3. Dynamic modeling of the DFIG

The dynamic equivalent circuit of DFIG can be expressed in an arbitrary reference frame rotating at a speed equal to ω [6, 8]:

$$w\_{sd} = \frac{d\lambda\_{sd}}{dt} - a\lambda\lambda\_{sq} + R\_s i\_{sd} \tag{4}$$

The basic scheme of a wind turbine is shown in Figure 2. A back-to-back power converter is necessary to send the required voltage to the rotor. In the grid side, it is also necessary a converter since the power flow must be bidirectional, from the electric machine to the grid at super-synchronous operation of the machine and from the electric grid to the machine at subsynchronous operation. Grid side converter is usually controlled by cascade controller, which maintains DC-link voltage constant by regulating the grid current. The reactive power can also be controlled; hence, the wind turbine operates at unitary power factor, or it can inject reactive power to the grid similarly to an electrically excited synchronous generator. For purposes of this work, let us consider that the DC-link voltage is maintained constant by the grid-side controller.

DFIG rotors are typically connected in star with the neutral connection isolated, the voltage if measured from phase a to the neutral point of the rotor star will be a combination of the phase

Each component of the vector S1,2,3 has only two valid states, 0 or 1. Then, each phase voltage

It is well known from vector control that orienting the machine model presented in Section 3 in the stator flux reference frame is an effective way for decoupling active power (or torque) and reactive power control by means of rotor current regulation. In normal operation, the stator

2 6 4

<sup>3</sup> VDC, � <sup>1</sup>

2 �1 �1 �1 2 �1 �1 �1 2

3 7 5

2 6 4

S1 S2 S3

zfflffl}|fflffl{ S<sup>123</sup>

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59

3 7 5

<sup>3</sup> VDC, 0: The rotor side converter can be analyzed

(14)

voltages, which can be summarized using the following matrix equation:

van vbn vcn 3 7 <sup>5</sup> <sup>¼</sup> VDC 3

2 6 4

as a discontinuous sign function with variable gain.

3.1. Rotor-side power converter model

Figure 2. Basic scheme of DFIG-based wind turbine.

has six different possible values: � <sup>2</sup>

4. Sliding mode controller design

$$w\_{sq} = \frac{d\lambda\_{sq}}{dt} + \omega\lambda\_{sd} + R\_s i\_{sq} \tag{5}$$

$$
\omega\_{rd} = \frac{d\lambda\_{rd}}{dt} - (\omega - \omega\_m)\lambda\_{rq} + R\_r i\_{rd} \tag{6}
$$

$$
\sigma\_{rq} = \frac{d\lambda\_{rq}}{dt} + (\omega - \omega\_m)\lambda\_{rd} + R\_r i\_{rq} \tag{7}
$$

$$
\lambda\_{sd} = L\_s i\_{sd} + L\_m i\_{rd} \tag{8}
$$

$$
\lambda\_{sq} = L\_s i\_{sq} + L\_m i\_{rq} \tag{9}
$$

$$
\lambda\_{rd} = L\_r i\_{rd} + L\_m i\_{sd} \tag{10}
$$

$$
\lambda\_{rq} = L\_r i\_{rq} + L\_m i\_{sq} \tag{11}
$$

The electromagnetic torque can be expressed as an interaction between rotor current and stator magnetic flux:

$$T\_{em} = \frac{\Im PL\_m}{2L\_s} \left( \dot{\imath}\_{rd} \lambda\_{sq} - \dot{\imath}\_{rq} \lambda\_{sd} \right) \tag{12}$$

A common expression of the stator reactive power in terms of the stator voltage and rotor current is:

$$Q\_s = \frac{3L\_m}{2L\_s} \left(\upsilon\_{sd}i\_{r\eta} - \upsilon\_{sq}i\_{rd}\right) + \frac{3}{2L\_s} \left(\lambda\_{sd}\upsilon\_{sq} - \lambda\_{sq}\upsilon\_{sd}\right) \tag{13}$$

The second term of Eq. (13) is the reactive power required to magnetize the electric machine.

Figure 2. Basic scheme of DFIG-based wind turbine.

<sup>Λ</sup> <sup>¼</sup> <sup>ω</sup>mr

Pitch angle is normally used for aerodynamically reduce power extraction when the wind speed is above the nominal value. For normal operation, it is maintained constant, while the rotor speed is controlled by the DFIG to maintain the tip speed ratio constant, for the blades model shown in Figure 1, the nominal pitch angle is zero and the nominal tip speed ratio is 8 for a maximum power coefficient of approximately 0.41. The parameters used to generate the

The dynamic equivalent circuit of DFIG can be expressed in an arbitrary reference frame

The electromagnetic torque can be expressed as an interaction between rotor current and stator

A common expression of the stator reactive power in terms of the stator voltage and rotor

The second term of Eq. (13) is the reactive power required to magnetize the electric machine.

irdλsq � irqλsd

3 2Ls

λsdvsq � λsqvsd

Tem <sup>¼</sup> <sup>3</sup>PLm 2Ls

vsdirq � vsqird <sup>þ</sup>

Qs <sup>¼</sup> <sup>3</sup>Lm 2Ls

vsd <sup>¼</sup> <sup>d</sup>λsd

vsq <sup>¼</sup> <sup>d</sup>λsq

vrd <sup>¼</sup> <sup>d</sup>λrd

vrq <sup>¼</sup> <sup>d</sup>λrq

displayed function are c<sup>1</sup> = 0.5 ; c<sup>2</sup> = 116 ; c<sup>3</sup> = 0.4 ; c<sup>4</sup> =0; c<sup>6</sup> =5; c<sup>7</sup> = 21:

3. Dynamic modeling of the DFIG

rotating at a speed equal to ω [6, 8]:

58 Adaptive Robust Control Systems

magnetic flux:

current is:

<sup>η</sup><sup>V</sup> (3)

dt � ωλsq <sup>þ</sup> Rsisd (4)

dt <sup>þ</sup> ωλsd <sup>þ</sup> Rsisq (5)

dt � ð Þ <sup>ω</sup> � <sup>ω</sup><sup>m</sup> <sup>λ</sup>rq <sup>þ</sup> Rrird (6)

dt <sup>þ</sup> ð Þ <sup>ω</sup> � <sup>ω</sup><sup>m</sup> <sup>λ</sup>rd <sup>þ</sup> Rrirq (7)

λsd ¼ Lsisd þ Lmird (8)

λsq ¼ Lsisq þ Lmirq (9)

λrd ¼ Lrird þ Lmisd (10)

λrq ¼ Lrirq þ Lmisq (11)

(12)

(13)

The basic scheme of a wind turbine is shown in Figure 2. A back-to-back power converter is necessary to send the required voltage to the rotor. In the grid side, it is also necessary a converter since the power flow must be bidirectional, from the electric machine to the grid at super-synchronous operation of the machine and from the electric grid to the machine at subsynchronous operation. Grid side converter is usually controlled by cascade controller, which maintains DC-link voltage constant by regulating the grid current. The reactive power can also be controlled; hence, the wind turbine operates at unitary power factor, or it can inject reactive power to the grid similarly to an electrically excited synchronous generator. For purposes of this work, let us consider that the DC-link voltage is maintained constant by the grid-side controller.

