Dynamic Modeling of a Six-To-Three-Phase
Indirect Matrix Converter and Performance
Evaluation Using Linear Quadratic Regulator

Chabi S. SANNI

UPJV MIS

ADEME

Amiens, France

salomon.sanni@unilasalle.fr

Augustin MPANDA MABWE

UniLaSalle B2R

UniLaSalle Amiens

Amiens, France

augustin.mpanda@unilasalle.fr

Ahmed EL HAJJAJI

UPJV MIS

UPJV

Amiens, France

ahmed.hajjaji@u-picardie.fr

***Abstract***— This work focuses on the interfacing of a hexaphase permanent magnet synchronous generator (PMSG) to a three-phase load network via an indirect matrix converter (IMC). Generally, three configurations can be deduced from a hexaphase system based on the voltage transfer ratio (VTR) defined between the converter output and input. This work concerns only the case of high VTR reaching 96.6%. The aim of this article is to extend this work by comparing PI and LQR controllers under severe dynamic conditions. Closed-loop simulations of the system feeding a 1.35 MW inductive load are performed, and an overall assessment of system performance for both controllers is presented.

***Keywords***— PMSG, Matrix converter (MC), Indirect Matrix Converter (IMC), Voltage Transfer Ratio (VTR), PI, Space Vector Modulation (SVM), Linear Quadratic Regulator (LQR).

# I. INTRODUCTION

With the aim of reducing dependence on fossil fuels to mitigate the effects of climate change, and to achieve carbon neutrality in the future, renewable energy sources are today booming in development and ubiquitous across the globe. In particular, wind energy conversion systems (WECS) make a substantial contribution to the renewable energy mix. Recently, direct-coupled, variable-speed wind power systems have emerged, incorporating advanced static converter technologies based on SIC MOSFETs. In the field of power electronics, matrix converters (MCs) have received a great deal of attention for interfacing to the power grid, as they are supplanting conventional back-to-back (B2B) converters. MCs offer numerous advantages, including sinusoidal input/output currents, unitary power factor, compactness and extended service life thanks to the elimination of passive elements such as capacitors.

Two types of matrix converters exist: the conventional, also known as direct configuration (DMC), and the indirect configuration (IMC). Numerous publications discuss the indirect modulation technique of the 3Φ - 3Φ matrix converter, which uses SVM control. In [1, 2] it is mentioned that the maximum VTR reaches 86%, enabling the IMC to operate effectively as a step-down voltage converter. In the development of SVM, it is essential to synchronize the rectifier and inverter stages to accurately determine the duty cycles of the switches.

Machines with more than three phases offer several inherent advantages, including a high power factor, improved fault tolerance, increased reliability, and reduced torque ripples. However, these polyphase systems necessitate

innovative converter structures for grid interfacing. Numerous research efforts have aimed at improving the VTR and overall efficiency of MCs to meet these requirements.

In the literature, few papers discuss the use of SVM on MCs for polyphase systems (3Φ-ΝΦ). In [3, 4] the VTR for a 3Φ-5Φ system is reported as 78.86%. In [5] for a 3Φ-7Φ system, a VTR of 76.93% is mentioned. For a 3Φ-9Φ system studied in [6], it showed a VTR ranging from 76% to 94.5%. Recent research on 3Φ-ΝΦ IMCs not only emphasizes topology and structural modifications but also focuses on suppressing common-mode voltage (CMV) to minimize electromagnetic interference that can disrupt the normal operation of other equipment [7, 8]. Additionally, there are studies on single or dual-stage phase to multiphase IMCs [9, 10]. Enhancing VTRs of MCs can be achieved by interchanging the rectifier and inverter stages or by using cascaded or series Z-source (ZS) networks, with series ZS showing better performance. The voltage across the cascaded ZS capacitor is substantial, necessitating the use of large capacitors [11-14]. Despite these developments, there is few ΝΦ-3Φ MCs. References [7, 8] focus on SVM development for a 5Φ-3Φ system applied to a DMC using the direct SVM approach, and in [17], a six-phase motor is controlled using a three-phase DMC.

