# General Design Method of Modified Damping Assignment-PBC in Shipboard Evolutive Microgrids

Emeric Vuillemin, Jean-Philippe Martin, Mohamed Machmoum, Serge

Pierfederici, Farid Meibody-Tabar, Mathieu Weber

► To cite this version:

Emeric Vuillemin, Jean-Philippe Martin, Mohamed Machmoum, Serge Pierfederici, Farid Meibody-Tabar, et al.. General Design Method of Modified Damping Assignment-PBC in Shipboard Evolutive Microgrids. IEEE Transactions on Industrial Electronics, In press, IEEE Transactions on Industrial Electronics, 10.1109/TIE.2025.3569906. hal-05120188

HAL Id: hal-05120188

https://hal.science/hal-05120188v1

Submitted on 19 Jun 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# General Design Method of Modified Damping Assignment-PBC in Shipboard Evolutive Microgrids

Emeric Vuillemin, Jean-Philippe Martin, Mohamed Machmoum, Serge Pierfederici, Farid Meibody-Tabar, Mathieu Weber

*Abstract*—Passivity Based Control (PBC) is an appealing solution to control power converters while ensuring the stability of more and more complex and modular electrical systems. However, the original IDA-PBC (Interconnection and Damping Assignment-PBC) controller design method generates singularities in the control-variables leading to their saturation and spikes of current. This paper highlights this non-desired phenomenon by applying the original IDA-PBC to the control of a boost converter. A novel controller design is then proposed to remove the singularities of the control-variables while preserving the proof of passivity of the whole controlled system. The proposed method introduces a new damping assignment approach to tune the performances of the system. This design can be applied to a specific sub-class of Port Controlled Hamiltonian (PCH) systems which includes the average models of different converters. To highlight the generalization of the proposed method, it is applied to a 5th-order 2kW 2-input Modular Three-Level Boost Converter (MTL-BC), and validated experimentally. The advantage of the controller is presented for a Microgrid System (MGS) application compared to other conventional control strategies and how it improves the stability margin in the presence of a Constant Power Load (CPL).

*Index Terms*—Interconnection and Damping Assignment-Passivity Based Control (IDA-PBC), Large-signal Stability, Modular Three-Level Boost Converter (MTL-BC), Port Controlled Hamiltonian (PCH).

## I. INTRODUCTION

THE project HYSysPEM of France 2030 aims to propose new power system architectures to decarbonize the maritime transport. Fuel Cell (FC) is considered as the main alternative to fossil fuels [1]–[3]. Based on the use cases presented in [4], [5], a typical DC-Microgrid System (MGS) structure for maritime application is presented in Fig. 1 with every elements interconnected together through power converters. Because of the high-power level required in such application, the power is segmented between several FCs in a multi-stack system [6]. One focus of the project HYSysPEM is

Received 6 September 2024; revised 28 November 2024, 30 January 2025, 11 April 2025; accepted 5 May 2025. This research was funded by the French National Research Agency (ANR) under France 2030, grant ANR-22-PEHY-0018. A CC-BY 4.0 public copyright license has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission, in accordance with the grant's open access conditions. ©① (Corresponding author: Emeric Vuillemin)

Emeric Vuillemin and Jean-Philippe Martin and Farid Meibody-Tabar and Mathieu Weber and Serge Pierfederici are with Université de Lorraine, CNRS, LEMTA, F-54000 Nancy, France (e-mail: emeric.vuillemin@univ-lorraine.fr; jean-philippe.martin@univ-lorraine.fr; serge.pierfederici@univ-lorraine.fr; farid.meibody-tabar@univ-lorraine.fr; mathieu.weber@univ-lorraine.fr)

Mohamed Machmoum is with Nantes University, IREENA Laboratory, 37 Boulevard de L'université BP 406, 44602 Saint-Nazaire, France (e-mail:mohamed.machmoum@univ-nantes.fr)

<figure><img src="image_2.png" /><figcaption>Fig. 1. Example of shipboard evolutive DC-MGS.</figcaption></figure>

to propose modular association of stacks either to give more degrees of freedom during the system's design phase, or to enable optimal power management during operation [7], [8]. Eventually, unplanned reconfiguration is possible if one of the sub-systems fails and has to be disconnected. Similar analysis can be done regarding other elements of the MGS. In all these conditions, the stability of the MGS is mandatory to ensure safe, reliable and optimal operations.

Many criteria and methods exist to evaluate the stability of MGS which are often non-linear systems. Small-signal approaches such as impedance specification and Lyapunov's indirect method only assess the local stability of a linearized system. Large-signal techniques such as Takagi-Sugeno approach [9] or Lyapunov's direct method enable the evaluation of global stability of non-linear systems. However, they are limited by their complexity for high-order systems. Moreover, a new study can be needed if the architecture of the model is modified. Passivity is a promising system's characteristic based on dissipative systems theory [10] as such systems are globally stable, minimizing an energy function. Furthermore, as long as passive sub-systems are interconnected together, the passivity of the complete system is ensured, proving its global stability. Nevertheless, the equilibrium point unlikely matches the design objectives with desired dynamics. Passivity Based Control (PBC) aims to control the system such that the energy function is reshaped to meet the design specifications and ensuring the passivity property of the controlled system.

The authors of [11] were ones of the first to apply PBC design method to control power converters using Port Controlled Hamiltonian (PCH) formalism which describes a class of non-linear systems. By using Interconnection and Damping Assignment (IDA) to design the closed-loop system it is possible to tune the desired dynamics, the damping level and, indeed, the stability. The attractive properties of IDA-PBC encouraged the use of this method for MGS applications either for DC [12], [13] and AC [14] grids.

As IDA-PBC is model based, it suffers from the unknown system's disturbances leading to steady-state errors. To solve
this issue, an integral action was first proposed in [15] by the original authors of PBC. Similar technique was later applied to DC-MGS in [16] and for FC control in [17]. PBC is often used in hybrid strategies with sliding mode [18] or proportional integral [19] to improve robustness and performances.

Nonetheless, the stability of a MGS, which interconnects several sources, Energy Storage System (ESS) and active loads through power converters individually controlled with the previously mentioned methods, is not always guaranteed for every type of interconnection. The authors of [14], [20] solve this concern with a modified IDA-PBC which preserves passivity as long as the power converters are controlled with their proposed method and are interconnected together via passive elements. The authors proposed to adapt the design method from [21] by firstly adding a disturbance observer [22] to remove steady-state error and secondly by including the electrical inputs and outputs of the system in the PCH formulation so the passivity propagates between interconnected subsystems. As long as power converters are controlled with the IDA-PBC proposed in [20], the stability of a modular MGS is ensured. However, when the method is applied to the model of power converters, the expression of the control-variables contains a singular region where saturations of duty-cycle and fast variations of state-variables are observed.

The main contribution of this paper is to propose a novel IDA-PBC design method which does not comprise the above mentioned singular region. A generalized calculation procedure is presented for application on many average model of power converters. The method is validated experimentally to control a high-order Modular Three-Level Boost Converter (MTL-BC) [23] featuring multi-input sources and multi-level DC-bus which is best suited in shipboard mid-voltage MGS applications. The advantage of the novel IDA-PBC in MGS stability is compared with a hybrid-PBC and a two-loop control strategy. Finally, the paper highlights how the damping of IDA-PBC improves the stability margin in the presence of a Constant Power Load (CPL). The paper is structured as follows:

Section II recalls the original IDA-PBC design method. It is then applied to control a boost converter, highlighting the impact of the singular region.

Section III presents the novel IDA-PBC design method removing the singular region while preserving the strong results on passivity propagation. Then the proposed IDA-PBC is applied in section IV on the model of a 2-input MTL-BC to prove its validity on a high order converter.

Experimental validation is performed in section V on a 2kW MTL-BC to show the efficient control of state-variables. The proposed controller is compared with the original IDA-PBC [20]. Robustness analysis is conducted.