#### 3.1. Rotor-side power converter model

DFIG rotors are typically connected in star with the neutral connection isolated, the voltage if measured from phase a to the neutral point of the rotor star will be a combination of the phase voltages, which can be summarized using the following matrix equation:

$$
\begin{bmatrix} v\_{\text{out}} \\ v\_{\text{bn}} \\ v\_{\text{cn}} \end{bmatrix} = \frac{V\_{\text{DC}}}{3} \begin{bmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{bmatrix} \begin{bmatrix} S\_1 \\ S\_2 \\ S\_3 \end{bmatrix} \tag{14}
$$

Each component of the vector S1,2,3 has only two valid states, 0 or 1. Then, each phase voltage has six different possible values: � <sup>2</sup> <sup>3</sup> VDC, � <sup>1</sup> <sup>3</sup> VDC, 0: The rotor side converter can be analyzed as a discontinuous sign function with variable gain.

#### 4. Sliding mode controller design

It is well known from vector control that orienting the machine model presented in Section 3 in the stator flux reference frame is an effective way for decoupling active power (or torque) and reactive power control by means of rotor current regulation. In normal operation, the stator voltage vector will lead the stator flux by approximately 90 degrees neglecting voltage drop due to the stator resistance. From Eqs. (12) and (13) oriented at the stator flux direction, i.e., <sup>λ</sup>sq = 0 and vsd = 0, we obtain a decoupled control system: Tem ¼ � <sup>3</sup>PLm <sup>2</sup>Ls <sup>λ</sup>sdirq and Qs ¼ � <sup>3</sup>Lm 2Ls vsqird <sup>þ</sup> <sup>3</sup> 2Ls λsdvsq: However, under unbalanced conditions, the phase shift between voltage and flux will not be constant; therefore, robust control is necessary to withstand this perturbation.

From Eq. (8), the direct axis component of stator current can be expressed as:

$$\dot{\lambda}\_{sd} = \frac{\lambda\_{sd} - L\_{ml}i\_{rd}}{L\_{s}} \tag{15}$$

Substituting Eq. (15) in Eq. (10):

$$
\lambda\_{rd} = L\_r' i\_{rd} + \frac{L\_m}{L\_s} \lambda\_{sd} \tag{16}
$$

where L<sup>0</sup> <sup>r</sup> <sup>¼</sup> Lr � <sup>L</sup><sup>2</sup> m Ls .

The quadrature component of rotor flux can be obtained in a similar way:

$$
\lambda\_{r\eta} = L\_r' i\_{r\eta} + \frac{L\_m}{L\_s} \lambda\_{sq} \tag{17}
$$

SMC is robust against bounded disturbances and parameter variations; hence, it is a good alternative to control perturbed plants as the DFIG model shown in Figure 3. As we want to

that the reference values are much slower than the dynamics of the system and that the stator

2Ls

On the other hand, the voltage vector may have a d component due to voltage perturbation:

irq � v\_

ird � � <sup>þ</sup>

ωrλsd þ ωrL<sup>0</sup>

�kvsqvrd zfflfflfflffl}|fflfflfflffl{ dQs

irq � v\_


\_

sq ird � vsq \_

rird

em; <sup>σ</sup>Qs <sup>¼</sup> Qs � <sup>Q</sup><sup>∗</sup>

<sup>s</sup> are the desired (reference) value of torque and reactive power, considering

irqλsd <sup>þ</sup> irqλ\_

3 2Ls

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�kPλsdvrq zfflfflfflfflfflffl}|fflfflfflfflfflffl{ dTem

> 3 2Ls

�kPL<sup>0</sup>

<sup>λ</sup>\_ sdvsq <sup>þ</sup> <sup>λ</sup>sdv\_sq � �

<sup>r</sup>irqλ\_ sd zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ PTem

<sup>s</sup> (20)

<sup>λ</sup>\_ sdvsq <sup>þ</sup> <sup>λ</sup>sdv\_sq � � (22)

(23)

(24)

sd � � (21)

control the torque and reactive power, the following sliding surfaces are selected:

<sup>σ</sup>\_ Tem <sup>¼</sup> <sup>T</sup>\_ em ¼ � <sup>3</sup>PLm

<sup>v</sup>\_sdirq <sup>þ</sup> vsd \_

Lm Ls

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ � � FTem

<sup>v</sup>\_sdirq <sup>þ</sup> vsd \_

sq ird � � � � <sup>þ</sup>

<sup>σ</sup>Tem <sup>¼</sup> Tem � <sup>T</sup><sup>∗</sup>

flux is perfectly aligned with the reference frame d axis:

Figure 3. Equivalent current plant including stator flux perturbation.

<sup>s</sup> <sup>¼</sup> <sup>3</sup>Lm 2Ls

Substituting Eqs. (18) and (19) in (22) and (23):

σ\_ Tem ¼ kPλsd Rrirq þ

σ\_ Qs ¼ kvsq Rrirq � ωrL<sup>0</sup>

Lm Ls λ\_ sd þ L0 r vsq

<sup>r</sup>irq � � zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ FQs

where T<sup>∗</sup>

em and Q<sup>∗</sup>

<sup>σ</sup>\_ Qs <sup>¼</sup> <sup>Q</sup>\_

þkvsq

Substituting Eqs. (16) and (17) in (6) and (7) and solving for dird dt and dirq dt :

$$\frac{d\mathbf{i}\_{rd}}{dt} = \frac{1}{L\_r'} \left( \sigma\_{rd} - R\_r \mathbf{i}\_{rd} - \frac{\mathbf{L}\_m}{\mathbf{L}\_s} \frac{d\lambda\_{sd}}{dt} + \omega\_r \mathbf{L}\_r' \mathbf{i}\_{rq} + \frac{\mathbf{L}\_m}{\mathbf{L}\_s} \omega\_r \lambda\_{sq} \right) \tag{18}$$

$$\frac{d\dot{l}\_{rq}}{dt} = \frac{1}{L\_r'} \left( \upsilon\_{rq} - R\_r \dot{i}\_{rq} - \frac{L\_m}{L\_s} \frac{d\lambda\_{sq}}{dt} - \omega\_r L\_r' \dot{i}\_{rd} - \frac{L\_m}{L\_s} \omega\_r \lambda\_{sd} \right) \tag{19}$$

where ω<sup>r</sup> = ω<sup>s</sup> � ωm. Since the reference frame selected rotate at a synchronous speed, ω<sup>r</sup> is equivalent to the rotor current angular speed and the slip angular frequency.

