Sound pressure level determination around the transformer using coherent and incoherent analytical summation methods

Abstract

In this paper, coherent and incoherent analytical summation methods from elementary sound sources on the transformer’s tank are presented. They aim to determine a one-third-octave spectrum of sound pressure levels (SPL) in points around the transformer. Each part of the transformer’s surface is treated as a separate plane source so the analytical calculation can be applied to find SPL in the surroundings. Reflections from the ground are also considered by putting an image of the plane sources on the other side of the reflecting plane and adding its contributions to the total pressure in the desired points. Grid-like vibration measurements of vibration velocity on the tank surfaces are used as input parameters. Vibrations and sound pressure levels are measured to validate the method on the 5 MVA transformer experimental object. The SPL around the transformer in short-circuit (SC) and open-circuit (OC) tests is measured in the semi-anechoic chamber to compare it with theoretical results. By analyzing the results, the coherent calculation with reflection provided the most accurate results. In the SC operating condition, the normalized root mean square error (NRMSE) is 17.2%, and in the OC operating condition, it is 10.9%. The novelty of the presented method is that it considers complicated transformer geometry where each surface is calculated as a separate noise source. It calculates noise at a distance from the elementary vibration sources, considering phase and reflection from the hard ground, and provides detailed noise maps that can be used for noise modeling around the substations.

Keywords

Transformer plane sound source noise radiation coherent and incoherent summation sound mapping estimation

Introduction

Increasing demand for energy in growing cities pushes the locations of transformers closer to the populated areas. A transformer in work produces specific noise at the first few harmonics of double the network frequency. The vibration sources in transformers are windings and the core. The sound pressure generated by the vibrations from the windings and the core is transmitted to the tank through the transformer oil and metallic paths.¹ Transmission through oil is the dominant path for winding vibrations, and transmission through structure depends on the tank’s modal properties.² Vibrations of the tank surface are radiated as noise to the surroundings. The radiated sound level depends on the tank wall’s radiation efficiency and the surrounding medium acoustic properties. Except for the SPL, the inhabitants’ annoyance depends on the sharpness, roughness, and tonality of the transformer noise.³ The process of vibration transmission and radiation is shown in Figure 1.

Figure 1.

Noise transmission paths and radiation of noise from the tank wall to the environment.

Measurement of the vibration of sources (windings and core) can be used as validation of finite element method (FEM) models that can be used as noise prediction tools.^4–8 On the other hand, measurements of the vibrations on the tank wall are more straightforward and accessible, and they can be used as a direct input for the noise modelling software.

The procedure of SPL measurements around the transformer in OC and SC tests is explained in detail in standards^9,10 to estimate sound power. In the OC test, the core is the dominant source. In the SC test, it is the windings. Therefore, precise prediction of noise radiation patterns around the vibrating transformer can be of significant interest when placing the transformer in substations and surrounding it with walls. An example of the usage of sound absorbers and insulators, as well as vibration insulators in indoor distribution substation, is given in the literature.¹¹ Positions of sound barriers can be determined in the substation project stage with the help of noise modelling software with appropriate input parameters. Two substations and their surroundings are analyzed and acoustically mapped.¹² Because of the large surrounding area of the substations, point or plane noise sources are used to replace the transformers depending on the distance of the emission point from the transformer. In another paper,¹³ transformers and reactors are simulated as an equivalent of several point sources based on the equivalent source method. The sound power of sources is determined by acoustic holography. In the following literature,¹⁴ the prediction of the substation noise is modelled so that the transformer and reactor are considered as sound sources consisting of a series of small plane sources.

The method developed to predict sound radiated by a power transformer based on vibration measurement is described in the literature .¹⁵ This approach idealizes the transformer tank as a rectangular box. It is based on modelling the transformer tank as a concatenation of rigid right-angled wedges. There is also approach,¹⁶ where acceleration on the transformer tank wall and the noise level at 0.3 m from the wall are used to predict the noise radiation. The transformer tank wall is treated as two flat planes and two semicylindrical shells.

