer short-circuit impedance (positive sequence).
|Vp /Vs |
|Vs /Vp |
0.0621 0.0625
16.1 15.9
opening which is connected between the source and the transformer, see Fig. 19a. The output voltage from the function generator is measured and realized as an ideal voltage source in a time domain simulation while the measured current through the switch is used to define the time instant where the switch is to open in the simulation. Fig. 19b shows that the simulated voltage response on the transformer side of the switch is highly accurate. The frequency is about 8.5 kHz, similar to Fig. 17b. It is remarked that the peak value of the 50 Hz feeding voltage is very small (0.12 V) compared to the observed peak in the recovery voltage (13 V) which results from the current chopping. The measurement and simulation starts about 20 ms before the breaker opens to ensure that there are no visible transients present in the simulation when the breaker opens. 10.2. Opening switch on LV side A similar measurement is performed on the LV side as shown in Fig. 20a and b. Again a very accurate result is obtained. The dominant frequency is about 50 kHz which is somewhat lower than the 60 kHz dominant oscillation in Fig. 15b.
Nominal Measured Y w./0.1 Measured Y w.o./0.1 Measured Y w.o./0.1 , and use of cable compensation
Zk,p []
Zk,s []
(0.0034 + j0.18) (0.14 + j0.18) (0.042 + j0.19) (0.010 + j0.18)
(0.88 + j47) (37 + j47) (9.8 + j47) (1.7 + j48)
in the measurements while the resistive part is much too large by a factor of 42. One major cause is that the measurements used 0.1 resistors that were placed in series with the LV terminals as explained in Section 4.2. These resistors appear on the HV side as R = 0.1·(137/8.5)2 = 26 . A set of frequency domain measurements were also performed without the 0.1 resistors (third row), but it is observed in Table 4 that the resistive part is still too high, by a factor 11. The accuracy can be further improved by eliminating the effect of the applied measurement cables which have a DC resistance of 0.0057 /m and lengths betwen 4.8 m and 6.2 m. The fourth row in Table 4 shows the effect of cable elimination using the S-parameterbased transmission line method described in Ref. [17]. The resistive part is now much closer to the nominal value. It is remarked that the 0.1 resistors have in general a negligible effect on the transient waveforms resulting from impinging voltage waves from the network.
12. Export to EMTP-type simulation programs 11. Model accuracy at 50 Hz The model is capable of reproducing the nominal voltage ratio with a high degree of accuracy as shown in Table 3. The model’s short-circuit impedance is however less accurate, in particular for its resistive part. Table 4 compares the nominal short-circuit impedance (first row) with the one that can be inferred from the measurements (second row). The reactive part is quite accurate
The simulation procedure [24] applied in the preceding sections is suitable for time domain validation with simple terminal conditions applied to the transformer terminals. For application in system level simulations with many different network components, it is more practical to include the model in one of the EMTP-type simulation platforms. The following describes the export of the model to leading EMTP-type programs.
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y1,2
1
y1,0
55
2
y2,0
Fig. 21. Admittance branches of two-node network.
12.1. PSCAD and EMTP-RV In PSCAD, the pole-residue model (14) can in the most recent program versions be directly included in the simulation, based on the procedure described in Ref. [24]. For use in EMTP-RV, the poleresidue model must be converted into a real-valued state-space model which can be directly loaded using the state-space component that is part of EMTP-RV.
Fig. 22. Simulated waveforms by alternative EMTP-type programs.
12.2. ATP ATP does not have the capability of directly including a poleresidue model or a state-space model. Therefore, one must instead convert the pole-residue model into an equivalent circuit as shown in Ref. [25]. However, the direct application of that approach to the proposed transformer model leads to a netlist which includes circuit elements with extreme numerical values that can compromise the accuracy of the simulation result. To see this, let us recall the relation between admittance matrix elements and admittance branches. Consider a model with two nodes as shown in Fig. 21. Using nodal analysis, the admittance matrix is established as
Y=
y1,0 + y12
−y12
−y12
y2,0 + y12
(15)
The reverse process, generalized to an n-conductor system gives the following procedure for calculating the branch admittances from Y,
yi i =
N
Yi j
(16)
j=1
yi j = −Yi j
(17)
One problem with this procedure arises for the pole-residue terms that are related to Ya . As described in Section 6.2, each 3 × 3 block of the associated residue matrices has a row-sum equal to zero which leads to zero values for the diagonal elements yii in (16). Due to the finite precision of the computations on a computer, the admittance elements are not exactly zero, leading to circuit elements with extreme values. This problem is overcome by simply bypassing the writing to file of the diagonal circuit elements associated with Ya (Fig. 22). 12.3. Comparative result For the example case in Fig. 15a, the transferred overvoltage to the LV side terminal 4 is simulated when applying a unit step voltage to terminal 1. The simulation is performed using PSCAD, EMTP-RV, ATP and the aforementioned Matlab implementation. The PSCAD simulation is with version v4.2 with the pole-residue model included as a user-defined component. Fig. 22 shows that the four simulation platforms give virtually identical results.
Fig. 23. Ground fault initiation by ideal switch.