In Figure 3, the equivalent system for current regulation is shown. Under normal operation, the stator flux induces a voltage in the quadrature axis loop. The induced voltage is proportional to the slip and affects only the active power (or torque) regulation loop. Under a grid fault, the stator flux is directly affected and can be analyzed as another perturbation affecting the current regulation. Analyzing the flux in positive, negative, and natural fluxes, we can see the influence of the induced voltage (perturbation) to the current regulation loop. The positive sequence flux has an induced voltage proportional to sωs, the slip is less than 0.3 for this type of machine; therefore, the induced voltage is low. The natural flux does not rotate; therefore, the induced voltage is proportional to the mechanical speed of the machine, and the negative flux rotates opposite to the reference frame orientation, the induced voltage is very large, proportional to near twice the synchronous speed (2 � s)ω<sup>s</sup> [8]. If the grid fault is very large, no control strategy could withstand this kind of perturbation, since the induced voltage may be larger than the rotor voltage. For that reason, external protection devices are required, and the simplest one is the crowbar.

Stator-Flux-Oriented Sliding Mode Control for Doubly Fed Induction Generator http://dx.doi.org/10.5772/intechopen.70714 61

Figure 3. Equivalent current plant including stator flux perturbation.

voltage vector will lead the stator flux by approximately 90 degrees neglecting voltage drop due to the stator resistance. From Eqs. (12) and (13) oriented at the stator flux direction, i.e.,

flux will not be constant; therefore, robust control is necessary to withstand this perturbation.

isd <sup>¼</sup> <sup>λ</sup>sd � Lmird Ls

> <sup>r</sup>ird þ Lm Ls

> <sup>r</sup>irq þ Lm Ls

> > dλsd dt <sup>þ</sup> <sup>ω</sup>rL<sup>0</sup>

dλsq dt � <sup>ω</sup>rL<sup>0</sup>

Ls

Ls

where ω<sup>r</sup> = ω<sup>s</sup> � ωm. Since the reference frame selected rotate at a synchronous speed, ω<sup>r</sup> is

In Figure 3, the equivalent system for current regulation is shown. Under normal operation, the stator flux induces a voltage in the quadrature axis loop. The induced voltage is proportional to the slip and affects only the active power (or torque) regulation loop. Under a grid fault, the stator flux is directly affected and can be analyzed as another perturbation affecting the current regulation. Analyzing the flux in positive, negative, and natural fluxes, we can see the influence of the induced voltage (perturbation) to the current regulation loop. The positive sequence flux has an induced voltage proportional to sωs, the slip is less than 0.3 for this type of machine; therefore, the induced voltage is low. The natural flux does not rotate; therefore, the induced voltage is proportional to the mechanical speed of the machine, and the negative flux rotates opposite to the reference frame orientation, the induced voltage is very large, proportional to near twice the synchronous speed (2 � s)ω<sup>s</sup> [8]. If the grid fault is very large, no control strategy could withstand this kind of perturbation, since the induced voltage may be larger than the rotor voltage. For that

λsdvsq: However, under unbalanced conditions, the phase shift between voltage and

<sup>2</sup>Ls <sup>λ</sup>sdirq and Qs ¼ � <sup>3</sup>Lm

λsd (16)

λsq (17)

dt and dirq dt :

<sup>r</sup>irq þ Lm Ls ωrλsq

<sup>r</sup>ird � Lm Ls ωrλsd 2Ls

(15)

(18)

(19)

<sup>λ</sup>sq = 0 and vsd = 0, we obtain a decoupled control system: Tem ¼ � <sup>3</sup>PLm

From Eq. (8), the direct axis component of stator current can be expressed as:

λrd ¼ L<sup>0</sup>

λrq ¼ L<sup>0</sup>

vrd � Rrird � Lm

vrq � Rrirq � Lm

equivalent to the rotor current angular speed and the slip angular frequency.

reason, external protection devices are required, and the simplest one is the crowbar.

The quadrature component of rotor flux can be obtained in a similar way:

Substituting Eqs. (16) and (17) in (6) and (7) and solving for dird

vsqird <sup>þ</sup> <sup>3</sup> 2Ls

60 Adaptive Robust Control Systems

where L<sup>0</sup>

Substituting Eq. (15) in Eq. (10):

<sup>r</sup> <sup>¼</sup> Lr � <sup>L</sup><sup>2</sup>

m Ls .

> dird dt <sup>¼</sup> <sup>1</sup> L0 r

> dirq dt <sup>¼</sup> <sup>1</sup> L0 r

SMC is robust against bounded disturbances and parameter variations; hence, it is a good alternative to control perturbed plants as the DFIG model shown in Figure 3. As we want to control the torque and reactive power, the following sliding surfaces are selected:

$$
\sigma\_{T\_{em}} = T\_{em} - T\_{em'}^\* \qquad \qquad \qquad \sigma\_{\mathbb{Q}\_s} = \mathbb{Q}\_s - \mathbb{Q}\_s^\* \tag{20}
$$

where T<sup>∗</sup> em and Q<sup>∗</sup> <sup>s</sup> are the desired (reference) value of torque and reactive power, considering that the reference values are much slower than the dynamics of the system and that the stator flux is perfectly aligned with the reference frame d axis:

$$\dot{\sigma}\_{T\_{cm}} = \dot{T}\_{cm} = -\frac{\Im PL\_m}{2\mathcal{L}\_s} \left( \dot{i}\_{r\eta}\lambda\_{sd} + i\_{r\eta}\dot{\lambda}\_{sd} \right) \tag{21}$$

