Advanced Terminal Voltage Control of Self-Excited Induction Generators in Variable-Speed Wind Turbines Using a Three-Level NPC Converter

The main elements of the studied system are: a wind turbine, a squirrel cage three-phase induction generator, autonomous load which is powered by a converter (3L_NPC) a DC side which includes a capacitor and a start-up battery provide the initial voltage to the capacitor and also to start the excitation process. The global scheme and control strategy of the system studied in this work is shows Figure 1.

1.png

Figure 1. System and DTC strategy scheme

2.1 Induction machine model

To implementation of DTC strategy, the dynamic model of model of the induction machine in the (α-β) frame should be established. The electrical equations of this model are written by following:

$\left[\begin{array}{c}V_{s \alpha} \\ V_{s \beta} \\ 0 \\ 0\end{array}\right]=\left[\begin{array}{cccc}R_s & 0 & 0 & 0 \\ 0 & R_s & 0 & 0 \\ -R_r & \omega_r \cdot l_r & R_T & -\omega_r \cdot\left(l_r+L_m\right) \\ -\omega_r \cdot l_r & -R_r & \omega_r \cdot\left(l_r+L_m\right) & R_r\end{array}\right] \cdot\left[\begin{array}{c}i_{s \alpha} \\ i_{s \beta} \\ i_{m \alpha} \\ i_{m \beta}\end{array}\right]+$$\left[\begin{array}{cccc}l_s & 0 & L_m+L_m^{\prime} \cdot \frac{i_{m \alpha}{ }^2}{\left|i_m\right|} & L_m^{\prime} \cdot \frac{i_{m \alpha} \cdot i_{m \beta}}{\left|i_m\right|} \\ 0 & l_s & L_m^{\prime} \cdot \frac{i_{m \alpha} \cdot i_{m \beta}}{\left|i_m\right|} & L_m+L_m^{\prime} \cdot \frac{i_{m \beta}{ }^2}{\left|i_m\right|} \\ -l_r & 0 & l_r+L_m+L_m^{\prime} \cdot \frac{i_{m \alpha}{ }^2}{\left|i_m\right|} & L_m \cdot \frac{i_{m \alpha} \cdot i_{m \beta}}{\left|i_m\right|} \\ 0 & -l_r & L_m^{\prime} \cdot \frac{i_{m \alpha} \cdot i_{m \beta}}{\left|i_m\right|} & l_r+L_m+L_m \cdot \frac{i_{m \beta}{ }^2}{\left|i_m\right|}\end{array}\right]$$\cdot$  $\left[\begin{array}{c}\frac{d i_{s \alpha}}{d t} \\ \frac{d i_{s \alpha}}{d t} \\ \frac{d i_{m \alpha}}{d t} \\ \frac{d i_{m \beta}}{d t}\end{array}\right]$   (1)

where, v_sα and v_sβ, representing the stator voltages, while i_sα and i_sβ represent the stator currents in in (α, β) reference frame.

The magnetizing currents, referred to as i_mα and i_mβ, are determined along the α and β axes respectively.

$\left\{\begin{array}{l}i_{m \alpha}=i_{s \alpha}+i_{r \alpha} \\ i_{m \beta}=i_{s \beta}+i_{r \beta}\end{array}\right.$   (2)

i_rα and i_rβ: represent the α-β rotor currents.

The magnetizing current i_m is defined as:

${{i}_{m}}=\sqrt{{{i}_{m\alpha }}^{2}+{{i}_{m\beta }}^{2}}$     (3)

ω_r: is the rotor angular speed.

With ${{\omega }_{r}}=p\Omega $     (4)

Besides, R_s, l_s, R_r and l_r are the stator and rotor phase resistances and leakage inductances respectively. L_m is the magnetizing inductance.

