A Comparison of Numerical Methods for Modeling Overpressure Effects from Low Impedance Faults in Power Transformers

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4th International Colloquium "Transformer Research and Asset Management” 4th International Colloquium "Transformer Research and Asset Management”

A Comparison of Numerical Methods for Modeling Overpressure A Comparison of Numerical Methods for Modeling Overpressure Effects from Low Impedance Faults in Power Transformers Effects from Low Impedance Faults in Power Transformers Jean-Bernard Dastousaa*, Ewald Taschlerbb, Sylvain Bélangeraa, Monther Saribb Jean-Bernard Dastous *, Ewald Taschler , Sylvain Bélanger , Monther Sari a a

Institut de recherche d’Hydro-Québec, 1800 boul. Lionel Boulet, Varennes, Québec, J3X 1S1, Canada b Siemens d’Hydro-Québec, AG Österreich, Transformers Weiz, Boulet, Elingasse 3, 8160 Québec, Weiz, Austria Institut de recherche 1800 boul. Lionel Varennes, J3X 1S1, Canada b Siemens AG Österreich, Transformers Weiz, Elingasse 3, 8160 Weiz, Austria

Abstract Abstract

Low-impedance faults in power equipment may lead to catastrophic tank ruptures, resulting in fires, oil spills and Low-impedance faults in power equipment may Up leadtotorecently, catastrophic tank for ruptures, in fires, oil or spills and projection of parts, as well as economic losses. the tank a largeresulting power transformer a shunt projection parts, asdesigned well as economic losses. Up filling, to recently, the tank for large power transformerSince or a 2008, shunt reactor wasoftypically to withstand vacuum transportation andasometimes earthquakes. reactor was typically designed to withstand vacuum filling, transportation sometimes earthquakes. Sincereactor 2008, Hydro-Quebec has implemented arc-containment requirements in its and power transformer and shunt Hydro-Quebec Tank has implemented arc-containment requirements in its power and shunt reactor specifications. rupture is a very complex phenomenon to investigate - arc transformer testing is cost-prohibitive and, specifications. Tank inrupture a very complex phenomenon arc testingsuch is cost-prohibitive moreover, random nature.is Therefore, mechanical designtoofinvestigate tanks to -withstand events, as welland, as moreover, nature. Therefore, mechanical design of methodologies. tanks to withstand as welland as investigationrandom of suchin phenomena, must rely primarily on numerical In this such paper,events, Hydro-Quebec investigation of such phenomena, must relyclasses primarily on numerical methodologies. In thisofpaper, Hydro-Quebec and SIEMENS present and compare two main of numerical methods, as well as some their possible variations, SIEMENS compare two main classes numerical asoverpressure well as someeffects of theirfrom possible variations, that may bepresent used toand obtain an optimized and safer of tank design tomethods, withstand low impedance that may be used to obtain an optimized and safer tank design to withstand overpressure effects from low impedance faults. faults. © 2017 The Authors. Published by Elsevier Ltd. © 2017 The Authors. Published by Elsevier Ltd. Peer-review under of the © 2017 The Authors. Published by Ltd. committee Peer-review under responsibility responsibility of Elsevier the organizing organizing committee of of ICTRAM ICTRAM 2017. 2017. Peer-review under responsibility of the organizing committee of ICTRAM 2017. Keywords: Tank rupture; low impedance faults; overpressure. Keywords: Tank rupture; low impedance faults; overpressure.

* Corresponding author. Tel.: 450-652-8341; fax: 450-652-8905. E-mail address:author. [email protected] * Corresponding Tel.: 450-652-8341; fax: 450-652-8905. E-mail address: [email protected] 1877-7058 © 2017 The Authors. Published by Elsevier Ltd. Peer-review the organizing committee 1877-7058 ©under 2017responsibility The Authors. of Published by Elsevier Ltd. of ICTRAM 2017. 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.2017.09.708