On the other hand, the voltage vector may have a d component due to voltage perturbation:

$$\dot{\sigma}\_{Q\_s} = \dot{Q}\_s = \frac{3L\_m}{2L\_s} \left( \dot{v}\_{sd}\dot{i}\_{rq} + v\_{sd}\dot{i}\_{rq} - v\_{sq}^\*\dot{i}\_{rd} - v\_{sq}\dot{i}\_{rd} \right) + \frac{3}{2L\_s} \left( \dot{\lambda}\_{sd}\sigma\_{sq} + \lambda\_{sd}\dot{v}\_{sq} \right) \tag{22}$$

Substituting Eqs. (18) and (19) in (22) and (23):

$$\dot{\sigma}\_{T\_{\rm un}} = \overbrace{kPM\_{sd}\left(R\_r\dot{\imath}\_{rq} + \frac{L\_m}{L\_s}\omega\_r\lambda\_{sd} + \omega\_r L\_r'\dot{\imath}\_{rd}\right)}^{\text{F}\_{T\_{\rm un}}} \underbrace{\omega\_{r\_{\rm un}}}\_{} \overbrace{kPM\_{sd}\omega\_{rq} - kPL\_r'\dot{\imath}\_{rq}\dot{\lambda}\_{sd}}^{\text{P}\_{T\_{\rm un}}} \tag{23}$$

$$\begin{split} \dot{\sigma}\_{Q\_{s}} &= \overbrace{k\upsilon\_{sq} \left(\mathcal{R}\_{r}\dot{i}\_{rq} - \omega\_{r}\mathcal{L}\_{r}^{\prime}\dot{i}\_{rq}\right)}^{F\_{Q\_{s}}} - k\upsilon\_{sq}\upsilon\_{rd}}\_{\begin{subarray}{c} \mathbf{k}\_{sd} \\ \mathbf{k}\_{sq} \end{subarray}} \\ \left\{ +k\upsilon\_{sq} \underbrace{\left[\mathcal{L}\_{m}\dot{\lambda}\_{sd} + \frac{\mathcal{L}\_{r}^{\prime}}{\upsilon\_{sq}} \left(\dot{\upsilon}\_{sd}\dot{i}\_{rq} + \upsilon\_{sd}\dot{i}\_{rq} - \dot{\upsilon}\_{sq}\dot{i}\_{rd}\right)\right]}\_{P\_{Q\_{s}}} + \frac{\mathcal{R}}{2\mathcal{L}\_{s}} \left(\dot{\lambda}\_{sd}\upsilon\_{sq} + \lambda\_{sd}\dot{\upsilon}\_{sq}\right) \end{split} \tag{24}$$

where <sup>k</sup> <sup>¼</sup> <sup>3</sup>Lm 2LsL<sup>0</sup> r , expressing Eqs. (23) and (24) in matrix form:

$$
\begin{bmatrix}
\dot{\sigma}\_{T\_{on}} \\
\dot{\sigma}\_{Q\_{\ast}}
\end{bmatrix} = \begin{bmatrix}
F\_{T\_{on}} \\
F\_{Q\_{\ast}}
\end{bmatrix} - k \begin{bmatrix}
0 & P\lambda\_{sd} \\
\upsilon\_{sq} & 0
\end{bmatrix} \begin{bmatrix}
\upsilon\_{rd} \\
\upsilon\_{rq}
\end{bmatrix} + \begin{bmatrix}
P\_{T\_{on}} \\
P\_{Q\_{\ast}}
\end{bmatrix} \tag{25}
$$

Under normal conditions, all the terms dependent on vsd, v\_sd, v\_sq and λ\_ sd are equal to zero. Under abnormal conditions those terms can be analyzed as perturbations PTem and PQs . The control signals vrd and vrq appears in the first derivate of the sliding surface, thus the relative degree of the control system is one. Then, finite time convergence to the sliding surface and robustness against bounded disturbance/uncertainties can be achieved using the control signal:

$$
\begin{bmatrix} \boldsymbol{\upsilon}\_{rd} \\ \boldsymbol{\upsilon}\_{rq} \end{bmatrix} = \begin{bmatrix} -M\_d \operatorname{sgn} \left( \boldsymbol{\sigma}\_{Q\_s} \right) \\ -M\_q \operatorname{sgn} \left( \boldsymbol{\sigma}\_{T\_m} \right) \end{bmatrix} \tag{26}
$$

where:

The constant term <sup>3</sup>

Figure 4. Basic scheme of SMC.

<sup>D</sup><sup>þ</sup> <sup>¼</sup> <sup>D</sup><sup>T</sup> DD<sup>T</sup> � ��<sup>1</sup> ¼ � <sup>3</sup>

4.1. Sliding mode existence condition

For a relative degree one system

With a scalar control

kVDC

The condition for satisfying the existence of the sliding mode is [9]:

1 Pλsd

from the transformation matrix, since the controller will only evaluate the sign of σabc, thus the controller is robust against parametric uncertainties; under normal conditions, the stator flux direct component (λsd) and stator voltage quadrature component (vsq) are constant. In Figure 4 is shown the basic scheme of the presented controller. The output of Eq. (29) is the equivalent abc sliding surface, then a function similar to sign is used to evaluate the switch state (0 means lower leg activated and 1 means upper leg activated). The resulting control system does not require modulation since the switching state is determined directly from the control system.

1 Pλsd

1 Pλsd sin θ<sup>d</sup>

sin ð Þ <sup>θ</sup><sup>d</sup> <sup>þ</sup> <sup>2</sup>π=<sup>3</sup> <sup>1</sup>

sin ð Þ <sup>θ</sup><sup>d</sup> � <sup>2</sup>π=<sup>3</sup> <sup>1</sup>

kVDC, which contains parameters of the electric machine, can be removed

1 vsq

vsq

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63

vsq

<sup>σ</sup>\_ <sup>¼</sup> F xð Þþ d xð Þu x<sup>∈</sup> <sup>R</sup><sup>n</sup> ð Þ (30)

u ¼ �M sgn ð Þ σ (31)

cos θ<sup>d</sup>

cos ð Þ θ<sup>d</sup> þ 2π=3

cos ð Þ θ<sup>d</sup> � 2π=3

where Md , Mq > 0. A detailed description of Md and Mq computation is given in section 4.1.

Remark: A discontinuous function is intentionally selected because the nature of the rotor-side power converter is discontinuous as well; therefore, the control signal can be easily used directly from the controller algorithm to the power converter without modulation. On the contrary, any continuous control; e.g. saturation, sigmoid function, etc. employed to smooth the discontinuous control in Eq. (26) must be modulated to be implementable in a power converter; therefore, chattering will be present no matter the control strategy used.

