{'fraction_non_alphanumeric': 0.02331111828777781, 'fraction_numerical': 0.009835295499251342, 'mean_word_length': 4.820574162679426, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Modelling of magnetostrictive vibration and acoustics in converter transformer

Suman Yadav, Department of Electrical Engineering, North Eastern Regional Institute of Science and Technology, Nirjuli, Arunachal Pradesh, India. Email: sumanyd12345@gmail.com Abstract The converter transformers are susceptible to more noise and vibration when compared to power transformers due to the presence of DC bias in the DC transmission line. DC bias occurs mostly due to inaccuracies in valve firing resulting in a small residual DC oscillating around zero. Measurement of magnetostriction becomes significant as it influences the vibration and noise from the core. Hence, a magnetostrictive model of a high‐voltage DC converter transformer has been developed. This work analyses the vibration and noise acoustics under such an occurrence. First, the core of the transformer model is designed in the stepped configuration for 240 MVA; then, magnetostrictive vibration is analysed by using suitable modules of COMSOL Multiphysics at different magnitudes of DC bias. The physics of noise has been interfaced using the Acoustics Module, and the results are recorded. Finally, artificial neural network model is developed for the prediction of vibration and noise characteristics of the model. The fitting process of neural network was then remodelled using various optimisation techniques, namely teaching–learning‐based optimisation, particle swarm optimisation, biogeography‐based optimisation, simulated annealing and binary coded genetic algorithm, and their results were compared to obtain the best‐suited method using % mean‐squared‐error evaluation.