In this publication, the system under study involves a hexaphase generator connected to an inductive load, featuring a 6Φ-3Φ IMC, as illustrated in Fig. 1. There is not ZS involved in the DC link as the objective is to suppress the capacitor which suffers multiple failures. Then the control of all stages rectifier and inverter are achieved through the application of the indirect SVM technique. In [18], a possible configuration derived from the hexaphase structure of the IMC is analyzed, along with the determination of the corresponding VTR. Then, in [19], the indirect technique of SVM is developed for all rectifier and inverter stages for a closed-loop system.

This paper introduces a dynamic model encompassing the entire system, consisting of the Permanent Magnet Synchronous Generator (PMSG), the IMC, input and output filters, and an inductive load. Additionally, it presents the development of the SVM technique tailored for the hexaphase IMC. In the closed-loop section, both PI and LQR control strategies are formulated. These control methods are specifically applied to the configuration exhibiting the highest VTR to assess their robustness, singularly under dynamic conditions such as voltage dips which can last less than a second and endure a voltage drop of 60% or less.
## II. SYSTEM DESIGN

The system understudy is depicted in Fig. 1 with a direct driven PMSG, and an inductive load connected to the IMC through the filters, LC on machine side and an LCL on grid or load side.

<figure><img src="image_1.png" /><figcaption>Fig. 1. System overview scheme without control part</figcaption></figure>

### A. PMSG Model

The dynamic model of the hexaphase PMSG in the d-q rotor reference frame is non-linear and modeled using the Vector Space Decomposition (VSD) approach. The VSD representation in (1) uses three orthogonal subspaces, namely the d-q subspace, ($z_1-z_2$) subspace and the zero-sequence subspace (omitted here).

$$ 
\left\{ 
\begin{aligned}
v_d &= -L_d \left( \frac{di_d}{dt} \right) - R_s i_d + \omega_c L_q i_q \\
v_q &= -L_q \left( \frac{di_q}{dt} \right) - R_s i_q - \omega_e (L_d i_d - \varphi_m) \\
v_{z_1} &= -L_0 \left( \frac{di_{z_1}}{dt} \right) - R_s i_{z_1} \\
v_{z_2} &= -L_0 \left( \frac{di_{z_2}}{dt} \right) - R_s i_{z_2} \\
\frac{d}{dt} \omega_m &= \left( \frac{1}{J} \right) (T_m - T_e - f \omega_m)
\end{aligned}
\right. 
\quad (1) 
$$

The PMSG's electromagnetic torque is calculated according to (2):

$$ T_e = 3p [(L_d - L_q)i_d + i_q \varphi_m] \quad (2) $$

With $p$ the number of pole pairs, $L_d, L_q$ the d-axis and q-axis stator total inductances, $L_0$ the stator leakage inductance, $\varphi_m$ is the permanent magnet flux linkage and $\omega_m$ is the rotor angular speed.

The generator torque can be controlled only by the stator q-axis current component. To achieve maximum output torque and highest efficiency, the d, $z_1$ and $z_2$ axis currents are set to zero.

### B. Indirect Matrix Converter Model

The investigated IMC structure is shown in Fig. 2. It has twelve bidirectional switches on rectifier side while on inverter stage, it has six single switches.

The system's input voltage and current vectors are described by expressions (3):

$$ 
\left\{ 
\begin{aligned}
V_{in} &= \left(\frac{1}{3}\right) \left( V_a + V_b e^{\frac{2\pi}{3}} + V_c e^{\frac{4\pi}{3}} + V_x e^{\frac{\pi}{6}} + V_y e^{\frac{5\pi}{6}} + V_z e^{\frac{3\pi}{2}} \right) \\
I_{in} &= \left(\frac{1}{3}\right) \left( I_a + I_b e^{\frac{2\pi}{3}} + I_c e^{\frac{4\pi}{3}} + I_x e^{\frac{\pi}{6}} + I_y e^{\frac{5\pi}{6}} + I_z e^{\frac{3\pi}{2}} \right)
\end{aligned}
\right. 
\quad (3) 
$$

The IMC rectifier stage is modeled based on (3) as shown in (4).