The experimental method for the determination of the sound radiation efficiency of the transformer (measurement of vibrations and noise) is presented in the literature.^17–19 Also, a validated boundary element method (BEM) model can be used to calculate the sound radiation efficiency of a transformer.²⁰

The motivation for this work was to develop a more accurate method for representing a transformer as a noise source from the known vibration velocity distribution on the tank surface. Applying the method can lead to a more favorable placement of the transformer itself and surrounding sound barriers at the place of installation. Compared to the previous research, the method presented in this paper considers complicated transformer geometry, and each transformer surface is treated as a separate noise source. The coherent method adds elementary surface vibration sources together considering the radiation phase. The calculation result is used to visualize noise maps in the surroundings in the OC test, SC test, and operation. In the paper, except for coherent, an incoherent analytical summation method is considered for accounting contributions of all vibration sources. In the incoherent method, radiation efficiency derived for a case of resonant vibrations of a considered surface²¹ is used as an input parameter for calculations instead of the time-consuming experimental methods used in previous research. Reflections are added as the image sources in the reflecting plane.

The paper consists of the theoretical background used to implement the method. The methodology of the SPL calculations is given, and the results consisting of comparisons with measurements and noise maps are presented. Finally, discussion and conclusions are given.

Theoretical background

In this section, the theoretical background of coherent and incoherent radiation from a plane sound source is presented, considering the transformer divided into surfaces.

Coherent sound addition requires that the difference in the sound phase from two different sources remains fixed with time. Adding sounds together coherently accounts for this phase difference from sources due to distance from the considered calculation point. On the other hand, incoherent sounds have random relative phases, and they sum as scalar quantities on an energy basis.²² This is often used to determine sound power levels according to standard ISO 3744²³ with higher measurement uncertainty at lower frequencies, assuming plane wave propagation at a short distance from the surface source.

Incoherent plane radiator

The radiated power of the elementary piston source in the near field²⁴ is:

W = 1 / 2 \cdot ρ c R_{R} π a^{2} U^{2}

(1)

where ρ is the air density, c is the speed of sound in air, ρcπa²R_R is the real part of the radiation impedance, a is a piston radius, and U is the piston velocity amplitude. R_R is defined as:

R_{R} (z) = \frac{z^{2}}{2 \cdot 4} - \frac{z^{2}}{2 \cdot 4^{2} \cdot 6} + \frac{z^{2}}{2 \cdot 4^{2} \cdot 6^{2} \cdot 8} - \dots

(2)

where z = 2ka and k is the wavenumber.²² In order to estimate the sound power that machines radiate due to their surface vibration, the mean square vibration velocity averaged over the surface needs to be determined.²⁵ Replacing U²/2 from equation (1) with the surface mean square velocity ⟨v²_S,t⟩ replacing the piston surface area πa² with the plate surface S, and R_R with surface radiation efficiency σ, equation (1) becomes:

W = ⟨ v^{2} ⟩_{S, t} S ρ c σ

(3)

where ⟨v²⟩_S,t is the surface mean square velocity, S is the plate surface, and σ is the surface radiation efficiency. Radiation efficiency is derived in,²¹ and it depends on the critical frequency f_c,²⁶ plate surface S, speed of sound in air c, and the propagation speed in plate c_B,eff. Critical frequency is dependent on the plate thickness and the material properties of the plate.

Radiation of a plane distribution of incoherent radiators²⁷ is shown in Figure 2. Cosine weighting on the radiation directivity of an elementary source is used. Weighting is implemented by multiplying the calculation that assumes uniform radiation with the directivity factor D_i, which is equal to the ratio of distances r and r_i from Figure 2.

Figure 2.

Rectangular noise source of length L and height H with field observation point O and used coordinate system.