| INTRODUCTION
The urbanisation of developing countries has led to an upward trend in the demand for electrical energy. High-voltage DC (HVDC) systems are widely being utilised owing to their qualities, namely transmission capacity, lower losses and improved stability for comparably longer distances. Despite the aforementioned virtues, they are vulnerable to many issues during practical implementation. One of them being introduction of DC bias in the converter transformer arising due to the coupling of AC/DC, imbalance in gating of converters, valve side harmonic disturbances and at times due to geomagnetic storms which may lead to permanent failure [1,2]. Thus, the analysis of the effect of DC bias in converter transformer has become indispensable with the growth of HVDC transmission in modern power systems.
The core is an important aspect in a transformer and not to mention it has to go through diverse circumstances, with magnetostriction being an inherent property of the material; vibration and noise are embedded with it [3,4]. The harmonic currents induced due to DC bias results in saturation, increased leakage flux, temperature rise in the core leading to local overheating, dielectric breakdown and vigorous electromagnetic vibration in the transformer [1,2]. The magnetic flux in the core varies with variation in the magnitude of DC bias injected to it, which ultimately leads to vibration. The vibration of core and windings in turn aids to the generation of noise, significantly impacting the environment [5,6]. A transformer has to fulfil certain requirements on noise levels as stated by Girgis et al. in their works on noise related to transformers [7]. The noise arising from the electromagnetic force and vibration in the core is referred to as ontic noise and that arising from vibration due to operation of cooling fans is known as cooling system noise. The objective here is the prediction of ontic noise which constitutes a standard multiphysics problem of electromagnetism, mechanics and acoustics as an addition.
On the application of an alternating voltage to the windings of a transformer, flux is induced in the core. The core, in turn, is made up of cold rolled grain oriented low loss Hi-B electrical steel which constitutes 3% of silicon with a density of 7.65 kg/dm 3 . This material bears a non-linear anisotropic property referred to as magnetostriction. The term magnetostriction could be introduced as alternating dimensional changes in the domains of the core due to the varying magnetic flux. Some quantifiable magnetostriction is exhibited by all ferromagnetic materials, and its mechanism on an atomic level is relatively complex as compared with microscopic level. The latter could be easily understood by domain wall theorem [6]. The forces on the core from current sources would act on a larger scale, whereas the forces exerted by magnetostriction are local as they are caused due to magnetisation. It is worth mentioning that both the external sources and magnetisation work simultaneously on the material, and both have to be considered as a whole. Many formulations are commonly used to describe the magnetostriction in relation to magnetic forces [8]. Normally, the orientation of the magnetic domain is random when there is a null magnetic field passing by; this is shown in Figure 1a. Conversely, the passage of an external magnetic field lets the domain react to the field and re-orient their sizes as explained in Ref. [9]. In this phenomenon, the domains with similar orientation with the applied magnetic field developed by varying the magnetic orientation on their discrete surface, thereby causing the magnetic domains with opposing orientation to the external field to deteriorate [10]. This leads to a resultant change in domain shape as seen from Figure 1b. Hence, magnetostriction inherits a change in shape and the dimensions of the material.
This phenomenon adds to the vibration of the core, which in turn traverses further to the tank mainly via the insulating medium and also through the mechanical coupling of the tank with the core. Most of the mechanical energy is exuded by the tank as wall noise. Owing to the complex structure and efficient coupling with insulation medium, the analytical model cannot be used to determine the radiation of sound. Broadly, empirical studies based on dimensional parameters and statistics are used by most manufacturers, which has shown limitations on application to newer designs making it unsuitable for parametric studies. Hence, this work utilises a finite element (FE)-based study to determine the interactions of magnetostrictive property of core with vibration and noise from the core with an insight of DC bias.
A significant amount of works in the past decade has reportedly been successful in eliminating harmonic currents in HVDC converter transformer implementing delta filtering windings, inductive filtering approach, among others [5,11,12]. In many of previous works, researchers have summarised on the effects of DC bias in the vibration of the core of transformers. In Ref. [13], the effect of nearness to the grounding substation to the vibration of the transformer has been studied in detail with significant variation in vibration fingerprint of the core. Besides, there have been attempts in evaluating the noise from the core of transformers with an emphasis on the effect of DC bias on the noise generated from the transformer. Research on the effects of the DC bias on different transformers using finite element method (FEM) was carried out by various authors. In Ref.
[14], a converter transformer is designed to study the voltage harmonics with the help of threedimensional (3D) FEM. Many authors have tried to study magnetising current waveform [15], magnetic saturation [16], losses and temperature rise [17] for various types of transformers. An Epstein frame-like core model was studied for power transformer by harmonic balance FEM [18]. Vibration and audible noise characteristic of power transformer caused by HVDC system under monopole operation was studied in Ref. [19]. The most recent work done by authors in Ref. [6] involved the study of magnetostriction using FEM, which is done on an electrical sheet of 160 kVA power transformer with a rated voltage of 2000/400 V. In Ref. [20], the vibration and noise of a 18.23-kVA transformer are studied in a rectangular core. Authors in Ref. [21] have studied the vibration in cores of transformer and reactor with an applied excitation of 136 and 268 V.
The work done so far is for lower rated transformers, but for the analysis of HVDC line, a converter transformer of a F I G U R E 1 Ferromagnetic domain (a) without external field and (b) with external field [6,10] YADAV AND MEHTA -333 higher rating is required. Hence a six-stepped core with a dual winding for a transformer of 240 MVA rated at 22/550 kV is designed for the analysis, and the simulations validated by designing artificial neural network (ANN)-based model for prediction of noise based on vibration and DC bias. The numerical method in this work employs the FEM of the COMSOL Multiphysics tool, which finds profound usage in engineering and proves to be an essential tool for acoustic, mechanical, electromagnetic field and many more multiphysics problems. Each of these fields is then solved by adequate adoption of differential equations. The method divides the model into simple geometric shapes [22]. A set of polynomial functions is defined in each element and is used to approximate the structural displacement field; further details on the solver could be read from Ref. [23]. The displacement field, thus obtained, helps in the computation of deformation. There have been several works on development of neural-network-based magnetic models [24][25][26]. In Ref. [25], the authors modelled the magnetostriction of electrical steel sheets using backpropagating neural network presenting an effective method of modelling the magnetic nature of electrical steel sheets. The present work tries to extend the process further by implementing a study of noise along with magnetostrictive vibration.
Also, a new algorithm for optimisation, namely teachinglearning-based optimisation (TLBO) has been implemented to optimally decide the ANN parameters [27] to reduce error and avoid local optimal solutions. The results using the TLBO algorithm has been compared with those using some of the other algorithms, namely particle swarm optimisation (PSO), biogeography-based optimisation (BBO), simulated annealing (SA) and binary coded genetic algorithm (BCGA).