13. Significance of transformer input impedance The frequency domain plot in Fig. 13 shows that there is a potential for transfer of large overvoltages from the HV side to the LV side, at frequencies around 2–4 MHz. For this to happen, an oscillating overvoltage in this frequency range must exist on the HV terminals. To investigate the scenario, we use the circuit in Fig. 23 using EMTP-RV. The transformer HV side is connected to an infinitelength uncoupled cable system with characteristic impedance ZC = 30 and propagation velocity v = 177 m/s which is fed from a three-phase ideal voltage source. The LV side terminals are open. An ideal single-phase ground fault occurs at a distance l = 22 m from the transformer HV side. This combination of length and velocity defines a quarter-wave resonance frequency of 2.0 MHz by (18), which coincides with one of the resonance peaks in Fig. 13. f/4 =
v 4l
(18)
Fig. 24 shows the waveforms on the HV and LV side terminals when the simulation has been started from 50 Hz initial conditions. It is observed that with ZC = 30 the presence of the transformer distorts and damps out the waveform on the transformer HV side, leading to a voltage on the LV side whose peak value is only 27 kV, being much lower than the value predicted by the 2 MHz resonance peak in Fig. 13. For comparison, the same plot includes the wave shapes that results when the 30 characteristic impedances are replaced with a very low (unrealistic) value ZC = 0.1 . In the latter case, the transformer does not damp out the square wave oscillation on the HV side, causing the voltage on the LV side to reach nearly 150 kV in the 5 s simulation time window. This result is not surprising since a unit step voltage that propagates on a line with characteristic impedance ZC and meets an open end terminated against a capacitance C to ground will initially give a voltage response at the end by (19). For instance, a capacitance of 1 nF will with ZC = 30 give a time constant T = 0.3 s, which is comparable
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to reproduce the transformer’s transient behavior with a high degree of accuracy, including voltage transfer between winding terminals with alternative loads connected to the terminals. The model can also be used for calculating transient recovery voltages due to interruption of currents. The model has been exported to PSCAD, EMTP-RV and ATP, for use in network interaction studies. Using EMTP-RV, it was shown that the transfer of resonant overvoltages from the HV side to the LV side can be strongly affected by the transformer input impedance. For oscillating frequencies in the MHz range, the considered transformer is capable of damping out the oscillation, thereby preventing resonant voltage build-up on the LV side.
Acknowledgments Fig. 24. Voltage response on terminals 1 and 4 (22 m cable).
The authors appreciate the assistance of Oddgeir Rokseth during the measurements, as well as Statkraft hydro station staff. Financial support was provided by the Research Council of Norway (RENERGI Programme), DONG Energy, EirGrid, Hafslund Nett, National Grid, Nexans Norway, RTE, Siemens Wind Power, Statkraft, and Statnett.
References
Fig. 25. Voltage response on terminals 1 and 4 (738 m cable).
to the travel time of the 22 m segment of = 0.12 s. One can therefore conclude that the transformer has a significant self-protective effect against very-high-frequency oscillations on the HV side terminals. For lower frequency components however, the transformer HV side is much less likely to damp out the overvoltage. This is shown in Fig. 25 when the cable length is increased to 738 m, giving a 60 kHz oscillation on the HV side which coincides with a peak in the voltage transfer in Fig. 13. u(t) = 2(1 − e−t/T ), T = ZC C
(19)
14. Conclusions A wideband, EMTP compatible terminal model has been developed for a 45-MVA generator step-up-transformer based on frequency sweep measurements and rational modeling via curve fitting. The approach is based on separate measurement and modeling of the zero system components in order to avoid error magnifications to result from the presence of a delta winding. The presented work improves the model extraction procedure in Ref. [4] by introducing a more rigorous approach for passivity enforcement of the model’s zero sequence component. In addition, an essential modification is made to the procedure for equivalent circuit representation associated with the model’s zero sequence component which avoids the occurrence of extreme circuit values. The accuracy of the model has been validated in both frequency domain and time domain. Comparison between time domain measurements and simulated waveforms shows that the model is able
[1] CIGRE Technical Brochure 577A, Electrical transient interaction between transformers and the power system. Part 1—Expertise, CIGRE JWG A2/C4.39, April 2014. [2] M.J. Manyahi, R. Thottappillil, Transfer of lightning transients through distribution transformers, in: International Conference on Lightning Protection, Cracow, Poland, 2002, pp. 435–440. [3] B. Gustavsen, Wide band modeling of power transformers, IEEE Trans. Power Deliv. 19 (1) (2004) 414–422. [4] B. Gustavsen, Frequency-dependent modeling of power transformers with ungrounded windings, IEEE Trans. Power Deliv. 19 (3) (2004) 1328–1334. [5] M. Popov, L. van der Sluis, R.P.P. Smeets, Evaluation of surge transferred overvoltages, Electr. Power Syst. Res. 78 (3) (2008) 441–449. [6] A. Borghetti, A. Morched, F. Napolitano, C.A. Nucci, M. Paolone, Lightninginduced overvoltages transferred through distribution power transformers, IEEE Trans. Power Deliv. 24 (1) (2009) 360–372. [7] B. Gustavsen, A hybrid measurement approach for wide-band characterization and modeling of power transformers, IEEE Trans. Power Deliv. 25 (3) (2010) 1932–1939. [8] A. Holdyk, B. Gustavsen, I. Arana, J. Holboell, Wide band modeling of power transformers using commercial sFRA equipment, IEEE Trans. Power Deliv. 