The mean square acoustic pressure at distance r from the pulsating spherical source is defined as²⁸:

< p^{2} > = \frac{W ρ c}{4 π r^{2}}

(4)

Only one-half of the sphere is considered because of the radiation in half-space only. That is done by replacing and dividing equation (4) by two and multiplying it by D_i. It follows that the mean square sound pressure at the observer location O due to elementary source i located at a distance r_i is ²⁸:

< p^{2} > = \frac{W_{i} ρ c D_{i}}{2 π r_{i}^{2}}

(5)

Elementary source i radiates power W_i defined as:

W_{i} = \frac{W d x d z}{H L}

(6)

Substituting (6) into (5), replacing D_i with the ratio of distances r and r_i, and integrating over the area of the plane, the mean square sound pressure at the observer’s location can be obtained²⁸ for incoherent plane radiation:

〈 p^{2} 〉 = \frac{ρ c W}{2 π H L} \int_{- h}^{H - h} d z \int_{d - L / 2}^{d + L / 2} {[x^{2} + z^{2} + r^{2}]}^{- 3 / 2} r d x

(7)

Coherent plane radiator

Using Green’s theorem, the Helmholtz equation can be transferred into an integral equation called the Kirchoff-Helmholtz integral.²⁹ The Kirchhoff-Helmholtz integral is then applied to the sound radiation of a piston in an infinite rigid baffle. It is required in that case that the edges are far enough from the middle of the surface where the velocity magnitude is highest so the diffraction effects can be neglected. In other words, all the sound power is radiated in the hemispherical half-space. It contains a limited region S with a given normal component of the velocity ${\overset{ˇ}{v}}_{n} (x, y)$ . Using Kirchhoff-Helmholtz integral in that case, the remaining relation is derived, called Rayleigh integral ²⁹:

\overset{ˇ}{p} (x, y, z, ω) = \frac{j ω ρ_{0}}{2 π} \int_{S} {\overset{ˇ}{p}}_{n} (x, y, ω) \frac{e^{- j k r}}{r} d S

(8)

Simplification is made by assuming a rectangular radiation surface instead of a circular. The area of the rectangular plate is divided into smaller rectangles, as shown in Figure 3. The normal component of vibration velocity is known only at one point for each of the smaller rectangles, and the same component (amplitude and phase) is assumed in the entire surface of each of the small rectangles. Because of the mentioned simplification and the discrete nature of the problem, the integral from equation (8) is written in the form of the sum:

\overset{ˇ}{p} (x, y, z, ω) = \frac{j ω ρ_{0}}{2 π} \sum_{i = 1}^{n} \sum_{j = 1}^{m} v_{n i j} (ω) \frac{e^{- j k r_{i j}}}{r_{i j}} x_{i} y_{j}

(9)

where v_nij is the normal component of vibration velocity at the surface of the plate.

Figure 3.

Rectangular noise source divided into smaller sources with defined distances r to field observation point P, normal components of velocity v_n, length x, and height y.

Applying equation (9) to the entire surface with known vibration velocities (measured or modelled) at one-third octave spectrum bands provides results that can be compared with measurements.

Reflection effect

A reflecting surface near the source will affect the directional properties of the source and the total power radiated by the source.³⁰ If the distance h_s between a source and a reflection plane is larger or comparable to distance r between a source and observation point (Figure 4) and larger than the wavelength, effective sound pressure is calculated by adding intensity contributions of the source and its image in the reflecting plane. The power of the image source is equal to the source power.

Figure 4.

Reflection from a rigid plane surface.

Taking reflection into account for the case of the incoherent plane radiation from small elementary sources can be done by simply putting an image of the incoherent plane source on the other side of the reflecting plane and adding its contribution to the contribution of the plane itself, considering new directivity factors, reflection coefficients and distances from the observer point. This approach is somewhat approximative because coherent summation is preferred for modeling the reflection effects. The reason is that the reflected waves maintain the same frequency and phase relationship as the original waves, and coherence allows constructive and destructive interference patterns between them. However, incoherent summation can be a reasonable approximation for broadband noise sources with complex sound fields, such as industrial machinery.

