5. Complex power and torque under unbalanced conditions

The active and reactive power can be obtained from electrical quantities seen from a stationary reference frame:

$$P\_s = \frac{3}{2} \text{Re} \left( \overrightarrow{v\_s} \,\, \overrightarrow{i\_s} \right) = \frac{3}{2} \left( \upsilon\_{s\alpha} \dot{\imath}\_{s\alpha} + \upsilon\_{s\beta} \dot{\imath}\_{s\beta} \right) \tag{41}$$

Tem <sup>¼</sup> <sup>P</sup> ωs

6. Simulation results

δQs

and air density (ρ = 1.25 kg\m3

= 128 � 103 and <sup>δ</sup>Tem = 811.

speed of the wind turbine Pr ≈ sPs.

the optimal value without measuring wind speed:

tions, the following condition must be met Qs<sup>12</sup> = � Qs21.

Ps<sup>11</sup> þ Ps<sup>12</sup> � Ps<sup>21</sup> � Ps<sup>22</sup> þ

The terms Ps<sup>12</sup> and Ps<sup>21</sup> are the cause of oscillation in torque and power when an unbalanced dip occurs. Since the condition for canceling torque oscillations (Ps<sup>12</sup> = Ps21) is opposite to the condition for canceling active power oscillations (Ps<sup>12</sup> = � Ps21), it is not possible to cancel both at the same time. It is preferable to cancel torque oscillations; otherwise the mechanical components may be severely damaged. On the other hand, to cancel reactive power oscilla-

To test the controller, a DFIG was simulated using the parameters displayed in Table 1. The blades model surface is shown in Figure 1 with gearbox ratio (η = 85.8) rotor radius (r = 40 m)

ratio Λopt = 7.9533 and the maximum power coefficient is Cp ,max = 0.4109. The pitch controller is ideal chopping the extracted aerodynamic power to the nominal power (2 MW) for wind speed above the nominal value. Considering that the maximum switching frequency of the converter is 7000 Hz; from Figure 6 the desired hysteresis width is δ = 90.04A; therefore,

The following electromagnetic torque reference is used for maintaining the tip-speed ratio at

Λ2 optη<sup>3</sup>

The wind speed profile is shown in Figure 7 II, the wind speed was taken from real measurements reported by the Department of Wind Energy, Technical University of Denmark [19] with

The power extracted by the blades is shown in Figure 7 III, during the high wind speed periods, the ideal pitch controller maintains the extracted power at the nominal value of 2 MW, while the mechanical speed is controlled during the rest of the time to optimize power extraction as shown in Figure 7 IV. The rotor converter nominal power limits the operational

The references are followed even under unbalanced grid conditions, a two-phase voltage dip is simulated at the terminals of the electric machine, the voltage dip is 20% of the nominal value. In Figure 7 I, it is displayed the detail of the voltage dip. The voltage dip starts at 95 seconds and ends at 98 seconds of the simulation, the time axis in Figure 7 I is chopped from 95.05 to

Cp,max

ω2

<sup>m</sup> (51)

1 <sup>2</sup> πρR<sup>5</sup>

Tref ¼ �

On the other hand, the reactive power reference is maintained at zero.

values oscillating in all the operational range of the wind speed.

3Rs <sup>2</sup> <sup>ı</sup> ! s2 2 � ı ! s1 

<sup>2</sup> (50)

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). The nominal wind speed of 12 m/s, the nominal tip speed

$$Q\_s = \frac{3}{2} \text{Im} \left( \overrightarrow{v\_s} \, \overrightarrow{i\_s} \right) = \frac{3}{2} \left( v\_{s\beta} \dot{i}\_{s\alpha} - v\_{s\alpha} \dot{i}\_{s\beta} \right) \tag{42}$$

where the operator x ! is the complex conjugate.

In case of unbalanced conditions, the symmetrical components methods can be used for simplifying analysis, since zero sequence does not produce complex power, only positive and negative sequences are analyzed:

$$
\overrightarrow{\boldsymbol{\upsilon}}\_{s} = \overrightarrow{\boldsymbol{\upsilon}}\_{s\_1} + \overrightarrow{\boldsymbol{\upsilon}}\_{s\_2} = \boldsymbol{\upsilon}\_{s\alpha 1} + \boldsymbol{\upsilon}\_{s\alpha 2} + j(\boldsymbol{\upsilon}\_{s\circ 1} + \boldsymbol{\upsilon}\_{s\circ 2}) \tag{43}
$$

$$
\overrightarrow{i\_s} = \overrightarrow{i\_{s\_1}} + \overrightarrow{i\_{s\_2}} = i\_{s\alpha 1} + i\_{s\alpha 2} + j(i\_{s\beta 1} + i\_{s\beta 2}) \tag{44}
$$

