Sliding Mode Control of a Wind Power System Based on a Self-Excited Asynchronous Generator

An aero-generator (Figure 1), more commonly known as a wind turbine, is a device that transforms part of the kinetic energy of the wind into mechanical energy available on a transmission shaft, then into electrical energy via an asynchronous cage generator and two static converters (rectifier, inverter) [12-14].

1.png

Figure 1. Schematic diagram of the system studied

2.1 Turbine modeling

The incoming wind power $p_{v}$:

$p_{v}=\frac{1}{2} \rho s V_{1}^{3}$     (1)

with:

$\rho$ : air density (kg/m³);

$C_{p}$ : power coefficient;

s: surface swept by the blades. (m²);

$V_{1}$: upstream wind speed (m/s).

The mechanical power $p_{m}$ available on the shaft of an aerogenerator is thus expressed [15-17].

$P_{m}=\frac{1}{2} \rho s C_{p}(\lambda) V_{1}^{3}$     (2)

with:

$C_{p}=\frac{P_{m}}{P_{v}}$       (3)

and:

$\lambda=\frac{\Omega_{t u r}{\quad} r}{V_{1}}$     (4)

with:

$\Omega_{t u r}$ : rotational speed of the wind turbine;

$\gamma$ : blade radius (mm).

The maximum value of $C_{p}$ is reached when:

$C_{P_{\max }}=\frac{16}{27}=0.593$

It is this theoretical limit, called the Betz limit, that fixes the maximum extractable power for a given wind speed [18]. Therefore, the maximum extracted power is:

$P_{m_{\max }}=\frac{1}{2} \rho S C_{P_{\max }} V_{1}^{3}$      (5)

The turbine torque is:

$C_{a e r}=\frac{P_{m}}{\Omega_{t u r}}=\frac{1}{2 \Omega_{t u r}} \rho S C_{p}(\lambda) V_{1}^{3}$     (6)

and:

$\Omega_{m e c}=G \Omega_{t u r}$       (7)

with:

$\Omega_{\text {mec }}$: angular speed of the generator(rad/s);

G: gain of the multiplier.

The mechanical torque on the generator shaft is:

$C_{m e c}=\frac{1}{G} C_{a e r}$      (8)

The generator tree is modeled by the following equation:

$J_{T} \frac{d \Omega_{m e ́ c}}{d t}=C_{T}-C_{V i s}$      (9)

The viscous friction torque is modeled by:

$C_{V i s}=f_{T} \Omega_{m e ́ c}$     (10)

The total torque of the wind turbine is given by:

$C_{T}=C_{m e ́ c}-C_{e m}$      (11)

$J_{T}=\frac{J_{\text {turbine }}}{G^{2}}+J_{g}$    (12)

with:

$C_{e m}$ : electromagnetic torque (N.m);

$J_{T}:$ total inertia of rotating parts $\left(k g \cdot m^{-3}\right)$;

$f_{T}$ : total viscous friction coefficient (N.m.s/rd);

$J_{g}$ : generator inertia (kg.m²).

2.2 The maximum power extraction techniques

The power captured by the wind turbine can be essentially maximized by adjusting the coefficient $C_{p}$. The use of a variable speed wind turbine allows maximizing this power by the Maximum Power Point Tracking (M.P.P.T). Let us consider the behavior of an MPPT for a wind energy conversion system. The MPPT command allows to position the system at the optimum power point regardless of wind speed (Figure 2) [19].

2.png

Figure 2. Extraction of maximum power for different wind speeds

To maximize power, when the wind speed is lower than its nominal value, the mechanical speed of the wind turbine corresponds to points 4, 5 and 6 [19] (Figure 3).