Section VI compares the stability effectiveness of IDA-PBC with other control strategies in the context of unmodeled architecture of MGS and in the presence of a CPL.

Section VII summarizes the achievements of this paper and introduces remaining research topics of the proposed IDA-PBC method for future work.

## II. ORIGINAL IDA-PBC DESIGN METHOD

### A. General method for PCH systems

This section introduces the IDA-PBC design method from [20] which provides a way to design the control of power converters allowing propagation of the passivity property. The PCH formalism used to describe non-linear systems is modified to include electrical inputs and outputs as in (1):

$$ egin{cases} \dot{x} = [J(x, u_{int}) - R(x)] \cdot \frac{\partial H}{\partial x}(x) + g_{ext} \cdot U_{ext} + g_{\gamma} \cdot \gamma \\ y_{ext} = g_{ext}^{\top} \cdot \frac{\partial H}{\partial x}(x) \\ H(x) = \frac{1}{2} x^{\top} Q x \end{cases} \quad (1) $$

where $x$ is the vector of state-variables of the system. $J(x, u_{int})$ is a skew-symmetric matrix defining interconnections of the state-variables. The internal control variables vector, $u_{int}$, is included inside the interconnection matrix. $R(x) \ge 0$ is the damping matrix modeling the losses. The function $H$ represents the energy stored in the system, here with $Q$ diagonal. $U_{ext}$ and $y_{ext}$ are vectors representing respectively electrical inputs and outputs used for external interconnections. It is assumed that all sub-system interconnections are done and modeled through these vectors. Additional disturbances are modeled in the vector $\gamma$. The matrices $g_{ext}$ and $g_{\gamma}$ link the corresponding vectors to the state-variables.

It is easily found, by assuming $\dot{\gamma} = 0$, that a PCH system respect the passivity criteria in (2). This inequality is closely linked to Lyapunov stability as it includes its direct method in the case of autonomous systems with $U_{ext} = 0$.

$$ \dot{H}(x) \le U_{ext}^{\top}(t) \cdot y_{ext}(t) \quad (2) $$

The IDA-PBC provides closed-loop equation of the controlled system to preserve the PCH formalism while achieving the desired damping and objectives by re-shaping the energy function and internal interactions such that:

$$ egin{cases} \dot{e}_x = [J_d - R_d] \cdot \frac{\partial H_d}{\partial e_x}(e_x) + g_{ext} \cdot (U_{ext} - U_{ext}^*) \\ y_{ext} - y_{ext}^* = g_{ext}^{\top} \cdot \frac{\partial H_d}{\partial e_x}(e_x) \\ H_d(x) = \frac{1}{2} e_x^{\top} Q e_x \end{cases} \quad (3) $$

where $e_x = x - x^*$ is the state error and $x^*$ is the value of state-variables in steady-state. $H_d$ is the new energy function which is minimum at $x^*$. The matrices $J_d$ and $R_d$ define the IDA of the closed-loop system. $U_{ext}^*$ represents the steady-state value of input signals and is extracted using a low-pass filter with a cut-off frequency largely slower than the system's dynamics. Finally, $\dot{\gamma}$ is used to observe the vector of additional disturbances $\gamma$. The disturbance observer is added to the system description with the following equation:

$$ egin{cases} \dot{\hat{x}} = [J - R] \cdot \frac{\partial H}{\partial x}(x) + g_{ext} \cdot U_{ext} + g_{\gamma} \cdot \dot{\hat{\gamma}} - k_s(\hat{x} - x) \\ \dot{\hat{\gamma}} = -\eta_1(\hat{x} - x) - k_p(\dot{\hat{x}} - \dot{x}) + \eta_2 \end{cases} \quad (4) $$
<figure><img src="image_3.png" /><figcaption>Fig. 2. Schematic of the studied boost converter.</figcaption></figure>

where $\hat{x}$ is the vector of observed state-variables. The gain matrices $k_s$, $k_p$, $\eta_1$ and $\eta_2$ are built such that the passivity with respect to the electrical inputs and outputs of the system is guaranteed using Lyapunov's function as proposed in [20]. The observer estimates the additional losses outside of model scope in order to remove the steady-state error. It is assumed that the disturbances have slow variations compared to state-variables.

Finally, by solving for the control variables vector $u_{int}$ in (3) the system behaves as designed in closed-loop. Moreover, based on the results from [20], the global system resulting in the interconnection of such controlled sub-systems and other passive elements remains passive.

## B. Application to boost converter control

The method described in [20] is applied to control a boost converter with the corresponding schematic on Fig. 2 and its associated state-equation in (5), where $D$ is the duty-cycle of the gate signal $u$ for the transistors.

$$ \underbrace{\begin{bmatrix} \dot{I} \\ \dot{V} \end{bmatrix}}_x = \underbrace{\begin{bmatrix} \frac{-r_L}{L^2} & \frac{D-1}{LC} \\ \frac{1-D}{LC} & 0 \end{bmatrix}}_{J-R} \underbrace{\begin{bmatrix} LI \\ CV \end{bmatrix}}_{Q \cdot x} + \underbrace{\begin{bmatrix} \frac{V_{in}}{L} \\ \frac{-I_{out}}{C} \end{bmatrix}}_{g_{ext} \cdot U_{ext}} + \underbrace{\begin{bmatrix} \frac{-\hat{\gamma}_v}{L} \\ \frac{-\hat{\gamma}_i}{C} \end{bmatrix}}_{g \cdot \gamma \cdot \gamma} \quad (5) $$

The objective is to control the output voltage $V$ of the boost converter with a single-loop control strategy so it converges to its reference value $V^*$. The reference inductor current $I^*$ is found by solving the state-equation (5) in steady-state.

In this application, the number of control variables is lower than the order of the system, which mean the size of $u_{int}$, is smaller than the number of state. It leads to an over-determined closed-loop equation. As proposed in [20], an additional coefficient $K$ is injected in $J_d$ which gives the following closed-loop matrices used in (3):

$$ J_d = \begin{bmatrix} 0 & \frac{-K}{LC} \\ \frac{K}{LC} & 0 \end{bmatrix}; R_d = \begin{bmatrix} \frac{R_i}{L^2} & 0 \\ 0 & \frac{R_v}{C^2} \end{bmatrix} \quad (6) $$

where $R_i$ and $R_v$ are coefficient for the damping assignment. The control-variable $D$ and the additional unknown $K$ are obtained by solving the closed-loop equation. The duty-cycle solution is given in (7):

$$ D = \frac{e_i \cdot [V - R_i e_i - V_{in}^* + r_L I + \hat{\gamma}_v + L \dot{I}^*]}{IV^* - VI^*} + \frac{e_v \cdot [I_{out}^* - R_v e_v - I + \hat{\gamma}_i + CV^*]}{IV^* - VI^*} \quad (7) $$

with $x^* = [I^* \ V^*]^T$, $e_i = I - I^*$, $e_v = V - V^*$ and $U_{ext}^* = [V_{in}^* \ I_{out}^*]^T$. A use case application is proposed with the parameters of Table I to visualize the duty-cycle function in Fig. 3. We observe that the line of equation $V = V^*/I^* \times$

<table><caption>TABLE I
PARAMETERS FOR THE BOOST CONVERTER SIMULATION.</caption><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>Electrical inputs</td><td>$V_{in} = 60V$; $I_{out} = 5A$</td></tr><tr><td>Inductor</td><td>$L = 300\mu H$; $r_L = 100m\Omega$</td></tr><tr><td>Output capacitor</td><td>$C = 235\mu F$</td></tr><tr><td>Damping assignment (6)</td><td>$R_i = 2\Omega$; $R_v = 10\mu\Omega^{-1}$</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Duty-cycle plot with the singular region.</figcaption></figure>

$I$ gives a zero denominator for the duty-cycle in (7). This singular region leads to a saturated duty-cycle, out of the $[0; 1]$ boundary, with the left and right side of the line diverging to $+\infty$ and $-\infty$ respectively. Moreover, the equilibrium point is part of the singular line, but the duty-cycle is determined using l'Hospital's rule [24] because the numerator also converges to zero. In all other cases, l'Hospital's rule cannot be applied.