The saturation effect is taken into account by the expression of the magnetizing inductance L_m with respects to i_m with using a polynomial approximation, of degree 12 [6]. The L_mwith respects to i_m, is given by the following equation:

$\left\{ \begin{matrix}   {{L}_{m}}\,=\,f\left( \left| i_{m}^{{}} \right| \right)\,=\,\sum\limits_{j=0}^{n}{a_{j}^{{}}\,.\,\left| i_{m}^{{}} \right|_{{}}^{j}}\begin{matrix}   {} & {} & {}  \\\end{matrix}\begin{matrix}   {} & {} & {}  \\\end{matrix}  \\   {}  \\   {{L}_{m}}_{{}}^{'}\,=\frac{d{{L}_{m}}}{d\left| i_{m}^{{}} \right|}\,=\,\frac{d}{d\left| i_{m}^{{}} \right|}f\left( \left| i_{m}^{{}} \right| \right)\,=\,\sum\limits_{j=0}^{n}{j\,.\,a_{j}^{{}}\,.\,\left| i_{m}^{{}} \right|_{{}}^{j-1}}  \\\end{matrix} \right.$      (5)

The different parameters of the studied squirrel induction machine are given in appendix.

2.png

Figure 2. Schematic of a three-level NPC inverter

2.2 Three-level NPC converter and DC link mode

Figure 2 displays the schematic diagram of a 3L_NPC. This converter comprises a DC-link capacitor, twelve fully-controlled switches, each accompanied by a freewheeling diode, as well as two power diodes situated in each phase leg. These components enable the connection of the phase output to the midpoint of the DC bus. The twelve switches, organized in four switches per leg, as specified in Table 1 [15], allow the generation of three levels of output voltages: U_dc/2, 0, and -U_dc/2, through their combinations.

Table 1. Switching states

A connection function S_ji is defined for every switch T_ji:

$\left\{ \begin{matrix}   {{S}_{xi}}=1\,\,\,\,if\,\,\,{{k}_{xi\,\,\,\,}}closed  \\   {{S}_{xi}}=0\,\,\,\,if\,\,\,{{k}_{xi\,\,\,\,}}open  \\\end{matrix} \right.$      (6)

With j=a, b, c and i=1,2,3,4.

The following equation defines a connection function $F_j^h$  which is associated with every state h of the arm x [15]:

$\left\{ \begin{matrix}   F_{x}^{2}={{S}_{x1}}\,{{S}_{x2}}  \\   F_{x}^{1}={{S}_{x1}}\overline{\,{{S}_{x2}}}  \\   F_{x}^{0}=\overline{{{S}_{x1}}}\overline{\,{{S}_{x2}}}  \\\end{matrix} \right.$      (7)

The connection functions are used to express the three voltage levels as depicted in the following equation:

${{v}_{xo}}={{V}_{dc}}(F_{x}^{2}\,-F_{x}^{0})$       (8)

The voltages of the legs can subsequently be expressed as stated in the study [16]:

$\left[\begin{array}{c}v_a \\ v_b \\ v_c\end{array}\right]=\frac{V_{d c}}{3}\left[\begin{array}{ccc}2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2\end{array}\right]\left[\begin{array}{c}F_a^2-F_a^0 \\ F_b^2-F_b^0 \\ F_c^2-F_c^0\end{array}\right]$     (9)

The input currents of the three-phase inverter are expressed as follows:

$\left\{ \begin{matrix}   {{i}_{d2}}=F_{a}^{2}\,{{i}_{a}}+F_{b}^{2}\,{{i}_{b}}+F_{c}^{2}\,{{i}_{c}}  \\   \begin{align}  & {{i}_{d1}}=F_{a}^{1}\,{{i}_{a}}+F_{b}^{1}\,{{i}_{b}}+F_{c}^{1}\,{{i}_{c}} \\ & {{i}_{d0}}=F_{a}^{0}\,{{i}_{a}}+F_{b}^{0}\,{{i}_{b}}+F_{c}^{0}\,{{i}_{c}} \\\end{align}  \\\end{matrix} \right.$     (10)

The following equations presents relationship between the capacitor currents and the alternating currents:

