# Control of three-phase PWM rectifier using multiple frame dq transform and harmonic modeling

Maxime Grosso, Pierre Riedinger, Jamal Daafouz, Serge Pierfederici, Blaise Lapôtre, Hicham Janati Idrissi

## ► To cite this version:

Maxime Grosso, Pierre Riedinger, Jamal Daafouz, Serge Pierfederici, Blaise Lapôtre, et al., Control of three-phase PWM rectifier using multiple frame dq transform and harmonic modeling. 50th Annual Conference of the IEEE Industrial Electronics Society, IECON 2024, Nov 2024, Chicago, United States. hal-04866357

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https://hal.univ-lorraine.fr/hal-04866357v1

Submitted on 6 Jan 2025

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# Control of three-phase PWM rectifier using multiple frame dq transform and harmonic modeling

Maxime Grosso, Pierre Riedinger, Jamal Daafouz, Serge Pierfederici, Hicham Janati Idrissi and Blaise Lapôtre

**Abstract**—We introduce an innovative harmonic-model-based control strategy tailored for a three-phase AC-DC converter. Our method involves a nonlinear harmonic domain model, which proves highly effective in formulating harmonic control laws with a global stability guarantee. This framework not only accommodates control objectives associated with harmonic distortion but also facilitates the precise tracking of periodic trajectories. Experimental results are presented to showcase the methodology's effectiveness in reducing Total Harmonic Distortion (THD), thereby validating the successful accomplishment of the targeted control objectives.

**Index Terms**—AC DC conversion, Harmonic Control, Active Filtering, Dynamic Phasor, Extended Harmonic Domain.

## INTRODUCTION

In recent years, the growing demand for high power quality in electrical systems emphasizes the critical necessity of addressing harmonic distortion [1]. While various control techniques have been developed to tackle harmonic distortion challenges [2]–[6], it is noteworthy that traditional methods employed for mitigating harmonic distortion may entail a compromise on system dynamics. Harmonic modeling, including Dynamic Harmonic Domain (DHD) [7], Dynamic Phasor Modeling (DP modeling) [8], Generalized State-Space Average (GSSA) [9], and Harmonic State Space [10], effectively transforms periodic systems into time-invariant systems of infinite dimension. These methods simplify analysis by leveraging time-invariant principles. Nevertheless, linear controls within the harmonic domain, as in [10], [11], fall short of incorporating crucial stability properties as outlined in [12].

Here, we present a control approach for three-phase grid-tied AC-DC converters based on harmonic modeling techniques with global stability guarantees. Building on the key properties emphasized by [12] and leveraging tools tailored for the harmonic domain from [13], we present a rigorous framework for designing harmonic control strategies. Our approach ensures time-domain stabilization for bi-linear time periodic systems, with a specific emphasis on the critical "coincidence condition" highlighted by [12]. As a result, we derive a nonlinear model within the harmonic domain that proves highly effective in achieving harmonic control while ensuring global stability guarantees. Furthermore, this harmonic control design seamlessly transitions into a periodic nonlinear control scheme

in the time domain, preserving stability. Our framework caters to diverse control objectives, encompassing harmonic distortion reduction and precise tracking of periodic trajectories. It integrates integral action to achieve precise attenuation of harmonic content in the phase current. Leveraging the natural tools of multiple sub-dq decomposition, akin to the approach outlined in [14], [15], facilitates effective implementation of integral action within the harmonic modeling framework. To assess practical applicability, we employ an experimental bench to implement our control techniques and demonstrate real-time reduction of Total Harmonic Distortion (THD). The obtained results validate the successful achievement of control objectives, thereby offering strong confirmation for the effectiveness of the proposed harmonic control approach.

The paper is organized as follows: mathematical preliminaries are covered in Section I, harmonic modeling is detailed in Section II, control objectives are discussed in Section III, harmonic control design is presented in Section IV, and the practical implementation and experimental results are expounded upon in Section V.

**Notations:** $C^a$ denotes the space of absolutely continuous function, $L^p([t_1 t_2], \mathbb{C}^n)$ (resp. $\ell^p(\mathbb{C}^n)$) denotes the Lebesgue spaces of $p$-integrable functions on $[t_1 t_2]$ with values in $\mathbb{C}^n$ (resp. $p$-summable sequences of $\mathbb{C}^n$) for $1 \le p \le \infty$. $L_{\text{loc}}^p$ is the set of locally $p$-integrable functions. To simplify the notation, $L^p([t_1 t_2])$ or $L^p$ will often be used instead of $L^p([t_1 t_2], \mathbb{C}^n)$. The identity matrix of dimension $n$ is denoted $\text{Id}_n$, while the matrix of dimension $n$ whose entries are all equal to 1 is denoted $\mathbf{1}_n$. The infinite identity matrix is denoted $\mathcal{I}$. Finally, $\otimes$ refers to the Kronecker product.