Substituting (43) and (44) in Eqs. (41) and (42) yields:

$$P\_s = \overbrace{\frac{3}{2} \left(v\_{\text{sa1}}i\_{\text{sa1}} + v\_{\text{s\S1}}i\_{\text{s\S1}}\right)}^{P\_{d1}} + \overbrace{\frac{3}{2} \left(v\_{\text{sa1}}i\_{\text{sa2}} + v\_{\text{s\S1}}i\_{\text{s\S2}}\right)}^{P\_{d2}} + \overbrace{\frac{3}{2} \left(v\_{\text{sa2}}i\_{\text{sa1}} + v\_{\text{s\S2}}i\_{\text{s\S1}}\right)}^{P\_{d2}} + \overbrace{\frac{3}{2} \left(v\_{\text{sa2}}i\_{\text{sa2}} + v\_{\text{s\S2}}i\_{\text{s\S2}}\right)}^{P\_{d2}} \tag{45}$$

$$Q\_s = \overbrace{\frac{3}{2} \left( v\_{s\text{fl}} i\_{\text{sa1}} - v\_{\text{sa1}} i\_{\text{g1}} \right)}^{Q\_{\text{att}}} + \overbrace{\frac{3}{2} \left( v\_{s\text{fl}} i\_{\text{sa2}} - v\_{\text{sa1}} i\_{\text{g2}} \right)}^{Q\_{\text{att}}} + \overbrace{\frac{3}{2} \left( v\_{s\text{fl}} i\_{\text{sa1}} - v\_{\text{sa2}} i\_{\text{g1}} \right)}^{Q\_{\text{att}}} + \overbrace{\frac{3}{2} \left( v\_{s\text{fl}} i\_{\text{sa2}} - v\_{\text{sa2}} i\_{\text{g2}} \right)}^{Q\_{\text{att}}} \tag{46}$$

On the other hand, electromagnetic torque can be obtained using the well-known equation:

$$T\_{em} = \frac{3P}{2} \text{Im} \left( \overrightarrow{\lambda\_s} \,\, \overrightarrow{l\_s} \right) \tag{47}$$

Using the symmetrical components theory, an unbalance condition can be modeled with invariant positive and negative sequence components; therefore, at steady state, the derivate term of Eqs. (4) and Eq. (5) are zero leading to the following positive and negative stator flux components:

$$
\overrightarrow{\dot{\lambda}}\_{s1} = \frac{\overrightarrow{\upsilon}\_{s1} - R\_s \overrightarrow{\iota}\_{s1}}{j\omega\_s}; \qquad \overrightarrow{\dot{\lambda}}\_{s2} = \frac{\overrightarrow{\upsilon}\_{s2} - R\_s \overrightarrow{\iota}\_{s2}}{-j\omega\_s} \tag{48}
$$

Substituting Eqs. (43) and (44) and (48) in (47):

$$T\_{\rm em} = \frac{3P}{2\omega\_s} \text{Re} \left[ \overrightarrow{\overline{v}\_{s1}} \overline{\overline{i\_{s1}}} + \overrightarrow{\overline{v}\_{s1}} \overline{\overline{i\_{s2}}} - \overrightarrow{\overline{v}\_{s2}} \overline{\overline{i\_{s1}}} - \overrightarrow{\overline{v}\_{s2}} \overline{i\_{s2}} - R\_s \left( \left| \overrightarrow{i\_{s1}} \right|^2 - \left| \overrightarrow{i\_{s2}} \right|^2 \right) \right] \tag{49}$$

Comparing Eq. (50) with Eq. (45), it is easy to see that the same terms appear in both equations:

$$T\_{em} = \frac{P}{\omega\_s} \left[ P\_{s11} + P\_{s12} - P\_{s21} - P\_{s22} + \frac{\Im R\_s}{2} \left( \left| \begin{array}{c} \overrightarrow{\mathbf{r}}\_{s2} \end{array} \right|^2 - \left| \begin{array}{c} \overrightarrow{\mathbf{r}}\_{s1} \end{array} \right|^2 \right) \right] \tag{50}$$

The terms Ps<sup>12</sup> and Ps<sup>21</sup> are the cause of oscillation in torque and power when an unbalanced dip occurs. Since the condition for canceling torque oscillations (Ps<sup>12</sup> = Ps21) is opposite to the condition for canceling active power oscillations (Ps<sup>12</sup> = � Ps21), it is not possible to cancel both at the same time. It is preferable to cancel torque oscillations; otherwise the mechanical components may be severely damaged. On the other hand, to cancel reactive power oscillations, the following condition must be met Qs<sup>12</sup> = � Qs21.