| FINITE ELEMENT ANALYSIS OF VIBRATION AND NOISE
Accurate modelling of magnetostriction in a transformer core requires the proper coupling of magnetic, electrical and mechanical domains [28]. A 3D model was generated using the CAD module of COMSOL Multiphysics. Coupling was implemented by adding appropriate sub-domain variables for stresses and magnetic fields. As seen from the literature, a single-phase three-limb transformer unit was reported to be most affected by DC bias as compared to three-phase transformer [29]. This work established a model of a 240-MVA transformer with single-phase four limbs and double windings, modelled as cylinders. The structure of the core composed of separate sections to incorporate the influence of flux at the joint sections of the yoke. This design is further enclosed in a simplified box-shaped geometry of the tank. Table 1 features the parameters of the transformer. The obtained model is shown in Figure 2.
An FE mesh is created for the developed model, and the element size parameters are to be selected in such a manner that there is an agreement between accuracy and time of computation. With the quality of mesh sufficing the task, a total of 737,567 tetrahedrons, 123,252 triangles, 9738 edge elements and 316 vertex elements with a minimum element quality of 0.05172 are developed in the mesh. The size parameters of the developed mesh are given in Table 2 showing the optimal element parameters for convergence, which have been verified by reducing the mesh size further with only an increase in computation time but no significant

| Analysis of electromagnetic fields
Transient analysis of the electromagnetic field is performed. The non-linearity of the core of the transformer has been considered, and the characteristic data are separately defined for the rolling and perpendicular to the rolling direction. As presented in Ref. [30], the B-H data are amended based on constitutive relations between relative permeability, stress and magnetic field intensity. This has been depicted as follows: Assuming that there would be an even distribution of current through the coil, a single-phase supply is fed to the HV windings. The strength of electromagnetic field built up around the windings can be varied by varying the exciting supply through the coil and its distribution largely depends on magnetic characteristics of the material in and around the coil, namely permeability of the core, air and surrounding material. Magnetic shielding is assumed to be present at the box-shaped boundary of the transformer. The simulation is performed using magnetic fields interface of the software, which performs the calculation of electromagnetic field by making use of Maxwell's Theory as presented below: Equations (3)-(5) of the magnetic field module are solved by setting the variable A x , A y , A z and ψ. Predefined materials from the COMSOL Multiphysics library are chosen for the domains and the properties are modified to match those obtained from Ref. [29]. The core material so chosen bears B max ¼ 2.4 T, the settings of the interface are attuned for analysis of complete field in the geometry. Homogenised numeric multi-turn coils (a lumped model of a packet of smaller insulated wires wound together, ensuring the flow of current only in the direction of wires) are used in the model with the conductivity of copper as 5.89 � 10 7 S/m. To obtain a unique and stable solution for the defined vector potential, gauge fixing for A-Field must be implemented which imposes the gauge ∇.A ¼ 0 by addition of a potential variable ψ. The variable ψ imposes a condition on the derivatives of the vector potential. A time range of 60 ms was subdivided into equal steps of 5 � 10 À 4 s for the analysis. The electromagnetic force exerted on the core is calculated for global stress exerted on the structure. For a better insight on the electrical aspects, the nominal electric and electromagnetic fields along with the flux lines are featured on the transformer as shown in Figure 4. It could be seen that the flux generated is sufficient to build secondary EMF in the transformer. The majority of flux is focussed in the central limbs with their magnitude ranging at 0.8 T and are uniformly distributed in other two limbs. It is worth noting that the fields vary with the input to the transformer.

| Harmonic analysis
The steady-state response of a structure to sinusoidally varying loads can be obtained using harmonic response analysis. This analysis involves the calculation of response at multiple frequencies and obtains its relation with the displacement of the structure. As the electromagnetic field study was conducted in the time domain, a fast Fourier transformation (FFT) is performed to obtain harmonics amplitude.

| Structural dynamics
The structural field analysis module utilises Equations (6) and (7); using these equations, the deformation components u, v and w are obtained. In this module, the coil domains are considered to be linear elastic material with the core taken as magnetostrictive material. Desired magnetoelastic properties for the computation are the compliance matrix S H , piezomagnetic decoupling matrix d HT , density and relative permeability of the material. The effective field in the core could be given by [31]  The coefficient of M in Equation (8) depicts the interaction of magnetism and stress. The magnetisation model can be defined as [31] The displacement results are obtained for every eigenmode of the fundamental frequency. A first few of the eigenmodes calculated by the FEM has been shown in Figure 5. These are the modes at which the system is prone to vibrate at certain frequencies and it deforms into a corresponding shape referred to as the eigenmode. The resultant displacement measured is an addition of all the eigenmodes of deformation of the core. This analysis provides the shape of deformations in the subsequent modes and their respective amplitudes of vibration are obtained with the help of structural analysis.