29 (3) (2014) 1446–1453. [9] A. Morched, L. Marti, J. Ottevangers, A high frequency transformer model for the EMTP, IEEE Trans. Power Deliv. 8 (3) (1993) 1615–1626. [10] M. Tiberg, D. Bormann, B. Gustavsen, C. Heitz, Generic and automated simulation modeling based on measurements, in: International Conference on Power Systems Transients, Lyon, France, 2007. [11] F. Delfino, R. Procopio, M. Rossi, High-frequency EHV/HV autotransformer model identification from LEMP test data, IEEE Trans. Power Deliv. 26 (2) (2011) 714–724. [12] B. Gustavsen, A.P. Brede, J.O. Tande, Multivariate analysis of transformer resonant overvoltages in power stations, IEEE Trans. Power Deliv. 26 (4) (2011) 2563–2572. [13] D. Filipovic-Grcic, B. Filipovic-Grcic, I. Uglesic, High-frequency model of the power transformer based on frequency-response measurements, IEEE Trans. Power Deliv. 30 (1) (2015) 34–42. [14] M. Popov, et al., Investigation of the overvoltage and fast transient phenomena on transformer terminals by taking into account the grounding effects, IEEE Trans. Ind. Appl. 51 (6) (2015) 5218–5227. [15] B. Jurisic, I. Uglesic, A. Xemard, F. Paladian, Difficulties in high frequency transformer modeling, Electr. Power Syst. Res. 138 (2016) 25–32. [16] B. Gustavsen, Removing insertion impedance effects from transformer admittance measurements, IEEE Trans. Power Deliv. 27 (2) (2012) 1027–1029. [17] B. Gustavsen, Eliminating measurement cable effects from transformer admittance measurements, IEEE Trans. Power Deliv. 31 (4) (2016) 1609–1617. [18] D. Deschrijver, L. Knockaert, T. Dhaene, Improving robustness of vector fitting to outliers in the data, Electron. Lett. 46 (17) (2010) 1200–1201. [19] B. Gustavsen, A. Semlyen, Rational approximation of frequency domain responses by vector fitting, IEEE Trans. Power Deliv. (1999) 1052–1061. [20] D. Deschrijver, M. Mrozowski, T. Dhaene, D. De Zutter, Macromodeling of multiport systems using a fast implementation of the vector fitting method, IEEE Microw. Wirel. Compon. Lett. 18 (6) (2008) 383–385.
B. Gustavsen, B. Tandstad / Electric Power Systems Research 142 (2017) 47–57 [21] B. Gustavsen, Fast passivity enforcement for pole-residue models by perturbation of residue matrix eigenvalues, IEEE Trans. Power Deliv. (2008) 2278–2285. [22] S. Grivet-Talocia, Passivity enforcement via perturbation of Hamiltonian matrices, IEEE Trans. Circuits Syst. I 51 (9) (2004) 1755–1769. [23] A. Semlyen, B. Gustavsen, A half-size singularity test matrix for fast and reliable passivity assessment of rational models, IEEE Trans. Power Deliv. 24 (1) (2009) 345–351.
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[24] B. Gustavsen, H.M.J. De Silva, Inclusion of rational models in an electromagnetic transients program—Y-parameters, Z-parameters, S-parameters, transfer functions, IEEE Trans. Power Deliv. 28 (2) (2013) 1164–1174. [25] B. Gustavsen, Computer code for rational approximation of frequency dependent admittance matrices, IEEE Trans. Power Deliv. 17 (4) (2002) 1093–1098.
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Wideband modeling of a 45-MVA generator step-up transformer for network interaction studies Bjørn Gustavsen a,∗ , Bjørn Tandstad b a b
SINTEF Energy Research, NO-7465 Trondheim, Norway Statkraft, NO-0216 Oslo, Norway
a r t i c l e
i n f o
Article history: Received 20 June 2016 Received in revised form 3 August 2016 Accepted 30 August 2016 Keywords: Transformer Black-box model Frequency dependency Simulation Transients Transient recovery voltage
a b s t r a c t A stable and passive six-terminal model is developed for a 45-MVA generator step-up transformer based on frequency sweep measurements in the range 5 Hz–10 MHz and curve fitting with rational functions. The modeling employs separate treatment of the zero sequence system in combination with a new variant of the fast residue perturbation method for passivity enforcement. For use in system level simulations, the model can be employed in PSCAD, EMTP-RV and ATP. With ATP, the representation with an equivalent electrical circuit requires some care to prevent the occurrence of near zero circuit elements. Comparison with measured time domain responses at reduced voltage shows that the model can accurately reproduce transient voltage transfer between the windings while taking into account the loading effect of the connected system. The model also reproduces the transient recovery voltage (TRV) when interrupting short-circuit currents. As an application example, the model is demonstrated for simulation of resonant voltage transfer from the high-voltage side to the low-voltage side. In this setting, it is shown that the transformer input impedance can at high frequencies greatly influence the shape of the impinging overvoltages, thereby giving a self-protective effect against incoming overvoltages. © 2016 Elsevier B.V. All rights reserved.
1. Introduction The simulation of high-frequency transformer-network interactions [1] requires transformer models that are sufficiently accurate within the considered frequency band in terms of voltage ratios and impedance characteristics. Relevant simulation studies include voltage transfer between windings, resonant overvoltage buildup, and circuit breaker transient recovery voltage. A number of works have been presented that demonstrate successful application of measurement-based black-box modeling of distribution transformers [2–8] and power transformers [9–14], using low-amplitude measurements. These works assume linearity for the transformer behavior, which is usually considered a valid assumption at higher frequencies. Despite the many positive reports, the successful modeling of a transformer is non-trivial and still requires improvements [15].
∗ Corresponding author at: SINTEF Energy Research, P.O. Box 4761 Sluppen, NO7465 Trondheim, Norway. E-mail addresses: [email protected] (B. Gustavsen), [email protected] (B. Tandstad). URL: http://www.sintef.no (B. Gustavsen). http://dx.doi.org/10.1016/j.epsr.2016.08.035 0378-7796/© 2016 Elsevier B.V. All rights reserved.