In the case of the coherent plane, the wave phase needs to be considered. It is derived²² that the complex amplitude reflection coefficient of plane waves is equal to:

R_{p} = \frac{Z_{N 2} \cos θ - Z_{N 1} \cos Ψ}{Z_{N 2} \cos θ + Z_{N 1} \cos Ψ} = \frac{ρ_{2} c_{2} \cos θ - ρ_{1} c_{1} \cos Ψ}{ρ_{2} c_{2} \cos θ + ρ_{1} c_{1} \cos Ψ}

(10)

where Z_N1 and Z_N2 are the characteristic impedances of medium 1 and medium 2, Θ is the angle of incidence and reflection, and Ψ is the angle of incidence and transmission (Figure 4). Characteristic impedances are equal to the product of the speed of sound and density of each medium.

The elementary surface is small (can be considered as a point source), and regarding that, spherical instead of plane waves can be assumed. It is derived³¹ that the complex amplitude reflection coefficient of a spherical wave is equal to:

R_{s} = R_{p} + B G (w) (1 - R_{p})

(11)

The term B is defined as:

B = \frac{B_{1} B_{2}}{B_{3} B_{4} B_{5}}

(12)

where B is a f (Θ, ρ₁, c₁, ρ₂, c₂, k₁, k₂), k₁ and k₂ are the propagation coefficients of each medium equal to the ratio ω/c. The total expression for B is given in.²² The term G(w) is defined as:

G (w) = 1 - j \sqrt{π} w e^{- w^{2}} e r f c (j w)

(13)

where erfc is the error function, and w is defined as:

w = \frac{1}{2} (1 - j) {[2 k_{1} (r_{1} + r_{2})]}^{1 / 2} \frac{B_{3}}{B_{6}^{1 / 2}}

(14)

The total sound pressure is then equal to³²:

p_{t o t} = p_{D} + p_{R} = \frac{A}{r} e^{- j k r} + R_{s} \frac{A}{r_{I} + r_{R}} e^{- j k (r_{I} + r_{R})}

(15)

where A is the sound pressure amplitude at the surface of the source and distances r, r_I, and r_R are labelled in Figure 4.

Methodology and measurements

SPL calculation from elementary sources

The measured object used to validate this method is the 5 MVA transformer experimental object. Vibrations are measured at 855 points on the tank wall surface using three PCB Piezotronics 608A11 accelerometers of sensitivity 100 mV/g. The acquisition card from National Instruments consisted of the chassis NI cDAQ-9171 and the module NI-9234, which were used to collect measurement data. LabVIEW software with the software module Sound and Vibration Assistant 2015 is used in the measurement process. The experiments are done in environmental conditions with minimum ambient noise. The accelerometers are calibrated using a calibrator, B&K 4294, to check the sensitivity (9.81 m/s2 at 159.15 Hz). MATLAB is used to process the results. The methodology of grid-like vibration measurements used for analytical SPL determination is explained in more detail in the literature.³³ For analytical calculation of SPL in a point around the transformer, each surface of the transformer tank is treated as a separate plane source. Two calculation methods are used to solve this problem. In the first method, the tank walls are treated as coherent, and in the second as an incoherent radiator. The summation of contributions at the calculation point is done only for the surfaces for which the y-coordinate of the calculation point is positive (coordinate system of Figure 2). Reflections from the ground are also considered by putting an image of those plane sources (surfaces) on the other side of the reflecting plane (floor). Finally, those contributions are summed up with the contribution of planes themselves. To better understand the method, an example of surfaces that contribute to the total SPL in the calculation point is shown in Figure 5(a).

Figure 5.(

a) Example of surfaces that contribute to the total SPL in calculation point (b) numbered surfaces of the transformer tank wall and geometry angles in a local coordinate system of surface 15.

