Details of the Fixed-speed SRWT Control

The control system that was applied to the fixed-speed SRWT model in this study is shown in Figure 3-25. Several guidelines were used to obtain the appropriate control parameters.

As shown in Table 3-1, the Proportional (P) and Integral (I) current control gains were set to 20 and 40, respectively. These gains were selected and set in the 𝐼𝑞𝑠 Current controller. Noting that the rotor flux and the magnetising

current (𝑖_𝑚) are proportional as shown in Equation (3-56), and considering that the magnetising inductance (𝐿_𝑀) is constant, similar gains of P = 20 and I = 40 were also selected for the 𝜓𝑑𝑟 Flux controller.

𝜓𝑑𝑟 = 𝐿𝑀𝑖𝑚 (3-56)

Figure 3-25 Schematic diagram of the fixed-speed SRWT control system

However, for the speed controller, the P and I gains were calculated based on the bandwidth that was appropriate for the natural frequency (𝑓_𝑛) of the turbine rotation. Using the natural frequency as a reference, the control system can be improved by choosing a suitable cut-off frequency (𝑓𝑐,𝑃𝐼) for the speed

controller. The value of 𝑓_{𝑐,𝑃𝐼} should be less than the natural frequency [149, 150] and should provide a sufficiently low bandwidth [151]. A bandwidth that is too large may cause the signal to be dominated by noise when the control system become too active [149], whereas a bandwidth that is too low may result in the control system becoming nearly inactive because the control system becomes

PI(s) Vds_ref Vqs_ref Frd wr Iqs [wr_ref] [Flux_dr_ref] PI(s) Vqs Vds Vds_ref Vqs_ref PI(s) 𝜓𝑑𝑟∗ 𝜔𝑟∗ PI Current Controller PI Flux Controller PI Speed Controller 𝑖𝑞𝑠∗ 𝑣𝑞𝑠 𝑣𝑑𝑠 𝑣𝑑𝑠∗ 𝑣𝑞𝑠∗ 𝑣𝑞𝑠∗ 𝑣𝑑𝑠∗ 𝑖𝑞𝑠 𝜔𝑟 𝜓𝑑𝑟 SCIG Decoupling circuit

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too sensitive to the disturbances [151]. The bandwidth of the speed controller can be derived as follows:

1- To determine the 𝑓𝑛 of the turbine rotor, it is common to consider the

lower frequency component of the frequency spectrum signal of the turbine vibration [152]

.

As can be seen in the power spectral density function versus the frequency signal of the Carter turbine, which was taken from [131, 153] and [115] and is depicted in Figures 3-26(a) and (b), respectively, the 𝑓_𝑛 of the rotor is approximately 4 Hz. Hence, the natural bandwidth (𝜔𝑛) of the turbine rotor is approximately 2𝜋𝑓𝑛= 25.13

rad/s.

(a)

(b)

Figure 3-26 Graphs of power spectral density versus vibration frequency of the Carter wind turbine (a) taken from [131, 153] (b) taken from [115].

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For the figure shown in Figure 3-26(a), the fast Fourier transform (FFT) spectrum of the accelerometer reflects to the generator frequency derived from the measured three-phase current signals. The generator frequency was then used to infer the blade rotor frequency as the high-speed shaft rotation frequency can be derived from the generator frequency data. Meanwhile, in Figure 3-26(b), the power spectrum reflects the drive train vibration frequency of the Carter wind turbine (as considered in Figure 3- 26(a)) in which the vibration frequency was derived using the data obtained for the variation of power in response to wind speed. Interestingly, from these figures, it can be observed that both power spectrums show an identical first power spectral peak component with the peak frequency of 4 Hz. This frequency was therefore assumed to be the natural frequency of the blade rotor.

2- The cut-off frequency (𝑓_{𝑐,𝑃𝐼}) of the speed controller should be less than _𝑝1 of 𝑓_𝑛 as shown in Equation (3-57) [154]

𝑓𝑐,𝑃𝐼 <𝑓_𝑝𝑛 (3-57)

where 𝑝 should be greater than 5. If 6 is chosen for 𝑝, 𝑓𝑐,𝑃𝐼 is equal to 0.67

Hz, which can be considered as the cut-off frequency of the speed controller. Hence, the 𝜔_{𝑐,𝑃𝐼} of the speed controller is approximately 4.21 rad/s. where 𝜔_{𝑐,𝑃𝐼}< 𝜔_𝑛. Good disturbance attenuation can be achieved using this rule.