This work presents exhaustive results from the measurementbased modeling of a 45-MVA three-phase generator step-up transformer. Its modeling is challenging because of its high voltage ratio (137 kV/8.5 kV) which leads to an undesirable scaling of the four sub-blocks of the terminal admittance matrix, and because its low-voltage winding is connected in delta. The latter results in the terminal admittance matrix having one very small eigenvalue at low frequencies which can lead to large error magnifications in applications if not handled properly. The measurement procedure is first described, emphasizing procedures for retaining the accuracy at low frequencies and noise removal. Following the procedure in Ref. [4], the zero sequence system is measured separately in order to capture the very small eigenvalue at low frequencies that pertains to the delta winding. A passive pole-residue model is extracted for the zero sequence system alone, and for the remaining admittance matrix with the zero sequence system excluded. The two models are finally combined into a single model. Special challenges are addressed and resolved for the passivity correction step which were not considered in Ref. [4]. The accuracy of the model is validated against measured voltage transfer functions in the frequency domain. Using convolution, the model is also validated in the time domain for voltage transfer between windings, with alternative loading conditions. Finally,
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Table 1 Transformer main data. Sn
Upn
Usn
er
ek
45 MVA
8.5 kV
137 kV
0.21%
11.2%
Table 2 List of Instruments. Current sensor Vector network analyzer
Ion Physics, model CM-100-6L Agilent E5061B-3L5
the model is also validated when applied in calculation of breaker transient recovery voltage (TRV). The accuracy at 50 Hz is also discussed. For use with general simulations, EMTP-type circuit simulators should be used. The export to the programs PSCAD, EMTP-RV and ATP is reviewed, and special challenges related to generation of ATP circuit netlist are described and resolved. The model is applied in EMTP-RV for simulation of resonant voltage transfer from the high-voltage side to the low-voltage side.
The unit is a three-phase two-winding YNd11 transformer of rated power Sn = 45 MV that is used for connecting a hydro power generator to the grid. The primary voltage has two settings: 8.5 kV or 7.5 kV while the secondary voltage is 137 kV. In this work, a model is developed with the secondary winding neutral point solidly grounded and with the 8.5 kV setting on the primary, leading to a component with six external terminals. The main electrical data for the transformer are given in Table 1, referenced to 20 ◦ C. Parameters er and ek denote respectively the real and imaginary part of the short-circuit impedance. 3. Problem statement The transformer behavior is to be characterized (measured) in the frequency domain with respect to its external terminals. Since linearity is assumed, the characterization will be based on the admittance matrix Y (6 × 6) which relates the terminal voltages v (6 × 1) with the terminal currents i (6 × 1), i(ω) = Y meas (ω) v(ω)
(1)
The (measured) admittance matrix Ymeas is to be approximated (fitted) by a pole-residue type rational model (2) which is stable, passive, and symmetrical. The modeling is to be performed in such way that the model can be utilized with different terminal conditions, i.e. without excessive error magnifications. N Rk k=1
jω − ak
(2)
The model is to be utilized in an EMTP-type simulation environment and be validated against measurements. 4. Measurement setup and processing 4.1. Measurement setup The admittance matrix Y was measured in the frequency range 5 Hz–10 MHz using a setup similar to the one described in Ref. [3]. The setup utilizes the gain-phase measurement capability of a Vector Network Analyzer (VNA) in combination with a wide-band current monitor (Table 2) that is integrated within a connection box, see Fig. 1. Manual reconnections on the connection box allows
a
N
A
2. Transformer main data
Ymeas (ω) ∼ = Y(ω) = R0 +
Fig. 1. VNA and connection box with built-in current monitor.
B
b
c
C
Fig. 2. Grounding plane (solid line) on top of transformer.
to measure the elements of Y one-by-one, and to perform voltage transfer measurements using voltage probes. The six transformer terminals were connected to the box using shielded coaxial cables which ranged from about five to six meters in length. The use of such long lengths was necessary as the measurement equipment had to be placed on a platform since the space on top of the transformer was too small to allow a suitable work environment. A braided wire flat was placed on top of the transformer and connected to the neutral point (“N”) and to the tank at several local points, see Fig. 2. The braided wire served as a local earth to which the cable screens were connected. Calibration was performed to account for the non-ideal frequency response and insertion impedance effect of the current monitor [16]. The setup gives the admittance matrix seen from the connection box, i.e. of the transformer with the measurement cables in series. The measurement cables do noticeably influence the measurements and thus the model that is extracted, and their effect should therefore be eliminated. However, in order to validate the accuracy of the measurement and model extraction procedure against additional measurements on the connection box, the results after cable elimination will not be shown. The connection points on the box are therefore referred to as transformer terminals, in both frequency domain and time domain measurements. The effect of cable elimination is presented in detail in Ref. [17] for this 45 MVA transformer where a new cable elimination method based on a transmission line equivalent is applied. 4.2. Series resistors on LV side At low frequencies, the admittance matrix elements are large in magnitude with a small real part. As a result, the insertion impedance from measurements cables and connections may affect the measured admittance elements significantly, thereby corrupting the behavioral information such as voltage ratio and passivity characteristics. Also, since the output impedance of the VNA is 50 , the voltage at the VNA reference input can become too low to be measured with sufficient accuracy. As a precaution, small resistors (0.1 ) were placed in series with the LV terminals. Their effect are discussed in Section 11.
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Fig. 3. Noise removal by fitting a rational model to the data using with L1 -norm minimization. The model uses 12 logarithmically spaced real poles between 1 Hz and 10 kHz.