$$ 
\begin{bmatrix} I_a \\ I_b \\ I_c \\ I_x \\ I_y \\ I_z \end{bmatrix} = \begin{bmatrix} S_1 & S_2 \\ S_3 & S_4 \\ S_5 & S_6 \\ S_7 & S_8 \\ S_9 & S_{10} \\ S_{11} & S_{12} \end{bmatrix} * \begin{bmatrix} I_{DC+} \\ I_{DC-} \end{bmatrix} 
$$

$$ 
\begin{bmatrix} V_{DC+} \\ V_{DC-} \end{bmatrix} = \begin{bmatrix} S_1 & S_3 & S_5 & S_7 & S_9 & S_{11} \\ S_2 & S_4 & S_6 & S_8 & S_{10} & S_{12} \end{bmatrix} * \begin{bmatrix} V_a \\ V_b \\ V_c \\ V_x \\ V_y \\ V_z \end{bmatrix} \quad (4) 
$$

Output voltage and current vectors are of the form below (5):

$$ 
\left\{ 
\begin{aligned}
V_{\text{out}} &= \left(\frac{2}{3}\right) \left( V_A + V_B e^{\frac{2\pi}{3}} + V_C e^{\frac{4\pi}{3}} \right) \\
I_{\text{out}} &= \left(\frac{2}{3}\right) \left( I_A + I_B e^{\frac{2\pi}{3}} + I_C e^{\frac{4\pi}{3}} \right)
\end{aligned}
\right. 
\quad (5) 
$$

The IMC inverter stage is based on (5) and expressed in (6).

$$ 
\begin{bmatrix} V_A \\ V_B \\ V_C \end{bmatrix} = \begin{bmatrix} S_{13} & S_{14} \\ S_{15} & S_{16} \\ S_{17} & S_{18} \end{bmatrix} * \begin{bmatrix} V_{DC+} \\ V_{DC-} \end{bmatrix} 
$$

$$ 
\begin{bmatrix} I_{DC+} \\ I_{DC-} \end{bmatrix} = \begin{bmatrix} S_{13} & S_{15} & S_{17} \\ S_{14} & S_{16} & S_{18} \end{bmatrix} * \begin{bmatrix} I_A \\ I_B \\ I_C \end{bmatrix} \quad (6) 
$$

<figure><img src="image_2.png" /><figcaption>Fig. 2. Complete configuration of the hexaphase IMC</figcaption></figure>

### C. Modeling the Output Filter

<figure><img src="image_3.png" /><figcaption>Fig. 3. Output filter with the load</figcaption></figure>

#### - Load voltage

With the output filter represented in Fig. 3 and using KCL at node A yields (7):

$$ 
\left\{ 
\begin{aligned}
C \frac{dV_{od}}{dt} &= I_{id} - I_{od} + C\omega V_{oq} \\
C \frac{dV_{oq}}{dt} &= I_{iq} - I_{oq} - C\omega V_{od}
\end{aligned}
\right. 
\quad (7) 
$$

#### - IMC output current

Neglecting the current through the capacitor and considering the overall system, KVL allows to find (8):
$$
\left\{
\begin{array}{l}
L_f \frac{dI_{id}}{dt} = V_{id} - r_f I_{id} + L_f \omega I_{iq} - V_{od} \\
L_f \frac{dI_{iq}}{dt} = V_{iq} - r_f I_{iq} - L_f \omega I_{id} - V_{oq}
\end{array}
\right.
\qquad (8)
$$

with L<sub>f</sub>=L<sub>1</sub>+L<sub>2</sub> and r<sub>f</sub>=r<sub>1</sub>+r<sub>2</sub>

### III. SVM TECHNIQUE FOR 6Ø-3Ø IMC

#### A. Active vectors of the inverter stage

The number of possible combinations of phase interconnections is eight (2³), with six active vectors and two zero vectors. The inverter complex plane is shown in Fig. 4.

<figure><img src="image_4.png" /><figcaption>Fig. 4. Complex plane of the inverter stage</figcaption></figure>

#### B. Active vectors of the rectifier stage

The number of available combinations of phase interconnections is thirty-six (6²), thirty active vectors and six null vectors.

With the rectifier active vectors, two complex planes are defined. The first complex plane is composed by 12 active vectors while the second complex plane has 18 active vectors (Fig. 5 and Fig. 6).

<figure><img src="image_5.png" /><figcaption>Fig. 5. Voltage Source Rectifier 12 active vectors complex plane</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 6. Voltage Source Rectifier 18 active vectors complex plane</figcaption></figure>

#### C. Rectifier and Inverter Configurations with corresponding VTR

As MCs operate in buck mode, it is crucial to determine the maximum voltage ratio achievable by the system using the complex planes defined for both the rectifier and inverter stages. The overall complex planes define three configurations for developing SVM control with each configuration corresponding to the association of inverter complex plane

and a set of vectors selected from the rectifier complex planes (case 1 to case 3):

**Doalty vectors (Case 1):** Use of the 12 active vectors of the Dodecagon (Fig.5) for the control with the magnitude of |D|=0.5774Vdc.

