Considerations

In practice, the amount of details in the model must be appropriately limited. Otherwise, the number of input parameters may become impractical, the performance of the model may suffer too much and the risk of unexpected conflicts such as numerical errors, unsolvable algebraic loops and convergence issues may increase. This chapter briefly discusses proba-bly the most significant simplifications of the generator model.

3.9.1 Time step

Differential equations are solved numerically step by step, which may cause very significant calculation errors if the time step is set to be too large. The selection of suitable time step is a trade-off between performance and accuracy. The highest frequency and the lowest time constant of the system can be used as a guideline. One method is to decrease the time step until no significant changes in the behaviour of the model can be observed. In Simulink^®, it is possible to use a variable-step solver and specify the suitable max. step size and tolerance.

The variable-step approach can significantly increase performance without losing accuracy.

The default solving method in the variable-step solver of the used Simulink^® version is Dor-mand-Prince.

3.9.2 Saturation

In ferromagnetic materials, magnetic domains that contain already perfectly aligned mag-netic dipole moments, tend to align with an external magmag-netic field allowing greatly in-creased magnetic field density compared to what is possible in vacuum. In the virgin state of the material, the magnetic domains cancel each other on average when viewed from out-side the object. The realignment of the domains depends on the external field strength, tem-perature and mechanical disturbance. Increasing the magnetic field density by increasing the external field strength becomes more and more difficult when approaching the point where all the domains are aligned, after which the added external field magnetizes only air, which has almost a negligible effect. This is called saturation and from the modelling point view this means that the inductances should actually be dependent on the currents. Furthermore, the saturation as a phenomenon contradicts one of the main principles of the two-axis theory that is the decoupling of direct and quadrature axes. In order to accurately present the satu-ration of a machine, the effect of satusatu-ration usually cannot be neglected. The cross-saturation can be interpreted so that the d-axis currents saturate the q-axis inductances and

vice versa. It should be noted that the cross-saturation characteristics are quite machine spe-cific and can be somewhat unpredictable without FEA results.

Starting point for implementing saturation in the model could be neglecting the effect on leakage inductances and considering only the magnetizing inductances because the flux path from which the leakage inductances are derived include more air, so they are not as heavily affected by the saturation. For a steady state operation, the static magnetizing inductances 𝐿_md,q could be updated based on the operating point and a look-up-table created from in-ductance surfaces 𝐿_md = 𝑓(𝑖_md, 𝑖_mq) and 𝐿_mq= 𝑓(𝑖_md, 𝑖_mq) computed with FEM. Also, the PM flux linkage 𝜓_PM should be adjusted accordingly to the 𝐿_md because the virtual PM current 𝑖_PM remains constant. According to the reference (Kaukonen J., 1999), new dynamic inductance parameters and differential equations are needed for transient operation. The air gap flux linkage differentials with the new parameters in that case are given as follows:

d𝜓_md where 𝐿_Md,q are the magnetizing inductances including an incremental inductance change during a transient, 𝐿_dq,qd are the cross-coupling inductances, 𝐿^d_md,q are the tangent slope dynamic magnetising inductances from a (𝑖_m, 𝜓_𝐦) magnetizing curve and 𝛾 is the angle between 𝒊_m and the d-axis (Kaukonen J., 1999).

3.9.3 Iron losses

In order to obtain more accurate efficiency numbers with the model, the iron losses should be taken into account. The iron losses usually refer to the losses in the stator core. However, from the perspective of more accurate thermal modelling, for example, losses of the similar type in the permanent magnets should also be evaluated.

It seems to be quite common to neglect the iron losses in synchronous machine models. The reason for this is that the iron losses in DOL SMs are typically small in comparison to the copper losses and the fluctuation is relatively small because of the almost constant fre-quency. Also, the iron losses as a phenomenon is not very well compatible with the two-axis theory. It is not completely clear what would be the best way to implement the iron losses to the dynamics. An equivalent iron loss resistance could be used. Another possible simple approach could be calculating the iron losses separately with some variant of Bertotti’s loss model based on the operating point and adding the result as an equivalent virtual friction to the mechanical model.

3.9.4 Harmonics

The harmonic distortion in DOL generators is typically not one of the main concerns at least from the stability point of view, especially when short pitching and skewing are used to cancel the most significant harmonic frequencies. However, one possible starting point for including harmonic frequencies to the model is shown in Figure 3.7.

Figure 3.7 Possible approach to modelling the effect of harmonics.

By simply summing the harmonics to the grid voltage reference, the effect passes down all the way to the electromagnetic torque creating torque ripple. This could enable interesting demonstrations with the two-mass model. However, the reliability of this approach might be a bit questionable. At least one concern is that the effect of stray capacitances become in-creasingly more significant with higher frequencies. The harmonic distortion in the current should become accordingly scaled based on the inductances, but this is not tested or studied in this thesis any further.