49
Fig. 5. Voltage transfer from HV side to LV side.
Fig. 6. Zero sequence measurement.
side and the low-voltage (LV) side via the partitioned admittance matrix
iH
=
iL
YHH
YHL
YLH
YLL
vH
(3)
vL
which gives the voltage transfers as Fig. 4. Admittance matrix elements.
4.3. Noise suppression It was found that some of the (small) measured admittance matrix elements had a significant amount of noise at around 50 Hz and at lower harmonics. This noise component is undesirable as it leads to spurious poles in the final model. The noise is suppressed by a signal processing step via rational approximation as follows. First, the number of samples is increased from 401 to 1601 by linear interpolation between the sample points, thereby reducing the probability that the least-squares rational approximation to be calculated will interpolate the sample values. Next, each admittance element is at low frequencies (below 1 kHz) replaced with the response of a low-order rational model with prescribed logarithmically spaced real poles. The residues are calculated using L1 -norm minimization [18] when solving the least-squares problem instead of the standard L2 -norm, further reducing the contribution from the outliers in the data. Fig. 3 shows the effect of the noise removal on the three small elements associated with the two off-diagonal blocks of Y. 5. Frequency domain measurements
v H = H HL v L ,
H HL = −Y −1 HH Y HL
(4a)
v L = H LH v H ,
H LH = −Y −1 LL Y LH
(4b)
The voltage transfers are compared with a direct measurement of these quantities using voltage probes. The agreement is found to be excellent for HHL , but for HLH the accuracy is poor at frequencies below 10 kHz as shown in Fig. 5. 5.2. Zero sequence admittance matrix The reason for the large errors in Fig. 5 is the presence of the ungrounded delta winding on the LV side which has only a weak capacitive coupling to ground at low frequencies. Following the procedure in Ref. [4], the accuracy problem is alleviated by making a separate measurement of the zero sequence system (on the connection box) as shown in Fig. 6, leading to a 2 × 2 matrix Y0 ,
iH iL
= Y0
vL
, Y0 =
yHH
yHL
yLH
yLL
(5)
The accuracy of this measurements is further improved in the lower frequency range by using the capacitive perturbation method described in Section 3 in Ref. [4]. With that method, one measures in addition the voltage transfer from terminal H to L, before (vLH ) and after (ˆvLH ) adding a suitably sized capacitor to the open terminal L. One next replaces elements yLH , yHL and yLL with the following,
5.1. Full admittance matrix yLH = yHL = Fig. 4 shows the measured elements of the admittance matrix Y, after noise removal. The quality of the measurement is assessed by calculating the voltage transfer between the high-voltage (HV)
vH
yLL = −
yLH
vLH
−jωC · vˆ LH 3(1 − vˆ LH /vLH )
(6) (7)
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The resulting model is subjected to iterative passivity enforcement using fast residue perturbation (FRP) [21] in combination with passivity checking. Some remarks are in order regarding the implementation. 6.3. Passivity enforcement of Ya Direct application of FRP to Ya results in a model where property (8) is lost for the sub-blocks of the residue matrices. Therefore, one must repeatedly reapply steps 1 and 2 to the residue matrices and perform the perturbation, leading to a time consuming and unreliable process. This difficulty is avoided by using a modified version of the FRP approach. It follows from (8) that the residue matrices Ri have two eigenvalues that are zero with associated eigenvectors that can be selected as
Fig. 7. Zero sequence admittance matrix.
t 0,a =
1(3×1)
,
t 0,b =
1(3×1)
(9)
Fig. 7 compares the elements of the zero sequence component of the original (6 × 6) Y with those of the directly measured Y0 , and those of Y0 obtained by the perturbation method. The zero sequence component of the original Y is obtained by replacing each of the four sub-blocks in (3) with a single element that equals the sum of the block-elements divided by three. The directly measured Y0 is replaced with the result by the perturbation method at frequencies below 8 kHz. The perturbation used a capacitance of 16.9 nF.
With the approach of FRP [21], only the eigenvalues of the residue matrices are perturbed. The procedure is now modified by excluding from the vector of free variables those two eigenvalues that are zero. That way, the two zero eigenvalues and associated eigenvectors (9) remain unchanged by the passivity enforcement. Also, property (8) remains unchanged.
6. Modeling with rational functions
6.4. Passivity assessment of Ya
The separately measured zero sequence system is utilized in the modeling procedure using the approach described in Ref. [4], followed by a new approach for passivity assessment and enforcement. Here, two rational models are extracted that are stable and passive. The first one (Ya ) is for the admittance matrix with the zero sequence system removed, and the second one (Yb ) is for the zero sequence system alone. Finally, the two models are concatenated into a single model, Y = Ya + Yb . This approach is motivated by the fact that Y has a large spread in its eigenvalues at low frequencies which makes it hard to create a model from Y which retains the accuracy of the small eigenvalues.
The passivity enforcement scheme is employed within an iterative loop where passivity violations are identified in each run. In order to safely determine all violations, the eigenvalues of the Hamiltonian test matrix (10) [22] or the half-size test matrix (11) [23] can be assessed for determining cross-over frequencies between passive and non-passive frequency ranges. Matrices A, B, C, and D represent the state-space model associated with the pole-residue model and are readily obtained as shown in Ref. [25].
6.1. Removal of zero-sequence system
M=
1(3×1)
A − B(D + DT ) −CT (D + DT )
−1
−1
−1(3×1)
B(D + DT )
C
C
The zero sequence system is removed from Y in (3) by applying to each sub-block the following procedure. 1. Subtract from each row the average value of the row elements; 2. Subtract from each column the average value of the column elements.