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1. Introduction Low-impedance faults in power equipment may lead to catastrophic tank ruptures, resulting in fires, oil spills and projection of parts. Such faults, initiated from a breakdown of the electrical insulation, result in the generation of an electrical arc which vaporizes the surrounding oil. The decomposition of oil leads to the formation of a high temperature and high pressure gas bubble, which expansion results in a rapid pressure rise within the transformer tank. Since 2008, Hydro-Quebec has implemented arc-containment requirements in its power transformer and shunt reactor specifications; these are detailed in [1]. These requirements specify that such equipment tanks must be able to withstand without opening a low impedance fault of a given amount of energy according to its voltage class, without also projection of parts. Furthermore, it is required that beyond the specified amount of energy, the tank must fail safely by opening only at the cover level in order to minimize oil spills and collateral damage to neighboring structures and equipment. The main design parameter resulting from these requirements is a uniform design pressure that the tank must withstand, which is primarily a function of the amount of energy to be contained and the tank flexibility. Tank rupture is a very complex phenomenon to investigate - arc testing is cost-prohibitive and, moreover, random in nature. Therefore, mechanical design of tanks to withstand such events, as well as investigation of such phenomena, must rely primarily on numerical methodologies. Towards these ends, there are two main methods that may be used: • non-linear static finite element analysis ; • explicit dynamic finite element/finite volume analysis. The first method is best suited for the mechanical design of tanks, and is currently used by manufacturers in order to comply with the arc containment requirements discussed before, based on a uniform design pressure. The second one is best suited towards investigative purposes, as it permits studying the effect of many parameters, such as the arc position and duration, and the resulting pressure distribution within the tank which is generally not spatially uniform within the tank for some time from the fault initiation to some later time after its termination [1]. Both methods have advantages and limitations, and both rely on a high level of knowledge and expertise in order to be used adequately. This paper presents the application of these methods in the design and investigation of the fault containment capacity of a 735 kV shunt reactor. A comparison between the results from both methodologies, as well as the effects of different modeling techniques within each, will be presented and discussed. Nomenclature C DLF EM E k Pa Pf Pd Po SMsh SMso V V0

Volumetric flexibility of the tank, ΔV/ΔP (m3/kPa) Dynamic load factor Young’s modulus (GPa) Arc energy of the fault (kJ) Gas quantity generated by kJ of arc energy in oil, 5.8x 10-4 m3/kJ of arc Applied pressure (kPa) Pressure to failure (kPa) Design pressure (kPa) Hydrostatic pressure at the center height of the the main tank due to the oil mass in the tank and conservator (kPa) Submodel, boundary condition from static shell model Submodel, boundary condition from static solid model Volume of the tank Volume of the tank at P0

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Voil Y0 ε εf ν σut

3

Volume of oil in the tank Yield strength at 0.2% strain (MPa) Equivalent plastic strain Equivalent plastic strain to failure Poisson ratio Ultimate tensile strength of the material, true stress value (MPa)

2. Tank under study 2.1. General description The tank under study is a single phase 110MVAr - 735kV shunt reactor delivered in 2011 by Siemens to HydroQuebec (Fig. 1). Its main dimensions and characteristics are given in Table 1. According to Hydro-Quebec requirements, a tank of this voltage class must be able to withstand without opening a fault of 20 MJ.

Fig.1 Detailed outline drawing Table 1. Tank characteristics Characteristic

Value

Length (bottom plate)

3700 (mm)

Width (bottom plate)

3470 (mm)

Height (bottom plate to cover)

3485 (mm)

Weight of active part

15.4 (to)

Amount of oil in the tank

33430 (l) / 29090 (kg)

2.1.1. Geometry used in modeling A finite element model is created by a subdivision of a given geometry into a given number of finite elements. This subdivision process is termed meshing, and results in a finite element mesh. A proper subdivision must be

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obtained in order to obtain representative results. In general, a finer mesh (that is one made of smaller finite elements) provides more detailed results than a coarser mesh, at the expense of longer calculation times. Therefore the size of the mesh must be adjusted such that it provides the level of details required for a given analysis and in a given part; it may vary locally where more or less details are required. As we will see under, the mesh size has been adjusted accordingly in the different models used. The geometry of the tank under study was taken from a parasolid file out of Creo/ProEngineer and simplified by removing all unnecessary parts that did not influence significantly the pressure withstand capacity of the tank. Two types of finite element models were used in this study: a shell model and a solid model (Fig.2). Both models were used in the nonlinear static analysis simulations but only the shell model was used in the explicit dynamic simulations. A shell model (made of shell elements) is one for which the thickness of the different parts is not modeled explicitly, while being taken into account implicitly in the calculations. A shell model represents adequately the general behavior of the tank, and is best suited to model structural parts that have flat surfaces and a small thickness relative to their other dimensions, such as plates, walls, cover and stiffeners. It is however not suited to perform detailed stress analysis on parts that have more complex tridimensional geometries, such as welds in particular. A solid model is one that represents directly the actual thickness and geometry of the different parts. It is therefore suited to model any parts, including welds, and is best used for detailed and accurate analysis in locations where stresses and deformations vary rapidly over small distances. a b c

Fig.2. (a) CAD-geometry of shunt reactor; (b) simplified shell model; (c) simplified solid model

2.1.2. Mechanical properties The walls, cover and stiffener of the tank are made of mild steel S355. The constitutive law describing the plastic true stress-true strain relationship used in modeling was obtained from of a bilinear curve approximating the general behavior of the material. The first point on this curve corresponds to the yield strength at a plastic strain of 0.2 %. The second point corresponds to the engineering ultimate tensile strength and the engineering strain occurring at the onset of necking. These values were converted to true stress and true strain as required by the finite element model. The weld metal Pittarc G6 was also modeled using a bilinear law for the stress-strain relationship. The mechanical properties used in modeling are given in Table 2. Table 2. Material properties Material