Surfaces numbered 1-7 contribute to the SPL in the calculation point. Contributions for the other surfaces are not considered. Local coordinate systems of each surface are oriented such that normal to the surface is in the y-direction. The x-axis is in a horizontal direction, and the z-axis is in a vertical direction. The origin is in the middle of the surface. The representation of not including surface 15 in the calculation is shown in Figure 5(b). In the local coordinate system of the surface, the calculation point is not in the range from 0° to 180° in the x-y plane, i.e., the point’s y coordinate in the local coordinate system of the surface is negative.

Measurements in a semi-anechoic chamber

The SPL around the transformer for OC and SC tests is measured in the semi-anechoic chamber using sound level meter B&K 2270 of the dynamic range above 123 dBA to compare summation and measurement results. The transformer experimental model in the chamber on rail transfer trolleys is shown in Figure 6(a).

Figure 6.

(a) Transformer experimental model on rail transfer trolleys (b) Location of 14 measurement points at 1 m height in a semi-anechoic chamber.

Measurements are done using a point-by-point procedure at 14 points around the transformer at 1 m height from the reflecting floor. The locations of the points on the prescribed contour are shown in Figure 6(b).

Results

Comparison of different modelling results with measurements

Analytically determined one-third octave spectrum and the total value of A-weighted SPL (L_p,A) are compared with measurements in 14 points around the transformer in SC and OC tests. A comparison of the spectrum and the total values of the two calculation methods is shown for selected points 4 and 11 in Figures 7 and 8, respectively. Both calculations, with and without reflections from the floor, are shown. Although calculation with reflections corresponds more to the actual situation, calculation without reflections is shown for comparison. In the incoherent method, reflections always add around 0.6 dBA to the SPL. That is the case for one-third of octave spectrum bands and total levels. The reason is that the image source in the reflective plane contributes to the noise in the calculation point around 7-9 dB less due to the divergence effect (depending on the surface) than the source itself. In the coherent method, the influence of reflections varies because of phase influence. At some one-third octave spectrum bands, reflections increase, and on some decrease the SPL. The same applies to total SPL levels. Some examples where total SPL levels are decreased are points 1, 4, 8, and 11 in the SC test and points 3 and 11 in the OC test, as shown in Figure 9. The coherent method with reflections is accurate within ±1.1 dBA for point 4 and ±11.5 dBA for point 11 for the overall value in both operating conditions, OC and SC tests. The actual cause of the significant underestimation of noise in point 11 and other points from the same transformer side (8, 9, 10, and 12), as shown in Figure 9, is difficult to determine. It could be a vibration of additional parts on the transformer tank (connection for coolers, untightened screws, etc.), unsymmetric placement of transformer experimental object in the semi-anechoic chamber (practical reason because of rails), insufficient size of the chamber regarding the measured object (around 50 times larger volume of the chamber, but ISO 3745³⁴ recommends it to be at least 200 times larger), etc.

Figure 7.

Comparison of measurement in point 4 with two proposed analytical methods with and without taking reflection into account in (a) SC test, (b) OC test.

Figure 8.

Comparison of measurement in point 11 with two proposed analytical methods with and without taking reflection into account in (a) SC test, (b) OC test.

Figure 9.

Comparison of measured values of total A-weighted SPL in all 14 points with two proposed analytical methods with and without taking reflection into account in (a) SC test, (b) OC test.

A comparison of measured total A-weighted SPL in all 14 measurement points around the transformer with coherent and incoherent analytical calculation methods with reflections considered is shown in Table 1. Comparison is shown for both operating conditions, OC and SC tests.

Table 1.

Comparison of measured values of total A-weighted SPL in points with the analytical calculation methods in OC and SC tests (M – measurement, CR – coherent with reflection, IR – incoherent with reflection).