An average model simulation is performed to highlight the impact of the singular region on Fig. 4 outside of the equilibrium point. The controller is tested on an increasing voltage reference trajectory. Since the derivative of the reference $\dot{V}^*$ is used to compute the duty-cycle, the step trajectory is made differentiable using a second-order filter. On Fig. 4a, the derivative is set to $\dot{V}^* = 0$ and good tracking performances are visible. A second simulation is performed including the reference derivative on Fig. 4b. Uncontrolled values of duty-cycle are observed during the reference transient leading to spikes of inductor current.

Theoretically, including the voltage reference derivative provides better trajectory tracking of the equilibrium point, while not being exactly on it. As seen on Fig. 3, the singular region is in the neighborhood of the equilibrium point. Then the duty-cycle oscillates between in and out of bounds values

<figure><img src="image_5.png" /><figcaption>Fig. 4. Average model simulation of a boost converter with IDA-PBC comparing usage of the derivative of voltage reference.</figcaption></figure>
during the transient of the reference as it tries to control state-variables. This paper proposes a novel approach detailed in section III which removes the discussed singularities. All the strong proof of stability for the disturbance observer and passive subsystems interconnection are preserved.

### III. NOVEL IDA-PBC DESIGN METHOD

#### A. Description of the sub-class of PCH systems

The novel method proposed in this article is based on a sub-class of PCH systems, in which many power electronics average model are included. This section describes the state-equation formalism considered. From (1) we can split the interconnection matrix as $J = J_0 + J_{u_{int}}$ giving the new open-loop equation:

$$ \dot{x} = [J_0 + J_{u_{int}} - R] Qx + g_{ext}U_{ext} + g_{\gamma}\gamma \quad (8) $$

where $J_0$ is the part of $J$ without control variables while $J_{u_{int}}$ is a linear combination of them. As such it is possible to write:

$$ J_{u_{int}} = \sum_{i} \frac{\partial J}{\partial u_{int,i}} u_{int,i} \quad (9) $$

The proposed method of this article bases its calculation on PCH systems written as follows:

$$ \dot{x} = \left( J_0 + \sum_{i} \frac{\partial J}{\partial u_{int,i}} u_{int,i} - R \right) Qx + g_{ext}U_{ext} + g_{\gamma}\gamma \quad (10) $$

#### B. Control variables splitting

By further splitting control variables as $u_{int} = u_{int}^* + u_{int}^c$, where $u_{int}^*$ is used to find steady-state variables and $u_{int}^c$ defines closed-loop dynamic, we can write:

$$ \left\{ \begin{aligned} \dot{x} &= JQe_x + (J_0 + J_{u_{int}}^* + J_{u_{int}}^c) Qx^* - RQx \\ & \qquad +g_{ext}U_{ext} + g_{\gamma}\gamma \\ J_{u_{int}}^* &= \sum_{i} \frac{\partial J}{\partial u_{int,i}} u_{int,i}^* \\ J_{u_{int}}^c &= \sum_{i} \frac{\partial J}{\partial u_{int,i}} u_{int,i}^c \end{aligned} \right. \quad (11) $$

Finally, by setting:

$$ (J_0 + J_{u_{int}}^*) Qx^* = RQx - g_{ext}U_{ext}^* - g_{\gamma}\hat{\gamma} + \dot{x}^* \quad (12) $$

$$ J_{u_{int}}^c Qx^* = -R_d Qe_x \quad (13) $$

where (12) is the new steady-state equation to solve to achieve the minimum of the energy function $H_d$ at $x^*$. Section IV gives an example by applying the method to a 2-input MTL-BC. Equation (13) is the damping assignment further detailed in section III-C. The final expression of the closed-loop system is similar to (3) but with $J_d = J$:

$$ \dot{e}_x = (J - R_d) Qe_x + g_{ext}(U_{ext} - U_{ext}^*) + g_{\gamma}(\gamma - \hat{\gamma}) \quad (14) $$

In a MGS application where each element shares the same close-loop formalism, with correct choice of the observer's parameters (4) according to [20], we can write the following passivity property:

$$ \dot{V}^i = \dot{H}_d^i < (U_{ext}^i - U_{ext}^{i*})^\top \cdot (y_{ext}^i - y_{ext}^{i*}) \quad (15) $$

where the superscript $i$ represents the i-th element of the MGS. Since all ports of the PCH are compatible in closed-loop as they are in open-loop (same unit with a shift of origin) we can write for an autonomous MGS ($U_{total} = 0$):

$$ V_{total} = \sum_{i} \dot{V}^i < \sum_{i} (U_{ext}^i - U_{ext}^{i*})^\top \cdot (y_{ext}^i - y_{ext}^{i*}) = 0 \quad (16) $$

which confirms that a MGS is stable as long as it includes passive components and converters controlled with the proposed IDA-PBC.

#### C. Generating a symmetric damping matrix

Directly solving the damping assignment of (13), especially if arbitrary values are selected, is not straight forward. This section proposes a design method that generates a symmetric damping matrix $R_d$ with easy setting of its eigenvalues to respect the definite-positiveness of the damping matrix for PCH model of the closed-loop system (3). We introduce for each control variable an expression which depends linearly on every state error as follows:

$$ \left\{ \begin{aligned} u_{int,i}^c &= r_i p_i^\top e_x \\ p_i &= [p_{i,1} \quad \dots \quad p_{i,n}]^\top \end{aligned} \right. \quad (17) $$

where the scalar $r_i$ is a freedom degree for damping assignment. The vector $p_i$ gives the coefficient for control dependence on error. Eventually, the damping assignment equation (13) can be written as:

$$ J_{u_{int}}^c Qx^* = \sum_{i} r_i \underbrace{\frac{\partial J}{\partial u_{int,i}} \cdot Qx^* \cdot p_i^\top Q^{-1} \cdot Qe_x}_{-R_{d,i} = r_i A_i (p_i)} \quad (18) $$

with $R_{d,i}$ the damping matrix associated to $u_{int,i}$ and $A_i$ an intermediate matrix for calculations. The complete damping matrix $R_d$ is the sum of the $R_{d,i}$. The objective is to make $R_{d,i}$ symmetric using the elements of the vector $p_i$. As such, we need to achieve $A_i = A_i^\top$:

$$ \frac{\partial J}{\partial u_{int,i}} \cdot Qx^* \cdot p_i^\top Q^{-1} = -Q^{-1} p_i x^{i*} Q \cdot \frac{\partial J}{\partial u_{int,i}} \quad (19) $$

with $Q$ diagonal. Solving this equation leads to a relation between each element of $p_i$ to make $A_i$ and $R_{d,i}$ symmetric. The method is tested on several power converters' model on which solving (19) leads always to at least one solution. Finally, by studying the eigenvalues of $R_{d,i} = -r_i A_i$, we can choose $r_i$ to obtain a definite-positive damping matrix.

### IV. VALIDATION OF THE PROPOSED METHOD ON A MTL-BC

#### A. MTL-BC system description

The proposed IDA-PBC design method is applied on the average model of a MTL-BC. The converter is presented in [23] proposing a modular association of Three-Level Boost Converters (TL-BCs) with a modified output-series connection by sharing a capacitor between two subsequent TL-BC. The MTL-BC share the advantages of the cascaded association of power converters [8], such as high voltage conversion ratio and reduced voltage stress, while improving both controllability in unbalanced input power scenarios and availability. These are
<figure><img src="image_6.png" /><figcaption>Fig. 5. Schematic of the studied 2-input MTL-BC.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6. Global control schematic of the proposed IDA-PBC for the MTL-BC.</figcaption></figure>

key features to fulfill for multi-stack maritime applications. The high order of the system make it a good candidate to show the validity of the proposed generalized method.