ρ (kg/m3)

EM (GPa)

ν (-)

Yo (MPa)

σut (MPa)

εf (mm/mm)

S355

7850

200

0.3

355

571

0.111

Pittarc G6

7850

200

0.3

497

630

0.100

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3. Nonlinear static analysis This section presents the general calculation procedure used in the verification of the tank arc containment capacity, and it details the two different models used (shell and solid), and their respective results. 3.1. General calculation procedure The general calculation procedure is made of two distinct steps. First a model of the entire tank is obtained to evaluate the design pressure according to Hydro-Quebec required procedure, as detailed under. The resulting model is called the global model and it permits to identify the location of the most stressed areas, while however not permitting an accurate analysis of such areas. Next a subsection of each most stress area of interest is modeled using the submodeling technique, which represents only the geometry around this area. This technique, using generally a much finer mesh than in the global model, permits an accurate and detailed analysis of these areas, and is therefore used to assess the tank arc containment capacity up to rupture 3.1.1. Design pressure evaluation using the global model The design pressure is evaluated by following the flowchart presented in Fig. 4 from Hydro-Quebec Std Sn14.1j. In this procedure, the pressure is applied incrementally and uniformly inside the tank, and the corresponding volume increase is computed. The first pressure increment corresponds to the hydrostatic pressure P0 from which the initial volume V0 is computed. This pressure may be represented accurately spatially by the pressure distribution according to height, or may be approximated by a unique value representing the average hydrostatic pressure at the mid-height of the tank. At each pressure increment, the volume change, flexibility and dynamic load factor (using Fig. 5) are evaluated using the corresponding applied pressure, to obtain the design pressure according to equation (1), until the latter equals the applied pressure. The intersection of the applied and design pressures sets the final design pressure of the tank. Pressure is then increased furthermore in the simulation until failure of the tank is reached. The relatively coarse finite element mesh of the global model accurately calculates the volume increase of the tank and accordingly, the displacements in the model, but is not suited to accurately predict the rupture as discussed. Convergence studies have shown that a mesh size of 1 to 2 mm is required to precisely predict the failure of the tank in the cover weld. A global model with such a fine mesh, even only in the cover weld and coarser elsewhere, would result in an unpractically long computing time, considering the size of a power transformer tank. Therefore, as discussed next, the submodeling technique is used to accurately assess the arc containment capacity of the tank and corresponding failure pressure. 3.1.2. Failure pressure evaluation using the submodeling technique A submodel is created and centered around the area of interest to perform a detailed analysis and predict the pressure to failure. In this method, the displacements calculated in the global model are applied as boundary conditions on the interfaces of the submodel chosen. The dimensions of the submodel must be large enough to obtain an adequate representation of the stresses and strains in the area of interest. These depend on the size of the elements in the global mesh, as well as their size in the submodel. These dimensions are best determined by a convergence analysis but its description falls outside the scope of the present paper. Note that the submodels are made of solid elements only, for adequately representing the weld at the cover. The ductile failure criterion selected in Hydro-Quebec requirements is based on the equivalent plastic strain (Mises strain) at its value at the onset of necking in a uniaxial tension test (ultimate strain). Failure is deemed to occur when the equivalent plastic strain reaches the ultimate strain throughout the throat of a weld, or throughout the thickness of a structural element at another location. Therefore, this criterion is not based on the attainment of the ultimate strain at a single point only in the mesh, but rather, on a given volume. Fig. 3 shows an example of the

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evolution of the volume deemed failed (in red) for an ultimate strain value of 0.10 mm/mm, as the pressure increases inside the tank. In this example, the weld bonds the cover and the flange together.

Fig. 3 Example of the evolution of the plastic strain until failure as the pressure increases inside a tank

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end

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7

  k•E Pd = DLF • 100 • 0.25 + − 50 + P0 100 • C  

(1)

2,5 DLF

2 1,5 1

0

20

40

60

80 100 120 140 160 180 200

C/Voil (10-5 kPa-1) Fig. 5. Dynamic load factor

3.2. Software and type of finite elements used In the nonlinear static analysis, both a shell and a solid model were used and compared. The static analysis was performed with ANSYS © Mechanical v18 software [2]. A list of the element types used in the global models and submodels can be found in Table 3 (their description can be found in ANSYS documentation[2]). The shell global model used only 4 nodes linear shell elements (SHELL181) for the tank and the weld and is designated as model-2D. The solid model used 8 nodes solid shell elements for all the plates of the tank and 20 nodes quadratic hexahedral elements for the welds (SOLSH190 and SOLID186 respectively) and is designated as model-3D. The parts are held together with contact elements. The volume is computed by the use of 3D fluid element (HSFLD242). These 5 nodes elements enable the global model to include a contained fluid inside the tank and capture the effect of fluid pressure. One of the nodes is centrally located in the tank and specifies the applied pressure while the other 4 nodes created a plane element that covers the inside wall of the tank. Table 3. Types of finite elements Types