## 6. Simulation results

Ps <sup>¼</sup> <sup>3</sup> 2 Re vs ! ı<sup>s</sup> ! � �

Qs <sup>¼</sup> <sup>3</sup> 2 Im vs ! ı<sup>s</sup> ! � �

! is the complex conjugate.

vs ! <sup>¼</sup> <sup>v</sup> ! <sup>s</sup><sup>1</sup> þ v !

> is ! ¼ i ! <sup>s</sup><sup>1</sup> þ i !

Substituting (43) and (44) in Eqs. (41) and (42) yields:

þ 3

þ 3

λ ! <sup>s</sup><sup>1</sup> <sup>¼</sup> <sup>v</sup> ! <sup>s</sup><sup>1</sup> � Rs ı ! s1

Substituting Eqs. (43) and (44) and (48) in (47):

Re v ! <sup>s</sup><sup>1</sup> ı ! <sup>s</sup><sup>1</sup> þ v ! <sup>s</sup><sup>1</sup> ı ! <sup>s</sup><sup>2</sup> � v ! <sup>s</sup><sup>2</sup> ı ! <sup>s</sup><sup>1</sup> � v ! <sup>s</sup><sup>2</sup> ı !

Tem <sup>¼</sup> <sup>3</sup><sup>P</sup> 2ω<sup>s</sup> <sup>2</sup> vsα<sup>1</sup>isα<sup>2</sup> <sup>þ</sup> vsβ<sup>1</sup>isβ<sup>2</sup> � � zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Ps<sup>12</sup>

<sup>2</sup> vsβ<sup>1</sup>isα<sup>2</sup> � vsα<sup>1</sup>isβ<sup>2</sup> � � zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Qs<sup>12</sup>

jω<sup>s</sup>

where the operator x

68 Adaptive Robust Control Systems

Ps <sup>¼</sup> <sup>3</sup>

Qs <sup>¼</sup> <sup>3</sup>

components:

negative sequences are analyzed:

<sup>2</sup> vsα<sup>1</sup>isα<sup>1</sup> <sup>þ</sup> vsβ<sup>1</sup>isβ<sup>1</sup> � � zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Ps<sup>11</sup>

<sup>2</sup> vsβ<sup>1</sup>isα<sup>1</sup> � vsα<sup>1</sup>isβ<sup>1</sup> � � zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Qs<sup>11</sup>

¼ 3

¼ 3

In case of unbalanced conditions, the symmetrical components methods can be used for simplifying analysis, since zero sequence does not produce complex power, only positive and

> þ 3

þ 3

<sup>2</sup> Im <sup>λ</sup><sup>s</sup> ! ıs ! � �

Using the symmetrical components theory, an unbalance condition can be modeled with invariant positive and negative sequence components; therefore, at steady state, the derivate term of Eqs. (4) and Eq. (5) are zero leading to the following positive and negative stator flux

; λ

Comparing Eq. (50) with Eq. (45), it is easy to see that the same terms appear in both equations:

! <sup>s</sup><sup>2</sup> <sup>¼</sup> <sup>v</sup> ! <sup>s</sup><sup>2</sup> � Rs ı ! s2

<sup>2</sup> � � � �

�jω<sup>s</sup>

<sup>s</sup><sup>2</sup> � Rs ı

! s1 � � � � � � 2 � ı ! s2 � � � � � �

On the other hand, electromagnetic torque can be obtained using the well-known equation:

Tem <sup>¼</sup> <sup>3</sup><sup>P</sup>

<sup>s</sup><sup>2</sup> ¼ vsα<sup>1</sup> þ vsα<sup>2</sup> þ j vsβ<sup>1</sup> þ vsβ<sup>2</sup>

<sup>s</sup><sup>2</sup> ¼ isα<sup>1</sup> þ isα<sup>2</sup> þ j isβ<sup>1</sup> þ isβ<sup>2</sup>

<sup>2</sup> vsα<sup>2</sup>isα<sup>1</sup> <sup>þ</sup> vsβ<sup>2</sup>isβ<sup>1</sup> � � zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Ps<sup>21</sup>

<sup>2</sup> vsβ<sup>2</sup>isα<sup>1</sup> � vsα<sup>2</sup>isβ<sup>1</sup> � � zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Qs<sup>21</sup>

<sup>2</sup> vsαis<sup>α</sup> <sup>þ</sup> vsβis<sup>β</sup>

<sup>2</sup> vsβis<sup>α</sup> � vsαis<sup>β</sup>

� � (41)

� � (42)

� � (43)

� � (44)

<sup>2</sup> vsα<sup>2</sup>isα<sup>2</sup> <sup>þ</sup> vsβ<sup>2</sup>isβ<sup>2</sup> � � zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Ps<sup>22</sup>