Schematic of the studied 2-input MTL-BC is given in Fig. 5 and its associated state-equation in (20). To simplify the system average model, the state-equation is expressed with $Q\dot{x}$:

$$Q\dot{x} = \underbrace{\begin{bmatrix} -r_{L1} & 0 & D_{1t} - 1 & D_{1b} - 1 & 0 \\ 0 & -r_{L2} & 0 & D_{2t} - 1 & D_{2b} - 1 \\ 1 - D_{1t} & 0 & 0 & 0 & 0 \\ 1 - D_{1b} & 1 - D_{2t} & 0 & 0 & 0 \\ 0 & 1 - D_{2b} & 0 & 0 & 0 \end{bmatrix}}_{Q \cdot (J-R) \cdot Q} \cdot x + \underbrace{\begin{bmatrix} V_{in1} \\ V_{in2} \\ -I_{out} \\ -I_{out} \\ -I_{out} \end{bmatrix}}_{Q \cdot g_{ext} U_{ext}} \cdot \begin{bmatrix} \gamma_{v1} \\ \gamma_{v2} \\ \gamma_{i01} \\ \gamma_{i12} \\ \gamma_{i23} \end{bmatrix} \quad (20)$$

where the state vector is $x = [I_1 \ I_2 \ V_{01} \ V_{12} \ V_{23}]^T$. The value of the passive elements are stored in the diagonal matrix $Q = \text{diag}(L_1 \ L_2 \ C_{01} \ C_{12} \ C_{23})$. The control-variables are the duty-cycles $u_{int} = [D_{1t} \ D_{1b} \ D_{2t} \ D_{2b}]^T$ for each switching cell with gate signals $(u_{1t} \ u_{1b} \ u_{2t} \ u_{2b})$. The proposed control strategy is summarized in Fig. 6 and further detailed in the following subsections.

## B. Damping assignment

Based on the method proposed in section III-C, we define the control vector $u_{int}^c$ as in (17) with $\{r_i\}_{i \in [1,4]}$ the damping coefficients associated to each control variable. Then, by solving (19) for each duty-cycle, we obtain their expression:

$$u_{int}^c = \begin{bmatrix} D_{1t}^c \\ D_{1b}^c \\ D_{2t}^c \\ D_{2b}^c \end{bmatrix} = \begin{bmatrix} r_1 & 0 & -r_1 \frac{I_1^*}{V_{01}^*} & 0 & 0 \\ r_2 & 0 & 0 & -r_2 \frac{I_1^*}{V_{12}^*} & 0 \\ 0 & r_3 & 0 & -r_3 \frac{I_2^*}{V_{12}^*} & 0 \\ 0 & r_4 & 0 & 0 & -r_4 \frac{I_{23}^*}{V_{23}^*} \end{bmatrix} \cdot e_x \quad (21)$$

It leads to a final matrix $R_d$ which is the sum of matrices $R_{d,i}$. For example, the damping matrix associated to $D_{1t}$ is:

$$R_{d,1} = r_1 \begin{bmatrix} \frac{-V_{01}^*}{L_1^*} & 0 & \frac{I_1^*}{C_{01}L_1} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \frac{I_1^*}{C_{01}L_1} & 0 & \frac{-I_1^{*2}}{V_{01}^*C_{01}^2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \quad (22)$$

for which the only non-zero eigenvalue $\lambda_1$ is:

$$\lambda_1 = -r_1 \frac{(L_1 I_1^*)^2 + (C_{01} V_{01}^*)^2}{V_{01}^* (L_1 C_{01})^2} \quad (23)$$

By simply setting $r_1 < 0$, it results in $R_{d,1} \ge 0$. Similar conclusion is done for other control-variables, by choosing $r_i < 0$, which finally gives the complete damping matrix $R_d \ge 0$ correctly designed.

## C. Steady-state equation solving

Reference control vector $u_{int}^* = [D_{1t}^* \ D_{1b}^* \ D_{2t}^* \ D_{2b}^*]^T$ is found by solving (12) leading to:

$$\left\{ \begin{aligned} D_{1t}^* &= 1 - \frac{I_{out}^* + \gamma_{i01} + C_{01} \dot{V}_{01}^*}{I_1^*} \\ D_{1b}^* &= 1 + \frac{V_{in1}^* - (1 - D_{1t}^*)V_{01}^* - \hat{\gamma}_{v1} - r_{L1}I_1 - L_1 \dot{I}_1^*}{V_{12}^*} \\ D_{2t}^* &= 1 + \frac{V_{in2}^* - (1 - D_{2b}^*)V_{23}^* - \hat{\gamma}_{v2} - r_{L2}I_2 - L_2 \dot{I}_2^*}{V_{12}^*} \\ D_{2b}^* &= 1 - \frac{I_{out}^* + \gamma_{i23} + C_{23} \dot{V}_{23}^*}{I_2^*} \end{aligned} \right. \quad (24)$$

In power electronics we are interested in accurate control of output voltage meaning that $V_{01}^*, V_{12}^*$ and $V_{23}^*$ are fixed. By adding a proportional relation between both reference currents, $g \cdot I_1^* = I_2^*$, it is possible to find the expression of $I_1^*$ which guarantees the steady-state equation (12):

$$L^* \cdot I_1^* \cdot \frac{dI_1^*}{dt} = I_1^* \cdot V^* - P^* \quad (25)$$

where $P^*$ and $V^*$ are introduced to simplify calculations:

$$\left\{ \begin{aligned} P^* &= \sum V_n^* \cdot (I_{out}^* + \hat{\gamma}_{i,n} + C_n \dot{V}_n^*), \quad n = \{0, 1, 2, 3\} \\ V^* &= V_{in1}^* - \hat{\gamma}_{v1} - r_{L1}I_1 + g \cdot (V_{in2}^* - \hat{\gamma}_{v2} - r_{L2}I_2) \\ L^* &= L_1 + g^2 \cdot L_2 \end{aligned} \right. \quad (26)$$

This differential equation presents an analytical solution, but its complex formulation is not suitable in practice and it is diverging with time. However, it is possible to approximate the
<figure><img src="image_8.png" /><figcaption>Fig. 7. Set of parameters area for a converging current reference sequence with several values of k.</figcaption></figure>

<table><caption>TABLE II
BOUNDARIES COEFFICIENTS FOR A CONVERGING SEQUENCE.</caption><thead><tr><th>k</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th></tr></thead><tbody><tr><td>C<sub>k</sub><sup>&oplus;</sup></td><td>1</td><td>1.4</td><td>1.8</td><td>2.3</td><td>2.7</td><td>3.2</td><td>3.7</td><td>4.1</td><td>4.6</td></tr><tr><td>C<sub>k</sub><sup>&ominus;</sup></td><td>0.5</td><td>1.4</td><td>1.5</td><td>2.3</td><td>2.5</td><td>3.2</td><td>3.5</td><td>4.1</td><td>4.4</td></tr></tbody></table>

time derivative of $I_1^*$ with fixed time step (27). This approximation is easily implemented on real-time power electronics controllers:

$$ \frac{dI_1^*}{dt}(t) \approx \frac{i_n^* - i_{n-k}^*}{k\Delta T} \qquad (27) $$

where $i_n^*$ and $i_{n-k}^*$ are the computed reference current at time steps n and $n-k$. The controller time step is $\Delta T$. The accuracy of the derivative is defined using $k \ge 1$. We can then provide a recursive sequence expression for the reference current such that:

$$ i_{n+1}^* = \beta + \frac{\alpha}{k\Delta T} \cdot i_n^* \cdot (i_n^* - i_{n-k}^*) \quad (28) $$

with $\beta = P^*/V^*$ and $\alpha = L^*/V^*$. Convergence of this sequence is not always achieved, depending on the parameters. Using numerical simulation, Fig. 7 shows the area defined by the set of values $\{\frac{\alpha}{\Delta T}, \beta\}$, for several values of k, leading to a converging sequence.