Points	SC			OC
Points	M	CR	IR	M	CR	IR
p1	40.4	33.3	39.5	43.7	38.7	36.8
p2	34.5	37.3	39.7	39.7	40.1	37.4
p3	40.3	39.0	40.6	45.0	40.9	38.4
p4	41.0	41.5	41.5	43.0	41.9	39.7
p5	42.8	40.6	40.2	38.3	42.6	38.7
p6	40.9	39.5	36.3	43.8	40.5	35.3
p7	37.5	39.0	30.9	46.7	38.2	32.5
p8	36.8	23.7	28.5	42.1	37.7	32.6
p9	38.4	27.4	30.1	46.5	39.6	32.2
p10	42.9	29.5	34.0	38.0	41.2	34.4
p11	43.2	31.6	37.9	42.0	35.7	37.0
p12	40.0	30.8	40.1	46.4	40.2	36.9
p13	32.7	36.0	40.8	44.8	38.7	36.4
p14	39.4	37.4	40.2	46.6	42.0	36.3

For comparison of calculations, normalized root mean square error (NRMSE) is used, which is defined as:

N R M S E (L_{p A C}) = \sqrt{\frac{1}{N} \cdot \frac{\sum_{i = 1}^{N} {({| L}_{p A C} (p_{i}) | - {| L}_{p A M} (p_{i}) |)}^{2}}{{{| L}_{p A M, \max} |}^{2}}}

(16)

where N is the number of the measurement points, L_pAM,max is the maximum measured value of L_pA in all 14 points (p) to which error is normalized, and L_pAC and L_pAM are values calculated with the proposed calculation and measured value of L_pA, respectively.

A graphical representation of the total A-weighted SPL in all 14 points with two proposed analytical methods for both operating conditions is shown in Figure 9. The NRMSE for each calculation is shown. For both calculations in both operating conditions, taking reflections into account decreases the NRMSE. The NRMSE is, on average, decreased by 1.5%.

Considering both operating conditions, the smallest error on average is in the coherent calculation with reflections considered. In the SC operating condition, it is 17.2%, and in the OC operating condition, it is 10.9%.

Spatially averaged results are those the manufacturers should guarantee and are used to calculate the sound power. Spatially averaged A-weighted SPL from point-by-point measurements is calculated as follows ⁹:

L_{p A t o t} = 10 \cdot \lg (\frac{1}{N} \sum_{i = 1}^{N} 10^{0.1 L_{p A i}})

(17)

where N is the number of microphone positions, and L_pAi is the measured total A-weighted SPL at the ith microphone position.

A comparison of spatially averaged measurements in 14 points with the spatially averaged calculations using two calculation methods is shown in Figure 10. Reflections add around 0.5-1 dBA for the coherent SPL calculation method. For the incoherent SPL calculation method, reflections add 0.6 dBA.

Figure 10.

Comparison of spatially averaged measurements (in 14 points around the transformer) with spatially averaged calculation using two proposed analytical methods with and without taking reflection into account in (a) SC test (b) OC test.

In the SC test, the average noise around the transformer is 3.1 dBA lower than the measurements for the coherent and 1.5 BA lower for the incoherent analytical calculation. In an OC test, the average noise around the transformer is 4 dBA lower than the measurements for coherent analytical calculation. For the incoherent calculation, it is 7.6 dBA lower than the measurement results.

Noise maps

Based on coherent calculation with reflections considered, noise maps are visualized around the transformer in free space. A-weighted SPL (L_pA) is calculated in an area of 20 m × 20 m at 1 m height from the floor. Calculation points are spaced from each other by 1 m in each dimension, which makes a total of 441 points. Additional 8 points near the transformer surface are manually added to better represent SPL in the vicinity of the transformer. The noise map of total L_pA in the SC test is shown in Figure 11. Most of the noise is radiated from the low voltage (LV) side of the transformer (the upper side in the following figures), and the least noise is radiated from the cooler side of the transformer (the right side in the following figures). In the SC test, noise is predominantly on a one-third octave frequency band of 100 Hz. Also, the 200 Hz band is significant, as shown in Figure 10(a).