| Acoustics analysis
FEM along with absorbing boundary condition of perfectly matched layers is implemented to obtain the sound field. Equations (10) and (11) present the equations involved in acoustics calculation: Variable p is obtained from Equation (10) for sound field module, where ρ 0 is the density of the fluid, p is the pressure, c is the velocity of sound, ω is the angular frequency and C c is the bulk modulus. Frequency domain, acoustics interface is used to study acoustics. This interface performs a bidirectionally coupled study with structural dynamics, the result from time-to-frequency (FFT) analysis as performed in the previous sections is applied to the acoustics study. The input boundary conditions are provided as displacement of the structure.

| Addition of DC bias to the model
The main objective of this work is the study of noise and vibration from the core of the transformer when subjected to DC bias. This has been achieved by injecting DC to the coils as explained by Chen et al. [6] shown in Figure 6. The analysis has been done by applying DC in the range of 0.5-3 A, and the respective vibration and noise fields are compared.

| Calculation of the vibration and noise of designed transformer
A 240-MVA four-pillar single-phase transformer is simulated and analysed using COMSOL Multiphysics software. The flowchart in Figure 7 shows the process of calculation of vibration and noise. The effect of magnetostriction in vibrational displacement can be observed from the plot shown in Figure 8, which shows the nature of displacement caused due to electromagnetic and magnetostrictive forces, respectively from left to right. The maximum magnitude of vibrational displacement due to electromagnetic forces is nearly 1/17th of that due to magnetostrictive forces working along under nominal operating condition of the transformer. The process depicted in Figure 7 is repeated for cases of varying DC bias and the respective vibration and noise field for all the cases were recorded. It can be seen from Figure 9 that there is a significant increase of vibrational displacement with increase in the magnitude of DC bias. The contour plot of the obtained sound field is shown in Figure 10. It is evident from the plots that the noise increases with an increase in applied DC bias continually until the saturation limit of the core is attained. After which, the sound pressure level ceases to increase further, this result arises in a strong agreement with Ref. [6] and is well below the prescribed DC injection limits as per regulation norms [32]. For this design, the acoustic pressure field stopped increasing on the application of DC bias beyond 3 A and thus the same range of DC bias has been taken as the upper limit for analysis.

| Prediction of transformer core vibration using artificial neural network
ANN is a mechanism of computation which is roughly inspired by the functioning of an animal brain. It learns by instances, without any specific protocol or rules for the set of task [33]. It has been in use for prediction of the relation between input and output, even in complex scenarios. An ANN is modelled on a bunch of interconnected artificial neurons. Each neuron can communicate with other neurons, thus forming a vast network of neurons hence the name. The information received by each neuron is then processed, which is further transmitted to other interconnected neurons. Each connection is assigned with weights representing its significance. These combinations of neurons are organised to form layers, with multiple neurons in each layer forming a network as a whole. The data is received by the input layer and after processing is supplied from the outer layer. Figure 11 shows a basic architecture of a four-layered neural network. As seen from figure there could be multiple hidden layers between the input layer and the output layer which defines the total number of layers in the network. The hidden layers apply the predicted function to the inputs obtained from the input layer, which are further passed on to the next layer. By choosing the number of layers, the number of neurons in each layer and also the type of function, a neural network for a problem can be obtained. The process of determination of weights is called as training of a network, and it requires outputs with known inputs to be fed to the network [34]. Equation (12) defines the relation between input and output data of the network [33]. The output vector is represented as Ω, ϕ is the activation function, ι k is the kth input from the input vector with its respective weight represented as ω k and β denotes the bias.
For this work, the inputs are derived from the measurements of magnetic induction, and the outputs from the vibrational displacements. A total of 131 samples of magnetic induction and respective vibrational displacement were taken.
The samples were then fed to the network, for training,validation and testing in the ratio of 7.5:1.5:1. The network was then trained and tested. Figure 12 shows the error plot of the designed neural network against the validation inputs.
The Lavenberg-Marquardt algorithm was used to attain the possible optimal solution in the design process of the neural network. Furthermore, a newer optimization algorithm, namely TLBO, was implemented for attaining the optimal model and the results were compared with various other optimisation algorithms.

