# A Computationally Effective Method for Iron Loss Estimation in a Synchronous Machine from a Static Field Solution

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TL;DR: The proposed iron losses computation method showed a fair accuracy and a considerable speed-up of the computations, and can be an excellent alternative for the iron losses estimation in the optimization procedure of synchronous machines, where a considerable amount of finite element solutions needs to be carried out.

Abstract: In this paper, a computationally effective iron loss calculation method for synchronous machines is presented. The method is based on a single static 2D finite element field solution in the machine cross-section, which makes it much faster than the one based on the time-stepping solution. The developed method is applied to a salient pole synchronous machine, and the computational accuracy is validated against the time-stepping method. The proposed iron losses computation method showed a fair accuracy and a considerable speed-up of the computations. It can be an excellent alternative for the iron losses estimation in the optimization procedure of synchronous machines, where a considerable amount of finite element solutions needs to be carried out. Besides the losses comparison, local reconstruction of the time dependency of other quantities such as the magnetic vector potential and the magnetic flux density is reported for a better understanding of the method.

## Summary (3 min read)

### Introduction

- This reprint may differ from the original in pagination and typographic detail.
- The method is based on a single static 2D finite element field solution in the machine cross-section, which makes it much faster than the one based on the time-stepping solution.
- Moreover, the proposed iron losses computation method is not limited to a particular synchronous machine and applicable to any kind of synchronous machine.

### II. METHODOLOGY

- The iron losses will be computed from the Fourier decomposition of the flux density waveform over one period.
- The empirical iron loss models are fast, easy to implement, and applicable for the iron losses estimation roughly [12].
- Chys and Ced expresses the hysteresis and eddy current loss coefficients, ωs stands for the angular frequency of the supply, Vc is the volume of the stator iron core, and B̂n is the peak flux density value at nth harmonic components.
- Moreover, the minor hysteresis loops were not taken into consideration properly.
- The computation of the excess loss coefficient through the Epstein frame test is difficult as it does not allow us to identify the difference between the eddy current due to the classical loss and excess loss [17].

### B. Time Dependence of the Flux Density from the Static Analysis

- In finite element method, the flux density B values are computed natively in the x-y coordinates; thus, the actual iron losses are calculated from the flux density components Bx and By values.
- In practice, the flux density B is solved by assuming the two-dimensional approximation where the flux density lies only in (x, y) plane in Cartesian coordinates or (r, φ) plane in cylindrical coordinates and does not depend on the z-axis.
- In FCSMEK, the stator finite element mesh is constructed by multiplying the slot pitch mesh.
- The time dependence of the flux density waveform can be achieved by assuming that the flux density waveform is moving with the fundamental angular frequency ω and time t.
- It is intrinsic within the time dependency reconstruction.

### C. Flux Density Waveform Formation in Static Analysis

- In such a case, the number of stator slots and the number of points in the flux density waveform should be equal.
- In such a way, elements to elements flux density waveform are constructed at each slot pitch interval over two pole pitches to calculate the average iron losses from a closed cycle of the flux density waveform using the Equations 1, and 2.
- As one observation point was specified in each element; hence, no elements were missed.
- Such variation over the solution region in Figure 1 provides the half cycle of the flux density waveform.
- For this reason, the total number of elements in this slot pitch is not equal compared to the other slot pitch meshes.

### A. Studied Synchronous Machine and Simulation Parameters

- The proposed method has been applied to a 12.5 MW salient pole synchronous machine, which consists of 90 stator slots and six rotor poles.
- Thus, the iron losses only in the stator core are computed by the proposed method and the conventional method for a fair comparison.
- The non-linearity of the system equations was solved by the Newton-Raphson iteration method.
- It can be seen in Table I that the operating parameters obtained from the static analysis and the dynamic analysis are in relatively good agreement.
- Therefore, the static computation may overestimate or underestimate these parameters slightly.

### B. Magnetic Flux Density and Harmonics Analysis

- Figure 3 shows the flux contour lines and the flux density distribution of the smallest symmetry section in the static analysis.
- The space distribution of the radial Br and tangential Bφ components of the flux density values in the stator yoke and teeth at given radial positions from the rotor midpoint over one period for both methods are presented in Figure 2.
- The Fourier transformation was performed to analyze the harmonic components present in the flux density waveform in Figure 2.
- Unevenly spaced samples can also be reconstructed as accurately as possible if the average sampling rate follows the Nyquist rule, but the signal may loss one or two samples.
- In the case of dynamic analysis, the most significant harmonic components, i.e., up to 9th are depicted in Figure 4, and all the harmonic components were taken into account for the iron losses calculation.

### D. Computational Cost

- The iron losses computation from a single static field solution is much faster than the time-stepping simulation.
- The iron losses computation from a single static simulation costs 0.512 s; on the other hand, the time-stepping simulation requires 35.07 s for the same number of elements and nodes using an Intel-Xeon 3.4-GHz 16-GB-RAM workstation.
- The computation time for both methods is listed in Table I.

### E. Effect of Damper Windings, Supply Voltage Phase Shift and Rotor Angle Variation on Iron Losses

- The effect of damper windings on the flux density harmonic reduction; consequently, the iron losses computation was studied for the dynamic analysis.
- The conductivity of the damper windings was set close to zero, in the case of dynamic analysis, so that no current can induce in the bars.
- Later, the phase angle φ was varied from 0o to 5o, and the rotor angle θ was varied from 31.9o to 36.9o at the one-degree interval.
- The stator iron losses are decreased slightly at each interval with increasing the supply voltage phase shift φ and the rotor angle θ as illustrated in Figure 7, which is maximum 2.5% compared to the computed iron losses at the initial position, i.e., φ = 0 and θ = 31.9o; hence, no significant effect was found.
- The authors are aware of the importance of computing the iron losses at multiple load points.

### IV. CONCLUSION

- A method of the iron losses computation in a synchronous machine from a static field solution has been proposed in this paper.
- The proposed method has relatively good accuracy compared to the dynamic analysis based method (8.94% maximum difference) and showed a high potentiality of the iron losses computation with a less computation cost over the conventional time-stepping method.
- Moreover, increasing the computational accuracy of the proposed method can substitute the time consuming conventional loss computation method in many applications.
- The accuracy of the developed loss computation technique can be improved by reconstructing the stator mesh, so that, every slot pitch meshes can be taken into account and forming a more densely mesh in order to attain more sampling points, which eventually increase the resolution of the flux density waveform.
- The utmost goal is to improve the accuracy of the Fourier decomposition of the flux density waveform, hence, the iron losses computation, which is their next step.

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