Figure 11.

Visualization of noise map of total L_pA around the transformer (area of 20 m × 20 m at 1 m height from the floor) based on coherent calculation with reflections in SC test.

The noise map of total L_pA in the OC test is shown in Figure 12. The radiation pattern differs from the one in the SC test. Although most of the noise is also radiated from the LV side of the transformer, the radiation pattern around the transformer is more uniformly distributed.

Figure 12.

Visualization of noise map of total L_pA around the transformer (area of 20 m × 20 m at 1 m height from the floor) based on coherent calculation with reflections in OC test.

Transformer noise in operation is defined as the logarithmic addition of the SC noise and OC noise. If the fans or pumps are in operation, which is not the case here, the noise of the cooling system should also be added. Noise in operation corresponds to the actual state at the substation. As explained in,⁹ that is valid when combining total sound levels and sound levels for individual frequency bands in different loading conditions. A-weighted SPL in operation is defined as follows:

L_{p A o p} = 10 \cdot \lg (10^{0.1 L_{p A S C}} + 10^{0.1 L_{p A O C}})

(18)

where L_pASC is A-weighted SPL in the SC test, and L_pAOC is A-weighted SPL in the OC test.

The total noise map in the operation of the transformer is shown in Figure 13. Except for the total noise maps, noise maps at specific one-third-octave frequency bands can be visualized. That can be especially interesting if there is a problem at a specific band. In that case, transformer placement and position of sound barriers can be more accurately determined. An example of visualization of the first three dominant frequency bands in operation is shown in Figures 15 and 16.

Figure 13.

Visualization of noise map of total L_pA around the transformer (area of 20 m × 20 m at 1 m height from the floor) based on coherent calculation with reflections in operation.

For 8 points distanced 10 m from the transformer and shifted by 45° (Figure 17), results calculated from the operating conditions at the noise maps from Figures 11–16 are compared in Table 2.

Figure 14.

Visualization of noise map of the one-third octave frequency band 100 Hz around the transformer (area of 20 m × 20 m at 1 m height from the floor) based on coherent calculation with reflections in operation.

Figure 15.

Visualization of noise map of the one-third octave frequency band 200 Hz around the transformer (area of 20 m × 20 m at 1 m height from the floor) based on coherent calculation with reflections in operation.

Figure 16.

Visualization of noise map of the one-third octave frequency band 315 Hz around the transformer (area of 20 m × 20 m at 1 m height from the floor) based on coherent calculation with reflections in operation.

Figure 17.

Chosen 8 points A-H around the transformer (10 m from the origin shifted by 45°) for which calculation results are compared in Table 2.

Table 2.

Results of coherent calculations with reflections considered in points A-H from Figure 17 for different operating conditions shown in Figures 11–16 (OP - operation).

	SC test	OC test	OP	OP 100 Hz	OP 200 Hz	OP 315 Hz
A	30.8	26.3	32.2	28.8	28.4	14.0
B	23.4	24.1	26.8	21.7	18.2	21.5
C	10.5	20.6	21.0	9.6	14.2	15.9
D	23.6	27.1	28.7	23.3	24.6	22.3
E	21.7	26.5	27.8	21.3	25.4	13.8
F	27.4	25.6	29.6	27.0	20.1	22.8
G	25.7	23.0	27.6	25.4	21.7	7.2
H	25.5	29.4	30.9	21.8	29.3	17.7

Discussion

Determining noise radiation patterns around the transformer is crucial when the transformer operates at the site. This paper presented coherent and incoherent methods for calculating SPL with reflections from the floor included.

In spatially averaged SPL, reflections from hard surface add around 0.5–1 dBA for the coherent calculation method, and for the incoherent method, reflections add around 0.6 dBA. Calculation with reflections from the floor is more accurate for both methods. By taking reflections into account, the NRMSE is, on average, decreased by 1.5%.