## I. PRELIMINARIES

Let's begin by introducing the Fourier decomposition over a sliding window of length $T$ of a complex-valued function of time $x \in L_{\text{loc}}^2(\mathbb{R}, \mathbb{C})$. It is defined by the infinite sequence $\mathcal{F}(x) = X \in C^a(\mathbb{R}, \ell^2(\mathbb{C}))$ and whose components satisfy:

$$X_k(t) = \frac{1}{T} \int_{t-T}^{t} x(\tau) e^{-j\omega k\tau} d\tau$$

for $k \in \mathbb{Z}$, with $\omega = \frac{2\pi}{T}$. If $x = (x_1, \dots, x_n) \in L_{\text{loc}}^2(\mathbb{R}, \mathbb{C}^n)$ is a complex valued vector function, then $X = \mathcal{F}(x) = (\mathcal{F}(x_1), \dots, \mathcal{F}(x_n))$. $X_k$ is called the $k$-th phasor of $X$.

Additionally, we consider the Toeplitz transformation of a scalar function $x \in L_{\text{loc}}^2$, which is defined as an infinite-dimensional Toeplitz matrix given by:

M. Grosso, P. Riedinger, J. Daafouz and S. Pierfederici are with Université de Lorraine, CNRS, F-54000 Nancy, France. M. Grosso is also with Safran Electronics & Defense, Massy, France (email: name.surname@univ-lorraine.fr).

H. Janati-Idrissi and B. Lapôtre are with Safran Electronics & Defense, Massy, France (email: name.surname@safrangroup.com)
$$ \mathcal{T}(x) = \begin{bmatrix} \vdots & & \vdots & & \vdots \\ & x_0 & x_{-1} & x_{-2} & \vdots \\ \vdots & x_1 & x_0 & x_{-1} & \vdots \\ & x_2 & x_1 & x_0 & \vdots \\ \vdots & & \vdots & & \vdots \end{bmatrix}, \quad (1) $$

where $x_i, i \in \mathbb{Z}$ refer to the components of $X = \mathcal{F}(x)$. For a matrix function $A = (a_{ij})_{i=1,\dots,m,\ j=1,\dots,n}$, this extends to

$$ \mathcal{A} = \mathcal{T}(\mathcal{A}) = \begin{bmatrix} \mathcal{A}_{11} & \cdots & \mathcal{A}_{1n} \\ \vdots & & \vdots \\ \mathcal{A}_{m1} & \cdots & \mathcal{A}_{mn} \end{bmatrix}, \quad \text{with } \mathcal{A}_{ij} = \mathcal{T}(a_{ij}) \quad (2) $$

By invoking the so-called coincidence condition derived in [12], any n-dimensional system:

$$ \dot{x} = f(t, x) \quad (3) $$

having Caratheodory solution has a harmonic representation

$$ \dot{X} = \mathcal{F}(f(t, x)) - \mathcal{N}X \quad (4) $$

$$ \text{where } \mathcal{N} = \text{Id}_n \otimes \text{diag}(j\omega k, k \in \mathbb{Z}). \quad (5) $$

Conversely, a solution $X$ of (4) has a representative $x$ that satisfies (3) only if $X$ belongs to a well-defined subspace $H$ of $C^a(\mathbb{R}, \ell^2(\mathbb{C}^n))$ [12]. Moreover, $x$ is recovered from $X$ using the formula [12]:

$$ x(t) = \mathcal{F}^{-1}(X)(t) = \sum_{k=-\infty}^{+\infty} X_k(t)e^{j\omega k t} + \frac{T}{2}\dot{X}_0(t) \quad (6) $$

For polynomial systems, the term $\mathcal{F}(f(t, x))$ in (4) can be easily and explicitly determined using the following rule:

1) For a $\mathcal{A}$ matrix function ($\in L_{\text{loc}}^{\infty}$) and a vector $x \in L_{\text{loc}}^{2}$:

$$ \mathcal{F}(Ax) = \mathcal{T}(A)\mathcal{F}(x) = \mathcal{A}X \quad (7) $$

where $X = \mathcal{F}(x)$ and where $\mathcal{A} = \mathcal{T}(A)$.