3.png

Figure 3. Characteristic (power - mechanical speed)

2.3 Modeling of the self-excited asynchronous generator

The Park equations of the machine in the frame of the rotating field [19, 20]:

$V_{s d}=R_{s} i_{s d}+\frac{d \psi_{s d}}{d t}-w_{s} \psi_{s q}$     (13)

$V_{s q}=R_{s} i_{s q}+\frac{d \psi_{s q}}{d t}+w_{s} \psi_{s d}$       (14)

$0=R_{r} i_{r d}+\frac{d \psi_{r d}}{d t}-w_{s l} \psi_{r q}$      (15)

$0=R_{r} i_{r q}+\frac{d \psi_{r q}}{d t}+w_{s l} \psi_{r d}$      (16)

With:

$\left(V_{s d}\right.$ and $\left.V_{s q}\right)$: stator voltages on the axes (d,q)(V);

$\left(i_{s d}\right.$ and $\left.i_{s q}\right)$: stator currents on the axes (d,q)(A);

$\left(i_{r d}\right.$ and $\left.i_{r q}\right)$: rotor currents on the axes (d,q)(A);

$\left(\psi_{s d}\right.$ and $\left.\psi_{s q}\right)$: stator fluxes on the axes (d,q)(Wb);

$\left(\psi_{r d}\right.$ and $\left.\psi_{r q}\right)$: rotor fluxes on the axes (d,q)(Wb);

$\mathrm{W}_{\mathrm{Sl}}=\mathrm{W}_{\mathrm{S}}-\mathrm{W}$: sliding speed(rad);

$\mathrm{W}_{\mathrm{S}}$: rotating field speed(rad/s);

w: rotor speed(tr/mn);

$R_{S}:$ stator resistance $(\Omega)$;

$R_{r}:$ rotor resistance $(\Omega)$.

The flux is given by:

$\left\{\begin{array}{l}\psi_{s d}=L_{s} I_{s d}+L_{m} I_{r d}=L_{\sigma s} I_{s d}+\psi_{m d} \\ \psi_{s q}=L_{s} I_{s q}+L_{m} I_{r q}=L_{\sigma s} I_{s q}+\psi_{m q}\end{array}\right.$     (17)

$\left\{\begin{array}{l}\psi_{r d}=L_{r} I_{r d}+L_{m} I_{s d}=L_{\sigma r} I_{r d}+\psi_{m d} \\ \psi_{r q}=L_{r} I_{r q}+L_{m} I_{s q}=L_{\sigma r} I_{r q}+\psi_{m q}\end{array}\right.$   (18)

where,

$\psi_{m}:$ magnetizing flux, $\psi_{m}=L_{m} I_{m}$;

$I_{m}$ : magnetizing current;

$\psi_{m d}=L_{m} I_{m d}, \psi_{m q}=L_{m} I_{m q}$;

$L_{S}$: stator inductance (mH);

$L_{r}$ : rotor inductance (mH);

$L_{m}$: magnetization inductance (mH);

$L_{\sigma S}$ : stator leakage inductance [mH);

$L_{\sigma r}$: rotor leakage inductance (mH).

$\left\{\begin{array}{l}I_{m d}=I_{s d}+I_{r d} \\ I_{m q}=I_{s q}+I_{r q}\end{array}\right.$     (19)

$I_{m}=\sqrt{I_{m d^{2}}+I_{m q^{2}}}$    (20)

The synchronous electromagnetic torque developed by the rotating field is given by the following expression [19]:

$C_{e m}=\frac{3}{2} P\left(\psi_{d s} I_{q s}-\psi_{q s} I_{d s}\right)$    (21)

$I_{s d}=\frac{\psi_{s d}-\psi_{m d}}{L_{\sigma s}}$     (22)

$I_{s q}=\frac{\psi_{s q}-\psi_{m q}}{L_{\sigma s}}$     (23)

$I_{r d}=\frac{\psi_{r d}-\psi_{m d}}{L_{\sigma r}}$    (24)

$I_{r q}=\frac{\psi_{r q}-\psi_{m q}}{L_{\sigma r}}$     (25)

Operation as a stand-alone generator requires a capacitor bank and a residual magnetic flux in the rotor iron is essential for the self-starting of the generator, it is used as a source of reactive energy for the machine [19]. So, this is applied self-excited machine.

By applying Ohm's law to each phase:

$\frac{d}{d t} V_{s d}=-\frac{1}{C} I_{s d}+w_{s} V_{s q}$     (26)

$\frac{d}{d t} V_{s q}=-\frac{1}{C} I_{s q}-w_{s} V_{s d}$     (27)

with:

C: capacitor for excitation (μF).

