## Results

The q-axis transfer function Yq(s) of the electromagnetic part of the Lagerwey LW-50/750 generator has been identified using an ARX model structure. A third order model turns out to be sufficient. The q-axis parameters (i.e. Lq(s) and Rs) of measurement M1 are deduced algebraically from Yq(s) using Eq. (4.10). From this equation it can be observed that the stator-winding resistance Rs is equal to the DC gain of the inverse of the quadrature-axis transfer function Yq. The resulting value of Rs is

Determining the values for Rs of the other two measurements gives similar results. The resulting mean value of Rs is 57.92 mO with a standard deviation of 0.02 mO. In addition, it can be concluded that the stator-winding temperature has not changed significantly during the experiments. It must be noted that, during normal operation, the temperature of the copper windings will increase from about 20° (temperature at which the MSR-measurements are performed) to about 100°. As a consequence, the stator-winding resistance will increase from 57.9 mQ to about 77.8 mO using the temperature resistance coefficient of copper: acu = 4.3 • 10-3 K-1. Interestingly, the value of the stator-winding resistance calculated from the machine design is equal to 81.8 mO [224]. Consequently, for time-domain simulation of the Lagerwey LW-50/750 generator, the estimated value of Rs have to be corrected for the differences in temperature between measurement and actual operation.

Finally, the resulting quadrature-axis synchronous inductance is given by

Lq q with the following roots s 2 + 157.3 • s + 2766

and DC-gain

Calculated was a value of 7.7 mH [224].

The structure of the symmetric d-axis transfer function matrix (i.e. common denominator) and the high signal-to-noise ratio calls for a MIMO ARX model structure. Numerical difficulties in deducing the d-axis parameters from the estimated transfer functions, however, forced us to use the following three-step approach:

• Step 1: Use the modified step-response test data to determine the stator-winding resistance Rs and the field winding resistance Rf ;

• Step 2: Generate the direct-axis stator flux 'd and direct-axis field winding flux if by running a simulation with the measured ud, uf, id, and if acting as input. The generated flux signals as well as the measured currents of measurement M18 are shown in Fig. 4.26;

• Step 3: Import the generated flux signals into SITB and identify the MIMO (multiple-input-multiple-output) ARX transfer function between the fluxes'd, if and the currents id, if (see Eq. (3.107)).

It should be noted that converting the individual transfer functions in Eq. (3.107 to state-space prior to combining the transfer functions did not solve the problem. Recall that the state-space representation is best suited for numerical computations.

For both inputs id and if, the percentage of the variations in the fluxes i d, i f that is reproduced by the (fourth order) model is larger than 99.5% (identification data set). The common denominator of the MIMO transfer function between the fluxes i d, i f and the currents id, if is given by

Ldo(s) • Lfo(s) - L2fdo(s) = s4 + 672 • s3 + 756.9 • s2 + 1795.1 • s + 1197.4

inputs

outputs inputs outputs

Figure 4.26: Left figures: simulated inputs ^d and ^f as function of time. Right figures: Measured outputs id and if as function of time from measurement M18.

with the following roots

«1 = |
-1.2878 |

«2 = |
-1.6416 |

«3 = |
-10.5381 |

«4 = |
-53.7496 |

The aforementioned q-axis parameters and d-axis transfer function matrix, as well as the field-winding resistance, are implemented in the block diagram shown in Fig. 4.22. The resulting inputs and outputs of the model are shown in Fig. 4.27 for a validation data set. Obviously, the simulated data matches the measured data very well. Observe that uf is not equal to zero because the short-circuit is not perfect due to the slip-rings. Fig. 4.28 shows the outputs once again, but now on a reduced time scale. The percentage of the output variations that is reproduced by the model is, in the case of the identification data set, 99.74%, 99.47%, and 99.89% for id, if, and iq respectively. For the validation data set the percentage is 99.16%, 92.78%, and 99.76% for id, if, and iq respectively. In addition, the quality of the model is also checked by examining the cross correlation function between inputs and output residuals. In both cases an almost zero cross-covariance exists.

inputs outputs

outputs

Figure 4.27: Left figures: measured inputs Ud, Uf and uq as function of time. Right figures: outputs id, if and iq as function of time. Solid limes: measured data (validation data set), and dashed-lines: simulated data.

outputs outputs

Figure 4.28: Outputs id, if and iq as function of time (zoom of right figures in Fig. 4.27). Solid limes: measured data (validation data set), and dashed-lines: simulated data.

Figure 4.28: Outputs id, if and iq as function of time (zoom of right figures in Fig. 4.27). Solid limes: measured data (validation data set), and dashed-lines: simulated data.

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