Shell 2D

Solid shell 3D

Hexahedral 3D

Contact 2D

Contact 3D

Fluid

Number of nodes

4

8

20

4

8

5

Interpolation function

Linear

Linear

Quadratic

Linear

Quadratic

Linear

ANSYS element type

SHELL181

SOLSH190

SOLID186

CONTA173

CONTA174

HSFLD242

3.3. Global and sub models description The initial CAD model contains many parts like valves and other devices, which are of no interest for the simulation, and these have therefore been removed in the models. However, the welds between tank and cover have been retained, and have been modeled differently in each global model as discussed under; the welds elsewhere have not been modeled and instead, contact elements were used between the welded parts to model them. The welds at the cover level must be retained in the models, as these turn out to be the most stress areas and are also necessary to model adequately the flexibility of the tank in this area. 3.3.1. Model-2D (shell model) Model-2D was meshed using a 30 mm mesh size for the tank (average length of each finite element side) and an 18 mm mesh size for the skirting. The resulting mesh consisted of 174,551 nodes and 178,322 elements (Fig. 6a). Note that the welds in this model are also modeled in a 2D manner using shell elements; this has been proven sufficient to adequately model the flexibility of the tank.

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The submodel (SMsh) derived from the global shell model was meshed using a 5mm size, for the skirting and for the welds a 2mm size was used. The overall dimensions of this submodel are 600 x 330 x 270 mm (Fig. 6b). a

b

Fig. 6. (a) mesh of the global shell model ; (b) mesh of the associated submodel SMsh

3.3.2. Model-3D (solid model) For Model-3D, advantage was taken of the planar symmetry of the structure, and therefore only half of the tank was modeled (Fig. 7a). In this model, the welds at the cover location were also modeled using solid elements. The overall mesh size is 40 mm for the tank, cover and stiffeners, and 5 mm for the welds. The resulting mesh consisted of 1,056,790 nodes and 262,762 elements. The overall dimensions of the submodel (SMso) are 600 x 540 x 440 mm (Fig. 7b). The mesh size is similar to the SHsh submodel. The SMso submodel contains 1,032,754 nodes and 330,127 elements (that is more elements than the global models !). Fig. 7c shows the finite element representation of the welds in the submodel. a

b

c

Fig. 7. (a) mesh of the global solid model ; (b) geometry of the submodel SMso; (c) detail of the mesh around the welds

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3.4. Load and boundary conditions The tank rests on an elastic support with a 2.5 N/mm3 stiffness and is fixed horizontally at two corners (Fig. 8a). Point masses are added to represent the weight of the active part (Fig. 8b). Standard earth gravity is active. The pressure is applied on all internal faces of the tank for the global models. The hydrostatic pressure is applied on all surfaces inside the tank; the 2D-model used the spatial variation of this pressure with height while the 3D-model used the average pressure at mid-height of the transformer uniformly. a

b

Fig. 8. (a) fixity at 2 corners of the bottom face ; (b) point mass simulating the active part.

The boundary conditions apply to the submodels are the pressure and the prescribed displacement from the global model. The submodel having a finer mesh, the displacement are interpolated onto the nodes. 3.5. Results The design pressure will be first be compared between Model-2D and Model-3D. Next , the computed strains in the corresponding submodels will be compared, along with the resulting pressure at failure. 3.5.1. Design pressure The volumetric flexibility and the determination of the design pressure for Model-2D and Model-3d are presented in Fig. 9. The difference of volumetric flexiblity between the 2 models initially increases with the pressure, but then stabilizes to approximatively 13 %. This translates in a difference of 6.1 % in the resulting design pressure. Previous comparisons of design pressures between solid and shell models for various sizes of tank have shown similar differences. Table 4 provides values of design pressure for both models.

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b shell model solid model

0.010

Design pressure (kPa)

Volumetric flexibility (m3/kPa)

a

0.008 0.006 0.004 0.002 0.000

0

100

200

300

400

500

600

900 800 700 600 500 400 300 200 100 0

Applied pressure (kPa)

211

applied pressure shell model solid model

0

100

200

300

400

500

600

Applied pressure (kPa)

Fig. 9. (a) comparison of the shell and solid models flexibility ; (b) comparison of the design pressure determination. Table 4. Comparison of design pressure and pressure to failure between the 2 models Models

Design pressure Pd (kPa)

Applied pressure at failure Pf (kPa)

Shell

508

569

Solid

540

710

Difference

6.1 %

22.0

3.5.2. Failure pressure Fig. 10a shows the plastic strain for the submodel SMso at 710 kPa, and the location of the maximum strain value where the rupture occurs. The plastic strain reaches the 10% criteria throughout the weld that connects the flange and the skirting, as displayed on a YZ plane at the location of the maximum plastic strain (Fig. 10b). a

b

Fig. 10. (a) plastic strain in the submodel SMso at 710 kPa ; (b) plastic strain on the plane of rupture at 710 kPa