Boundaries of these areas can be defined by the following constraints:

$$ -C_k^{\ominus} \cdot \frac{\Delta T}{\alpha} < \beta < C_k^{\oplus} \cdot \frac{\Delta T}{\alpha} \qquad (29) $$

where $C_k^{\ominus}$ and $C_k^{\oplus}$ are given in Table II for several values of k. The convergence criteria is computed in real-time and the value k is changed accordingly to extend the convergence. An hysteretic algorithm is provided in Fig. 8 with $m^{\oplus}$ and $m^{\ominus}$ used as margins.

## V. EXPERIMENTAL VALIDATION

### A. Novel proposed design method

Experimental validation is performed on a 2-input MTL-BC with the controller design method proposed in this article. The controller is implemented on a dSPACE SCALEXIO with DS6001 processor board. The DS6202 board is used to generate the gate signals of the transistors (G3R75MT12J MOSFET SiC), and the DS6221 to measure output of the current and voltage sensors. Switching and sampling frequencies are set

<figure><img src="image_9.png" /><figcaption>Fig. 8. Hysteretic algorithm to dynamically choose k.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9. Experimental set-up.</figcaption></figure>

at $f_{sw} = 20kHz$. The parameters of the studied system are gathered in Table III. The system is fed with two regulated constant voltage power supplies, TDK Lambda G300-17. A Höcherl & Hackl ZSAC electronic load is used on the output side. Waveform acquisition is done through SCALEXIO and ControlDesk 2023B.

Picture of the experimental bench is presented in Fig. 9 with highlight of the converter and controller. Fig. 6 summarizes the proposed control strategy. As presented in (3), steady-state

<table><caption>TABLE III
EXPERIMENTAL PARAMETERS FOR THE MTL-BC.</caption><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>Input voltage</td><td>V<sub>in1</sub> = V<sub>in2</sub> = 60V<br>V<sub>01</sub> + V<sub>12</sub> + V<sub>23</sub> = 130 – 180V</td></tr><tr><td>Inductors</td><td>L<sub>1</sub> = 425μH; L<sub>2</sub> = 300μH<br>r<sub>L1</sub> = 300mΩ; r<sub>L2</sub> = 250mΩ</td></tr><tr><td>Output capacitors</td><td>C<sub>01</sub> = C<sub>12</sub> = C<sub>23</sub> = 230μF</td></tr><tr><td>Damping matrix (21)</td><td>r<sub>1</sub> = r<sub>2</sub> = r<sub>3</sub> = r<sub>4</sub> = -0.02A<sup>-1</sup></td></tr><tr><td>Damping matrix (30)</td><td>R<sub>1</sub> = R<sub>2</sub> = 1Ω<br>R<sub>3</sub> = R<sub>4</sub> = R<sub>5</sub> = 0.01Ω<sup>-1</sup></td></tr><tr><td>Observer parameters</td><td>k<sub>p</sub> = -100 · I<sub>5</sub> · Q<br>k<sub>s</sub> = diag (20, 20, 10, 10, 10) · 10<sup>3</sup></td></tr><tr><td>Inner-loop (Fig. 15)</td><td>G<sub>p</sub> = diag (8 · 10<sup>3</sup>, 8 · 10<sup>3</sup>, 100, 100)<br>G<sub>i</sub> = diag (16 · 10<sup>6</sup>, 16 · 10<sup>6</sup>, 2500, 2500)</td></tr><tr><td>Outer-loop (Fig. 15)</td><td>G<sub>p</sub> = 800, G<sub>i</sub> = 16 · 10<sup>4</sup></td></tr><tr><td>Decoupling filter</td><td>ω<sub>in</sub> = 2000rad/s</td></tr><tr><td>Input filter (Fig. 13a)</td><td>L<sub>in</sub> = 300μH – 0.01Ω, C<sub>in</sub> = 100μF</td></tr><tr><td>Output filter (Fig. 13b)</td><td>L<sub>out</sub> = 1000μH – 0.01Ω, C<sub>out</sub> = 100μF</td></tr><tr><td>Output filter (Fig. 18a)</td><td>L<sub>out</sub> = 10mH, C<sub>out</sub> = 160μF</td></tr></tbody></table>
<figure><img src="image_11.png" /><figcaption>Fig. 10. Experimental validation of the proposed IDA-PBC.</figcaption></figure>

values of the electrical inputs $U_{ext}^*$ are extracted using a low-pass filter with bandwidth $\omega_{ext} = 2000rad/s$.

Fig. 10 shows the controlled state-variables of the MTL-BC using IDA-PBC designed with the proposed method. The current of each inductor is set such that $0.9 \cdot I_1^* = I_2^*$. At $t = 0ms$ the output voltage reference is increased from 130V to 180V with the voltage shared between the three capacitors. It is followed at $t = 25ms$ by a load step from $I_{out} = 2.5A$ to $7.5A$. Finally at $t = 40ms$, the load comes back to $I_{out} = 2.5A$ showing a slower voltage balance equalization which is expected at lower operating power. Voltage over- and under-shoots are correctly predicted with model simulation and can be tuned thanks to output capacitors sizing.

## B. Comparison with the original design method

To compare with the design method used in [20], based on solving the closed-loop equation (3) for the duty-cycles, the control is applied on the same MTL-BC. Interconnection and Damping matrices used are given in (30), with K an additional coefficient introduced because the system is of the 5th order with 4 control variables. The damping coefficients $R_i$ allows to tune the dynamics of the closed-loop system. Table III gathers the parameters used experimentally.

$$ \left\{ \begin{array}{l} Q \cdot J_d \cdot Q = \begin{bmatrix} 0 & 0 & -K & -K & 0 \\ 0 & 0 & 0 & -K & -K \\ K & 0 & 0 & 0 & 0 \\ K & K & 0 & 0 & 0 \\ 0 & K & 0 & 0 & 0 \end{bmatrix} \\ Q \cdot R_d \cdot Q = \text{diag}(R_1 \ R_2 \ R_3 \ R_4 \ R_5) \end{array} \right. \quad (30) $$

Both methods are compared on a constant load current $I_{out} = 1.4A$ as shown on Fig. 11. At the beginning, the method proposed in this article is used. At $t = 1s$, the controller switches to the original IDA-PBC control, resulting in spikes of current and duty-cycle saturation visible on the right half part of the waveform. On the zoomed area we can see the current converging again to the steady state after a spike.

As discussed in section II-B, duty-cycles solution converges to a finite value in steady-state. However, the noise of the

<figure><img src="image_12.png" /><figcaption>Fig. 11. Proposed IDA-PBC compared experimentally with the original one.</figcaption></figure>

measured state-variables leads to duty-cycles moving around this equilibrium point, entering a singular region in the neighborhood of the equilibrium point as in Fig. 3, leading to the observed spikes. In practice, it is possible to design an algorithm to filter fast changing values of duty-cycles to better converge to a stable steady-state.

## C. Robustness analysis

In this section, a robustness analysis is performed in the case of uncertain passive elements value. Capacitors value can change in time because of aging, resulting in a decrease of their initial capacity. Inductors value strongly depends on the magnetic material used. For instance, ferrite cores provide almost constant value on a large range of current before an abrupt saturation while powder cores have a smooth decrease on the operating range.