<sup>2</sup> vsβ<sup>2</sup>isα<sup>2</sup> � vsα<sup>2</sup>isβ<sup>2</sup> � � zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Qs<sup>22</sup>

(45)

(46)

(47)

(48)

(49)

þ 3

þ 3 To test the controller, a DFIG was simulated using the parameters displayed in Table 1. The blades model surface is shown in Figure 1 with gearbox ratio (η = 85.8) rotor radius (r = 40 m) and air density (ρ = 1.25 kg\m3 ). The nominal wind speed of 12 m/s, the nominal tip speed ratio Λopt = 7.9533 and the maximum power coefficient is Cp ,max = 0.4109. The pitch controller is ideal chopping the extracted aerodynamic power to the nominal power (2 MW) for wind speed above the nominal value. Considering that the maximum switching frequency of the converter is 7000 Hz; from Figure 6 the desired hysteresis width is δ = 90.04A; therefore, δQs = 128 � 103 and <sup>δ</sup>Tem = 811.

The following electromagnetic torque reference is used for maintaining the tip-speed ratio at the optimal value without measuring wind speed:

$$T\_{r\!f} = -\frac{\frac{1}{2}\pi\rho R^5 \mathbb{C}\_{p,\max}}{\Lambda\_{opt}^2 \eta^3} \omega\_m^2 \tag{51}$$

On the other hand, the reactive power reference is maintained at zero.

The wind speed profile is shown in Figure 7 II, the wind speed was taken from real measurements reported by the Department of Wind Energy, Technical University of Denmark [19] with values oscillating in all the operational range of the wind speed.

The power extracted by the blades is shown in Figure 7 III, during the high wind speed periods, the ideal pitch controller maintains the extracted power at the nominal value of 2 MW, while the mechanical speed is controlled during the rest of the time to optimize power extraction as shown in Figure 7 IV. The rotor converter nominal power limits the operational speed of the wind turbine Pr ≈ sPs.

The references are followed even under unbalanced grid conditions, a two-phase voltage dip is simulated at the terminals of the electric machine, the voltage dip is 20% of the nominal value. In Figure 7 I, it is displayed the detail of the voltage dip. The voltage dip starts at 95 seconds and ends at 98 seconds of the simulation, the time axis in Figure 7 I is chopped from 95.05 to

Then, the safe operation of the power converter regarding commutation frequency is ensured. Furthermore, the proposed SMC is capable to reject stator flux variations providing a desired operation even under unbalanced grid conditions. Compared with classical control techniques as DTC, the presented SMC does not require the modification of the control loop. The equations for compute the control gains that ensure robustness and existence of SMC are given.

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Simulations validating the advantages of the proposed SMC are shown.

d, q Direct, quadrature reference frame axis component

α, β Alpha, beta reference frame axis component

Ps, Qs Stator active, reactive power [W, VA]

Tem Electromagnetic torque [N∙m]

ω<sup>s</sup> Synchronous speed [rad/s]

θsf Stator flux angular position [rad]

θ<sup>m</sup> Rotor mechanical angle [rad] ω<sup>r</sup> Rotor electrical speed [rad/s]

Pwind Aerodynamic power [W]

CP Power coefficient γ Pitch angle [deg] Λ Tip-speed ratio

Nomenclature

v Voltage [V] i Current [A]

λ Magnetic flux [Wb]

s,r Stator, rotor sub index

Lm Magnetic inductance [H]

Ls Stator inductance [H] Lr Rotor inductance [H] P Number of pole pairs.

VDC DC-link voltage [V]

ω<sup>m</sup> Rotor speed [rad/s]

s Slip

Figure 7. Simulation results: I stator voltage, II wind speed profile, III extracted aerodynamic wind power, IV rotor speed, V electromagnetic torque, VI stator active and reactive power, VII rotor current, VIII rotor filtered voltage.

97.95 s, in order to better see the stator voltage waveform during the fault. As demonstrated in Section 5, it is not possible to maintain both torque and reactive power constant during unbalanced conditions; therefore, the stator reactive power is affected (see Figure 7 VI) since the controller forces the electromagnetic torque to be constant. As a result, the controller injects a negative sequence current to the rotor in order to cancel torque oscillations out (see Figure 7 VII), it is worth mention that other control strategies as vector control requires a double control loop for controlling positive and negative sequence separated [13], which not only complicates the current regulation algorithm but also requires sequence decoupling of rotor current, a very complex issue that is avoided by using a robust control strategy.

Finally, the filtered rotor voltage is displayed in Figure 7 VIII. The controller does not require modulation and automatically injects negative sequence voltage to regulate torque and reactive power under unbalanced conditions.