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The plastic strain at failure for submodel SMsh is presented in Fig. 11a. Details of the failure area in this model show that the failure occurs in the weld connecting the cover and skirting (Fig. 11b), that is at a different location than for submodel SMso. a

b

Fig. 11 (a) plastic strain in the submodel SMsh at 622 kPa ; (b) plastic strain on the plane of rupture at 622 kPa at the center of the submodel

3.5.3. Displacement at selected locations The displacements were obtained and compared between both global models at 4 distinct locations in areas of large displacement on each face of the tank modeled, save for the bottom. The corresponding results are provided in Table 5 under, along with the results from the explicit dynamic analyses. It is observed that displacements from Model-2D are larger at all locations than the ones from Model-3D. 3.5.4. Discussion Comparison of results in the previous section between Model-2D and Model-3D shows that a good agreement is obtained for the design pressure (6.1% diff.), while the difference is more pronounced for the pressure at failure (22% diff.). The difference observed here for the design pressure may be explained by one and/or the following factors: • Model-2D does not have any weld that bind the skirting to the cover and the flange, and this translates in a larger displacement of the cover, as exemplified by the displacement evaluated at this location (see Table 5), and therefore, in a more flexible tank than Model-3D. • Model-3D uses bonded contact elements; accuracy of the interaction between parts in this case is dependent on the mesh size to a certain extent. • The meshing of Model-2D in the weld area is continuous between the different parts involved, without use of contact elements, since the different parts share the same nodes at their common boundaries. This generally translates in a more accurate representation of the interaction between such parts. As the displacements from the global models are applied to the boundaries of submodels, the same factors as above will be at play in the observed difference for the pressure at failure from the submodels. In addition, the following factors may contribute to explain the observed differences:

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• The error in the prescribed displacements used at the boundaries of a submodel increases as the size of the submodel decreases; the dimensions of SMsh are slightly smaller here than for SMso (see section 3.3). • The solid global model includes the weld attached to the skirting and these welds are meshed with 5 mm hexahedral elements. The nodal displacements prescribed to the same welds in the SMso submodel will then introduce small errors as these are meshed with 2 mm element size in the corresponding submodel. • Besides the pressure to failure value, the failure develops in different welds. The failure happens in the weld between the skirting and the flange for the SMso submodel, and in the weld between the cover and the skirting in the SMsh submodel. 4. Explicit dynamic analysis This section presents the application of the explicit dynamic analysis method to the study of the same shunt reactor tank, for an arc of 20 MJ, and of 50 ms duration, occurring within the tank at a given location. Two different models were investigated and compared in this study. 4.1. General description of the method Explicit dynamic analysis permits to represent adequately the different phenomena involved in low impedance faults within transformers. It permits the modeling of the interaction between the transformer tank, its internal oil and the gas bubble generated by the oil decomposition. It permits to study the spatial variation in time of the pressure and stresses in the tank, as a function among other things of time, arc energy and its duration, as well as arc location, as would occur in a ‘real’ low impedance fault event. A detailed description of this method and its application to the study of low impedance faults in different types of transformers has been presented in [3, 4]. An application of the methodology to study the release of overpressure by venting is also presented in [1]. In summary, in this method, the tank is modeled in the same way as was done in nonlinear static analysis, using a finite element mesh. The fluids at hand are modeled using finite-volume elements in a volumetric mesh englobing the finite element mesh. The fluid-structure interaction between both types of mesh is modeled using a strong coupling algorithm that allows for the calculation of their interaction in ‘real time’. As discussed in [4], this method is very computationally intensive, due in part to the large number of finitevolume element required in the modeling of the fluids, as well as due to the small time step required for stability of calculation (in the order of microseconds). This method is therefore not well adapated to detailed stress analysis of transformer tanks, especially their welds, which accurate modeling using this method would probably translate into many week of computer time. This method is therefore more suited for understanding effects of the different parameters involved in low impedance faults, as well as to elaborate design methodologies taking account of the possible variation of these parameters. It has been used here to demonstrate its capacities, as well as to assess the adequacy of the Hydro-Quebec requirements for the tank under study. Two separate investigations have been performed independently here, using two different explicit dynamic programs, each with a different way of modeling the introduction of arc energy generated by oil decomposition. Also different mesh sizes have been used in the two investigations. The first investigation led to the numerical model identified as Model-D1, while the second to the model identified as Model-D2. 4.2. Software and type of finite elements used Model-D1 was based on the use of ANSYS LS-DYNA, while Model-D2 was based on the use of ANSY Autodyn [2]. The transformer tank in both programs has been modeled using 4 nodes shell elements, while the fluids (oil and gas from oil decomposition) have been modeled using finite-volume hexahedral elements (their description can be found in ANSYS documentation for both software[2]).