Because of the discretized current reference calculation in (28), the robustness analysis is performed in discrete time at

<figure><img src="image_13.png" /><figcaption>(a) Varying capacitor $C_{12}$.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(c) Simulation validation of the model used for robustness analysis.</figcaption></figure>

<caption>Fig. 12. Robustness analysis of the controlled system.</caption>
<figure><img src="image_15.png" /><figcaption>Fig. 13. Connection of the MTL-BC with LC filters.</figcaption></figure>

the sampling frequency. The controlled system is expressed as:

$$X[n+1] = f(X[n]) \quad (31)$$

Using small-signal stability analysis it is possible to infer the stability around an operating point of this system by studying the eigenvalues of the jacobian of f. The value of k used for current reference calculation is assumed constant in the neighborhood of the linearization.

According to the experimental parameters, the operating point considered is at $I_{out} = 9A$ to highlight the limit of stability, resulting in $k = 4$. Both $C_{12}$ and $L_1$ are changed separately between 160% - 40% of their nominal value. Eigenvalues are displayed in Fig. 12 along with the unit-circle which defines the limit of stability for discrete linear systems.

As observed in Fig. 12a, two eigenvalues exit the unit-circle for $C_{12} < 100\mu F$. In order to validate the discretized model used for the analysis, two simulations are performed with $C_{12} = 90\mu F$ and $C_{12} = 110\mu F$ in Fig. 12c. The simulations confirm the accuracy of the model as its stability is correctly predicted by eigenvalues study.

# VI. IDA-PBC IN UNKNOWN MGS ARCHITECTURE

## A. Comparison with other non-linear controllers

To highlight the interest of the proposed IDA-PBC strategy for modular MGS, this section proposes to compare it with other conventional control strategies. To model an unknown architecture of MGS, it is proposed to add an input LC filter as shown Fig. 13a, where the MTL-BC is presented Fig. 5. The control strategies studied in this part are not using the LC filter's parameters as they are unknown.

First, a hybrid sliding mode control-PBC strategy proposed in [18] is implemented on the MTL-BC to compare its performance without LC filter. As simulated in Fig. 14a, the main advantage of such strategy is to improve the dynamic performances of the controller especially during load steps resulting in less voltage drops or overshoots. These hybrid controls feature good robustness and large-signal stability.

Nevertheless, in the context of unmodeled elements on one port, the stability is not guaranteed. Such large bandwidth controllers act as a CPL from the MGS's point of view [25]. The Fig. 14b shows a simulation with the hybrid-PBC and an input LC filter as in Fig. 13a with a resistive load. The load power is increased gradually leading to instability for $\{I_1, I_2\} > 2A$ or $P_{out} > 300W$.

This phenomenon is expected as the ports used for the proof of stability are not compatible with other passive elements of a MGS, hence the passivity property does not propagate. The

<figure><img src="image_16.png" /><figcaption>Fig. 14. Simulation of the MTL-BC control with a hybrid-PBC strategy.</figcaption></figure>

problem was solved thanks to [20] by preserving the natural electrical port in the closed-loop IDA-PBC:

$$\tilde{U}_{ext} = U_{ext} - U_{ext}^{*} \quad \text{and} \quad \tilde{y}_{ext} = y_{ext} - y_{ext}^{*} \quad (32)$$

In a similar way, a non-linear two-loop control strategy [26], Fig. 15, is implemented experimentally with an input LC filter, Fig. 13a. The outer-loop controls the total output voltage and the faster inner-loop controls both inductor currents as well as the voltage balance between capacitors. The general law defining the error's dynamic is given as:

$$\left\{ \begin{array}{l} \dot{\varepsilon} + G_p \varepsilon + G_i \int \varepsilon dt = 0 \\ \varepsilon = x^* - x \end{array} \right. \quad (33)$$

The state-variables are given in (34) with $E_c$ the electro-static energy of the capacitive bus $\{V_{01}, V_{12}, V_{23}\}$. $P_{in} = I_1V_{in1} + I_2V_{in2}$ is the power provided to the capacitors through the converter with assumed unitary efficiency. $P_{out} = I_{out}V_{out}$ is the load power. A 2nd order low-pass filter is used to make a differentiable inner-loop's reference. Controller's parameters are given in Table III.

$$x_{in} = \begin{bmatrix} I_1 \\ I_2 \\ V_{01} - V_{12} \\ V_{12} - V_{23} \end{bmatrix}, \quad \begin{cases} x_{out} = E_c = \frac{1}{2} \sum C_i V_i^2 \\ \dot{E}_c \approx P_{in} - P_{out} \end{cases} \quad (34)$$

The experimental results of the proposed IDA-PBC and the two-loop control are shown on Fig. 16. The output power is increased gradually over time. The IDA-PBC in Fig. 16a achieves the maximum power capability of the converter at $t = 30ms$, without system's instability. The two-loop controller in Fig. 16b shows unstable behavior with increasing oscillation of input voltage $V_{in1}$ and inductor current $I_1$ for $I_1 > 6A$. This test confirms the increased stability of the IDA-PBC in the context of a MGS where external elements of a sub-system are unknown.

<figure><img src="image_17.png" /><figcaption>Fig. 15. Two-loop control strategy for the MTL-BC.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>Fig. 16. Experimental comparison of the proposed IDA-PBC and a conventional two-loop control with an input LC-filter and a resistive load.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>(a) High damping $r_1 = -0.02A^{-1}$. (b) Low damping $r_1 = -0.004A^{-1}$. Fig. 17. Simulation of the MTL-BC with an output LC filter before a CPL.</figcaption></figure>

## B. IDA-PBC in the presence of a CPL

In this part, we focus on the benefits of IDA-PBC in the presence of a CPL. As presented in [27], converters controlled with PBC in a MGS add virtual damping, stabilizing the system. While global stability is not proven in the case of unknown line parameters, the additional damping improves the stable region compared to other controllers. This behavior is highlighted in simulation with the MTL-BC, Fig. 5, supplying an ideal CPL through a LC filter, with $L_{out} = 1mH$ and $C_{out} = 100\mu F$, as shown Fig. 13b. First on Fig. 17a, the proposed control is used with well designed damping coefficients. Then on Fig. 17b, the damping is greatly reduced. We observe that the maximum power before instability can be more than doubled thanks to the virtual damping introduced by IDA-PBC.

In theory, the damping can be increased to ensure stability depending on the system's characteristics. However, in practice, the damping coefficient is acting as a proportional gain and is typically limited by the sampling frequency if all other parameters are kept constant. In our application, the limiting coefficient is around $r_1 \approx -0.11A^{-1}$ for a sampling frequency of $20kHz$. Increasing the sampling frequency provides a larger damping coefficient range, and thus a larger potential stability margin for a CPL.

To further validate these results in the context of shipboard MGS, an inverter is designed and controlled to drive a permanent magnet synchronous motor. The inverter is connected to the MTL-BC through a LC-filter as presented in Fig. 18a. Two Proportional Integral (PI) controllers with large bandwidth are used to control the currents of the motor in the dq-frame and an outer-loop PI to control the mechanical speed. The shaft of

<figure><img src="image_20.png" /><figcaption>Fig. 18. Experimental bench with the motor drive.</figcaption></figure>

the motor is connected to a permanent magnet generator with a constant resistive load, simulating a high friction coefficient on the rotor. A picture of the bench is presented in Fig. 18b. The mechanical angle is sampled thanks to a resolver and the DS6121. In the following tests, the LC-filter is selected with $L_{out} = 10mH$ and $C_{inv} = 160\mu F$, to highlight the limiting power of a CPL without requiring high current values. It is worth to note that no information is shared between the MTL-BC and the inverter controllers.

A first experiment is performed in Fig. 19a where the MTL-BC is replaced by a constant voltage source at $V_{out} = 180V$. The mechanical speed of the motor is increased which increases the power drawn from the power supply. As observed in Fig. 19a, increasing oscillations of the LC-filter's voltage and current are present starting around 2300RPM or $\approx 380W$. This test highlights the CPL behavior of the control used for the inverter.