The coherent calculation with reflections provided the most accurate results. Considering measurements in all 14 points, NRMSE in SC operating condition is 17.2%, and in OC operating condition, it is 10.9%. For the mentioned calculation, spatially averaged SPL is 3.1 dBA lower than the measurements in the SC test. In the OC test, the spatially averaged SPL is 4 dBA lower than the measurements.

Visualization of noise maps is shown in Figures 11–16, and the results in 8 points at a 10 m distance from the transformer are compared in Table 2. The cause of SPL in operation is predominantly from the core noise (OC test). The average difference between operation and OC noise in 8 points is 2.7 dBA. Because of that, a similar SPL distribution is obtained by comparing Figures 12 and 13. Operational noise at the one-third octave band of 100 Hz is caused dominantly by winding vibrations (SC test). In Table 2, the 100 Hz band of operation noise is lower than the noise in the SC test. The reason for that is the 200 Hz component in the SC test, which also contributes to the noise. By comparing Figure 14 with Figure 11, a similar noise distribution is visible because of the dominance of the 100 Hz band.

The advantages of this method in comparison to the research conducted hitherto are as follows. It considers complicated transformer geometry instead of simpler, as in previous research,^12,13,15,16 and noise modeling programs (rectangular surface sound sources). Each transformer surface is calculated as a separate noise source. Sound radiation efficiency is calculated with sufficient accuracy using expressions for resonant plates instead of time-consuming experimental methods. It uses a sum of the elementary surface vibration source’s contributions for coherent calculation by considering the radiation phase. Finally, it provides detailed noise maps calculated using the presented method. Maps can be used for noise modelling around the substations in the project stage.

A disadvantage of the method is that it neglects the diffraction of the sound waves around the transformer’s corners, which can be significant at longer wavelengths, i.e., at lower frequencies, which introduces a larger uncertainty at lower frequencies.

Conclusion

In this paper, coherent and incoherent calculation methods, with reflections considered, are used to calculate the transformer’s one-third-octave spectrum of SPL. The calculation is compared with measurements in a semi-anechoic chamber in SC and OC tests. At the calculation point, the summation is done only for the surfaces for which that point is in front of the radiation surface because of dominance, i.e., from 0° to 180°.

In the incoherent method, reflections always add a value of around 0.6 dBA to the SPL. In the coherent method, the influence of reflections varies depending on the point location and band considered. For example, in spatially averaged results in 14 points, for the coherent SPL calculation method, reflections add around 0.5–1 dBA.

Comparing the presented two methods with reflections considered, the coherent method provides more accurate results, and the NRMSE is the smallest. For example, in the SC operating condition, the spatially averaged SPL is 3.1 dBA lower than the measurements, and the NRMSE is 17.2%. On the other hand, in the OC operating condition, the spatially averaged SPL is 4 dBA lower than the measurements, and the NRMSE is 10.9%.

Based on coherent calculation with reflections considered, noise maps around the transformer in free space are visualized. Because of different mode shapes at different resonances, the transformer tank does not radiate noise uniformly in all directions. This phenomenon can be captured using the described approach of L_pA calculation from vibration measurements. In the SC test, most noise is radiated from the transformer’s low voltage (LV) side. In the OC test, the radiation pattern around the transformer is more uniformly distributed. Noise in operation corresponds to the actual state at the transformer substation. Total noise maps and noise maps at specific frequency bands of the transformer are visualized. Using those maps, the placement of transformers in complicated surroundings with more reflection surfaces and sound barriers can be more accurately determined in the substation project stage.

Footnotes

Acknowledgments

Special thanks to Končar—Power Transformers Ltd, a Joint Venture of Siemens Energy and Končar, for providing the transformer experimental model for the purpose of vibrations and noise measurement.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Karlo Petrovic

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