The hexagon with 18 active vectors (Fig. 6) leads to 3 main groups of active vector amplitudes which are:

Large: |L|=0.6440Vdc, Medium: |M|=0.4714Vdc and Small: |S|=0.1725Vdc.

**Sx vectors (Case 2):** Use of the six large active vectors.

**Tw vectors (Case 3):** Use of the six large and six medium active vectors.

Table I summarises the VTR for the three configurations.

<table><caption>TABLE I
VTR RELATED TO CONFIGURATIONS</caption><thead><tr><th>Configuration</th><th>m<sub>c</sub></th><th>m<sub>v</sub></th><th>VTR</th></tr></thead><tbody><tr><td>3φ-3φ</td><td>I<sub>ref</sub>/I<sub>d</sub></td><td></td><td>86.6%</td></tr><tr><td>6φ-3φ<br>Do vectors (Case 1)</td><td>3.464 I<sub>ref</sub>/I<sub>d</sub></td><td></td><td>50%</td></tr><tr><td>6φ-3φ<br>Sx vectors (Case 2)</td><td>1.793 I<sub>ref</sub>/I<sub>d</sub></td><td>√3 V<sub>ref</sub>/V<sub>d</sub></td><td>96.6%</td></tr><tr><td rowspan="3">6φ-3φ<br>Tw vectors (Case 3)</td><td>m<sub>c</sub>=2.196 I<sub>ref</sub>/I<sub>d</sub></td><td></td><td rowspan="3">78.9%</td></tr><tr><td>m'<sub>c</sub>=2.45 I<sub>ref</sub>/I<sub>d</sub></td><td></td></tr><tr><td>m''<sub>c</sub>=1.793 I<sub>ref</sub>/I<sub>d</sub></td><td></td></tr></tbody></table>

For further work the Sx vectors (Case 2) configuration is used, as its maximum transfer ratio is close to 100%.

### IV. CLOSE LOOP CONTROL OF THE SYSTEM

In this section, the PI and LQR controls are discussed.

#### A. PI corrector

The PI corrector is the simplest and most widely used in control engineering [20 to 22].

##### a) Rectifier stage

The external loop is controlled by the MPPT method to extract the maximum energy supplied by the turbine. The reference currents of the generator's d, z₁ and z₂ axes are set to zero, while that of the q axis is determined by MPPT torque control. The transfer function of the system in each axis is in the form of a first-order system (9), and the closed-loop control system corresponding is shown in Fig. 7 to Fig. 9.

$$H(s)=\frac{1}{sL+R} \qquad (9)$$

<figure><img src="image_7.png" /><figcaption>Fig. 7. PI control system for the d axis of the inner loop</figcaption></figure>
- *Quadrature Axis*

<figure><img src="image_8.png" /><figcaption>Fig. 8. PI control system for the q axis of the inner loop</figcaption></figure>

- *z₁ and z₂ Axis*

<figure><img src="image_9.png" /><figcaption>Fig. 9. PI control system for the z axis of the inner loop</figcaption></figure>

b) Inverter stage

The General inverter control is cascaded with an external loop and an internal loop. With the system being in islanding mode, the grid forming control is performed. The grid-forming power converter is controlled in closed loop to work as ideal ac voltage source with a set of references V* for the voltage, and ω* for the frequency.

The set of relation (7) is used to control the external loop which regulates the load voltage and (8) for the internal loop which regulates the output currents of the IMC, and whose references are obtained from the output of the external loop as shown in Fig. 10 and Fig. 11.

<figure><img src="image_10.png" /><figcaption>Fig. 10. External stage loop control</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 11. Internal stage loop control onduleur</figcaption></figure>

B. LQR control development

In the control field, LQR is a method allowing to calculate
the gain matrix of a state feedback control system. Articles [23
to 26] explain LQR control implementation.