2) For two matrix functions $A$ and $B$ ($\in L_{\text{loc}}^{\infty}$):

$$ \mathcal{T}(AB) = \mathcal{T}(A)\mathcal{T}(B) = \mathcal{A}B \quad (8) $$

For a $T$-periodic vector $x$ or matrix function $A$, both $X = \mathcal{F}(x)$ and $\mathcal{A} = \mathcal{T}(A)$ are constant. Consequently, if system (3) is $T$-periodic, the infinite-dimensional system (4) is time-invariant.

When considering a lowercase vector-valued function $x \in L_{\text{loc}}^2(\mathbb{R}, \mathbb{C}^n)$, we adopt the following convention: $X = \mathcal{F}(x)$ represents the Fourier transform of $x$, while $\mathcal{X} = \mathcal{T}(x)$ represents the Toeplitz form of $x$. By abuse of notation, we also use $\mathcal{T}(X)$ in place of $\mathcal{T}(x)$ to denote the function that constructs a Toeplitz matrix using the elements of $X = \mathcal{F}(x)$.

## II. HARMONIC MODELING OF THE CONVERTER

We recall the dynamical equations of the AC-DC converter depicted in Figure 1 in both the abc and dq reference frames. We consider a single $L_g$ filter with a line resistance $r_g$. The total current of the load is denoted $i_{\text{load}}$. We define $s_m$ as the switching variable for arm $m \in \{a, b, c\}$, taking values of 0 or 1. Hence, the associated duty cycle $d_m \in [0, 1]$ is the average value of $s_m$ over a switching period.

<figure><img src="image_2.png" /><figcaption>Fig. 1. A grid-connected AC/DC converter with a controlled non linear load</figcaption></figure>

### A. Average model

From [16], under abc reference frame, the model is given by:

$$ \left\{ \begin{array}{l} L_g \frac{d}{dt} i_{abc} = -r_g i_{abc} - C_{33} d_{abc} v_{dc} - e_{abc} \\ C \frac{d}{dt} v_{dc} = d'_{abc} i_{abc} - i_{\text{load}} \end{array} \right. \quad (9) $$

where the generic notation $x_{abc}$ stands for $x_{abc} = [x_a, x_b, x_c]^T$ and where $C_{33} = \text{Id}_3 - \frac{1}{3}\mathbf{1}_3$. This system reads:

$$ \dot{x} = Ax + G(x)d + Bv \quad (10) $$

where $x = [i_{abc}, v_{dc}]^T$, $v = [e_{abc}, i_{\text{load}}]^T$, $d = d_{abc}$ and

$$ A = \begin{bmatrix} -\frac{r_g}{L_g} & I_3 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \quad G(x) = \begin{bmatrix} \frac{C_{33}}{L_g} v_{dc} \\ \frac{i_{abc}}{C} \end{bmatrix}, \quad B = \begin{bmatrix} \frac{I_3}{L_g} & 0 \\ 0 & -\frac{1}{C} \end{bmatrix}. $$

Since the system is bilinear, we also define $A(d)$ such that $G(x)d = A(d)x$, that is:

$$ A(d) = \begin{bmatrix} 0 & \frac{C_{33}}{L_g} d \\ \frac{1}{C} d' & 0 \end{bmatrix}. \quad (11) $$

### B. Multiple frequency rotating frame

For the sake of harmonic analysis, it is useful to extend the dq transformation [16] to multiples of the fundamental grid frequency, as in [14]. Let us define $T_k$ the transformation matrices as:

$$ T_k = \frac{2}{3} \begin{bmatrix} \cos(k\theta) & \cos(k(\theta - \frac{2\pi}{3})) & \cos(k(\theta + \frac{2\pi}{3})) \\ \sin(k\theta) & \sin(k(\theta - \frac{2\pi}{3})) & \sin(k(\theta + \frac{2\pi}{3})) \end{bmatrix} \quad (12) $$

where $\theta = \omega t$ is the electric angle. Multiplying the 3 phase current $i_{abc}$, and assuming balanced operation, we obtain $i_{\text{dqk}} = T_k i_{abc}$. Notably if $i_{abc}$ has a non zero $k$-th phasor, then $\langle i_{\text{dqk}} \rangle_0 \neq 0$. We now extend system (9) with the output $i_{\text{dqk}}$, for $\mathbb{K}$ a finite list of positive integer corresponding to the order of targeted harmonics:

$$ i_{\text{dqk}} = T_k(t) i_{abc}, \quad k \in \mathbb{K} \subset \mathbb{N} \quad (13) $$