2.4 The cage generator simulation

We simulated the asynchronous squirrel cage machine used as a generator according to the block diagram model with a stator reference frame using MATLAB software.

Figure 4 shows the stator current response of the asynchronous no-load generator, one notices that the current response initially takes an increasing form in an exponential way as in the linear case (effect of starting), then converges towards a constant value. Also, the stator current takes a perfectly sinusoidal and balanced form (Figure 5).

Figure 6 shows the stator voltage response of the asynchronous no-load generator. Initially, the voltage increases exponentially as in the linear case, then it converges towards a constant value depending on the condenser value and the wind speed. Also, one notes that the stator voltage has a sinusoidal form (Figure 7).

4.png

Figure 4. Evolution of stator current at no load

5.png

Figure 5. Detail of the no-load stator current

6.png

Figure 6. No-load stator voltage evolution

7.png

Figure 7. Detail of the no-load stator voltage

Figure 8 shows the stator voltage curve of an asynchronous generator operating under load at the instant t = 5s. It can be observed that the amplitude of the voltage decreases when the load is applied (Figure 9).

8.png

Figure 8. Stator voltage with load

9.png

Figure 9. Detail of the stator voltage with load

The outputs to be set are:

$y=\left[\psi_{r} U_{d c}\right]^{T}$     (42)

$\left\{\begin{array}{c}S_{1}\left(\psi_{r}\right)=\psi_{r}-\psi_{r}^{{ }^{*}} \\ S_{2}\left(U_{d c}\right)=U_{d c}-U_{d c}{ }^{*} \\ S_{3}\left(i_{s d}\right)=i_{s d}-i_{s d}{ }^{*} \\ S_{4}\left(i_{s q}\right)=i_{s q}-i_{s q}{ }^{*}\end{array}\right.$      (43)

Then:

$i_{s d}^{*}=\frac{\tau_{r}}{L_{m}}\left(\frac{1}{\tau_{r}} \psi_{r}+\dot{\psi}_{r}^{*}\right)-K_{1} \operatorname{sign}\left(s_{1}\right)$       (44)

$i_{s q}=\frac{c U_{d c}}{w \psi_{r}} \frac{L_{r}}{L_{m}}\left(\frac{U_{d c}}{c R}+\dot{U}_{d c}^{*}\right)-K_{2} \operatorname{sign}\left(s_{2}\right)$     (45)

$U_{s d}^{*}=\sigma L_{s}\left(y_{1} i_{s d}-w_{s} i_{s q}-y_{2} \psi_{r}+\dot{I}_{s d}{ }^{*}\right)$ $-K_{3} \operatorname{sign}\left(s_{3}\right)$       (46)

$U_{s q}^{*}=\sigma L_{s}\left(w_{s} i_{s d}+y_{3} i_{s q}+w_{s} y_{4} \psi_{r}+\dot{I}_{s q}{ }^{*}\right)$$-K_{4} \operatorname{sign}\left(s_{4}\right)$      (47)

c: DC bus capacitor;

R: line resistance.

The detailed scheme of a sliding mode controller of a self-excited asynchronous generator integrated is illustrated in Figure 10, where the schema presents the control of the system using a sliding mode controller implemented with indirect field orientation control (IFOC) is applied. So, this command is consists of a voltage regulator of the contained bus, current regulators (Isd-Isq) and a block for calculating the angle θS. From the contained bus voltage regulator, we get the reference current Isq, so the current reference Isd is the image of the rotor flux. The reference voltages Vsd and Vsq are the images of the direct and quadrature currents Isd and Isq obtained by a sliding mode controller.

10.png

Figure 10. Control diagram by sliding mode applied to the asynchronous cage generator

In this work, the modeling and the control of a wind power system based on a self-excited asynchronous generator has been proposed. So, the proposed modeling of self-excited asynchronous generator for the purpose of optimizing the output of a wind turbine. The proposed sliding mode control method by controlling the dc link voltage, which is well known for its robustness, stability, simplicity and very low response time. Indeed, the obtained simulation results confirm that the sliding mode control has shown an improvement in the performance of the mechanical and electrical outputs of a self-excited asynchronous generator.