Power Transformer Winding Model For Lightning Impulse Testing.pdf

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Procedia Engineering 202 (2017) 297–304

4th International Colloquium "Transformer Research and Asset Management”

Power transformer winding model for lightning impulse testing Tomislav Župana*, Bojan Trkuljab, Željko Štihb Končar Electrical Engineering Institute, Fallerovo šetalište 22, 10002 Zagreb, Croatia University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia a a

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Abstract This paper presents a method for calculating the internal voltage transients useful for power transformer winding modelling. The method is based on lumped parameter model of a transformer winding and the transient response is obtained using time-domain analysis. Lumped circuit parameters are calculated using self-developed solvers which are benchmarked using professional finite element method (FEM) software. The results show that the presented approach gives satisfactory results and is computationally very fast. © 2017 The Authors. Published by Elsevier Ltd. © 2017 The under Authors. Published by Ltd. committee of ICTRAM 2017. Peer-review responsibility of Elsevier the organizing Peer-review under responsibility of the organizing committee of ICTRAM 2017. Keywords: power transformers; numerical simulation; electromagnetic transients; time-domain analysis; coils; internal overvoltages

1. Introduction Power transformers are an essential part of every electrical power system and are typically designed to be in operation almost continuously for several decades. During their lifetime, they are exposed to high frequency transient overvoltages due to switching operations and lightning strikes [1-3]. Therefore, when determining the geometry and type of the winding, one of the typical design criteria is the winding’s ability to withstand transient voltage surges. Under steady-state operating conditions, voltage is distributed uniformly along the winding of a transformer [3]. However, winding’s behavior during high-frequency transients is much more complicated. Capacitances between the winding conductors, which are negligible at nominal frequency, become significant at higher frequencies. They are generally responsible for the initial nonuniform voltage distribution along the winding during transient conditions [1]. Moreover, capacitive connections together with the inductive connections inside the winding give rise to voltage

* Corresponding author. E-mail address: [email protected] 1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of ICTRAM 2017.

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of ICTRAM 2017. 10.1016/j.proeng.2017.09.717

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oscillations along the winding. These oscillations become more pronounced if the frequency spectrum of the transient input signal contains frequencies which are close to fundamental resonant frequencies of the winding under consideration [3]. Nomenclature BEM DC FEM FRA LI εr σ

boundary element method direct current finite element method frequency response analysis lightning impulse relative permittivity of the surrounding medium electrical conductivity of a conductor

Because of the abovementioned winding behavior under transients, it is of interest to transformer manufacturers to be able to predict the winding’s transient response and optimize its geometry and insulation during the design stage. 1.1. Scope of the paper The analysis of fast transient overvoltages and the behavior of the transformer winding under such conditions are of great interest and has been a topic of various papers [1-6]. There are two main approaches, one based on the winding modelling and the other one on measurements. The former is of interest to transformer manufacturers since they have the insight into winding’s geometry needed for detailed numerical models [2,6]. The latter approach is of use to power system operators which usually use FRA methods in obtaining the transformer behavior under transient conditions [7,8]. When modelling the transformer winding, it is usually represented with an electrical circuit consisting of a number of resistances (R), inductances (L) and capacitances (C). The winding models can be broadly divided in two groups: models based on lumped parameters [2,4,6] and models based on distributed parameters [5]. The most daunting task in both approaches is in determining the mentioned RLC parameters. Since the power transformer usually consists of several hundreds to couple of thousands of turns, it is very complicated and time-consuming to obtain all the RLC parameters for every turn. Therefore, they are either calculated using simple analytical expressions which cannot take detailed winding geometry into account, or the turns are grouped and then a reduced winding model is constructed [9]. This paper presents a numerical method for determining the RLC parameters of the winding’s lumped parameter model which is both computationally fast and yields more precise results compared to analytical formulations. The developed model is useful in analysis of the winding when subjected to high-frequency overvoltages (up to a couple of MHz). The emphasis is given on capacitance and inductance calculation, for which numerical procedures have been developed. Each turn of the winding is treated individually and no reduction of turns is made. The conductors are assumed to be of a rectangular cross section since they are typically used in power transformers. The obtained results from the presented approach are benchmarked against the results calculated using the professional FEM-based software Infolytica® MagNet® and Infolytica ElecNet®. The winding’s design and its ability to withstand atmospheric discharges is usually tested using the 1.2/50 µs voltage signal, as defined in IEC 60076-3. For that reason, the response of the model of the winding presented in this paper will be simulated using the same input signal. The analysis will be done using a transient solver developed for this purpose based on formulations made by Dommel [10].