3- Equation (3-58) [155] was used to verify that the 𝑓_𝑛 of the Carter turbine is approximately 4 Hz

𝑓_𝑛 = _2𝜋1 √𝐾𝑠

𝐽𝑟 (3-58)

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The low and high speed shafts of the Carter turbine were measured during maintenance. The shaft was divided into several sections. Each section has a different length and diameter. The stiffness of the low speed shaft is estimated using Equation (3-59)

𝐾 = 𝐺 ∑ 𝑑𝑖4

32𝐿𝑖

𝑛

𝑖=1 (3-59)

where 𝑑 is the shaft’s diameter, 𝐺 is the element’s modulus of rigidity (90 Gpa for steel) and 𝐿 is the element’s length. Using Equation (3-59), the estimated stiffness is equal to 2.35 x 105 _{N.m/rad. This value was confirmed by Harson}

[115], who calculated a stiffness of 2.36 x 105 _{N.m/rad. Using this value of 𝐾, the}

natural frequency of the turbine rotor is 4 Hz.

𝑓𝑛=_2𝜋1 √_𝐽𝐾 𝑟= 1 2𝜋√ 2.35𝑥105 372 = 4 𝐻𝑧

The final values of the P and I gains for the speed controller are thus equal to 4.6927 and 10.0396, respectively, using a phase margin of 65º from the rule in Equation (3-57). Using the PI gains that were selected for the three controllers as depicted in Figure 3-25, the generated power, flux, stator current and speed when the SRWT model is run with realistic wind speeds for 600 seconds are shown in Figure 3-27. Though no actual time series data can be compared with the simulation results in this figure, observations have made based on the theoretical approach.

Figure 3-27 shows that all of the signals behave normally. The generated power behaves as expected between zero and 22 kW, and the other simulated signals track their reference signals closely. Note that, to make the machine operate efficiently, the rotor flux should work at rated flux. As shown in Figure 3- 27(b), the simulated rotor flux, Frd is successfully maintained to work around the reference rotor flux. The reference rotor flux can be calculated using Equations (2-52) or (2-56). Because the rotor flux can be maintained around the reference rated flux, the direct-axis component of stator current also is limited at a constant current. See Equation 3-56. Using vector control, the magnitude and the phase angle of the stator current could be varied to control the actual stator current. Thus, the actual stator current can be controlled as required, by

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controlling the quadrature-axis component; meanwhile maintaining the direct- axis current component. As depicted in Figure 3-27(c), the simulated quadrature-axis of the stator current varies smoothly as estimated by its reference. The reference of the quadrature-axis component of the stator current can be calculated using Equations (2-57).

Figure 3-27 Generated electrical power, flux, stator current and speed responses when the fixed-speed SRWT system model is controlled by the 𝜓_𝑑𝑟

flux controller, the 𝐼_𝑞𝑠 current controller and the speed controller

0 100 200 300 400 500 600 0 1 2 x 104 P g ( W a tt )

(a) Generated power

0 100 200 300 400 500 600 0.5 1 1.5 F rd ( W b )

(b) Reference and simulated Flux

0 100 200 300 400 500 600 0 100 200 Iq s ( A m p )

(c) Reference and simulated stator current

0 100 200 300 400 500 600 120 130 140 Time (s) w r (r a d /s )

(d) Reference and simulated generator speed Reference Frd Frd Reference Iqs Iqs Reference wr wr

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Figure 3-27(d) depicts the electrical speed of the generator. It can be seen that the simulated electrical rotor speed can be controlled to rotate at the rated speed of approximately 1305 rpm or 136.65 rad/s. This reference electrical rotor speed can be calculated using Equations (2-23) and (2-26), respectively. The positive and negative deviations of the electrical rotor speed from its reference are considered to be due to the slip factor under normal conditions.

3.7 Chapter Summary

This chapter explained the steps involved in the development of a model of the stall-regulated, fixed-pitch, fixed-speed Carter wind turbine and provided a detailed explanation of the development of the SIMULINK model of the fixed- speed SRWT model from the first sub-model to the end sub-model. The method used to estimate the unknown parameters for the drivetrain and SCIG models and the approach specified to design the controllers’ gains were also explained, and the method used to improve the complete fixed-speed SRWT model with a dynamic stall algorithm was described. Finally, the dynamic system of this turbine was validated using the manufacturer’s specifications for operations at a fixed speed (50 Hz) and using data measured during normal operations (at a frequency of 43.5 Hz). The good responses from the developed fixed-speed SRWT with the dynamic stall model under these two operating conditions greatly increased the confidence in the validity of the model. The fixed-speed SRWT with the dynamic stall model will be then used as the complete SRWT model in later chapters with the addition of designed IVC and SLIVC model algorithms to develop a stall-regulated variable speed wind turbine (SRVSWT) model.

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4

Description of Controllers, Tuning Procedure and

In document Internal Model Control (IMC) design for a stall-regulated variable-speed wind turbine system (Page 121-127)