1 Introduction

In the mid − 1980s, the direct torque control strategy of induction machine was introduced by Takahashi to overcome the problems of conventional controls such as scalar control (SC) and field oriented control (FOC) [1, 2]. DTC is based on the direct regulation of electromagnetic torque and flux of the machine. This control is characterized by good torque dynamic response, high robustness and less complexity than other controls [3]. It allows minimizing the influence of parametric variations in the machine and calculating the control variables which are stator flux and electromagnetic torque from the stator current measurements without using mechanical speed sensors [4].

Recently, many authors have applied the DTC technique to the Doubly Fed Induction Motor connected to two-level voltage source inverters (2LVSIs) [2, 5, 6]. The researchers have pointed out the remarkable dynamic performance of this technique as well as its robustness to parametric variations in the motor. However, DTC suffers from major disadvantages such as (a) high torque and flux ripples that generate mechanical vibrations and undesirable acoustic noise, (b) variable switching frequency which causes switching losses, (c) and the negligence of stator and rotor resistances lead problems at low speed.

Several solutions have been proposed to minimize the ripples and to keep the frequency at constant value, for example: fuzzy logic controller (FLC), artificial neural network (ANN), space vector modulation (SVM) technique and predictive control (PC) have been applied to improve the conventional DTC of DFIM [7,8,9,10]. Nevertheless, the practical implementation of these methods is more complicated.

The voltage inverter is the most essential part of the DTC. It is well known that increasing the levels number of the inverter is a better solution in DTC drives [11]. The three-level voltage source inverter with NPC structure is more efficient in terms of its lower switching frequency, reduced stress across the semiconductors, less harmonic content, and lower voltage distortion compared to 2LVSI [12].

Although many studies about DTC for permanent magnet synchronous motors (PMSMs) and DTC for induction motors (IMs) using the multi-level inverter to increase the voltage vector number of switching table have been conducted individually [13, 14]. However, to the author’s knowledge, no study on DTC for doubly fed induction motor powered by three-level inverters is available. This gives us the opportunity to propose and design a DTC based on the use of 3LVSI with NPC structure to control the doubly fed induction motor for the first time in the literature.

The main contributions of this work are as follows:

  • The new DTC switching tables of the DFIM connected to two 3LVSIs with NPC structure is designed to give better performance, while retaining the merits of the conventional DTC namely robustness and simplicity.

  • The torque and flux ripples are reduced and the inverter switching frequency is mastered to limit the different problems of DFIM (mechanical vibration, acoustic noises, heating, ageing…).

  • The performance of the DFIM connected to two 3LVSIs is compared to that of the DFIM connected to two 2LVSIs so as to illustrate the improvements made.

This paper is structured as follows: section 2 presents the dynamic modeling of DFIM. Section 3 introduces the mathematical model of the three-level inverter with NPC structure and describes the DTC technique of DFIM based on two switching tables and hysteresis controllers. In section 4, the simulation results using the MATLAB/SIMULINK environment are presented and analyzed. Finally, Section 5 concludes the paper giving some comments and future directions.

2 Modeling of the DFIM

The power supply of the doubly fed induction motor is assured by two 3LVSIs [15]. Figure 1 illustrates the synoptic schema of the studied system.

Fig. 1
figure 1

Synoptic schema of studied system

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In the literature, for a better representation of the behavior of DFIM, it is necessary to use a specific and simple model. The two-phase model (α,β) given by the Concordia transformation is largely used for direct torque control [16].

The electric equations of DFIM in the reference frame (α,β) are given by [17]:

  • Stator voltages:

$$\left\{\begin{array}{c}{v}_{s\alpha }={R}_s.{i}_{s\alpha }+\frac{d{\psi}_{s\alpha }}{dt}\\ {}{v}_{s\beta }={R}_s.{i}_{s\beta }+\frac{d{\psi}_{s\beta }}{dt}\end{array}\right.$$
(1)

  • Rotor voltages:

$$\left\{\begin{array}{c}{v}_{r\alpha }={R}_r.{i}_{r\alpha }+\frac{d{\psi}_{r\alpha }}{dt}+{\omega}_m.{\psi}_{r\beta}\\ {}{v}_{r\beta }={R}_r.{i}_{r\beta }+\frac{d{\psi}_{r\beta }}{dt}-{\omega}_m.{\psi}_{r\alpha}\end{array}\right.$$
(2)
$$\mathrm{With}:{\omega}_s-{\omega}_r={\omega}_m$$

The magnetic equations of DFIM in the reference frame (α,β) are expressed by:

  • Stator flux:

$$\left\{\begin{array}{c}{\psi}_{s\alpha }={L}_s{i}_{s\alpha }+{L}_m.{i}_{r\alpha}\\ {}{\psi}_{s\beta }={L}_s{i}_{s\beta }+{L}_m.{i}_{r\beta}\end{array}\right.$$
(3)