B
−AT + CT (D + DT )
S = A(BD−1 C − A)
−1 T −1 T
(10)
B
(11)
Both approaches fail as D is singular due to the removed zero sequence system. The singularity is removed by adding to D a contribution (12) associated with the zero sequence system, before performing the passivity assessment. This modification does not change the cross-over frequencies that are to be determined.
2 · 1(3×3)
−1(3×3)
−1(3×3)
2 · 1(3×3)
As a result, each block of the modified admittance matrix Ya gets one eigenvector with unity entries and associated eigenvalue equal to zero,
D→D+
(0 , t0 ) = (0, 1(3×1) )
Fig. 8 shows the resulting model for Ya using N = 100 poleresidue terms, after passivity enforcement. It is observed that the matrix elements are in general fitted with a high degree of accuracy.
(8)
6.2. Pole-residue modeling of Ya Ya is subjected to pole-residue modeling (2) using vector fitting (VF) [19,20] with inverse magnitude weighting. Each residue matrix Ri of the model is subjected to steps 1 and 2 described in Section 6.1 to remove the zero sequence component from the subblocks of Ri . As a result, the model’s admittance matrix Y inherits the same properties (8) as Ya in terms of the zero sequence eigenvalues and eigenvectors.
(12)
6.5. Zero sequence modeling The 2 × 2 zero sequence admittance matrix Y0 (5) is fitted with a pole-residue model using VF and subjected to passivity enforcement using iterative FRP. Fig. 9 shows the resulting model using N = 40 pole-residue terms, demonstrating an accurate result for all elements.
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Fig. 8. Passive model of Ya .
51
Fig. 10. Elements of Y, for combined model.
Fig. 11. Eigenvalues of Y, for combined model. Fig. 9. Passive model of Y0 .
Each element of Y0 is expanded into a 3 × 3 block through (13).
i,j Y0
1 = 1(3×3) · 3
i,j r0
+
N i,j rk k=1
=
s − ak
i,j R0
+
N0 i,j Rk k=1
s − ak
(13)
The four sub-models are combined into a single model by stacking the residue matrices,
Yb =
R1,1 0
R1,2 0
R2,1 0
R2,2 0
+
N0 1 i=1
s − ak
R1,1 k
R1,2 k
R2,1 k
R2,2 k
(14)
6.6. Model combination Finally, the model of Ya and of Yb are combined into a single model by appending the two sums of pole-residue terms into a single sum Y = Ya + Yb , giving a model with N + N0 pole-residue terms. Figs. 10 and 11 show respectively the elements of the final Y and its eigenvalues, demonstrating excellent agreement. An exception is for two of the small eigenvalues in Fig. 11 at frequencies below 1 kHz. These two eigenvalues are found to be associated with the transformer no-load current which are unimportant in most transient simulation studies. Fig. 12 verifies that the model is passive within the 5 Hz–10 MHz frequency range as the eigenvalues of (Y + YH ) are positive. The procedure ensures passivity at all frequencies, ω ∈ [0, ∞).
Fig. 12. Eigenvalues of (Y + YH )/2, for combined model.
7. Frequency domain validation The model accuracy is validated in the frequency domain by comparing the voltage transfers (4a) and (4b) as calculated by the model’s Y with those of a direct measurement. Figs. 13 and 14 show that the voltage transfers are highly accurate in both directions, within the full frequency range. Comparison between Figs. 13 and 5 shows that the previous inaccuracies below 10 kHz are not present in the final model, which is due to the separate measurement and modeling of the zero sequence system. It
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Fig. 13. Voltage transfer from high to low.
Fig. 14. Voltage transfer from low to high.
is noted that in the case of the three small elements in Fig. 14, the relative errors are large below 10 kHz. However, these errors are inconsequential for a simulation since the small magnitudes imply low voltages. 8. Time domain validation: voltage excitation on HV side As a final validation, a set of time domain measurements are performed at reduced voltage and compared against time domain simulations using the pole-residue model. Again, all measurements are performed on the connection box. The setup consists of a function generator with 50 output impedance (Tektronix AFG3052C), a storage oscilloscope (Tektronix DPO 4054B), and a 0.5 m coaxial cable for connecting the function generator to the connection box. When applying the voltage to the LV side, a wide-band amplifier with low output impedance (Toellner 7608) was placed between the function generator and the transformer. The amplifier was necessary as the function generator output voltage would otherwise collapse. The voltage measurements were made using passive voltage probes with 10 M input impedance. 8.1. Unloaded transformer A step-like voltage is applied to terminal 1 with terminals 2 and 3 grounded (on the connection box) and the remaining terminals open, see Fig. 15a. The voltage response on the terminals are measured and the measured voltage on terminal 1 is applied to the model in an EMTP-like time domain simulation with model interfacing based on a Norton equivalent and recursive convolution [24].
Fig. 15. (a) Voltage application to HV side of unloaded transformer. (b) Open circuit voltages on LV side. (c) Open circuit voltages on LV side. Initial response.
Fig. 15b and c (zoomed view) shows the measured and simulated responses. It is observed that highly accurate results are obtained, both for the 60 kHz oscillation in Fig. 15b and the 2 MHz oscillation in Fig. 15c. These oscillations are also found as dominant peaks in the ditto frequency domain responses in Fig. 13. 8.2. Loaded transformer The transformer is loaded with resistors on the connection box as shown in Fig. 16a. The selected values correspond approximately to the characteristic impedance of cables (30 ) and overhead lines (400 ). Fig. 16b and c compares measured and simulated voltage responses at the terminals. The resistors are in the simulation represented by circuit elements and the measured voltage on node 0 is represented by an ideal voltage source. Again, highly accurate results are obtained. It is further observed that the oscillations are strongly damped compared to the previous unloaded transformer case, in particular the 60 kHz oscillation in Fig. 15b. This result implies that the model interacts correctly with the connected circuit.