Regulating these outputs involves diminishing the harmonic content of the grid current. However, managing such a system is intricate, mainly due to the superposition of periodic terms $T_k(t)$. The approach proposed in [15] is based on PI controllers independently designed in each rotating frame. Subsequently, the contributions from these independent controllers are summed to collectively perform harmonic rejection.
### C. Harmonic model of the converter

Building upon the previous section, the harmonic model of (9) with output (13) and period $T = \frac{2\pi}{\omega}$ can be expressed as a time invariant bilinear system:

$$ egin{cases} \frac{d}{dt} I_{abc} = I_3 \otimes \left(-\frac{r_g}{L_g} \mathcal{I} - \mathcal{N}\right) I_{abc} + \frac{(C_{33} \otimes \mathcal{V}_{dc})}{L_g} D + \frac{E_{abc}}{L_g} \\ \frac{d}{dt} V_{dc} = -\mathcal{N}V_{dc} + \frac{\mathcal{I}_{abc}^*}{C} D - \frac{I_{load}}{C} \\ I_{dq_k} = \mathcal{T}_k I_{abc} \quad k \in \mathbb{K} \end{cases} \quad (14) $$

where $I_{abc} = \mathcal{F}(i_{abc})$, $D = \mathcal{F}(d_{abc})$ the harmonic domain switching control, $E_{abc} = \mathcal{F}(e_{abc})$, $I_{load} = \mathcal{F}(i_{load})$, $I_{dq_k} = \mathcal{F}(i_{dq_k})$, $\mathcal{I}_{abc} = \mathcal{T}(i_{abc})$, $\mathcal{V}_{dc} = \mathcal{T}(v_{dc})$, $\mathcal{T}_k = \mathcal{T}(T_k)$, constructed according to definitions in Section I. Notably, since $T_k$ are periodic matrices, only their $\pm k$-th phasors are non zero and:

$$ \langle T_k \rangle_k = \frac{1}{3} \begin{bmatrix} 1 & e^{-jk\frac{2\pi}{3}} & e^{jk\frac{2\pi}{3}} \\ e^{-\frac{\pi}{2}} & e^{j(-k\frac{2\pi}{3} - \frac{\pi}{2})} & e^{j(k\frac{2\pi}{3} - \frac{\pi}{2})} \\ e^{-\frac{\pi}{2}} & e^{j(-k\frac{2\pi}{3} - \frac{\pi}{2})} & e^{j(k\frac{2\pi}{3} - \frac{\pi}{2})} \end{bmatrix} $$

making $\mathcal{T}_k = \mathcal{T}(T_k)$ easy to express from (1) and (2).

This demonstrate how harmonic modeling allows us to build a single unified model that simultaneously integrates all $T_k$ transformations and enabling the synthesis of a single controller simultaneously addressing multiple harmonic objectives.

We will refer to the compact expression, as in (10), as:

$$ \dot{X} = (\mathcal{A} - \mathcal{N}) X + \mathcal{G}(X) D + \mathcal{B}V \quad (15) $$

$$ Y = CX \quad (16) $$

with $\mathcal{A} = \mathcal{T}(\mathcal{A}) = \begin{bmatrix} -\frac{r}{L} I_3 \otimes \mathcal{I} & 0 \\ 0 & 0 \end{bmatrix}$, $\mathcal{B} = \begin{bmatrix} \frac{I_3 \otimes \mathcal{I}}{L} & 0 \\ 0 & -\frac{\mathcal{I}}{C} \end{bmatrix}$, $\mathcal{G}(X) = \mathcal{T}(\mathcal{G}(x)) = \begin{bmatrix} \frac{C_{33} \otimes \mathcal{V}_{dc}}{\mathcal{I}_{abc}} \\ \frac{\mathcal{I}_{abc}}{C} \end{bmatrix}$, $X = (I_a, I_b, I_c, V_{dc}) = \mathcal{F}(x)$, $V = (E_{abc}, I_{load}) = \mathcal{F}(e_{abc}, i_{load})$. $Y$ and $\mathcal{C}$ in (16) are harmonic output and output matrices, stacking the sets of selected $I_{dq_k}$ and $\mathcal{T}_k$, further expended in next section.