The state system is in the form:

$$
\begin{cases}
\dot{x} = Ax + Bu + Ew \\
y = Cx
\end{cases}
\tag{10}
$$

With controllable matrices *A* and *B*.

x the state variable, u the control, w the disturbance and y the
output.

A performance index (11) is defined to achieve optimum state
variable feedback gain.

$$
J = \int_{0}^{\infty} x^{\mathrm{T}} Q x + u^{\mathrm{T}} R u \, \mathrm{d}t \quad (11)
$$

With $Q \in \mathbb{R}^{n*n}$ et $R \in \mathbb{R}^{m*m}$, $n$ the number of state variables and $m$ the number of commands.

The Q and R matrices are used to adjust the weights of the x-
states and the u-control. They are real symmetrical matrices,
positive semidefinite and positive definite respectively.

The optimal control which stabilizes and minimizes J is
expressed by the relation:

$$
K = R^{-1} B^T P \quad (12)
$$

With matrix P a solution of the Riccati matrix equation:

$$
A^T P + PA - PBR^{-1} B^T P + Q = 0 \quad (13)
$$

Optimum control is obtained as:

$$
u = -Kx \tag{14}
$$

with $K \in \mathbb{R}^{m*n}$

1) Determining optimal control gain

To provide a zero steady-state error for the controlled
variables, the system is synthesized using LQR optimal
control with a system augmented by the state vector.

By adding the error system to the control, we obtain:

$$
\begin{cases}
\dot{x} = Ax + Bu + W\varphi_m \\
\dot{\varphi} = y_{\text{ref}} - y = y_{\text{ref}} - Cx
\end{cases}
\tag{15}
$$

The augmented state vector is:

$$
\dot{X} = \begin{bmatrix} \dot{x} \\ \dot{\epsilon} \end{bmatrix} = \begin{bmatrix} Ax+Bu+Ew \\ y_{\text{ref}}-Cx \end{bmatrix} \quad (16)
$$

The new state matrix form is as follows:

$$
\dot{X} = A_1 X + B_1 U + E_1 W + B_2 Y_{\text{ref}} \quad (17)
$$

With:

$$
A_1 = \begin{bmatrix} A & 0 \\ -C & 0 \end{bmatrix}; B_1 = \begin{bmatrix} B \\ 0 \end{bmatrix}; E_1 = \begin{bmatrix} E \\ 0 \end{bmatrix}; B_2 = \begin{bmatrix} 0 \\ I \end{bmatrix}
$$

And optimal control:

$$
\begin{align}
U &= -KX \nonumber \\
U &= -K_1 \int e \, dt + K_p x \tag{18}
\end{align}
$$

To obtain the values for the optimum gain K, we use
MATLAB's LQR function.

$$
K = \operatorname{lqr}(A_1, B_1, Q, R) \quad (19)
$$

2) Applying LQR to our system

a) Rectifier stage

The state system of the stator equations in (1) is in the form:

$$
\begin{cases}
\dot{x} = Ax + Bu + W\varphi_m \\
y = Cx
\end{cases}
\tag{20}
$$

$$
\text{With: } x = [i_d \ i_q \ i_{z1} \ i_{z2}]^T; u = [v_d \ v_q \ v_{z1} \ v_{z2}]^T
$$

$$
A = \begin{bmatrix}
-\frac{R_s}{L} & \omega_e & 0 & 0 \\
-\omega_e & -\frac{R_s}{L} & 0 & 0 \\
0 & 0 & -\frac{R_s}{L_0} & 0 \\
0 & 0 & 0 & -\frac{R_s}{L_0}
\end{bmatrix}; \quad C = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix};
$$
$$B = \begin{bmatrix} -\frac{1}{L} & 0 & 0 & 0 \\ 0 & -\frac{1}{L} & 0 & 0 \\ 0 & 0 & -\frac{1}{L_0} & 0 \\ 0 & 0 & 0 & -\frac{1}{L_0} \end{bmatrix}; \quad W = \begin{bmatrix} 0 \\ \frac{1}{L} \\ 0 \\ 0 \end{bmatrix}$$

b) Inverter stage

Using (7) and (8), the system of state equations of the inverter stage is in the form:

$$\begin{cases} \dot{x} = Ax + Bu + Ew \\ y = Cx \end{cases} \quad (21)$$

With:

$$x = [i_d \ i_q \ v_{od} \ v_{oq}]^T; \quad u = [v_{id} \ v_{iq}]^T; \quad w = [i_{od} \ i_{oq}]^T$$