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4.3. Global model description 4.3.1. Model-D1 The geometry of the tank model is the same as shown in Fig. 2a. The active part is also modeled to represent its interaction with the oil and gas from oil decomposition. This part has been modeled using a simplified representation as shown in Fig. 12a. A cross-section of the resulting structural model is shown in Fig. 12b. The location of the arc is also shown in Fig 12a: just over the active part under the high voltage turret supporting the corresponding bushing. The welds between the cover and flange have been modeled using shell elements of constant height around the cover perimeter as shown in Fig. 12b. The tank structure consists of 164,708 shell elements in this model. The fluid domain is shown in Fig. 13a; it englobes the structure. The fluid domain showing only the oil within the transformer is shown in Fig. 13b. The fluid domain consists of 2,567,788 finite-volume elements. a

b

Fig. 12 (a) active part ; (b) cross-section of structural Model-D1

a

b

Fig. 13 (a) fluid domain of Model-D1; (b) oil in fluid domain of Model-D1

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4.3.2. Model-D2 The geometry and arc location are similar to Model-D1, save for the fact that only half of the tank has been modeled due to its symmetry (Fig. 14a). Also, the geometry of Model-D2 has been slightly simplified as compared to Model-D1 by suppressing some parts within the transformer, among others the tube holding the active part in between the two smaller opposite walls of the tank (see Fig. 12a and Fig. 14a). The corresponding mesh consists of 67,251 elements; it is a bit coarser than for Model-D1 by comparison of half of the elements used for the latter. The fluid domain is shown in Fig. 14b, where the arc location is in red. The fluid domain consists of 288,800 finite-volume elements; it is noteworthy that the mesh discretization used for the fluids is coarser here as compared to Model-D1. a b

Fig. 14 (a) finite element mesh of Model-D2 ; (b) fluid domain of Model-D2

4.4. Loads and boundary conditions Loads and boundary conditions are slightly different between both models. In Model-D1, gravity is included while it is not the case in Model-D2. Previous investigation with Model-D2 has shown that this load has a minor influence on the simulation results. Regarding the boundary conditions, the entire bottom of Model-D1 is fixed while only the nodes of the structure supporting the active part are fixed in Model-D2. Again, previous investigations have shown that this results in minor differences in the simulation results. 4.5. Arc modeling The methods of arc modeling are different between both models. In Model-D1, the internal energy is provided by a predefined energy curve, introduced in the model by a gas which properties are adjusted by trial and error, such that the required amount of energy is introduced in the mesh during the arc duration of 50 ms at the location shown in Fig. 12a. In Model-D2, the arc energy is introduced at the same location by a flowing gas of constant temperature and internal energy, which density and velocity are calculated at each time step, such that the corresponding energy is introduced at a linear rate, representing an arc of constant power over the fault duration. The corresponding methodology and parameters have been detailed in [3, 4]. It is noteworthy that in this methodology, the physical parameters used to describe the introduced gas are based on actual measurements from oil decomposition experiments. For a fault of 20 MJ, the resulting mechanical energy that should theoretically be introduced in the model for a gas generation rate of 85 cm3/kJ of arc energy (see [3]) is 4.86 MJ, which is 24.3 % of the total arc energy. In Model-D1, the actual energy introduced was 5.13 MJ, while it was 4.96 MJ in Model-D2. We can therefore conclude that the introduction of energy in both models is similar and close to the theoretical value.

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4.6. Comparison of results between both explicit dynamic models Due to time limitations and limited computer resources, Model-D1 results were evaluated for a physical duration of 80 ms; Model-D2, results were obtained for a 200 ms duration. The following subsections present a comparison between both models over the 80 ms duration. 4.6.1. Change of tank volume As the oil can be considered nearly incompressible over the range of pressure variation here, the change of tank volume can be obtained by evaluating the variation of the gas bubble resulting from the arc decomposition. Fig. 15 shows the corresponding time variation of tank volume. It is observed that the results are in good agreement between both models.

Fig. 15 time variation of tank volume from explicit dynamic analysis models

4.6.2. Displacements The spatial distribution of total displacement over the tank at 80 ms is presented in Fig. 16 for both models. A fairly good agreement is obtained between the models, although Model-D2 shows slightly larger displacements. It is noteworthy that the displacements and tank deformations are larger on the high voltage turret side (larger turret), which is on the same side as the arc location. This exemplifies the physical observation from real events that the tank deformations are often more pronounced on the side of the arc location [1]. a

b

Fig. 16 (a) displacements for Model-D1 at 80 ms ; (b) displacements for Model-D2 at 80 ms

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4.6.3. Stresses The spatial distribution of Von Mises stresses over the tank at 80 ms is presented in Fig. 17 for both models. A relatively good agreement is obtained between the models, although Model-D1 shows slightly larger stresses. a

b

Fig. 17 (a) Von Mises stresses for Model-D1 at 80 ms ; (b) Von Mises stresses for Model-D2 at 80 ms