  • Rotor flux:

$$\left\{\begin{array}{c}{\psi}_{r\alpha }={L}_r{i}_{r\alpha }+{L}_m.{i}_{s\alpha}\\ {}{\psi}_{r\beta }={L}_r{i}_{r\beta }+{L}_m.{i}_{s\beta}\end{array}\right.$$
(4)

The electromagnetic torque expression of DFIM as a function of the stator flux and stator currents is written as follows [18, 19]:

$${T}_{em}=p.\left({\psi}_{s\alpha }{i}_{s\beta }-{\psi}_{s\beta }{i}_{s\alpha}\right)$$
(5)

The fundamental equation of dynamics is:

$$J.\frac{d\Omega}{dt}+f.\Omega ={T}_{em}-{T}_r$$
(6)

3 Direct torque control strategy

3.1 DTC strategy for the DFIM connected to two 2LVSIs

The principle of DTC is based on the direct regulation of the torque and flux of DFIM, by applying the different voltage vectors of inverters (tow-level) [5]. The choice of these vectors is made using two switching tables and the hysteresis regulators whose role is to control the electromagnetic torque and flux of the motor in a decoupled manner. The output of these switching tables determines the optimal voltage vector of inverter to be applied at each switching instant.

The vector voltage expression of each two-level inverter can be given in the form below [2]:

$$V=\sqrt{\left(2/3\right).}{U}_{dc}.\left({S}_a+{S}_b{e}^{\left(j2\pi /3\right)}+{S}_c{e}^{\left(j4\pi /3\right)}\right)$$
(7)

With: Sa, Sb, Sc are switching logic states.

Figure 2 presents the set of voltage vectors delivered by each inverter.

Fig. 2
figure 2

Voltage vectors delivered by the two-level inverter

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In DTC, the accuracy of electromagnetic torque and flux estimation is very important to ensure satisfactory performance. The stator and rotor flux are estimated from the following eqs. [5]:

$${\widehat{\psi}}_s=\sqrt{{\widehat{\psi}}_{s\alpha}^2+{\widehat{\psi}}_{s\beta}^2}$$
(8)
$${\widehat{\psi}}_r=\sqrt{{\widehat{\psi}}_{r\alpha}^2+{\widehat{\psi}}_{r\beta}^2}$$
(9)

The electromagnetic torque is estimated from the measured stator currents:

$${T}_{em}=p.\left({\psi}_{r\alpha }{i}_{r\beta }-{\psi}_{r\beta }{i}_{r\alpha}\right)$$
(10)

With:

$$\left\{\begin{array}{c}{\widehat{\psi}}_{s\alpha }=\underset{0}{\overset{t}{\int }}\left({v}_{s\alpha }-{R}_s.{i}_{s\alpha}\right).dt\\ {}{\widehat{\psi}}_{s\beta }=\underset{0}{\overset{t}{\int }}\left({v}_{s\beta }-{R}_s.{i}_{s\beta}\right).dt\end{array}\right.\kern0.5em \left\{\begin{array}{c}{\widehat{\psi}}_{r\alpha }=\underset{0}{\overset{t}{\int }}\left({v}_{r\alpha }-{R}_s.{i}_{r\alpha}\right).dt\\ {}{\widehat{\psi}}_{r\beta }=\underset{0}{\overset{t}{\int }}\left({v}_{r\beta }-{R}_s.{i}_{r\beta}\right).dt\end{array}\right.$$
(11)

The positions of stator and rotor flux are determined by:

$${\theta}_s= arctg\left(\frac{{\widehat{\psi}}_{s\beta }}{{\widehat{\psi}}_{s\alpha }}\right)$$
(12)
$${\theta}_r= arctg\left(\frac{{\widehat{\psi}}_{r\beta }}{{\widehat{\psi}}_{r\alpha }}\right)$$
(13)

The estimated values of the electromagnetic torque and flux are compared respectively to their reference values; the comparison results form the inputs of hysteresis comparators [20]. The stator and rotor flux are controlled using the two-level hysteresis comparators, while the electromagnetic torque is controlled using a three-level hysteresis comparator. The flux space is divided into six sectors of 60° each. The selection of the appropriate voltage vector is based on the control table shown in Table 1 [7]. The inputs of this table are the flux sector number and the outputs of hysteresis comparators.