### III. CONTROL OBJECTIVES FOR THE AC/DC RECTIFIER

We consider the following numerical values: $r_g = 1.15\Omega$, $L_g = 122\mu H$, $C = 100\mu F$, $R_L = 120\Omega$. The converter is fed with a 50Hz 45Vrms voltage on its AC side, and it is regulated to 150 Vdc for a nominal DC load current $i_{load} = \frac{150}{R_L} = 1.25A$.

#### A. Control objectives

The control objective is to maintain the DC bus voltage mean value at the given reference $v_{dc_{ref}}$ while simultaneously maximizing the power factor. This entails ensuring that the mean value of $i_q$ remains at 0, even in the presence of T-periodic DC or AC perturbations. Furthermore, to showcase the controller's ability to reject harmonics, we deliberately introduce a 150 Hz component (3-rd harmonics) in the DC side and tailor our controller to eliminate the resulting 2-nd and 4-harmonics in $i_{abc}$. To evaluate the effectiveness of the proposed control strategy, we analyze both the absolute value of the k-th phasor and the Total Harmonic Distortion (THD) (34). It is noteworthy that in steady state, the phasor aligns with the harmonics derived from classical Fourier analysis.

#### B. Nominal setpoint

The nominal setpoint is readily determined using dq model [16] in steady state, where arbitrary values for $v_{dc_{ref}}$ and $i_q = 0$ are considered. This yields the equilibrium values $(i_{dq}, v_{dc})^e$ and $d_{dq}^e$. Consequently, we easily deduce the corresponding periodic setpoint in the abc frame for both the time and harmonic domains as follows:

$$ V_{dc,0} = v_{dc,ref} \quad \text{and} \quad V_{dc,k} = 0 \quad \text{for} \ k \neq 0 \quad (17) $$

$$ I_{a,0} = 0, \quad I_{a,1} = \frac{1}{\sqrt{6}}(i_d - j i_q), \quad I_{a,k} = 0 \quad \text{for} \ k > 1 \quad (18) $$

$$ D_{a,0} = 0.5, \quad D_{a,1} = \frac{1}{\sqrt{6}}(d_d - j d_q), \quad D_{a,k} = 0 \quad \text{for} \ k > 1 \quad (19) $$

and $X_b = S_{-\frac{2\pi}{3}} X_a$, $X_c = S_{\frac{2\pi}{3}} X_a$, where $X$ stands for $I$ or $D$ and where the phase-shift operator is $S_\alpha = \text{diag}(e^{jk\alpha}, k \in \mathbb{Z})$. Similarly, the nominal harmonic inputs are given by:

$$ E_{a,0} = 0, \quad E_{a,1} = -\frac{E_{\text{rms}}\sqrt{2}}{2}, \quad E_k = 0 \quad \text{for} \ k > 1 \quad (20) $$

$$ I_{\text{load},0} = i_{\text{load}}, \quad I_{\text{load},k} = 0 \quad \text{for} \ k \neq 0 \quad (21) $$

Since the defined setpoint represents a T-periodic trajectory, it corresponds to an equilibrium $(X^e, D^e, V^e)$ for the harmonic system (15). Therefore, it satisfies the following relation:

$$ 0 = (\mathcal{A} - \mathcal{N})X^e + \mathcal{G}(X^e)D^e + \mathcal{B}V^e \quad (22) $$

Equivalently, in time domain, we have:

$$ \dot{x}^e = Ax^e + G(x^e)d^e + Bv^e \quad (23) $$

Having defined the control objectives and the setpoint, we define the relevant output $y = [v_{dc} \ i_q \ i_{dq_2} \ i_{dq_4}]'$ and the output

matrix $C(t) = \begin{bmatrix} 0 & 1 \\ T_1|_q & 0 \\ T_2 & 0 \\ T_4 & 0 \end{bmatrix}$, with $T_1|_q$ the second line of $T_1$. Let $\bar{x} = x - x^e$, $\bar{d} = d - d^e$, $A$ and $A(d)$ from (10) and (11). The error dynamics reads:

$$ \begin{cases} \dot{x} = [A + A(d^e(t))]\bar{x} + g(x)\bar{d} \\ y = C(t)\bar{x} \end{cases} \quad (24) $$

### IV. HARMONIC MODEL BASED CONTROL

We consider an integral action on $y$, with a (possibly periodic) gain matrix $L$, and reference signal $r$:

$$ \begin{cases} \dot{x} = [A + A(d^e(t))]\bar{x} + g(x)\bar{d} \\ \dot{z} = L(C(t)\bar{x} - r) \end{cases} \quad (25) $$

This system exhibits both nonlinearity and periodicity, making it non trivial to control in time domain [17]. Here we turn it into a time invariant infinite dimensional non truncated model, that allow us to use simple tools to design control.