| A brief on TLBO and other optimisation algorithms
The TLBO was introduced first in the year 2011 and it finds applications in many of the engineering fields. The working of this algorithm is divided into two phases, namely the teaching and the learning phase. As per the authors in Ref. [27], 'This algorithm imitates the teaching-learning ability of teacher and learners in a classroom'. In this algorithm, the learners are analogous to the population, the different subjects taught in the classroom are similar to the design variables and the results of the learning process are the fitness value of the problem. In the first part of the algorithm, the population is taught by the teacher, and the objective here is to improve the mean result of the population depending on the ability. At any nth iteration, let there be 'm' design variables, the population size be p ¼ 1, 2, 3, …, k and the mean result be 'R jn ' for a particular design variable 'j'. The best result considering all the design variables obtained in the entire population or set of learners is chosen to be a teacher by the algorithm. All the desirable values after this stage act as input to the next stage, wherein the learners or the population performs interactions among themselves and try to improve their performance. A learner from the population interacts with other learners and tries to increase its knowledge. If two learners A and B are chosen such that X 0 P,n ≠ X 0 Q,n at the end of the previous stage, then their resultant function values would be updated as Equation (13), where X 00 j;A;n and X 00 j;B;n denote the updated function values after the completion of the second phase: If X 0 A;n < X 0 B;n ; X 00 j;A;n ¼ X 0 j;A;n þ r n X 0 j;A;n À X 0 j;B;n À � and if X 0 B;n < X 0 A;n ; X 00 j;A;n ¼ X 0 j;A;n þ r n X 0 j;B;n À X 0 j;A;n The TLBO algorithm could be summarised through the steps shown in Algorithm 1. For detailed understanding on TLBO, readers are advised to go through [35] among many others. , is an evolutionary algorithm that optimises a function with respect to a given measure of quality [36]. BBO imitates relationships between different habitats, simulating the evolution of ecosystem. A set of random solutions are generated initially, which are then evaluated in terms of fitness values over a prefixed number of iterations and the best of the solutions are chosen. The population can switch from its own environment to other depending on its suitability index, the process continues till the solutions meeting up the fitness criteria are obtained [37].

| Binary coded genetic algorithm
This algorithm is inspired by Darwin's theory of natural evolution. Solution vectors are termed as chromosomes, parents are the solutions from which new solutions are generated further, offsprings are the newly generated solutions. These newer solutions are generated either by reproduction or crossover and mutation. As per the theory of evolution, it is the survival of the fittest of the solutions in an environment which bears limited resources. Thus, the algorithm is designed in such a way that the worse or unfit solutions are discarded. In BCGA, real variables are encoded into binary variables. If a bit length l is defined for a variable, then totally there are 2 l possible solutions between the variable bounds. A population is generated and the dimension of the problem is chosen. The number of bits representing each variable is chosen and thence the size of population matrix is determined (population size � no. of bits � dimension of problem). The next step involves the generation of the initial population at random with 0 or 1. Good solutions are identified from the solution and it has to be ascertained that these have multiple presences and the bad solution must have a weaker representation in the population. The selection operation then chooses the best of solutions, which would then undergo crossover thereby reducing the diversity of the population. The process is repeated until the best fitness is achieved in the population [38].

| Simulated annealing
This is a random search method developed by Kirkpatrick et al. in 1983. This technique is a variant of the hill-climbing method, wherewith every iteration, we try to achieve a higher value and discard lower values. The technique is inspired by the process of annealing in which the materials are heated to a very F I G U R E 1 3 Test points depicted in front and overall geometry view in top to bottom order YADAV AND MEHTA -341 F I G U R E 1 4 Predicted vibrational displacement (μm) using artificial neural network model optimised by teaching-learning-based optimisation, biogeography-based optimisation, binary coded genetic algorithm, simulated annealing and particle swarm optimisation for DC bias of (a) 0 A, (b) 0.5 A, (c) 1 A, (d) 1.5 A, (e) 2 A, (f) 3 A

-
high temperature for melting and are then cooled down to solid-state [39]. The objective in a physical annealing process is to line up all atoms on crystal lattice sites without any defects such that they entertain the lowest energy state. The process involves the generation of a random solution string and evaluation of fitness function for all chosen solution strings. Initial and final control parameters are chosen. Perturbation is introduced to the set of solutions and the newly formed strings are then evaluated and accepted if the fitness has been achieved. The process terminates once the global optima have been obtained.

| Particle swarm optimisation
PSO is a metaheuristic swarm intelligence technique, which may be defined as an attempt to design distributed problemsolving framework which is inspired by the collective behaviour of animal societies. PSO was proposed by J. Kennedy and R. Eberhart in 1995. In this technique, each particle is a bird or an animal which has a position and velocity associated with it. Particles change their position by adjusting their velocities to seek food, or to identify optimised parameter or even to stay away from predators. It is the task of the particle to remember the best location which is identified by it. Initial position and velocity of particles are generated randomly within the search region, the position and velocity of the particle are determined and finally, the objective function of the particle is evaluated. This is not a greedy search algorithm hence the update of the population is done at any cost, without any care of the fitness. The global and personal best is updated for every iteration depending on the fitness, the process terminates once the number of iterations get completed.