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Fig. 17. (a) Voltage application to LV side of unloaded transformer. (b) Open circuit voltages on HV side.
Fig. 16. (a) Voltage application to HV side of loaded transformer. (b) Voltage response on LV side of loaded transformer. (c) Voltage response on LV side of loaded transformer. Initial transient.
9. Time domain validation: voltage excitation on LV side A similar investigation is performed when placing the excitation on the transformer low-voltage side. 9.1. Unloaded transformer In the unloaded transformer case (Fig. 17a), the model accurately reproduces the transient response on the high-voltage side as can be seen in Fig. 17b. The oscillation frequency is about 8.5 kHz and is clearly observable by a dominant peak in the corresponding frequency responses in Fig. 14.
Fig. 18. (a) Voltage application to LV side of loaded transformer. (b) Voltage response on HV side of loaded transformer.
10. Time domain validation: TRV applications 9.2. Loaded transformer 10.1. Opening switch on HV side With 400 resistors connected to the HV side (Fig. 18a), the oscillations in Fig. 17b become completely damped out in both the measurement and the simulation, see Fig. 18b.
As a final validation, the accuracy in transient recovery voltage simulation is investigated. In this test a mechanical switch starts
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Fig. 19. (a) Current interruption on HV side. (b) Voltage response on terminal 1. Table 3 Transformer voltage ratio (positive sequence).
Nominal Model
Fig. 20. (a) Current interruption on LV side. (b) Voltage response on terminal 4.
Table 4 Transformer short-circuit impedance (positive sequence).
|Vp /Vs |
|Vs /Vp |
0.0621 0.0625
16.1 15.9
opening which is connected between the source and the transformer, see Fig. 19a. The output voltage from the function generator is measured and realized as an ideal voltage source in a time domain simulation while the measured current through the switch is used to define the time instant where the switch is to open in the simulation. Fig. 19b shows that the simulated voltage response on the transformer side of the switch is highly accurate. The frequency is about 8.5 kHz, similar to Fig. 17b. It is remarked that the peak value of the 50 Hz feeding voltage is very small (0.12 V) compared to the observed peak in the recovery voltage (13 V) which results from the current chopping. The measurement and simulation starts about 20 ms before the breaker opens to ensure that there are no visible transients present in the simulation when the breaker opens. 10.2. Opening switch on LV side A similar measurement is performed on the LV side as shown in Fig. 20a and b. Again a very accurate result is obtained. The dominant frequency is about 50 kHz which is somewhat lower than the 60 kHz dominant oscillation in Fig. 15b.
Nominal Measured Y w./0.1 Measured Y w.o./0.1 Measured Y w.o./0.1 , and use of cable compensation
Zk,p []
Zk,s []
(0.0034 + j0.18) (0.14 + j0.18) (0.042 + j0.19) (0.010 + j0.18)
(0.88 + j47) (37 + j47) (9.8 + j47) (1.7 + j48)
in the measurements while the resistive part is much too large by a factor of 42. One major cause is that the measurements used 0.1 resistors that were placed in series with the LV terminals as explained in Section 4.2. These resistors appear on the HV side as R = 0.1·(137/8.5)2 = 26 . A set of frequency domain measurements were also performed without the 0.1 resistors (third row), but it is observed in Table 4 that the resistive part is still too high, by a factor 11. The accuracy can be further improved by eliminating the effect of the applied measurement cables which have a DC resistance of 0.0057 /m and lengths betwen 4.8 m and 6.2 m. The fourth row in Table 4 shows the effect of cable elimination using the S-parameterbased transmission line method described in Ref. [17]. The resistive part is now much closer to the nominal value. It is remarked that the 0.1 resistors have in general a negligible effect on the transient waveforms resulting from impinging voltage waves from the network.
12. Export to EMTP-type simulation programs 11. Model accuracy at 50 Hz The model is capable of reproducing the nominal voltage ratio with a high degree of accuracy as shown in Table 3. The model’s short-circuit impedance is however less accurate, in particular for its resistive part. Table 4 compares the nominal short-circuit impedance (first row) with the one that can be inferred from the measurements (second row). The reactive part is quite accurate
The simulation procedure [24] applied in the preceding sections is suitable for time domain validation with simple terminal conditions applied to the transformer terminals. For application in system level simulations with many different network components, it is more practical to include the model in one of the EMTP-type simulation platforms. The following describes the export of the model to leading EMTP-type programs.
B. Gustavsen, B. Tandstad / Electric Power Systems Research 142 (2017) 47–57
y1,2
1
y1,0
55
2
y2,0
Fig. 21. Admittance branches of two-node network.
12.1. PSCAD and EMTP-RV In PSCAD, the pole-residue model (14) can in the most recent program versions be directly included in the simulation, based on the procedure described in Ref. [24]. For use in EMTP-RV, the poleresidue model must be converted into a real-valued state-space model which can be directly loaded using the state-space component that is part of EMTP-RV.
Fig. 22. Simulated waveforms by alternative EMTP-type programs.