The desired voltage need to be send through the rotor-side power converter. The voltage seen from the stator flux reference frame is obtained using park transform:

$$
\begin{bmatrix} v\_{rd} \\ v\_{rq} \end{bmatrix} = \begin{bmatrix} \cos\theta\_d & -\sin\theta\_d \\ \sin\theta\_d & \cos\theta\_d \end{bmatrix} \frac{2}{3} \begin{bmatrix} 1 & -1/2 & -1/2 \\ 0 & \sqrt{3}/2 & -\sqrt{3}/2 \end{bmatrix} \begin{bmatrix} v\_{an} \\ v\_{bn} \\ v\_{cn} \end{bmatrix} \tag{27}
$$

where the reference frame angle is θ<sup>d</sup> = θ<sup>m</sup> � θsf.

Using Eqs. (14) (25) and (27), it is possible to establish a relationship between dq and abc quantities:

$$
\begin{bmatrix} \sigma\_{T\_{m}} \\ \sigma\_{Q\_{c}} \end{bmatrix} = \overbrace{-k \begin{bmatrix} 0 & P\lambda\_{sd} \\ \sigma\_{sq} & 0 \end{bmatrix} T \frac{V\_{DC}}{3} \begin{bmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{bmatrix}}\_{\begin{bmatrix} 0 & \\ -1 & 2 \end{bmatrix}} \begin{bmatrix} \sigma\_{d} \\ \sigma\_{b} \\ \sigma\_{c} \end{bmatrix} \tag{28}
$$

Since the matrix D is not square, the Moore-Penrose pseudo-inverse is used:

$$
\begin{bmatrix} \sigma\_{\tt a} \\ \sigma\_{\tt b} \\ \sigma\_{\tt c} \end{bmatrix} = D^{+} \begin{bmatrix} \sigma\_{T\_{\tt c\tt}} \\ \sigma\_{Q\_{\tt}} \end{bmatrix} \tag{29}
$$

Figure 4. Basic scheme of SMC.

where:

where <sup>k</sup> <sup>¼</sup> <sup>3</sup>Lm

2LsL<sup>0</sup> r

62 Adaptive Robust Control Systems

, expressing Eqs. (23) and (24) in matrix form:

Under normal conditions, all the terms dependent on vsd, v\_sd, v\_sq and λ\_

vrd vrq � � � k

Under abnormal conditions those terms can be analyzed as perturbations PTem and PQs

control signals vrd and vrq appears in the first derivate of the sliding surface, thus the relative degree of the control system is one. Then, finite time convergence to the sliding surface and robustness against bounded disturbance/uncertainties can be achieved using the control signal:

<sup>¼</sup> �Md sgn <sup>σ</sup>Qs

where Md , Mq > 0. A detailed description of Md and Mq computation is given in section 4.1.

converter; therefore, chattering will be present no matter the control strategy used.

from the stator flux reference frame is obtained using park transform:

<sup>¼</sup> cos <sup>θ</sup><sup>d</sup> � sin <sup>θ</sup><sup>d</sup> sin θ<sup>d</sup> cos θ<sup>d</sup> � � 2

vrd vrq � �

quantities:

where the reference frame angle is θ<sup>d</sup> = θ<sup>m</sup> � θsf.

σTem σQs � �

¼ �k

0 Pλsd vsq 0 � �

Since the matrix D is not square, the Moore-Penrose pseudo-inverse is used:

2 6 4

σa σb σc

zfflffl}|fflffl{ <sup>σ</sup>abc

3 7 5

Remark: A discontinuous function is intentionally selected because the nature of the rotor-side power converter is discontinuous as well; therefore, the control signal can be easily used directly from the controller algorithm to the power converter without modulation. On the contrary, any continuous control; e.g. saturation, sigmoid function, etc. employed to smooth the discontinuous control in Eq. (26) must be modulated to be implementable in a power

The desired voltage need to be send through the rotor-side power converter. The voltage seen

3

Using Eqs. (14) (25) and (27), it is possible to establish a relationship between dq and abc

<sup>T</sup> VDC 3

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ D

> <sup>¼</sup> <sup>D</sup><sup>þ</sup> <sup>σ</sup>Tem σQs � �

2 6 4

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ T

> 0 ffiffiffi 3 <sup>p</sup> <sup>=</sup><sup>2</sup> � ffiffiffi

�Mq sgn σTem

" #

0 Pλsd vsq 0 � � vrd

vrq � �

� �

� �

1 �1=2 �1=2

� �

2 �1 �1 �1 2 �1 �1 �1 2

3 <sup>p</sup> <sup>=</sup><sup>2</sup>

> 3 7 5

2 6 4

σa σb σc 3 7

van vbn vcn 3 7

<sup>5</sup> (27)

<sup>5</sup> (28)

(29)

2 6 4

þ

PTem PQs � �

(25)

. The

(26)

sd are equal to zero.

<sup>¼</sup> FTem FQs � �

σ\_ Tem σ\_ Qs � �

$$D^{+} = D^{T} \left( D D^{T} \right)^{-1} = -\frac{3}{k V\_{DC}} \begin{bmatrix} \frac{1}{P \lambda\_{sd}} \sin \theta\_{d} & \frac{1}{v\_{sq}} \cos \theta\_{d} \\ \frac{1}{P \lambda\_{sd}} \sin \left( \theta\_{d} + 2\pi/3 \right) & \frac{1}{v\_{sq}} \cos \left( \theta\_{d} + 2\pi/3 \right) \\ \frac{1}{P \lambda\_{sd}} \sin \left( \theta\_{d} - 2\pi/3 \right) & \frac{1}{v\_{sq}} \cos \left( \theta\_{d} - 2\pi/3 \right) \end{bmatrix}.$$

The constant term <sup>3</sup> kVDC, which contains parameters of the electric machine, can be removed from the transformation matrix, since the controller will only evaluate the sign of σabc, thus the controller is robust against parametric uncertainties; under normal conditions, the stator flux direct component (λsd) and stator voltage quadrature component (vsq) are constant. In Figure 4 is shown the basic scheme of the presented controller. The output of Eq. (29) is the equivalent abc sliding surface, then a function similar to sign is used to evaluate the switch state (0 means lower leg activated and 1 means upper leg activated). The resulting control system does not require modulation since the switching state is determined directly from the control system.