Two tests are performed with the MTL-BC controlled by the proposed IDA-PBC, with different damping assignment. A high and low damping are used respectively in Fig. 19b and Fig. 19c with $r_1 = -0.02A^{-1}$ and $r_1 = -0.004A^{-1}$. The low damping controller presents oscillation around 3150RPM or $\approx 600W$ while the high damping controller reaches 4000RPM or $\approx 840W$, which is the maximum allowed power of the motor bench, without any oscillation.

This experimental test validate the results of the simulation Fig. 17. It highlights the increased stability provided by the virtual damping introduced by IDA-PBC even in the context of MGS with unknown parameters and in the presence of a CPL.

## VII. CONCLUSION

IDA-PBC is one of the most promising controller design for power converters to ensure the stability of modular MGS. This paper proposes a novel controller design method to improve
<figure><img src="image_21.png" /><figcaption>(a) Constant voltage source.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>(b) High damping IDA-PBC $r_1 = -0.02A^{-1}$.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>(c) Low damping IDA-PBC $r_1 = -0.04A^{-1}$.</figcaption></figure>

<caption>Fig. 19. Experimental results of a MTL-BC and a motor drive interconnected through a LC-filter.</caption>

the IDA-PBC used in [20] while preserving the strong results on passivity propagation. The previous method aims to find the expression of the duty-cycle such that the controlled system exhibits the exact desired behaviors. However, this expression presents singular regions which lead to saturations of duty-cycle which causes spikes of current. Nevertheless the method is extremely general for systems written using PCH formalism.

A novel method is proposed achieving the same closed-loop behaviors while avoiding the singular region. The method is generalized to a sub-class of PCH systems, which includes wide range of power converter average models. Then, it is applied to the controller design of a 2-input MTL-BC. Steady-state equation solving introduces a differential equation with an unpractical solution for experimental implementation. Since

power electronics controllers work with fixed time-steps, the discretized differential equation convergence is characterized and an algorithm is proposed for real-time control. Finally, experimental validation is performed on the MTL-BC. Voltage and current reference tracking performances are confirmed. Comparison with the conventional IDA-PBC from [20] is done on the same converter highlighting the removed singular region of the proposed method. Robustness analysis confirms the stability of the proposed IDA-PBC under uncertain parameters. The interconnection advantage of IDA-PBC is underlined in an unknown MGS architecture context compared to other conventional controllers. Finally, the paper highlights the improved stability margin of IDA-PBC in the presence of a CPL. Further work will focus on a more in depth study of the reference current calculation and on the inclusion of input current's constraints specific to FC application while preserving passivity of interconnected systems.

## REFERENCES

[1] S. Guo, Y. Wang, L. Dai, and H. Hu, "All-electric ship operations and management: Overview and future research directions," *eTransportation*, vol. 17, DOI 10.1016/j.etran.2023.100251, p. 100251, Jul. 2023.

[2] E. Nivolianiti, Y. L. Karnavas, and J.-F. Charpentier, "Energy management of shipboard microgrids integrating energy storage systems: A review," *Renewable and Sustainable Energy Reviews*, vol. 189, DOI 10.1016/j.rser.2023.114012, p. 114012, Jan. 2024.

[3] Ballard Power Systems. Ballard - Marine. (2024). [Online]. Available: https://www.ballard.com/marine/

[4] C. Lyu and V. Dinavahi, "Zero-Emission Marine Vessels: Multidomain Modeling and Real-Time Hardware-in-the-Loop Emulation on Adaptive Compute Acceleration Platform: Zero-emission marine vessels: Modeling and real-time emulation," *IEEE Electrification Magazine*, vol. 11, DOI 10.1109/MELE.2023.3320509, no. 4, pp. 54–63, Dec. 2023.

[5] Z. Wang, B. Dong, J. Yin, M. Li, Y. Ji, and F. Han, "Towards a marine green power system architecture: Integrating hydrogen and ammonia as zero-carbon fuels for sustainable shipping," *Int. J. Hydrogen Energy*, vol. 50, DOI 10.1016/j.ijhydene.2023.10.207, pp. 1069–1087, Jan. 2024.

[6] Y. Yan, Q. Li, W. Chen, W. Huang, J. Liu, and J. Liu, "Online Control and Power Coordination Method for Multistack Fuel Cells System Based on Optimal Power Allocation," *IEEE Trans. Ind. Electron.*, vol. 68, DOI 10.1109/TIE.2020.3016240, no. 9, pp. 8158–8168, Sep. 2021.

[7] N. Marx, L. Boulon, F. Gustin, D. Hissel, and K. Agbossou, "A review of multi-stack and modular fuel cell systems: Interests, application areas and on-going research activities," *Int. J. Hydrogen Energy*, vol. 39, DOI 10.1016/j.ijhydene.2014.05.187, no. 23, pp. 12 101–12 111, Aug. 2014.

[8] Y. Qiu, T. Zeng, C. Zhang, G. Wang, Y. Wang, Z. Hu, Meng Yan, and Z. Wei, "Progress and challenges in multi-stack fuel cell system for high power applications: Architecture and energy management," *Green Energy and Intelligent Transportation*, vol. 2, DOI 10.1016/j.geits.2023.100068, no. 2, p. 100068, Apr. 2023.

[9] Y. Blanco, W. Perruquetti, and P. Borne, "Stability and Stabilization of Nonlinear Systems and Takagi-Sugeno's Fuzzy Models," *Math. Probl. Eng.*, vol. 7, DOI 10.1155/S1024123X01001624, Jan. 2001.

[10] B. Brogliato, Ed., *Dissipative Systems Analysis and Control: Theory and Applications*, 2nd ed., ser. Communications and Control Engineering. London: Springer, 2007.

[11] R. Ortega, A. van der Schaft, B. Maschke, and G. Escobar, "Interconnection and damping assignment passivity-based control of port-controlled hamiltonian systems," *Automatica*, vol. 38, DOI https://doi.org/10.1016/S0005-1098(01)00278-3, no. 4, pp. 585–596, 2002.

[12] M. Cupelli, S. K. Gurumurthy, S. K. Bhanderi, Z. Yang, P. Joebges, A. Monti, and R. W. De Doncker, "Port Controlled Hamiltonian Modeling and IDA-PBC Control of Dual Active Bridge Converters for DC Microgrids," *IEEE Trans. Ind. Electron.*, vol. 66, DOI 10.1109/TIE.2019.2901645, no. 11, pp. 9065–9075, Nov. 2019.

[13] M. Hassan, C.-L. Su, F.-Z. Chen, and K.-Y. Lo, "Adaptive Passivity-Based Control of a DC–DC Boost Power Converter Supplying Constant Power and Constant Voltage Loads," *IEEE Trans. Ind. Electron.*, vol. 69, DOI 10.1109/TIE.2021.3086723, no. 6, pp. 6204–6214, Jun. 2022.
[14] M. Li, H. Geng, and X. Zhang, "Distributed Coordinated Control for Stabilization of Multi-Inverter Power Plant," *IEEE Trans. Ind. Electron.*, vol. 70, DOI 10.1109/TIE.2023.3237894, no. 12, pp. 12421-12430, Dec. 2023.

[15] R. Ortega and J. G. Romero, "Robust integral control of port-Hamiltonian systems: The case of non-passive outputs with unmatched disturbances," in *2011 50th IEEE Conference on Decision and Control and European Control Conference*, DOI 10.1109/CDC.2011.6160543, pp. 3222-3227, Dec. 2011.

[16] N. Khefifi, A. Houari, M. Machmou, A. Saim, and M. Ghanes, "Generalized IDA-PBC Control Using Enhanced Decoupled Power Sharing for Parallel Distributed Generators in Standalone Microgrids," *IEEE Journal of Emerging and Selected Topics in Power Electronics*, vol. 9, DOI 10.1109/JESTPE.2020.3034464, no. 4, pp. 5069-5082, Aug. 2021.

