4. The electrical machine (PMSG)
4.4 Simulations and results obtained
The simulations have been run in MATLAB-Simulink, with the Hydrodynamic WEC model of the simulations of the previous chapters and the Simulink model illustrated in figure 46 (which allows to control the PMSG). A general overview of the WEC model simulated is shown in figure 37. The control strategy implemented is the passive loading control with two constraints: the power limit (100 kW) and the torque limit (the module of the torque is limited to 850 Nm). The data of the PMSG, that have been used, are shown in table 8. The low, medium and high energy sea state (for 900 s) have been tested, each of them with the BL coefficient which involves the highest average mechanical power. The damping coefficients are shown in the next table and are the same of table 7.
Table 9 - Damping coefficients used in the simulations
BL coefficient [kg/s] Energy sea state
400000 low
4300000 medium
3100000 high
An ideal three phase inverter (with ideal switches) has been used to control the generator with the switching frequency (fs) equal to 2000 Hz. In this paragraph the efficiency of the electrical machine is calculated taking into account: the copper losses, the iron losses and the mechanical losses. The Simulink models used to calculate the losses of the generator are illustrated in appendix B. Subsequently the equations, that have been used, are shown and explained. From [29] equation (57) is obtained, it allows to calculate the electrical power (in real time) produced by the generator taking into account only the copper losses ( ).
(57) The iron and mechanical losses are not included in the previous equation and have to be calculated separately. As there is written in [30] [31] the iron losses are usually calculated considering the flux density (B) sinusoidal in the core material ( B is not perfectly sinusoidal, therefore it has been done an approximation that should not significantly affect the results). piron is the total iron-loss density (in real time) considering that the flux density varies sinusoidally with the angular velocity of the generator ωm [30]:
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(58) The total iron-loss density is composed by the hysteresis loss density ( ) and by the eddy-current loss density ( ). The following coefficients depend on the lamination material: β, and . They are respectively the Steinmetz, the hysteresis and eddy current constants. Typical values using silicon iron laminations, with given in rad/s, are in the following ranges [30]: = 40-55, β = 1.8-2.2 and = 0.04-0.07.
Finally the total iron-loss (Piron) is obtained multiplying the volume of the iron machine for the total iron-loss density. The mechanical losses (Padd ) (in real time) are calculated as in [32]:
[W] (59) The losses are directly proportional to the nominal apparent power (An) and to the root of the velocity of the generator expressed in revolutions per minute (n). c is the mechanical constant and it has to be in the range 0.4-0.6 [32], cosφ is the load factor.
As it can be seen in equation (59), differently of the iron losses, the mechanical losses are obtained directly from the equation and they are represented in W. In table 10 the parameters used in the model to calculate the generator losses are shown.
Table 10 – Parameters of the PMSG losses.
At this point the electrical power Pel (in real time) produced by the PMSG can be calculated subtrahend to the the iron and mechanical losses:
[W] (60)
Hence the total losses Ploss (in real time) of the PMSG can be calculated and they are equal to:
[W] (61)
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The total losses of the generator are equal to the total losses of the transformation from the mechanical power to the electrical power, because the inverter used in the Simulink-model is ideal. As it has been said at the beginning of this paragraph the three representative energy sea states have been tested, each of them for 900 s.
During this time the average value of the electrical power (Pel-average), of the mechanical power (Pmecc-average) and of the total losses (Ploss-average) has been measured. Finally the average efficiency (ηaverage ) in 900 s of simulation can be calculated for each energy sea state using the following equation:
(62) The results of the simulations are shown in the next table.
Table 11 – Results of the simulations simulations the electrical machine was considered ideal capable to follow the reference torque perfectly in each instant of the simulation). The average mechanical power obtained is more or less the same in the low energy sea state ( about 17 kW) instead, in the high and medium energy sea state, is decreased about of the 3 % . This is due to the fact that, as it has been explained in previous paragraph, the PMSG in the field-weakening operation region doesn’t follow perfectly the torque reference.
With the aim to respect the circle of the limit of current ( equation (43)), when during the 900 s of simulation, the module of the torque of the generator is equal to the maximum available torque in the field-weakening operation region which is lower than the torque reference (Tref). The reduction of the average mechanical power is considerable (3 %) in the medium and high energy case, and almost null in the low energy case because the generator works very few times in the
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field-weakening region in the low energy sea state. An increment of the energy content in the sea state implies an increase of the average efficiency ( ) of the PMSG. is equal to 0.881 in the low energy sea state and achieves the value 0.904 and 0.908 respectively in the medium and high energy sea state. This can be justified observing figure 52, wherein the efficiency map of a common PMSM (permanent magnet synchronous motor) with SMPMs (surface mounted permanent magnets) and a wide field-weakening region is illustrated.
Figure 52 – [33] Example of a PMSM efficiency map with SMPMs and a quite high overspeed ratio
Obviously the values of efficiency, torque and speed in figure 52 should be ignored because they are referred to a different electrical machine compared that one used in the simulations. But the way that varies the efficiency in the plane T- n (torque- rotor velocity) is similar for each PMSM with SMPM and a wide field-weakening region.
It is known that the generator in the medium and high energy sea state works most of the time in the field-weakening operation region on the line of the maximum available torque. As it can be seen in figure 52 on that line the efficiency is not the maximum achievable, but is quite high. Instead in the low energy sea state the generator works very few time in the field-weakening operation region and almost always in the region where the rotor velocity is lower than the nominal. In this area the efficiency is lower especially if the rotor velocity is very low or the torque is quite high. This involves a lower average efficiency for the low energy case.
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4.5 Final considerations
The case study simulated (shown in figure 37) is convenient to be applied if, in the scatter diagram of the location, the medium energy sea states and the high energy sea states prevail over the low energy ones. This is justified by the peak to average power ratios of table 7. The average efficiency, obtained using the damping coefficients of table 9 for each energy sea state, is higher in the medium and high energy sea state compared the low one. This is another advantage in favour of a location with many high and medium energy sea states. The conversion from the mechanical to the electrical power has been made successfully up to the DC side of the inverter that control the PMSG. The next step of this thesis regards the modelling of the grid connection. It allows to connect the wave-to-wire model to the electric grid and then it allows to produce electric energy by the sea waves to the electric grid.
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