Grid connection with a phase opposition

In the incident, a PMSG was reportedly connected to a grid with a phase difference of 180 degrees between the grid voltage and the induced emf. This is a quite extreme transient event because the momentary voltage across the stator resistance is at maximum with a given ro-tational speed. The reported machine parameters are shown in Table 4.1.

Table 4.1 Reported parameters of the PMSG that was connected with a phase difference of 180 degrees.

Quantity SI unit value Per unit value

Rated active power 𝑃_n 600 kW

Rated frequency 𝑓n 50 Hz

Rated rotational speed 300 rpm

PM induced voltage (line-to-line) 𝐸PM 430 V

Rated current 𝐼_n 890 A

DC resistance 𝑅_s 3.642 mΩ 0.0140

d-axis synchronous inductance 𝐿d 345 μH 0.4177

q-axis synchronous inductance 𝐿_q 360 μH 0.4359

Sustained short-circuit current 𝑖ssc 3077 A

Sub-transient short circuit current 𝑖_s^′′ 8235 A

Sub-transient reactance 𝑋_d^′′ 0.030 Ω 0.1156

Sub-transient time constant 𝜏_d^′′ 0.05 s

Damper winding reactance referred to stator 0.081 Ω 0.3122

Damper winding resistance referred to stator 8.97 mΩ 0.0346

Generator moment of inertia 𝐽 157 kgm²

The given parameter list is not entirely complete and directly compatible for the created model. The stator leakage inductance and separate damper winding parameters for the d and q axes are not available. Also, it seems that the 𝑖_ssc is a peak value rather than rms value considering the d-axis total inductance can be approximated (𝑅_𝑠 neglected) as

𝐿_d ≈

If the reported 𝑖_ssc was rms value, the 𝐿_d would contradict too much. On the other hand, the 𝑋_d^′′ seems to correspond to the 𝑖_s^′′ as follows Some manipulation and possibly trial and error are required in order to reproduce the grid connection incident. The problem is that the parameters listed correspond to the test and the conditions under which they were determined. The accuracy of the same parameters in dif-ferent situations vary. Moreover, the error margins of the parameters are unknown. For ex-ample, if a sudden short-circuit test were used, it is not clear whether temperature corrections were made and whether the slight speed fluctuation in the test were taken into account.

According to the reference (Kinnunen J., 2007), the sub-transient inductances of the model can be expressed as a series connection of stator leakage and parallel connection of magnet-izing and damper leakage inductances: sub-transient time constants can be calculated as the ratio of inductance to resistance in a closed-circuit stator winding:

The damper winding inductances are calculated from the listed damper winding reactance with a 50 Hz frequency

𝐿_D,Q = ^{0.081 Ω}

2π∙50 Hz= 258 µH. (4.7)

If the stator leakage is solved from (4.3), it is obtained that 𝐿_sσ= 91.37 µH, 𝐿_md= 253.63 µH, 𝐿_mq= 268.63 µH, 𝐿_Dσ= 258 µH − 253.63 µH = 4.20 µH and 𝐿_Qσ= 258 µH − 268.63 µH = −10.80 µH. As the 𝐿_Dσ is quite small and the 𝐿_Qσ becomes negative, in this case the damper winding total inductances are set equal to the magnetizing inductances, i.e. 𝐿_Dσ= 𝐿_Qσ = 0. This might be a bit coarse assumption, but must suffice because no more detailed information is available and using the sub-transient constant equation also produce negative inductances. Now it is obtained that 𝐿_sσ= 𝐿^′′_d =^{0.030 Ω}

2∙π∙50 = 95.49 µH, 𝐿_md = 345 µH − 95.49 µH = 249.51 µH, 𝐿_mq = 360 µH − 95.49 µH = 264.51 µH.

Without further information the same damper winding resistance could be used on the d and q axes, but according to the manufacturer, the d-axis damper winding resistance is most likely higher that the q-axis resistance. Based on a brief testing in which the damper winding resistances were varied around the initial value 𝑅_D,Q = 8.97 mΩ while comparing the recorded and simulated waveforms, the resistances were set to 𝑅_Q = 8.97 mΩ and 𝑅_D = 9.60 mΩ. The calculated time constants according to (4.5) and (4.6) are then 𝜏_d^′′ = 0.0072 s, which differs from the listed value quite a lot, and 𝜏_q^′′ = 0.0078 s. With regard to the stator resistance, skin and proximity effects make the AC-resistance larger than the DC-resistance.

The significancy of these effects depend on the frequency, the conductors used and their arrangement. In this case, the effect is assumed negligible, i.e. the AC resistance is equal to the DC resistance.

The turbine moment of inertia in this demonstration is set to 60 % of the inertia of the generator so the total moment of inertia is 0.6 ∙ 157 kgm²+ 157 kgm² = 251.2 kgm². The total friction is set to 3 Nms/rad. The reported grid voltage and frequency were 405 V line-to-line and 50 Hz. In the simulation the temperature is set to 20 °C because the machine is probably cool when synchronized. The recorded incident and simulation of the same case is shown in Figure 4.1.

Figure 4.1 Recorded (top) and simulated PMSG grid connection with a phase difference of 180 degrees between the grid voltage and the induced emf. 𝐿sσ= 95.49 µH (0.1156 pu.), 𝐿md= 249.51 µH (0.3021 pu. ), 𝐿_mq= 264.51 µH (0.3202 pu. ), 𝐿_Dσ= 𝐿_Qσ = 0, 𝐽_tot= 251.2 kgm², 𝐷_tot= 3 Nms/rad, 𝑅_s= 0.0036 Ω (0.0140 pu. ), 𝑅_D= 9.60 mΩ (0.0370 pu. ) 𝑅Q= 8.97 mΩ (0.0346 pu. ).

It can be seen, that both the recorded and simulated current have a similar pattern where the amplitudes of the currents vary periodically. In the recorded figure the scale is limited be-cause current probes with a rated value of 2 kA were used. Comparing the peaks in the beginning is not possible. In the simulation the first two swings are not as clearly distin-guishable from each other. The simulation attenuates roughly as fast as the recorded wave-form even though the calculated time constant differs from the listed value. Unfortunately,

the sudden short circuit test curves of the machine are not available, so it is not possible to verify the listed value.

Even better resemblance could likely be achieved by fine tuning the parameters, but it is not worth it with the limited available information. Also, it should be remembered that the model has simplifications. For example, the recorded waveform is distorted by the harmonics and the simulated is not. Also, saturation, temperature changes during the event and grid prop-erties affect the waveform.