$$A = \begin{bmatrix} -\frac{r_f}{L_f} & \omega & -\frac{1}{L_f} & 0 \\ -\omega & -\frac{r_f}{L_f} & 0 & -\frac{1}{L_f} \\ \frac{1}{C_f} & 0 & 0 & \omega \\ 0 & \frac{1}{C_f} & -\omega & 0 \end{bmatrix}; \quad C = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix};$$

$$B = \begin{bmatrix} \frac{1}{L} & 0 \\ 0 & \frac{1}{L} \\ 0 & 0 \\ 0 & 0 \end{bmatrix}; \quad E = \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ -\frac{1}{C_f} & 0 \\ 0 & -\frac{1}{C_f} \end{bmatrix}$$

V. SIMULATION RESULTS

For closed-loop control, the PI and LQR correctors are applied to the hexaphase system with the S<sub>x</sub> vectors configuration for a duration of t=1.5s. When the system reaches steady state, a 50% voltage dip is applied at t=0.8s for 100ms to test the controllers robustnesses.

<table><caption>TABLE II
SIMULATION PARAMETERS</caption><tr><td>PMSG</td><td colspan="2">S<sub>n</sub>=1.5MVA, Poles=80, R<sub>s</sub>=3.17mΩ, L=3.07 mH, J=10000 kg.m<sup>2</sup>, ϕ<sub>m</sub>=7.0172Wb, f=0</td></tr><tr><td>Turbine</td><td colspan="2">Rayon=35.25m, ρ=1.225 (kg/m<sup>3</sup>), V<sub>ω</sub>=11m/s</td></tr><tr><td>Input filter</td><td>L<sub>f</sub>=0.534mH</td><td>C<sub>f</sub>=0.0053F</td></tr><tr><td>Load</td><td>R<sub>c</sub>=0.463Ω</td><td>L<sub>c</sub>=713.37μH</td></tr><tr><td>Output filter</td><td>L<sub>o</sub>=6.1mH</td><td>C=278μF &nbsp;&nbsp;&nbsp; L<sub>o</sub>=0.165mH</td></tr></table>

Fig. 12 to Fig. 14 show the wave forms of signals. Fig. 15 compares system performance with the PI and LQR controllers under a voltage dip fault. The THDs of the input current and output voltage are illustrated in Fig. 16 and Fig. 17 for both PI and LQR controllers with the latest outperforming.

<figure><img src="image_12.png" /><figcaption>Fig. 12. PMSG side input signals</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 13. Load side output signals</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 14. d and q axis load voltage control</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>(b)</figcaption></figure>

<caption>Fig. 15. Performance evaluation of the system for PI and LQR controllers
(a) Voltage Rms and (b) Current and Voltage THDs</caption>

The analysis of Fig. 15 (a) shows that the LQR control gives better results compare to PI with rms values closer to the reference voltage of 400V despite the dip voltage. According to the recovery time, the PI corrector looks faster. However, Fig. 14 shows that this corrector is inoperative during the disturbance. Fig. 15 (b) proves the robustness of the LQR controller, which totally mitigate the fault with zero voltage overshoot, and current and voltage outputs THDs below 5%.

<figure><img src="image_17.png" /><figcaption>Fig. 16. Peak and THD values of input current</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>Fig. 17. Peak and THD values of output voltage</figcaption></figure>

CONCLUSION

In this study, we conducted dynamic modeling of a
hexaphase PMSG and its six-to-three phase Indirect Matrix
Converter (IMC). Utilizing the SVM method, which achieved
a peak VTR of 96.6%, we compared the performance of PI
and LQR controls under voltage drop conditions. The
simulation results revealed a notable difference in THDs
between the PI corrector and the LQR, with the latter
demonstrating greater robustness with lower drop and by
maintaining THDs below 5%. Although the LQR control
exhibited a longer recovery time, this can be adjusted through
the Q and R parameters, albeit with a risk of voltage overshoot
and increased system instability. Future work will focus on the
real-time implementation of the PI, LQR, and Sliding Mode
Control (SMC) controllers, with results to be published in
upcoming papers.

ACKNOWLEDGMENT

This work was supported by the French Environment
and Energy Management Agency (ADEME) and the
Hauts-de-France Region.

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