| Objective function
In order to model magnetostrictive vibration of the transformer, the unknowns in Equations (8) and (9) have to be estimated. Thus the parameters to be estimated are where K c , M c and C c are the stiffness, mass and damping coefficient of the material. Thus, the objective function is defined as minimise ∑ T n¼1 e 2 ðn; ϑÞ where n is the nth sample and T denotes the total samples. e(n, ϑ) is the displacement error. The stated objective function is minimised using TLBO algorithm as explained in the previous section. The comparison of the results is done by choosing specific test points in the geometry as shown in Figure 13. It is to be noted that the points denoted on the geometry in figure are scaled up. Certain parts of the model are hidden for the purpose of clear insight of the points. The results obtained using TLBO algorithm alongside that of the ANN model are compared with those using other available algorithms and are shown in Figure 14. It is evident from the figure that the vibration displacement obtained using TLBO method taking its best and worst of performances on an average is 5 where A i and C i are the actual and estimated values, i is the number of data points used for validation. The TLBO algorithm produces the best nearness to the designed ANN model for vibrational displacement which can be seen from the mean squared error (MSE) values of all the implemented optimisation methods as shown in Table 3. The displacements obtained using all the methods are depicted in Table 4.

| Noise prediction and comparison
The work emphasises on the nature of noise related to the magnetic source in transformer, that is, due to the vibration of the core. It has already been established that the presence of DC bias in the transformer causes a significant increase in vibration. This section aims to analyse the noise caused due to the effects established earlier. Acoustics analysis has been performed by considering the vibrations of the core as the source. As mentioned earlier, Figure 10 shows the effect of DC bias on noise generated from the transformer. An ANN model was developed to successfully predict the sound pressure level at different points of the transformer which was then compared with the predictions using other optimisation methods. The objective function for obtaining the optimal solution for the network is defined as The acoustic pressure level along with the percentage MSE using all the methods have been tabulated in Table 5. The lowest %MSE has been obtained to be of TLBO algorithm which is about 4.40% comparatively much better than other implemented techniques. The results from BBO too were in comparative nearness to those of TLBO, but the overall performance of TLBO algorithm considering the vibration and noise in its entirety has been found to be very encouraging.
It could be established from Figure 15 that the TLBO algorithm produces the best match with the ANN developed model.

| CONCLUSION
This work modelled the effects of DC bias in a converter transformer concerning the vibrations and noise generation from the transformer. A 240-MVA transformer has been modelled, DC bias of various magnitudes are applied to the transformer model and resulting vibration and acoustic noise are recorded. The data collection has been done by choosing multiple points in the design. Furthermore, ANN has been implemented to model the vibration and acoustics of the transformer which has been further verified by using optimisation algorithms. The proposed method efficiently models the effects of DC bias in magnetostrictive vibration and noise of the transformer. The following are the major inferences-: � Even a small amount of DC bias entering the transformer results in a substantial increase in magnetostriction of the core, which in turn increases the vibration of the core leading to excessive noise. � Under absolute conditions. that is, without DC bias, the magnetic vibrations of the core ranged to a maximum of 0.19,875 μm which further increased to 1.38,945 μm on the application of 3 A of DC bias. The consequent increase of vibrational displacement was observed on the application of DC biases in the order of [0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 A]. � The ANN model developed for prediction of vibration and noise characteristics of the transformer is meeting up the requirements to a very precise nearness. � The initial optimal selection of designed neural network was replaced by various optimisation algorithms and their performances were compared. It was found that out of various existing optimisation schemes, the TLBO proved its superiority over other tested algorithms with higher nearness to the prediction of the ANN model and FEbased results.
The converter transformer operating at the centre of the HVDC transmission system is vulnerable to DC bias. The analysis performed in present work would be of profound use in studying the nature of acoustics and mechanical vibrations of converter transformer subjected to DC bias. The results achieved could possibly provide the subsequent reference for improving noise cancellation schemes and DC bias restraint measures. Considering the minimal magnitude of bias as taken here, improved noise reduction methods must be incorporated for HVDC transformers. Therefore, the precise information of vibration and acoustics of a converter transformer will concede design development in future.

ACKNOWLEDGEMENT
The authors are thankful to North Eastern Regional Institute of Science and Technology, Nirjuli, for providing a suitable work environment and to all the faculties of the Department of Electrical Engineering for their help and support.