12.2. ATP ATP does not have the capability of directly including a poleresidue model or a state-space model. Therefore, one must instead convert the pole-residue model into an equivalent circuit as shown in Ref. [25]. However, the direct application of that approach to the proposed transformer model leads to a netlist which includes circuit elements with extreme numerical values that can compromise the accuracy of the simulation result. To see this, let us recall the relation between admittance matrix elements and admittance branches. Consider a model with two nodes as shown in Fig. 21. Using nodal analysis, the admittance matrix is established as
Y=
y1,0 + y12
−y12
−y12
y2,0 + y12
(15)
The reverse process, generalized to an n-conductor system gives the following procedure for calculating the branch admittances from Y,
yi i =
N
Yi j
(16)
j=1
yi j = −Yi j
(17)
One problem with this procedure arises for the pole-residue terms that are related to Ya . As described in Section 6.2, each 3 × 3 block of the associated residue matrices has a row-sum equal to zero which leads to zero values for the diagonal elements yii in (16). Due to the finite precision of the computations on a computer, the admittance elements are not exactly zero, leading to circuit elements with extreme values. This problem is overcome by simply bypassing the writing to file of the diagonal circuit elements associated with Ya (Fig. 22). 12.3. Comparative result For the example case in Fig. 15a, the transferred overvoltage to the LV side terminal 4 is simulated when applying a unit step voltage to terminal 1. The simulation is performed using PSCAD, EMTP-RV, ATP and the aforementioned Matlab implementation. The PSCAD simulation is with version v4.2 with the pole-residue model included as a user-defined component. Fig. 22 shows that the four simulation platforms give virtually identical results.
Fig. 23. Ground fault initiation by ideal switch.
13. Significance of transformer input impedance The frequency domain plot in Fig. 13 shows that there is a potential for transfer of large overvoltages from the HV side to the LV side, at frequencies around 2–4 MHz. For this to happen, an oscillating overvoltage in this frequency range must exist on the HV terminals. To investigate the scenario, we use the circuit in Fig. 23 using EMTP-RV. The transformer HV side is connected to an infinitelength uncoupled cable system with characteristic impedance ZC = 30 and propagation velocity v = 177 m/s which is fed from a three-phase ideal voltage source. The LV side terminals are open. An ideal single-phase ground fault occurs at a distance l = 22 m from the transformer HV side. This combination of length and velocity defines a quarter-wave resonance frequency of 2.0 MHz by (18), which coincides with one of the resonance peaks in Fig. 13. f/4 =
v 4l
(18)
Fig. 24 shows the waveforms on the HV and LV side terminals when the simulation has been started from 50 Hz initial conditions. It is observed that with ZC = 30 the presence of the transformer distorts and damps out the waveform on the transformer HV side, leading to a voltage on the LV side whose peak value is only 27 kV, being much lower than the value predicted by the 2 MHz resonance peak in Fig. 13. For comparison, the same plot includes the wave shapes that results when the 30 characteristic impedances are replaced with a very low (unrealistic) value ZC = 0.1 . In the latter case, the transformer does not damp out the square wave oscillation on the HV side, causing the voltage on the LV side to reach nearly 150 kV in the 5 s simulation time window. This result is not surprising since a unit step voltage that propagates on a line with characteristic impedance ZC and meets an open end terminated against a capacitance C to ground will initially give a voltage response at the end by (19). For instance, a capacitance of 1 nF will with ZC = 30 give a time constant T = 0.3 s, which is comparable
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to reproduce the transformer’s transient behavior with a high degree of accuracy, including voltage transfer between winding terminals with alternative loads connected to the terminals. The model can also be used for calculating transient recovery voltages due to interruption of currents. The model has been exported to PSCAD, EMTP-RV and ATP, for use in network interaction studies. Using EMTP-RV, it was shown that the transfer of resonant overvoltages from the HV side to the LV side can be strongly affected by the transformer input impedance. For oscillating frequencies in the MHz range, the considered transformer is capable of damping out the oscillation, thereby preventing resonant voltage build-up on the LV side.
Acknowledgments Fig. 24. Voltage response on terminals 1 and 4 (22 m cable).
The authors appreciate the assistance of Oddgeir Rokseth during the measurements, as well as Statkraft hydro station staff. Financial support was provided by the Research Council of Norway (RENERGI Programme), DONG Energy, EirGrid, Hafslund Nett, National Grid, Nexans Norway, RTE, Siemens Wind Power, Statkraft, and Statnett.
References
Fig. 25. Voltage response on terminals 1 and 4 (738 m cable).
to the travel time of the 22 m segment of = 0.12 s. One can therefore conclude that the transformer has a significant self-protective effect against very-high-frequency oscillations on the HV side terminals. For lower frequency components however, the transformer HV side is much less likely to damp out the overvoltage. This is shown in Fig. 25 when the cable length is increased to 738 m, giving a 60 kHz oscillation on the HV side which coincides with a peak in the voltage transfer in Fig. 13. u(t) = 2(1 − e−t/T ), T = ZC C
(19)
14. Conclusions A wideband, EMTP compatible terminal model has been developed for a 45-MVA generator step-up-transformer based on frequency sweep measurements and rational modeling via curve fitting. The approach is based on separate measurement and modeling of the zero system components in order to avoid error magnifications to result from the presence of a delta winding. The presented work improves the model extraction procedure in Ref. [4] by introducing a more rigorous approach for passivity enforcement of the model’s zero sequence component. In addition, an essential modification is made to the procedure for equivalent circuit representation associated with the model’s zero sequence component which avoids the occurrence of extreme circuit values. The accuracy of the model has been validated in both frequency domain and time domain. Comparison between time domain measurements and simulated waveforms shows that the model is able
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