[17] S. Pang, Z. Mao, X. Li, Y. Huangfu, M. Bahrami, J.-P. Martin, S. Pierfederici, and B. Nahid-Mobarakeh, "Hamiltonian Energy Control With Energy Management Strategy for Fuel Cell Hybrid Power System," *IEEE Trans. Ind. Appl.*, vol. 59, DOI 10.1109/TIA.2023.3304620, no. 6, pp. 7716-7724, Nov. 2023.

[18] M. N. Huynh, H. N. Duong, and V. H. Nguyen, "A Passivity-based Control Combined with Sliding Mode Control for a DC-DC Boost Power Converter," *J. of Robotics and Control (JRC)*, vol. 4, DOI 10.18196/jrc.v4i6.20071, no. 6, pp. 780-790, Nov. 2023.

[19] H. Cheng, D. Yang, C. Wang, and C. Tian, "The Hybrid Passivity-Based Control Strategy of a Novel Unidirectional Five-Level Rectifier Under the Unbalanced Load," *IEEE Trans. Ind. Electron.*, vol. 71, DOI 10.1109/TIE.2023.3340192, no. 9, pp. 10546-10555, Sep. 2024.

[20] M. Lapique, S. Pang, J.-P. Martin, S. Pierfederici, M. Weber, and S. Zaim, "Enhanced IDA-PBC applied to a 3-phase PWM-rectifier for stable interfacing between AC and DC microgrids embedded in More Electrical Aircraft," *IEEE Trans. Ind. Electron.*, DOI 10.1109/TIE.2022.3150079, pp. 1-1, 2022.

[21] S. Pang, B. Nahid-Mobarakeh, S. Pierfederici, M. Phattanasak, Y. Huangfu, G. Luo, and F. Gao, "Interconnection and Damping Assignment Passivity-Based Control Applied to On-Board DC-DC Power Converter System Supplying Constant Power Load," *IEEE Trans. Ind. Appl.*, vol. 55, DOI 10.1109/TIA.2019.2938149, no. 6, pp. 6476-6485, Nov. 2019.

[22] H. Renaudineau, J.-P. Martin, B. Nahid-Mobarakeh, and S. Pierfederici, "DC-DC Converters Dynamic Modeling With State Observer-Based Parameter Estimation," *IEEE Trans. Power Electron.*, vol. 30, DOI 10.1109/TPEL.2014.2334363, no. 6, pp. 3356-3363, Jun. 2015.

[23] M. Afkar, R. Gavagsaz-Ghoachani, M. Phattanasak, A. Siangsanoh, J.-P. Martin, and S. Pierfederici, "Generalization of a DC-DC Modular Converter Topology for Fuel Cell Applications," *IEEE Trans. Ind. Appl.*, vol. 58, DOI 10.1109/TIA.2022.3142659, no. 2, pp. 2255-2267, Mar. 2022.

[24] G. R. Lawlor, "A L'hospital's rule for multivariable functions," Aug. 2012.

[25] S. Singh, A. R. Gautam, and D. Fulwani, "Constant power loads and their effects in DC distributed power systems: A review," *Renew. Sustain. Energy Rev.*, vol. 72, DOI 10.1016/j.rser.2017.01.027, pp. 407-421, May. 2017.

[26] M. Afkar, R. Gavagsaz-Ghoachani, M. Phattanasak, J.-P. Martin, and S. Pierfederici, "Proposed System Based on a Three-Level Boost Converter to Mitigate Voltage Imbalance in Photovoltaic Power Generation Systems," *IEEE Trans. Power Electron.*, vol. 37, DOI 10.1109/TPEL.2021.3105571, no. 2, pp. 2264-2282, Feb. 2022.

[27] M. A. Hassan, C.-L. Su, J. Pou, D. Almakhles, T.-S. Zhan, and K.-Y. Lo, "Robust Passivity-Based Control for Interleaved Bidirectional DC-DC Power Converter With Constant Power Loads in DC Shipboard Microgrid," *IEEE Trans. Transport. Electrific*, vol. 10, DOI 10.1109/TTE.2023.3307878, no. 2, pp. 3590-3602, Jun. 2024.

**Emeric Vuillemin** received the Engineer's degree in electrical engineering from the École Nationale Supérieure d'Électrotechnique, d'Électronique, d'Informatique, d'Hydraulique et des Télécommunications (ENSEEIHT), Toulouse, France in 2020. He is working toward the Ph.D. degree in electrical engineering with the Laboratoire Energie & Mécanique Théorique et Appliquée, CNRS, Université de Lorraine, Nancy, France. His research focuses on design and control of power converters in fuel-cell systems for transportation.

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

**Jean-Philippe Martin** Jean-Philippe Martin received the graduation degree in electrical engineering from the University of Nancy, Nancy, France, in 1999 and the Ph.D. degree in electrical engineering from the Institut National Polytechnique de Lorraine (INPL), Nancy, France in 2003. Since 2004, he is an Assistant Professor at University of Lorraine and member of the LEMTA since January 2018. His research interests include DC and AC microgrid with centralized or decentralized control, multi-vector microgrid combining electrical, thermal and

hydrogen vectors, stability study of distributed power system, static converter architectures and their interactions with fuel cell and photovoltaic system.

<figure><img src="image_26.png" /></figure>

**Mohamed Machmou** received the Engineering degree from the Institut Supérieur Industriel, Liège, Belgium, in 1984, and the master's and Ph.D. degrees from the l'Institut National Polytechnique de Lorraine (INPL), Lorraine, France, in 1985 and 1989, respectively, all in electrical engineering. In 1991, he joined the École Polytechnique de l'Université de Nantes, Nantes, France, as an Assistant Professor, where he has been a Full Professor since September 2005. He is currently the Head of the Institut de Recherche en Energie Électrique de

Nantes Atlantique (IREENA), University of Nantes, Saint-Nazaire, France. His main area of interest includes power electronics, power quality, wind and tidal energy conversion systems, and microgrids. In these fields, he has supervised more than 34 Ph.D. students and published one book related to modeling of electrical machines, several chapter books, and nearly 300 journals and conferences in his areas of expertise.

<figure><img src="image_27.png" /></figure>

**Serge Pierfederici** received the Engineering degree from the École Nationale Supérieure d'Electricité et Mécanique, Nancy, France, in 1994, and the Ph.D. degree in electrical engineering from the Institut National Polytechnique de Lorraine, Nancy, in 1998. Since 2009, he has been a Full Professor with the Université de Lorraine, Nancy. He has authored or co-authored over 200 international peer-reviewed journals. His research interests include the stability study of distributed power systems, the modeling and control of power electronic systems,

and since recent years the distributed control of multisources and multicarrier microgrids. Dr. Pierfederici has been a recipient of several IEEE awards.

<figure><img src="image_28.png" /></figure>

**Farid Meibody-Tabar** received the Engineering degree from École Nationale Supérieure d'Electricité et de Mécanique (ENSEM), Nancy, France, in 1982, and the Ph.D. and Habilitation à diriger des recherches degrees from Institut National Polytechnique de Lorraine (INPL), Nancy, in 1986 and 2000, respectively. Since 2000, he has been a Professor with Université de Lorraine (ex-INPL), Nancy. His research activities with the LEMTA Laboratory, CNRS, Université de Lorraine, deals with architecture, modeling and control of power systems and

microgrids and electrical machines, their supply, and control and diagnosis.

<figure><img src="image_29.png" /></figure>

**Mathieu Weber** was born in Dieuze, France, in 1964. He received the master's degree in electronic electrotechnic automatic from the University of Henry Poincare, Nancy, France, in 1987. He is currently an Electronic Engineer responsible for laboratory instrumentation with the Laboratoire d'Energétique et de Mécanique Théorique et Appliquée (LEMTA), University of Lorraine, Nancy, France. His research activities focus on energy conversion for the electrical engineering team of LEMTA.