Let $\bar{X} = \mathcal{F}(\bar{x}) = X - X^e$, $\bar{D} = \mathcal{F}(\bar{d}) = D - D^e$ and $\bar{D}^e = \mathcal{T}(D^e)$. The system (25) is expressed in harmonic domain as:

$$ \begin{cases} \dot{\bar{X}} = (\mathcal{A} + \mathcal{A}(\mathcal{D}^e) - \mathcal{N})\bar{X} + \mathcal{G}(X)\bar{D} \\ \dot{\bar{Z}} = -\mathcal{N}\bar{Z} + \mathcal{L}\mathcal{C}(\bar{X} - R) \end{cases} \quad (26) $$

$$ \text{where } \mathcal{A}(\mathcal{D}^e) = \begin{bmatrix} 0 & \frac{(C_{33} \otimes \mathcal{I}) \mathcal{D}^e}{L} \\ \frac{\mathcal{D}^e}{C} & 0 \end{bmatrix}. $$
Our goal is to design a harmonic domain state feedback law for the system described in (26), ensuring the fulfillment of all previously specified requirements.

### A. Control design

Drawing inspiration from [18], we leverage the concept of forwarding control in the harmonic domain. Consequently, we present the subsequent control law aimed at achieving global and asymptotic stability for the system (26) [12]:

$$D = D^e - \mathcal{H}_1 G^*(X) (\mathcal{P}\bar{X} - \mathcal{M}^*\mathcal{H}_2(Z - \mathcal{M}\bar{X})) \quad (27)$$

The quantites $\mathcal{P}$, $\mathcal{M}$, $\mathcal{H}_1$ and $\mathcal{H}_2$ are defined as follows. In our scenario, the matrix $(\hat{A} + \hat{A}(\hat{D}^e) - \hat{N})^*$ is Hurwitz. Given a symmetric and positive definite matrix function $Q \in L^\infty([0, T])$ and its transformation $\hat{Q} = \mathcal{T}(Q)$, $P$ is a Hermitian, positive definite, and bounded operator on $\ell^2$ solution of the Lyapunov equation:

$$(\mathcal{A} + \mathcal{A}(\mathcal{D}^e) - \mathcal{N})^* \mathcal{P} + \mathcal{P}(\mathcal{A} + \mathcal{A}(\mathcal{D}^e) - \mathcal{N}) + \mathcal{Q} = 0 \quad (28)$$

$\mathcal{M}$ is the Toeplitz solution of the harmonic Sylvester equation:

$$-\mathcal{N}\mathcal{M} - \mathcal{M}(\mathcal{A} + \mathcal{A}(\mathcal{D}^e) - \mathcal{N}) + \mathcal{L}\mathcal{C} = 0 \quad (29)$$

It is important to acknowledge that equations (28) and (29) are of infinite dimension. To address this, we employ efficient methods for solving such infinite-dimensional equations with arbitrary precision, as outlined in [13].

Moreover, employing $\hat{Q} = \mathcal{T}(Q)$ as a design parameter to regulate the relative aggressiveness of the transient for each state of the system, the variables $\hat{H}_{1,2} = \mathcal{T}(H_{1,2})$, where $H_{1,2}$ is any positive definite symmetric matrix function, introduce an additional layer of design parameters. This layer enables the fine-tuning of control aggressiveness, facilitating a balance between the importance of state-feedback and integral action in the feedback loop.

The global and asymptotic stability of the closed-loop system is established in the harmonic domain through the utilization of the Lyapunov Function:

$$W(\bar{X}) = \bar{X}^* \mathcal{P} \bar{X} + (Z - \mathcal{M} \bar{X})^* \mathcal{H}_2 (Z - \mathcal{M} \bar{X})$$

Ultimately, the suggested harmonic control (27) possesses a time-domain counterpart $\bar{d} = \mathcal{F}^{-1}(D)$ that maintains stability properties, as indicated by [12]. This ensures the achievement of control objectives outlined in III-A for the system (25). The expression for the time-domain counterpart is provided below:

$$d = d^e - \mathcal{H}_1 G'(x) (P\bar{x} - M'\mathcal{H}_2(z - M\bar{x})) \quad (30)$$

Notice that $\mathcal{H}_{1,2} = \mathcal{T}^{-1}(\mathcal{H}_{1,2})$ $P = \mathcal{T}^{-1}(\mathcal{P})$ and $M = \mathcal{T}^{-1}(\mathcal{M})$ are T-periodic gain matrices.