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2. Calculation of RLC parameters In order to represent the transformer winding using the lumped parameter circuit model, the resistive, inductive and capacitive parameters have to be obtained. 2D axially symmetric approximation is assumed and each of the winding’s turns is modeled as a thick ring. The following subsections briefly explain how each of the parameters has been calculated. 2.1. Resistance calculation The winding’s resistance dampens the oscillations that arise during the fast transient overvoltages. The observed transients typically have high-frequency components. Moreover, conductors inside the winding of a transformer are densely packed. Therefore, influence of both skin and proximity effects on the effective resistance will not be insignificant. However, in order to take those effects into account, complex numerical methods have to be used. Such approach makes sense when frequency-domain analysis is used since the implementation of frequency dependent resistance in time-domain analysis is not trivial. The analysis of the winding’s model in here presented research is done in time-domain, hence the complex resistance calculation will not be covered in this paper. The frequency dependent resistance analysis has been done by the authors in [6]. The resistance matrix is calculated trivially using the basic DC resistance calculation. The resulting matrix is a diagonal one and neither skin nor proximity effects are considered. The skin effect influence on the effective resistance for the conductors with rectangular cross section can be implemented using [11]. 2.2. Capacitance calculation Capacitances are calculated using a BEM-based method. In such approach only the boundaries are discretized, and therefore the dimension of the problem is reduced by one. Capacitances of the transformer winding can be assumed to be independent of frequency and are calculated using the electrostatic analysis.    Electric field potential ϕ ( r ) at any point r due to the distribution of the surface charge density σ ( r ') is represented by [4]:



 ϕ (r ) =

 σ ( r ' ) dS '

  , 4πε r − r '

S'

(1)

  where r is the vector distance of a calculation point, r ' is the vector distance of a referent point on a source, and S ' is the surface of two-dimensional elements on which the charges are distributed. Constant surface charge density is assumed on each segment. The expression for the electric field potential on the surface segments can be written as: N

  σ ( r ' ) dS ' k

 ΔS ϕ (r ) =  k   , k =1 4πε r − r '

(2)

where N is the number of segments, and Δ S k is the surface of each segment. Unknown coefficients σ k are obtained from known potentials using the collocation method which results in a densely-populated system matrix. Total charge on the j-th conductor influenced by the charge on the i-th conductor Qij is:

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Qij =  σ j dS j =  σ kj Skj ,

(3)

k =1

Sj

where σ kj is the surface charge density on the k-th segment of the j-th conductor and S kj is its surface, while N is the number of the finite segments of j-th conductor. If ϕi and ϕ j are the potentials of i-th and j-th conductor, the elements of the capacitance matrix are calculated using the equation:

Cij =

Qij

i ≠ j.

ϕi − ϕ j

(4)

2.3. Inductance calculation Inductance matrix is obtained using the method for calculating inductances of coaxial circular coils in air with rectangular cross section and uniform current densities presented in [12]. The influence of the iron core on the inductance is essentially negligible since the secondary winding is short-circuited during the lightning impulse test. The problem is therefore considered linear. Assuming that the current densities are uniform over windings’ conductor cross sections, energy W stored in the magnetic field is: 2π

μ0 I 2

W=

2

4( Z 2 − Z1 ) (R 2 − R1 )

2

Z2

Z2

R2

R2

     ϕ

= 0 z = Z1 Z = Z1 r = R1 R = R1

cos ϕ rR dr dR dz dZ dϕ r + R 2 − 2rR cos ϕ + ( z − Z ) 2 2

.

(5)

Using quintuple integration, both height and thickness of the observed conductor are taken into account. Here, ( Z 2 − Z1 ) represents the height of the coil, R1 is the inner and R2 the outer radius of the coil, and ϕ is the angular coordinate in a cylindrical coordinate system. Equation (5), when compared with W = coil with rectangular cross section: L=

μ0 2

π

( Z 2 − Z1 ) (R 2 − R1 )

2

Z2

Z2

R2

1 2 LI , gives the term for the self-inductance L of the coaxial circular 2 R2

cos ϕ rR dr dR dz dZ dϕ

     ϕ

r + R 2 − 2rR cos ϕ + ( z − Z ) 2 2

= 0 z = Z1 Z = Z1 r = R1 R = R1

.