$\left[ \begin{align}  & {{i}_{c1}} \\ & {{i}_{c2}} \\\end{align} \right]=\frac{1}{2}\left[ \begin{matrix}   F_{a}^{1} & F_{b}^{1} & F_{c}^{1}  \\   -F_{a}^{1} & -F_{b}^{1} & -F_{c}^{1}  \\\end{matrix} \right]\,\,\left[ \begin{matrix}   {{i}_{a}}  \\   {{i}_{b}}  \\   {{i}_{c}}  \\\end{matrix} \right]$      (11)

The equation provides the expression for the i_n current:

${{i}_{dc}}={{i}_{b}}-{{i}_{R}}-{{i}_{c}}$       (12)

With ${{i}_{c}}=C\frac{d{{V}_{dc}}}{dt}$       (13)

The equation below represents the DC voltage:

${{V}_{dc}}=-\int{\frac{1}{C}\left( {{i}_{dc}}+{{V}_{dc}}\left( \frac{1}{R}+\frac{1}{{{r}_{b}}} \right)-\frac{{{V}_{0}}}{{{r}_{b}}} \right)}$      (14)

where, V₀ denotes the initial voltage across the capacitor, equivalent to the voltage of the battery.

Table 2. DTC switching table 27 vectors

Upon the diode being in a blocked state, the DC voltage attains a value of U_dc≥U₀. Consequently, the DC current and voltage become, respectively:

${{i}_{dc}}=-{{i}_{R}}-{{i}_{c}}$     (15)

${{V}_{dc}}=-\int{\frac{1}{C}\left( {{i}_{dc}}+\frac{{{V}_{dc}}}{R} \right)}$      (16)

The selection of converter switching states can be made from a switching table (refer to Table 2) [11, 23]. This table determines the set of 27 optimal voltage vectors that need to be applied to the converter at each switching instant.

2.3 Direct torque control strategy

Direct torque control (DTC) aims to directly regulate the torque of the machine, by applying the various voltage vectors of the inverter. The controlled variables are the stator flux and the electromagnetic torque which are usually controlled by hysteresis regulators. It is a matter of keeping these two instantaneous quantities with in a band around the desired value [15]. The output of these regulators determines the optimal inverter voltage vector to be applied at each switching instant. The improvement of the DTC with the use of NPC is shown in this part, because this command in case the two-level converters are used, the error information of torque and flux are directly implemented to choose the switching state without distinguishing the degree between very large or relatively small error. This obviously produces an imprecise response, the performance of the system can be improved if the level degree of the inverters used is increased in order to have a wide range of selection of the voltage vectors based on the level of variation observed in the error values of torque and flux [15, 24]. Thus, the partition of the position of the flux under numerous zones (sectors), allows us to have a considerable efficiency of control at the level of the new switching algorithm. To ensure a more precise control, the space of flux evolution is divided into twelve sectors (1...12), each spanning 30 degrees. This selection is made with the intention of enhancing the overall control accuracy:

$\frac{-\pi }{12}+(i-1)\frac{\pi }{6}\le S(i)\prec \frac{\pi }{12}+(i-1)\frac{\pi }{6}$       (17)

The five-level hysteresis comparator is using for controlled the electromagnetic torque, while the stator flux is controlled by the three-level hysteresis comparators as shown in the following Figure 3.

3.png

Figure 3. The hysteresis controllers used to control the electromagnetic torque and stator flux

The inputs of hysteresis controllers are errors (eφ_sd & e_Tem) which are found after the comparison between the estimated values (φ_sd & T_em) of the stator flux amplitude and the electromagnetic torque respectively and their reference signals (φ_{sd_ref} & T_{em_ref}). while the outputs variables of the controllers are combined while the outputs variables of the controllers are combined with the position of the stator flux vector (Z_n) to form out the inputs to the switching table.

The set of voltage vectors delivered by a three-level converter (NPC) as well as the sequences of corresponding phase levels are represented by the space vector diagram as shown in Figure 4.