4.6.4. Effective plastic strains The spatial distribution of Von Mises strains over the tank at 80 ms is presented in Fig. 18 for both models. A relatively good agreement is obtained between the models, although Model-D1 shows slightly larger strains on the side stiffeners. a

b

Fig. 18 (a) effective plastic strains for Model-D1 at 80 ms ; (b) effective plastic strains for Model-D2 at 80 ms

4.6.5. Summary of comparisons between both explicit dynamic models It is observed that a very good agreement was obtained between both models regarding the change of volume of the tank. For the other quantities compared, it was observed that the agreement was relatively good, although not as close as for the change of volume. Among the possible explanations for the observed differences are the following:

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• the quantity of energy introduced in Model-D1 is slightly more important (5.13 MJ) than in Model-D2 (4.96 MJ), a 3.4 % difference; the way the arc is modeled is also different ; • the mesh sizes used in both models are fairly different; the mesh sizes used in Model-D1 being much finer, especially for the fluid volume domain ; • the boundary and load conditions were also slightly different between both models ; • the models were also slightly different, with some internal elements suppressed in Model-D2. Finally, it is worth stating that the explicit dynamic analysis method is in general sensitive to the variation of different modeling and operating parameters, such as the safety factor on the time step, the different parameters used in the specification of the coupling, etc. Considering these and the possible explanations above, we can conclude that despite the many differences between the two models compared here (and possibly between the functioning of both programs), a fairly good agreement, although not perfect, was obtained between them. 5. Further investigations with explicit dynamic Model-D2 Since the calculation with Model-D2 was performed over a physical time of 200 ms (4 times the arc duration of 50 ms), and since explicit dynamic analysis has the ability to approximate in time the behavior of the phenomena involved with low-impedance faults, some noteworthy observations can be made from this simulation as presented next. 5.1.

Average pressure variation at the tank inner surfaces

Averaging the time variation of pressure variation across 131 locations uniformly distributed in the oil near the tank walls, bottom and cover, we obtained the average pressure rise at the tank inner surfaces as shown in Fig. 19, where it is plotted along with the pressure variation within the gas bubble. It is observed that initially, the pressure within the gaz bubble is much higher than the average pressure within the tank, with a maximum value of 5427 kPa (absolute). However later on, both pressures converge to the equilibrium value within the tank which is around 413 kPa (absolute) or 313 kPa gauge at 200 ms. It is noted that the peak value of the average pressure at the inner surfaces is 532 kPa (absolute) or 432 kPa gauge at 59 ms.

Fig. 19 Average pressure variation at the tank inner surfaces and in the gas buble

5.2. Pressure variation within the tank While the average pressure rise at the tank inner surfaces rises in a monotonic fashion, the pressure varies spatially inside the tank with time, where locations close to the fault initially experience higher pressure peaks than locations farther away. However with time, the pressure converges to an equilibrium value everywhere within the tank as shown in Fig. 19. It is observed that from approximately between 120 to 160 ms, the pressure becomes fairly uniform within the tank.

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t=20 ms

t=40 ms

t=60 ms

t=120 ms

t=160 ms

t=200 ms

Fig. 20. Spatial distribution of pressure at varying times for Model-D2

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5.3. Distribution of the total arc energy with time The energy input from the arc (24.3 % of the total electrical energy input as shown earlier) is transmitted to the tank, oil and gas bubble, under different form of energies. Figure 21 shows how the energy is distributed in these different materials with time.

Fig. 21. time distribution of the arc energy to the different materials involved (in the graph legend, ‘huile’ stands for oil)

At 200 ms, the distribution of the total energy is the following: • 67% is stored in the gas bubble from oil decomposition, mainly under the form of internal energy; • 17% is stored in the tank, under the form of plastic deformation energy (76 % ) and under the form of elastic deformation energy (24 %); • 16% is stored in the oil, mainly under the form of internal energy. It is noteworthy in the case studied here that the major part of the energy is stored in the gas bubble, while only 17% is actually stored in tank deformation. Considering the total electrical energy input (here 20 MJ), the actual proportion of the total electrical energy stored in the tank deformation is actually only 4.1 % of this value ! (e.g. 24.3% of 17%). 6. Comparison between static and dynamic analysis In this section, we compare different quantities in order to assess the differences between the nonlinear static analysis method and the explicit dynamic analysis results. The explicit dynamic analysis results from Model-D2 are used here at 200 ms. The nonlinear static analysis results at the design pressure are used, since the dynamic simulation is based on the required energy to withstand, from which the design pressure is evaluated. 6.1. Displacements at selected locations The total displacements at the locations shown in Fig. 22 are given in Table 5. It is observed that, save for location B on the tank wall next to the HV turret (the closest to the arc location), the non linear static method yields substantially larger displacements than generated in the actual dynamic phenomena. For other locations, the displacement obtained from the dynamic simulation is not in turn, substantially higher than the ones obtained from static analysis.