(6)

Similarly, the equation for the total energy stored in the magnetic field gives the expression for the mutual inductance M of a pair of coaxial circular turns [12]:

M=

μ0

π

R2

R4

Z2

Z4

cos ϕ rR dr dR dz dZ dϕ ( Z 2 − Z1 )( Z 4 − Z 3 )(R 2 − R1 )(R 4 − R3 ) ϕ= 0 r =R1 R =R3 z =Z1 Z =Z3 r 2 + R 2 − 2rR cos ϕ + ( z − Z ) 2

(7)

where R1 and R2 are the inner and the outer radii of the first coil, R3 and R4 are the inner and the outer radii of the second coil, ( Z 2 − Z1 ) represents the height of the first coil, and ( Z 4 − Z 3 ) represents the height of the second coil. When calculating the diagonal elements (self-inductance), the bounds of the two integrals in radial direction (over r and R ) are the same since both points defined by the coordinates refer to the same coil. This is true for the bounds of the two integrals in axial direction (over z and Z ) as well. The integrals in (6) and (7) are solved analytically, except for the integral over ϕ . Use of the L’Hopital rule resolves the problem with singularities in two points when numerical integration over ϕ is performed. The obtained

[ L ] matrix is a fully populated matrix consisting of self-inductances as diagonal and mutual inductances as off-

diagonal elements.



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3. Winding model and analysis

After RLC matrices have been calculated, the model of the transformer winding can be obtained. Fig.1 shows an example of a four-turn coil equivalent circuit to illustrate the way the model is established. One end of the winding is grounded, while the impulse signal is injected at the other end. Time-domain analysis is used to simulate the winding response to high-frequency transient overvoltages. Analyzing the eigenvalues of the system matrix obtained using state space analysis, the resonant frequencies of the winding can be obtained [13]. C13

C24

C13

C12 1

C23 L11

R11

2

L22

R22

L12 C1g

C2g

C34 3

L33

R33

L23 L13

C3g

4

R44

L44

L34 L24

C4g

L14

Fig. 1. Equivalent circuit for a four-turn transformer coil.

3.1. Voltage distribution calculation Transient voltage distribution is obtained solving the established transformer winding equivalent circuit using the method developed by Hermann Dommel [10]. This method replaces inductances and capacitances by an equivalent electric circuit using conductances and current sources. By using such substitution, the ordinary differential equations of lumped inductances and capacitances are converted to algebraic ones, integration of which is solved using the numerically A-stable trapezoidal rule. Unknown node voltages are obtained by solving the system of linear algebraic equations in time domain in fixed time steps. The details of the implementation of the method used for time-domain analysis in this paper are given in [4] and will be omitted here. 4. Winding example and results

The presented methodology was tested on a simple continuous disc winding example (Fig.2), data of which is given in Table 1. The same winding was modeled and analyzed using Infolytica® MagNet® 7.7.1 and Infolytica ElecNet® 7.7.1. 4.1. Numerical results Calculation of RLC matrices based on the presented methods for a given winding example was done in FORTRAN parallel code using 12-core HP ProLiant server. The same server was used for FEM calculations as well. Python programming language was used for time-domain computation of lightning voltage distribution over transformer winding.

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Fig. 2. Disc winding example used for verification of the presented method. Table 1. Disc winding example data. number of discs number of radial groups number of turns in a group total number of turns winding's inner diameter double-sided insulation thickness

54

conductor’s height

8 [mm]

4

conductor’s width

3 [mm]

5

radial canal height

6 [mm]

axial canal width

8 [mm]

1080 1200 [mm]

εr

2.5

1 [mm]

σ

5.77x107 [S/m]

Fig.3 and Fig.4 show the visualizations of capacitance and inductance matrices, respectively. Relative error in comparison to FEM solutions is given as well. As can be seen, the presented method gives satisfactory results.

Fig. 3. Visualization of the capacitance matrix calculated using the presented method and comparison with FEM results.



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Fig. 4. Visualization of the inductance matrix calculated using the presented method and comparison with FEM results.

Voltage distribution for simulated LI testing is presented in Fig.5. The results obtained using the presented method and FEM results are practically the same. The basic oscillating frequency is around 410 kHz.

Fig. 5. Calculated voltage distribution for simulated LI test obtained using the presented method (full line) and FEM benchmark (dashed line).

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Presented results for a 1080-turn example disc winding show that the results obtained using the calculation approach given in this paper are in a good agreement with the results obtained using professional FEM software. The biggest difference is in the time needed to calculate the RLC matrices. Using the presented approach, the results are obtained in 9 minutes. In order to have a valid computation time comparison, the same amount of processor cores has to be used in FEM calculation. Scaling to a 12-core parallel computation gives the total time for obtaining the RLC matrices using Infolytica® MagNet® and Infolytica ElecNet® to 410 minutes which is almost 7 hours. 5. Conclusion

The method for calculating the high-frequency transient response of power transformer winding is presented. The method is shown to be in a good agreement with professional FEM-based tools. However, the computational time needed to obtain the results is greatly reduced. Presented methodology can be of interest to transformer manufacturers during the design stage of winding production for optimizing the insulation with regards to transient overvoltages. Acknowledgements

This work was supported in part by the Croatian Science Foundation under the project number IP-2013-1118. References [1]

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