4.png

Figure 4. Space voltage vector of 3L-NPC inverter with their switching states [23]

The vector representation of the SEIG serves to highlight the dynamic control requirements for the electromagnetic torque of the induction machine. In order to achieve this, we present the electrical equations of the machine within the spatial vector [7]:

$\begin{align}  & \overline{{{V}_{s}}}={{R}_{s}}\overline{{{I}_{s}}}+\frac{d\overline{{{\Phi }_{s}}}}{dt} \\ & 0={{R}_{r}}\overline{{{I}_{r}}}+\frac{d\overline{{{\Phi }_{r}}}}{dt}-j\omega \overline{{{\Phi }_{r}}} \\\end{align}$     (18)

The voltage vector Vs, which is supplied by a 3L_NPC converter, can be expressed using the connection functions in the following form [7]:

$\overline{{{V}_{s}}}=\sqrt{\frac{3}{2}}{{V}_{dc}}({{S}_{a}}+{{S}_{b}}^{j\frac{2\pi }{3}}+{{S}_{c}}^{j\frac{4\pi }{3}})$      (19)

2.4 Stator flux and torque estimation

It is worthy noted that the stationary reference frame has using for estimating the stator flux and Tem from the expressions of stator current and stator voltage, which are given by the following equations [7]:

$\left\{ \begin{align}  & {{i}_{s\alpha }}=\sqrt{\frac{3}{2}}{{i}_{sa}} \\ & {{i}_{s\beta }}=\sqrt{\frac{1}{2}}({{i}_{sb}}-{{i}_{sc}}) \\\end{align} \right.$       (20)

$\left\{ \begin{align}  & {{V}_{s\alpha }}=\sqrt{\frac{3}{2}}{{V}_{dc}}({{S}_{a}}-\frac{1}{2}({{S}_{b}}-{{S}_{c}})) \\ & {{V}_{s\beta }}=\sqrt{\frac{1}{2}}{{V}_{dc}}({{S}_{b}}-{{S}_{c}}) \\\end{align} \right.$      (21)

The magnitude of the stator flux can be expressed as follows:

${{\Phi }_{s}}=\sqrt{\Phi _{s\alpha }^{2}+\Phi _{s\beta }^{2}}$      (22)

where,

$\left\{ \begin{align}  & {{\Phi }_{s\alpha }}=\int\limits_{0}^{t}{({{V}_{s\alpha }}-{{R}_{s}}{{i}_{s\alpha }})dt} \\ & {{\Phi }_{s\beta }}=\int\limits_{0}^{t}{({{V}_{s\beta }}-{{R}_{s}}{{i}_{s\beta }})dt} \\\end{align} \right.$     (23)

The equation of the electromagnetic torque is given from the stator flux $\left(\Phi_{s \alpha}, \Phi_{s \beta}\right)$ components and the stator current $\left(I_{s \alpha}, I_{s \beta}\right)$ as [7]:

${{T}_{em}}=p({{\Phi }_{s\alpha }}{{i}_{s\beta }}-{{\Phi }_{s\beta }}{{i}_{s\alpha }})$       (24)

In this work two strategies are studied, depends on the reference value of stator flux. The stator flux reference is effectively taken constant, equal to the nominal value, in the first strategy, i.e.:

${{\Phi }_{s\_ref}}={{\Phi }_{s\_nom}}=0.7Wb$     (25)

In the second technique, the reference of the stator flux is taken that is inversely proportional to the rotational speed, as determined by the following relationship:

${{\Phi }_{s\_ref}}=\frac{{{\omega }_{nom}}}{\omega }{{\Phi }_{s\_nom}}$     (26)

The objective of this paper is to enhance the performance of the SEIG in an autonomous wind system with variable speed. This is achieved by implementing direct torque control with the utilization of a three-level rectifier. The diphase analytical model of the induction generator, in the reference frame (α, β), is introduced taking into account the saturation effect by means of a variable magnetizing inductance. Two strategies of flux control are proposed; in the first strategy the flux is maintain constant. In the second strategy the stator flux is variable according to the driving speed.

The obtained simulation results demonstrate the effectiveness of the proposed (3L_NPC-DTC) control applied to the studied system. This control ensures better dynamic performance of a stand-alone induction generator and guarantees good regulation of the DC voltage and flux. Also, the studied system ensures significant reduction in torque and flux ripples with the two strategies, the THD of stator currents in the second strategy is considerably reduced in comparison with the first strategy.