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Fig. 22. Point locations used in displacement comparison Table 5. Comparison of total displacement at 4 locations on tank wall (mm) at design pressure A

B

C

D

Model-2D (static – shell)

Location:

164

153

197

97

Model-3D (static – solid)

159

135

171

66

Model-D2 (dynamic – shell)

44

163

116

68

6.2. Change of tank volume The total change of tank volume for the different models is presented in Table 6. It is observed that the static analysis method yields to substantially larger changes of volume than the dynamic analysis method. Table 6. Comparison of tank volume change at design pressure Model

Volume change (m3)

Model-2D (static – shell)

4.25

Model-3D (static – solid)

3.35

Model-D2 (dynamic – shell) @ 200 ms

2.80

Model-D2 (dynamic – shell) – maximum

2.88

6.3. Effective plastic strain distribution in the tank The distribution of effective plastic strain is shown at design pressure for Model-2D and at t=200 ms for ModelD2 in Fig. 23. Note that the level of strains are here only in the tank as the strains in the welds at the cover levels cannot be adequately identified using a global model with a coarser mesh as discussed. It is also to be noted that the maximum scale values for both models is 0.02 or 2 % strains; values in red are therefore at a strain level equal or higher to this value. Note also that as observed in Fig. 18, the strain levels do not reach substantially higher values than 2 % on the tank walls and cover. It is observed from Fig. 23 that the strain areas exceeding 2% are generally larger for the static analysis model, save for the dynamic model in the vertical stiffener on the HV turret side (close to the arc location).

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a

21

b

Fig. 23. (a) plastic strain at design pressure (540 kPa) for Model-3D; (b) plastic strain at 200 ms for Model-D2

6.4. Design and average pressure The design pressures for both static models and the maximum value of the average dynamic pressure are provided in Table 7. It is observed that the design pressure from the static models provides larger values than the maximum average pressure in time from the dynamic model. It it also observed by comparison with Fig. 20 that locally, the dynamic pressure in the tank may far exceed the design pressure. For example, it was noted that the pressure in the gas bubble reached a maximum value of over 5000 kPa. However such pressure spikes are of short duration and do not necessarily generate substantial damage, as exemplified by the previous quantities compared. Indeed it was observed that generally, the static analysis methodology yields larger results than seen in dynamic analysis, save for the tank area close to the arc where these results may be of the same order of magnitude as the static ones. Table 7. Comparison of design pressure from static analysis with maximum average pressure from dynamic analysis Model

Pressure (kPa - gauge)

Model-2D (static – shell)

508

Model-3D (static – solid)

540

Model-D2 (dynamic – shell) – maximum average pressure

432

7. Discussion and conclusion The present study compared different static and dynamic methodologies that can be used to assess the capacity of a transformer tank to withstand overpressure effects from low impedance faults. These methodologies were exemplified and compared through the analysis of a single shunt reactor tank. The static methods studied, based on Hydro-Quebec requirements, are best suited to assess the overpressure withstand ability of a given tank design. The design pressure calculated by these methods with shell and solid models, is within acceptable engineering tolerance between these two types of model. The submodeling technique as exemplified here, must be used for both types of models to adequately assess the pressure at rupture of a given tank design. The discrepancy exemplified here for the failure pressure between the two types of submodels studied, needs further investigation on others tanks of various sizes and flexibilities, in order to establish if such a trend is real.

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The explicit dynamic analysis method presents the advantage of simulating in real time the phenomena involved in low impedance faults. It is however computationally intensive, and is therefore best suited for investigative and research purposes. It was used here to show, among other things, how the pressure resulting from the fault varies spatially and in time within the tank. It also exemplified the observed fact that damage is more localized in the tank areas closer to the fault location. A comparison between the static and dynamic analysis methods tends to show that the static analysis method, based on Hydro-Quebec requirements, is adequate, as it provides results that are representative of the maximum ones obtained locally near the arc by the dynamic method. However, further investigation on several different transformer tank designs is necessary, in order to conclude on this observation. References [1] M. Foata, J.B. Dastous, “Power Transformer Tank Rupture Prevention,” CIGRE S2010, paper A2_102_2010, August 2010 [2] ANSYS Inc., Canonsburg, PA, USA. ANSYS Help System Version 18.0 [3] J.-B. Dastous, M. Foata, "Analysis of Faults in Distribution Transformers with MSC/PISCES-2DELK," Proceedings of the 1991 MSC World Users' Conference, Los Angeles, California. [4] J.B. Dastous, J. Lanteigne and M. Foata “Numerical Method for the Investigation of Fault Containment and Tank Rupture of Power Transformers,” IEEE Transactions on Power Delivery, Vo. 25, No.3, July 2010.