#### 4.1. Sliding mode existence condition

For a relative degree one system

$$
\dot{\sigma} = F(\mathbf{x}) + d(\mathbf{x})u \qquad \qquad (\mathbf{x} \in \mathbb{R}^n) \tag{30}
$$

With a scalar control

$$
\mu = -M\operatorname{sgn}(\sigma) \tag{31}
$$

The condition for satisfying the existence of the sliding mode is [9]:

$$d(\mathbf{x})M > |F(\mathbf{x})|\tag{32}$$

select the number of harmonics to be considered, and it can be applied to estimate oscillations in relative degree one systems. Therefore, the DFIG system with SMC presented in this chapter

;

ωmLrLs � ωsL<sup>0</sup>

1 0 0 1 " #.

x\_ ¼ Ax þ Bu; y ¼ Cx (34)

Stator-Flux-Oriented Sliding Mode Control for Doubly Fed Induction Generator

http://dx.doi.org/10.5772/intechopen.70714

RrLm ωmLmLr

�RrLs �ωmLrLs þ ωsL<sup>0</sup>

<sup>r</sup>Ls �RrLs

B þ D (35)

Im½ � L1ð Þ jnω (37)

rLs

65

;

(36)

�ωmLmLr RrLm

is analyzed in the frequency domain using Tsypkin's method.

; <sup>u</sup> <sup>¼</sup> vsd vsq vrd vrq � �<sup>T</sup>

<sup>m</sup> þ ωsL<sup>0</sup>

<sup>r</sup>Ls �RsLr

RsLm � ωmLsLm

�Lm 0

0 �Lr

Ls 0

0 Ls

diagonal elements of the following matrix equation:

systems with four poles and three zeros:

Tsypkin's locus is defined as [12]:

Lð Þ¼ s

Irdð Þs Vrdð Þ<sup>s</sup> <sup>¼</sup> Irqð Þ<sup>s</sup>

T jð Þ¼ <sup>ω</sup> <sup>X</sup><sup>∞</sup>

n odd

ωmLsLm RSLm

rLs

; <sup>C</sup> <sup>¼</sup> 0 0 0 0

The system presents a nonlinearity due to the mechanical speed ωm, which is dependent on the electromagnetic torque and the mechanical equation; however, since the time scales of the electrical quantities is much smaller than the time scale of the mechanical system, we are going to study system of Eq. (34) as a set of linear systems varying the mechanical speed in a range of �30% about the synchronous speed. Other supposition is that the system is fully decoupled, that is, vrd controls ird and vrq controls irq. So, we can obtain both transfer functions from the

<sup>G</sup>ð Þ¼ <sup>s</sup> <sup>C</sup>ð Þ <sup>s</sup><sup>I</sup> � <sup>A</sup> �<sup>1</sup>

Re½ �þ L1ð Þ jnω j

The conditions required to predict a limit cycle oscillating at an angular frequency ω<sup>0</sup> are [12]:

where I is a 4 x 4 identity matrix. The diagonal transfer functions will be relative degree one

Vrqð Þ<sup>s</sup> <sup>¼</sup> <sup>a</sup>3s<sup>3</sup> <sup>þ</sup> <sup>a</sup>2s<sup>2</sup> <sup>þ</sup> <sup>a</sup>1<sup>s</sup> <sup>þ</sup> <sup>a</sup><sup>0</sup>

X∞ n odd

1 n

d4s<sup>4</sup> þ d3s<sup>3</sup> þ d2s<sup>2</sup> þ d1s þ d<sup>0</sup>

Expressing Eqs. (4)–(11) in matrix form:

�RsLr <sup>ω</sup>mL<sup>2</sup>

<sup>m</sup> � ωsL<sup>0</sup>

where:

<sup>A</sup> <sup>¼</sup> <sup>1</sup> L0 rLs

<sup>B</sup> <sup>¼</sup> <sup>1</sup> L0 rLs

<sup>x</sup> <sup>¼</sup> isd isq ird irq � �<sup>T</sup>

�ωmL<sup>2</sup>

Lr 0

0 Lr

0 �Lm

�Lm 0

The stator-flux–oriented SMC has a very similar form of system of Eq. (30). Then, existence of SMC is ensured if the following conditions are met:

$$M\_{q,min} > |(F\_{T\_{cm}} + P\_{T\_{cm}})/d\_{T\_{cm}}| \colon \mathfrak{gl}M\_{d,min} > |(F\_{Q\_s} + P\_{Q\_s})/d\_{Q\_s}|\tag{33}$$

Therefore, choosing Md and Mq using Eq. (33) ensures finite convergence to the sliding variable and insensitivity to bounded disturbance/uncertainties. The only requirement to compute the gains Md and Mq is the knowledge of the bounds of the system and the disturbance/uncertainties, ∣FQs , Tem∣, ∣dQs , Tem∣ and ∣PQs , Tem∣. Note that real implementation of control gains in Eqs. (33) depends of DC-link voltage VDC, which is variable in practice, and it must be ensured that VDC is regulated correctly to ensure a robust performance of the rotor side of the DFIG system.

#### 4.2. Switching frequency limitation

In Figure 4 it is shown the scheme of an ideal sliding mode controller; however, it requires infinite switching frequency, which is not possible in real physical systems, therefore, the most common solution for this issue is to include a hysteresis loop to the ideal sign function [10] since hysteresis makes the switching frequency finite. Note that sigmoid functions and saturation can be implemented to reduce switching frequency as well as attenuate chattering. However, these functions are continuous and/or contain linear parts requiring modulation for its application to power electronics, while a sign function with hysteresis can be injected without modulation.

A widely used method to determine limit cycles and the oscillation frequency is the sinusoidal describing function (DF), which can be used for segmented nonlinear system composed by a linear system and a nonlinear part [12], that is the case of a linear plant controlled by a relaybased control system (see Figure 5). A requirement for applying DF is that the linear system L(s) must have a low-pass filter behavior. Furthermore, only the first harmonic is considered in the analysis. However, for relative degree-one systems as the one presented in this chapter, this technique is not suitable [12]. DF will predict no oscillations because it ignores the contribution of the harmonics. On the other hand, Tsypkin's method is an exact method in which one can

Figure 5. Relay-based control of a linear system.

select the number of harmonics to be considered, and it can be applied to estimate oscillations in relative degree one systems. Therefore, the DFIG system with SMC presented in this chapter is analyzed in the frequency domain using Tsypkin's method.