Table 1 Switching states of the two-level inverter
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3.2 DTC strategy for the DFIM connected to two 3LVSIs

The development of speed control and DTC of doubly fed induction motors has favored the use of three-level inverters. The increase in levels number of the latter proves to be a better solution in high power drives. The inverter is made up of switching cells, generally with transistors or GTO thyristors for large powers [21]. In this section, we present the study DFIM associated with two 3LVSIs with neutral point camped structure controlled by the DTC algorithm. Figure 3 illustrates the general schema of 3LVSI with NPC structure; it is one of the structures of three-level inverter. It has a lot of advantages, such as the higher number of voltage vectors generated, less harmonic distortion and low switching frequency [22]. Each arm of the inverter consists of 4 switches: Sk1, Sk2, Sk3, Sk4. The Sk1 and Sk2 have complementary operation.

Fig. 3
figure 3

Schema of three-level inverter with NPC structure

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On the control side, this converter topology offers the following advantages: high number of freedom degrees compared to the two-level inverter and reduced output current ripples.

The mathematical model of the 3LVSI is represented by the following matrix [23]:

$$\left[\begin{array}{l}{V}_a\\ {}{V}_b\\ {}{V}_c\end{array}\right]=\frac{1}{3}.{U}_{dc}.\left[\begin{array}{ccc}2& -1& -1\\ {}-1& 2& -1\\ {}-1& -1& 2\end{array}\right].\left[\begin{array}{l}{S}_{11}.{S}_{12}-{S}_{13}.{S}_{14}\\ {}{S}_{21}.{S}_{22}-{S}_{23}.{S}_{24}\\ {}{S}_{31}.{S}_{32}-{S}_{33}.{S}_{34}\end{array}\right]$$
(14)

With:

Va, Vb, Vc: Phase voltages.

Ski: Switching state.

Udc: DC bus voltage.

Each arm of the three-level inverter has three switching states shown in Table 2.

Table 2 Switching states of the three-level inverter
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The set of voltage vectors delivered by a three-level inverter are shown in Table 3, they are divided into 4 groups.

Table 3 Voltage vectors associated with switching states
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The 3LVSI produces 27 voltage vectors; some of them apply the same voltage vector. There are two possible configurations for each small vector and three for the zero vectors. Therefore, 19 different vectors are available in a three-level inverter [22]. The distribution of these voltage vectors in the reference frame (α, β) is shown in Fig. 4.

Fig. 4
figure 4

Voltage vectors of three-level NPC inverter

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The estimated values of the torque and flux are respectively compared to their reference values. The comparison results form the hysteresis regulators inputs. The stator and rotor flux are controlled using the three-level hysteresis comparators, while the electromagnetic torque is controlled using a five-level hysteresis comparator (Fig. 5). The flux space is divided into 12 sectors of 30° each.

Fig. 5
figure 5

Hysteresis comparators used to control flux and torque

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In order to realize the direct control of flux and torque of the DFIM connected to two three-level voltage inverters with NPC structure, we need to develop the switching table that optimizes the inverter possibilities. The construction of this table (Table 4) depends on the choice of voltage vector applied to increase or decrease the flux modulus and electromagnetic torque value.

Table 4 Switching table
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DTC scheme for the doubly fed induction motor connected to two 3LVSIs is illustrated in Fig. 6.

Fig. 6
figure 6

DTC strategy of DFIM connected to two 3LVSIs

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4 Simulation results and discusions

The direct torque control strategy of doubly fed induction motor was validated by numerical simulation using MATLAB/SIMULINK environment. The DFIM parameters used in the simulation are given in the Appendix.

The main features of this simulation are summarized as follows:

  • The widths of the hysteresis bands:

ΔTem1 = ±0.02 N.m, ΔTem2 = ±0.04 N.m, Δψs = ±0.001 N.m and Δψr = ±0.001 N.m.

  • The sampling frequency: fs = 10 kHz.

The simulation results of the DTC strategy of DFIM powered by two 2LVSIs are shown in Fig. 7.

Fig. 7
figure 7

Simulation results of DTC using 2LVSIs: (a) Rotation speed (b) Electromagnetic torque (c) Stator flux (d) Rotor flux (e) Stator flux trajectory (f) Rotor flux trajectory (g) stator and rotor flux sectors (h) Stator and rotor currents (i) FFT analysis of stator current isa (j) FFT analysis of rotor current ira (k) Switching state Sa

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Figure 7a shows the motor rotation speed. The speed reference changes from zero to 100 rad/s with specific acceleration rate (500 rad/s2), then it reduces to − 100 rad/s at t = 1 s. It can be seen that the speed reference has been properly tracked without any overshoot.

The load torque, torque reference and motor torque are represented in Fig. 7b. Here, the load torque is considered as disturbance, and the control method must track the speed reference independent of the load torque. The load torque changes from 0 Nm to 10 Nm at t = 0.5 s, then it decreases to 5 Nm at t = 1.5 s. The electromagnetic torque tracks its reference. Moreover, this torque has more ripples of 2.632 Nm.