### B. Tuning of the control

In this section, we give insights into how the design parameters have been determined in our specific application.

* $Q(t)$ is selected as the constant positive definite matrix

$$Q = \begin{bmatrix} I_3 & 0 \\ 0 & \alpha \end{bmatrix} \text{ with } \alpha = 10^{-4}$$

* $\mathcal{H}_1$ is selected as a constant scalar value $\frac{1}{50}\bar{\sigma}(\mathcal{G}(X^e)^*\mathcal{P}) = 0.613$ where $\bar{\sigma}$ refers to the maximal singular value.

* C is defined in (24)

* $L$ is constructed as $\text{blkdiag}(\ell_{v_{dc}}, \ell_{i_q}, I_2\ell_2, I_2\ell_4)$ with $\ell_{v_{dc}} = 0.1$, $\ell_{i_q} = 0.8165$ and $\ell_{2,4} = 0.1143$

* $\mathcal{H}_2$ is chosen as the diagonal matrix $\text{blkdiag}(1, 0.1, I_4) \times \beta$ with $\beta = \frac{1}{H_1} \frac{1}{50} \bar{\sigma}(\mathcal{G}(X^e)^*\mathcal{M}^*\mathcal{M}) = 6.5531$

To gain a deeper understanding of the dynamic behavior resulting from the implemented control, we can examine the eigenvalues of the harmonic state matrix within the linearized harmonic closed-loop model, as in [19] or [13].

## V. PRACTICAL IMPLEMENTATION

The experimental validation is conducted using a setup consisting of a 3-phases grid connected to a controlled DC load. A 3-phase grid is replicated using a Chroma 61704 controllable three-phase power supply connected to Line inductance (122µH) and 3-phase 2-level voltage inverter. The control is implemented using MicrolabBox and SIMULINK. A TopCon Reversible DC Power Supply is connected on the DC side of the converter in parallele to a 120 Ω tunable resistance. The DC power supply is controlled through the Dspace MicroLabBox to act as a programmable current load/power sink. The high bandwidth of the TopCon allows for injecting chosen harmonics. Testec TT SI 9001 tension probes are used for direct measurements of AC voltage relative to the neutral point from the AC source.

We obtain an estimation $\hat{\theta}$ of the electric angle $\theta$ through a Phase-Locked Loop (PLL) with a sufficiently fast dynamic, allowing us to neglect its impact. This enable us to shift the matrix expression of $P$ (and $M$) from time/frequency dependency ($P(t) = \sum_{k \in \mathbb{Z}} P_k e^{jk\omega t}$) to a more appropriate phase angle description. Additionnaly, $P$ and $M$ are expressed as a finite sum (fixed by $h_{max}$) of cosine and sine terms:

$$P = P_0 + \sum_{k=1}^{h_{max}} P_{c,k} \cos(k\theta(t)) + P_{s,k} \sin(k\theta(t)) \quad (31)$$

$$P_{c,k} = 2 \text{Re}(P_k) \quad \text{and} \quad P_{s,k} = -2 \text{Im}(P_k)$$

The control is implemented in discrete time at a frequency of 20 kHz to remain in industrial standard as:

$$d_k = d^e(\hat{\theta}_k) - \mathcal{H}_1 G(x_k)^T P(\hat{\theta}_k)(x_k - x^e(\hat{\theta}_k)) + \mathcal{H}_1 G(x_k)^T M(\hat{\theta}_k)^T \mathcal{H}_2 M(\hat{\theta}_k) \times [z_k - M(\hat{\theta}_k)(x_k - x^e(\hat{\theta}_k))] \quad (32)$$

$$z_{k+1} = z_k + T_s LC(\hat{\theta}_k)(x_k - x^e(\hat{\theta}_k)) \quad (33)$$

A comparative analysis is conducted by implementing a PI-type controller in cascade form with a reference filter and a factor of 10 between the cutoff frequency of each closed loop. (Reference filter bandwidth: 60&600rad/s, voltage loop BW: 627.1rad/s, current loop BW: $6.2643 \times 10^3$rad/s).