Expressing Eqs. (4)–(11) in matrix form:

$$
\dot{\mathbf{x}} = A\mathbf{x} + Bu; \qquad y = \mathbf{C}\mathbf{x} \tag{34}
$$

where:

d xð ÞM > ∣F xð Þ∣ (32)

Þ=dQs

j (33)

The stator-flux–oriented SMC has a very similar form of system of Eq. (30). Then, existence of

Therefore, choosing Md and Mq using Eq. (33) ensures finite convergence to the sliding variable and insensitivity to bounded disturbance/uncertainties. The only requirement to compute the gains Md and Mq is the knowledge of the bounds of the system and the disturbance/uncertainties, ∣FQs , Tem∣, ∣dQs , Tem∣ and ∣PQs , Tem∣. Note that real implementation of control gains in Eqs. (33) depends of DC-link voltage VDC, which is variable in practice, and it must be ensured that VDC is regulated correctly to ensure a robust performance of the rotor side of the DFIG

In Figure 4 it is shown the scheme of an ideal sliding mode controller; however, it requires infinite switching frequency, which is not possible in real physical systems, therefore, the most common solution for this issue is to include a hysteresis loop to the ideal sign function [10] since hysteresis makes the switching frequency finite. Note that sigmoid functions and saturation can be implemented to reduce switching frequency as well as attenuate chattering. However, these functions are continuous and/or contain linear parts requiring modulation for its application to power electronics, while a sign function with hysteresis can be injected without

A widely used method to determine limit cycles and the oscillation frequency is the sinusoidal describing function (DF), which can be used for segmented nonlinear system composed by a linear system and a nonlinear part [12], that is the case of a linear plant controlled by a relaybased control system (see Figure 5). A requirement for applying DF is that the linear system L(s) must have a low-pass filter behavior. Furthermore, only the first harmonic is considered in the analysis. However, for relative degree-one systems as the one presented in this chapter, this technique is not suitable [12]. DF will predict no oscillations because it ignores the contribution of the harmonics. On the other hand, Tsypkin's method is an exact method in which one can

Mq,min > jðFTem þ PTem Þ=dTem j; Md,min > jðFQs þ PQs

SMC is ensured if the following conditions are met:

4.2. Switching frequency limitation

Figure 5. Relay-based control of a linear system.

system.

64 Adaptive Robust Control Systems

modulation.

$$\begin{aligned} \mathbf{x} &= \begin{bmatrix} i\_{sd} & i\_{sq} & i\_{rd} & i\_{rq} \end{bmatrix}^{\top}; \mathbf{u} = \begin{bmatrix} \upsilon\_{sd} & \upsilon\_{sq} & \upsilon\_{rd} & \upsilon\_{rq} \end{bmatrix}^{\top}; \\\\ & - \mathbf{S}\_{s}L\_{r} & \omega\_{m}L\_{m}^{2} + \omega\_{s}L\_{r}^{\prime}L\_{s} & R\_{r}L\_{m} & \omega\_{m}L\_{m}L\_{r} \\\\ -\omega\_{m}L\_{m}^{2} & -\omega\_{s}L\_{r}^{\prime}L\_{s} & -R\_{s}L\_{r} & -\omega\_{m}L\_{m}L\_{r} & R\_{r}L\_{m} \\\\ \mathbf{S}\_{s}L\_{m} & & -\omega\_{m}L\_{s}L\_{m} & -R\_{r}L\_{s} & -\omega\_{m}L\_{r}L\_{s} + \omega\_{s}L\_{r}^{\prime}L\_{s} \\\\ \omega\_{m}L\_{s}L\_{m} & & R\_{S}L\_{m} & \omega\_{m}L\_{r}L\_{s} - \omega\_{s}L\_{r}^{\prime}L\_{s} & -R\_{r}L\_{s} \\\\ \mathbf{S} & \begin{bmatrix} L\_{r} & \mathbf{0} & -L\_{m} & \mathbf{0} \\\\ \mathbf{0} & L\_{r} & \mathbf{0} & -L\_{r} \\\\ -L\_{m} & \mathbf{0} & L\_{s} & \mathbf{0} \\\\ \mathbf{0} & -L\_{m} & \mathbf{0} & L\_{s} \end{bmatrix}; \mathbf{C} = \begin{bmatrix} \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} \end{bmatrix}. \end{aligned}$$

The system presents a nonlinearity due to the mechanical speed ωm, which is dependent on the electromagnetic torque and the mechanical equation; however, since the time scales of the electrical quantities is much smaller than the time scale of the mechanical system, we are going to study system of Eq. (34) as a set of linear systems varying the mechanical speed in a range of �30% about the synchronous speed. Other supposition is that the system is fully decoupled, that is, vrd controls ird and vrq controls irq. So, we can obtain both transfer functions from the diagonal elements of the following matrix equation:

$$\mathbf{G(s)} = \mathbf{C(sI - A)}^{-1}\mathbf{B} + \mathbf{D} \tag{35}$$

where I is a 4 x 4 identity matrix. The diagonal transfer functions will be relative degree one systems with four poles and three zeros:

$$L(\mathfrak{s}) = \frac{I\_{rd}(\mathfrak{s})}{V\_{rd}(\mathfrak{s})} = \frac{I\_{\eta\mathfrak{s}}(\mathfrak{s})}{V\_{\eta\mathfrak{l}}(\mathfrak{s})} = \frac{a\_3\mathfrak{s}^3 + a\_2\mathfrak{s}^2 + a\_1\mathfrak{s} + a\_0}{d\_4\mathfrak{s}^4 + d\_3\mathfrak{s}^3 + d\_2\mathfrak{s}^2 + d\_1\mathfrak{s} + d\_0} \tag{36}$$

Tsypkin's locus is defined as [12]:

$$T(j\omega) = \sum\_{n \text{ odd}}^{\text{on}} \text{Re}[L\_1(jn\omega)] + j \sum\_{n \text{ odd}}^{\text{on}} \frac{1}{n} \text{Im}[L\_1(jn\omega)] \tag{37}$$

The conditions required to predict a limit cycle oscillating at an angular frequency ω<sup>0</sup> are [12]:

$$\operatorname{Im}[T(j\omega\_0)] = \frac{\pi}{4} \left[ L(\simeq) - \frac{\delta}{M} \right] \tag{38}$$

$$\left[\text{Re}[T(j\omega\_0)]\right] < \frac{\pi}{4\omega\_0} \lim\_{\mathfrak{s}\to\infty} \left[\mathfrak{sl}L\_1(\mathfrak{s})\right] \tag{39}$$

<sup>δ</sup>Qs <sup>¼</sup> <sup>3</sup>Lmvsq 2Ls

of chattering while it is inversely proportional to the frequency.