Figure 7c and d show that the stator and rotor flux modulus follows perfectly its reference (1 Wb for stator flux, 0.5 Wb for rotor flux), as well as the evolution of these flux in (α,β) plan illustrated in Fig. 7e and f are perfectly circular.

Figure 7g represents the sectors repartition of stator and rotor flux in (α,β) plan, which shows that the DTC technique principle using 2LVSIs is ensured.

The stator and rotor currents of the DFIM are illustrated in Fig. 7h, the results show that the motor currents are sinusoidal. The harmonic spectra analysis of these currents is shown in Fig. 7i and j. Besides, the THD of stator current isa is equal to 8.75% and THD of rotor current ira is equal to 9.87%. The switching state of switch (Sa) of the two-level inverter is given in Fig. 7k. The switching frequency is variable, its average value is equal to 4 kHz.

The simulation results of the DTC strategy of DFIM connected to two 3LVSIs are shown in Fig. 8.

Fig. 8
figure 8

Simulation results of DTC using 3LVSIs: (a) rotation speed (b) electromagnetic torque (c) Stator flux (d) Rotor flux (e) Stator flux trajectory (f) Rotor flux trajectory (g) Stator and rotor flux sectors (h) Stator and rotor currents (i) FFT analysis of stator current isa (j) FFT analysis of rotor current ira (k) Switching state

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The results show the high performance of proposed method. From Fig. 8a, it can be seen that the rotation speed track its reference without overshoot, with a very low relative fall during the torque inversion, the release time is equal to 50 ms.

Figure 8b clearly shows good torque dynamics which perfectly follow its steady state reference with a fast response time. Moreover, the torque has small ripples of 0.972 Nm. From the analysis of Fig. 8c and d it follows that the stator and rotor flux modulus follows perfectly its reference with less ripples, as well as the evolution of these flux in (α,β) plan shown in Fig. 8e and f are perfectly circular.

Figure 8g represents the sectors repartition of the stator and rotor flux in (α,β) plan, there are 12 sectors which shows that the proposed method is ensured.

The components of the stator and rotor currents in reference frame (a,b,c) are shown in Fig. 8h, the motor currents are sinusoidal with a frequency proportional to the reference speed. These currents respond effectively to the variations imposed by load torque. Furthermore, Fig. 8i and j present the harmonic spectra analysis of the stator and rotor currents absorbed by DFIM, the results indicate that the total harmonic distortions of these currents are considerably reduced compared to the results obtained by DTC using 2LSVI (THD = 1.57% for stator current isa, THD = 1.52% for rotor current ira). The switching state of three-level inverter is given in Fig. 8k, the switching frequency is almost constant around 2.9 kHz, lower than that of DTC using 2LVSIs, which reduces switching losses.

Table 5 summarizes the comparative study between the two methods. The results of comparison show remarkable improvements obtained by DTC using 3LVSIs. These improvements include an important reduction in flux and torque ripples, as well as a minimization in the harmonics of the stator and rotor currents. Therefore, the proposed method provides better performance compared to DTC using 2LVSIs.

Table 5 Comparison between our proposal and DTC using 2LVSIs
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5 Conclusion

In this paper, we have presented the direct torque control for a doubly fed induction motor connected to two three-level voltage source inverters with NPC structure. The objective is to improve the motor performance. The DFIM modeling and DTC technical using 3LVSIs is developed in detail. Simulation results of the proposed method are comparatively analyzed against conventional DTC using 2LVSIs to confirm the effectiveness and superiority of the proposed control. The results analysis shows that the DTC using 3LVSIs provides better performance in terms of minimization of torque and flux ripples and optimization of currents distortion. This provides an opportunity for motor operation under minimum switching loss and noise. The experimental validation of the proposed method on a dSPACE controller board is considered future work.

5.1 Methods/experimental

This study presents a method for improving the performance of the direct torque control. The doubly fed induction machine is used as a motor connected to two three-level voltage source inverters. The system modeling and control are implemented in Matlab/Simulink environment. Comparison analysis of the proposed method with conventional DTC illustrates the advantages of this method.

6 Nomenclature

vs(α,β),vr(α,β) Stator and rotor voltages in the reference frame (α,β).

is(α,β), ir(α,β) Stator and rotor currents in the reference frame (α,β).

ψs(α,β), ψr(α,β) Stator and rotor flux in the reference frame (α,β).

Rs, Rr Stator and rotor resistances.

Ls, Lr Stator and rotor inductances.

M Mutual inductance.

P Pole pair number.

ωs, ωr Stator and rotor pulsations.

ω Mechanical pulsation.

Tr Load torque.

Tem Electromagnetic torque.

Ω Rotation speed of the machine.

J Moment of inertia.

f Coefficient of viscous friction.

θs, θr Position of the stator and rotor flux.

Udc DC bus voltage.