In addition, to mitigate the undesired harmonic behavior a notch filter is incorporated with the PI controller, its purpose is to prevent harmonic propagation from the input toward current
<figure><img src="image_3.png" /><figcaption>Fig. 2. Transient from three-phase diode rectifier mode to reference setpoint: $v_{dc}$, $i_a$ (time and THD), $d_a$, two reference filter are compared</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Step input on $I_{load}$, signals $v_{dc}$, $i_a$, $i_{dq}$.</figcaption></figure>

reference and voltage feedforward signals. Here, we present 3 experimental situations:

1) Transient from three-phase diode rectifier mode toward three-phase controlled voltage rectifier mode (Figure 2);

2) Response to a step current input (Figure 3);

3) Response to a step load current with an additional 150 Hz harmonic component (Figure 4).

The presentation of the experimental results includes time domain acquisitions. For AC signals ($i_{abc}$, $d_{abc}$), THD calculation is performed using 25 post-computed phasors. For DC signals, we compute an Harmonic content quantity

$$THD(x) = \sqrt{\frac{\sum_{k=2}^{25} |X_k(t)|^2}{|X_1(t)|^2}} \quad HC(x)(t) = \sqrt{\sum_{k=1}^{k=25} |X_k(t)|^2}, \quad (34)$$

where $X_k(t)$ is the k-th phasor of x.

a) Start: Figure 2 illustrates the transient response from three-phase diode rectifier mode to controlled mode, toward the nominal DC Voltage. Figure 2 displays 1) $V_{dc}$, 2) $I_a$ and its THD, 3) the applied duty cycle $d_a$. Figure 2 displays the same level of dynamic performance, accompanied by significantly lower harmonic distortion on the phase current $i_a$. An additional noteworthy observation is that during transients, the PI loop tends to saturate, whereas our proposed control strategy achieves rapid regulation without any saturation. This highlights the effectiveness of harmonic control in achieving fast and stable regulation.

b) Constant perturbation rejection: As depicted in Figure 3, the system is initially operated with a constant current deviation of 1A from the nominal setpoint. From this new equilibrium point, a step current load of 4A is applied. As illustrated in Figure 3, the implementation of a PI-type control scheme leads to significantly higher harmonic content observed in both $I_a$ and $i_{dq}$, along with comparatively lower

performance in rejecting DC voltage disturbances compared to the harmonic control approach. Notably, due to unmodeled dynamics and non-linearity, some harmonics content exist on the DC bus.

c) Harmonic rejection: Finally, in the 3-rd experiment, again a deviation of 1A is initially applied and we introduce a step current to the load side addition with a third harmonic (150 Hz) component (4A DC, 1A rms at 150 Hz, see Figure 4a) and its multiples, this correspond to the design of section III with objectives on harmonics regulation of phasors 2-4. We consider a total of four control cases:

* PI: The conventional PI control scheme.

* PI Notch: The PI augmented with a 150 Hz Notch filter on the $i_{dq}$ reference and $u_{dq}^*$ feedforward signals

* Hmc 03: The harmonic control scheme of IV-B, $h_{max}$ is set to 4 in (31).

* Hmc 036: An augmented Hmc 03 targeting additionally 5-th and 7-th phasors on $i_{abc}$ (6-th phasor of $i_{load}$).

The current load is depicted in Figure 4a. We ensured consistent injection of approximately the same 0th and 3rd phasors in all four experiments. However, due to coupling and non-idealities in the sources, there are some multiples of 150 Hz present. Regarding the regulation of the DC bus voltage in Figure 4b, both harmonic control approaches exhibit a faster response. Finally, concerning the regulation of the harmonic component of phase current in Figure 4c, the Notch filter slightly attenuates the 2nd and 4th components. In contrast, Hmc03 effectively eliminates them, while Hmc036 goes a step further by completely eliminating phasors of orders 2, 4, 5, and 7. This underscores the effectiveness of Hmc036 in safeguarding the AC bus against DC bus disturbances.

d) Robustness issues: No significant degradation in experimental performance was observed when the parameters ($r_g$, $L_g$, $C$) in the model (10) were deviated by ±40% from their actual experimental values and synthesizing the controller
<figure><img src="image_5.png" /><figcaption>Fig. 4. Load side harmonic injection: DC BUS (a) $I_{dc,sink}$, (b) $V_{dc}$, (c) $I_a$</figcaption></figure>

based on these deviated parameters.

## CONCLUSION

This paper presents a novel control strategy for an AC/DC converter. We derived a harmonic model and designed a globally stabilizing periodic state feedback controller. Our method effectively tracks periodic trajectories and mitigates harmonic disturbances, as demonstrated in the experiments. The results show significant reductions in harmonic distortion and promising applications in energy management.

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