4.3. Step-by-step design

Nominal power = 2 MW Voltage = 690 V / 50 Hz DC-link voltage = 1200 V Stator/rotor turns ratio = 1/2 Mutual inductance (Lm) = 2.5 mH Stator inductance (Ls) = 2.58 mH Rotor inductance (Lr) = 2.58 mH Stator resistance (Rs) = 2.6 mΩ Rotor resistance (Rr) = 2.9 mΩ|

Pole pairs (P)=2

Eq. (33) to guarantying robustness.

Table 1. Machine parameters used in simulation.

algorithm:

reference frame:

<sup>δ</sup>; <sup>δ</sup>Tem <sup>¼</sup> <sup>3</sup>PLmλsd

Moreover, as can be seen from Figure 6 and Eqs. (38) and (39), the addition of hysteresis can be used to attenuated chattering since the hysteresis width is directly proportional to amplitude

The design of the SMC proposed in this section can be summarized in the next step-by-step

2. Choose the gains, Md and Mq, of the controller in Eq. (26) using the conditions given in

3. Compute a hysteresis value for the torque and reactive power controllers (sign functions)

Then, following the step-by-step process described in this subsection, a robust SMC to regulate torque and reactive power in DFIG systems can be designed. Furthermore, practical implementation is considered in the design since a method to compute a hysteresis value, limiting

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

using Eqs. (40) to ensure an acceptable switching frequency in the power electronics.

undesired high frequency commutation in power electronics, is provided.

5. Complex power and torque under unbalanced conditions

1. Select the sliding mode surfaces for torque and reactive power using Eq. (20).

2Ls

Stator-Flux-Oriented Sliding Mode Control for Doubly Fed Induction Generator

http://dx.doi.org/10.5772/intechopen.70714

67

δ (40)

In Figure 6 it is shown the graphical solution for a DFIG with the characteristics shown in Table 1, the mechanical speed has minor influence at high frequencies; therefore, it is valid to consider the DFIG as a linear system (note that the number of pole pairs is not considered in (34); therefore, the mechanical speed reported is the one of an equivalent two pole machine). Supposing that the power inverter has a maximum switching frequency of f max ¼ 4000 Hz ! <sup>ω</sup><sup>0</sup> <sup>¼</sup> <sup>8000</sup><sup>π</sup> rad <sup>s</sup> , from Eq. (38), we can calculate the hysteresis width. We know that the maximum gain the system can have is Mmax <sup>¼</sup> <sup>2</sup> <sup>3</sup>Vdc, then the hysteresis loop that will ensure a maximum switching frequency of 4000 Hz is:

$$\delta = \frac{(0.3094)(4)}{\pi} \ast \frac{2}{3} 600 = 157.57 \text{ A}$$

However, the control system was designed to direct control torque and reactive power, so we need to calculate the hysteresis width of those quantities, which can be easily done since the control system is decoupled, and the current direct and quadrature components are directly related with reactive power and torque, respectively.

Figure 6. Graphical solution using Tsypkin's method.

Nominal power = 2 MW Voltage = 690 V / 50 Hz DC-link voltage = 1200 V Stator/rotor turns ratio = 1/2 Mutual inductance (Lm) = 2.5 mH Stator inductance (Ls) = 2.58 mH Rotor inductance (Lr) = 2.58 mH Stator resistance (Rs) = 2.6 mΩ Rotor resistance (Rr) = 2.9 mΩ| Pole pairs (P)=2

Im½ �¼ T jð Þ ω<sup>0</sup>

Re½ � T jð Þ ω<sup>0</sup> <

<sup>δ</sup> <sup>¼</sup> <sup>ð</sup>0:3094Þð4<sup>Þ</sup>

<sup>π</sup> � <sup>2</sup> 3

However, the control system was designed to direct control torque and reactive power, so we need to calculate the hysteresis width of those quantities, which can be easily done since the control system is decoupled, and the current direct and quadrature components are directly

<sup>ω</sup><sup>0</sup> <sup>¼</sup> <sup>8000</sup><sup>π</sup> rad

66 Adaptive Robust Control Systems

mum gain the system can have is Mmax <sup>¼</sup> <sup>2</sup>

maximum switching frequency of 4000 Hz is:

related with reactive power and torque, respectively.

Figure 6. Graphical solution using Tsypkin's method.

π

π 4ω<sup>0</sup>

In Figure 6 it is shown the graphical solution for a DFIG with the characteristics shown in Table 1, the mechanical speed has minor influence at high frequencies; therefore, it is valid to consider the DFIG as a linear system (note that the number of pole pairs is not considered in (34); therefore, the mechanical speed reported is the one of an equivalent two pole machine). Supposing that the power inverter has a maximum switching frequency of f max ¼ 4000 Hz !

<sup>4</sup> <sup>L</sup>ð Þ� <sup>∞</sup> <sup>δ</sup>

<sup>s</sup> , from Eq. (38), we can calculate the hysteresis width. We know that the maxi-

600 ¼ 157:57 A

M

lim<sup>s</sup>!<sup>∞</sup> ½ � <sup>s</sup>L1ð Þ<sup>s</sup> (39)

<sup>3</sup>Vdc, then the hysteresis loop that will ensure a

(38)

Table 1. Machine parameters used in simulation.

$$
\delta\_{Q\_s} = \frac{3L\_m \upsilon\_{sq}}{2L\_s} \delta; \qquad \qquad \delta\_{\Gamma\_m} = \frac{3PL\_m \lambda\_{sd}}{2L\_s} \delta \tag{40}
$$

Moreover, as can be seen from Figure 6 and Eqs. (38) and (39), the addition of hysteresis can be used to attenuated chattering since the hysteresis width is directly proportional to amplitude of chattering while it is inversely proportional to the frequency.

#### 4.3. Step-by-step design

The design of the SMC proposed in this section can be summarized in the next step-by-step algorithm:

1. Select the sliding mode surfaces for torque and reactive power using Eq. (20).

2. Choose the gains, Md and Mq, of the controller in Eq. (26) using the conditions given in Eq. (33) to guarantying robustness.

3. Compute a hysteresis value for the torque and reactive power controllers (sign functions) using Eqs. (40) to ensure an acceptable switching frequency in the power electronics.

Then, following the step-by-step process described in this subsection, a robust SMC to regulate torque and reactive power in DFIG systems can be designed. Furthermore, practical implementation is considered in the design since a method to compute a hysteresis value, limiting undesired high frequency commutation in power electronics, is provided.
