# Decentralized Synthesis of Passivity-Based Distributed Controllers for DC
Microgrids

Mohammad Javad Najafirad  Dept. of Electrical Engineering
Stevens Institute of Technology
Hoboken, USA
<EMAIL_ADDRESS>Shirantha Welikala  Dept. of Electrical Engineering
Stevens Institute of Technology
Hoboken, USA
<EMAIL_ADDRESS>
###### Abstract

Microgrids (MGs) provide a promising solution for exploiting renewable energy
resources (RESs). Most of the existing papers on DC MGs focused on controller
design. However, in this paper, we proposed a co-design passivity-based
control (PBC) which design the controller and communication topology
simultaneously. In addition, the plug-and-play feature is maintained without
compromising the stability and re-designing of the remaining controllers. To
this end, the local controller for distributed generators (DGs) and lines are
obtained using smaller decentralized LMI problems and then the global
controller is formulated using the passivity indices of the local controller
to restore voltages to the reference value.

###### Index Terms:

Distributed control, passivity-based control, DC microgrids, networked system
configurations, Decentralized control

##  I Introduction

Microgrids, localized power systems with the ability to generate and manage
their loads independently, are gaining recognition, especially in the realm of
Direct Current (DC) microgrids, driven by challenges in frequency
synchronization in AC MGs and the rising demand for DC loads. Hierarchical
control structures have emerged as a standard approach for managing these
microgrids, stratified into primary, secondary, and tertiary levels. At the
primary level, known as the local controller, local data is utilized to
regulate the DC bus voltage and ensure equitable power sharing among
distributed generators (DGs). The secondary controller, operating with a
slower response time, compensates for voltage deviations caused by the primary
level. Its primary objective is to restore the voltage on the DC bus and
ensure appropriate current sharing among DGs. The tertiary control level
oversees power flow management and energy planning.

These control levels can be implemented in centralized, decentralized, or
distributed manners. Centralized secondary control, governed by a central
controller, offers precise, flexible, and reliable control performance.
However, it is susceptible to single points of failure and lacks plug-and-play
capability. In contrast, decentralized secondary control operates without
direct communication. Although it eliminates the need for global information,
it may face challenges in effectively coordinating DGs due to the lack of
inter-device communication. On the other hand, distributed control relies on a
sparse communication network to facilitate information exchange among
neighboring DGs.

In [1], the authors considered the application of passivity theory to the
voltage stabilization of DC microgrids including non-linear ZIP loads and
dynamic RLC loads. They considered the PnP decentralized multivariable PI
controllers and showed that they passivated the generation units and
associated loads under certain conditions. The proposed control synthesis
requires only the knowledge of DGs’ local parameters, which allows the PnP
scenario. Compared with [2], they provide explicit inequalities for computing
control gain instead of Linear Matrix Inequality (LMI). These inequalities
only depend on the electrical parameters of the corresponding DG. In [2], a
decentralized control method is designed where the primary control is
developed in a PnP fashion in which the only global quantity used in the
synthesis algorithm is a scalar parameter and the design of the local
controller is line-independent. Therefore, there is no requirement to update
the controllers of neighboring DGs.

In [3], the problem of designing distributed controllers to guarantee
dissipativity of dynamically coupled subsystems. They provide distributed
subsystem-level dissipativity analysis conditions and then use these
conditions to synthesize controllers locally. They also make this synthesis
composition to ensure the dissipativity of the system while adding new
subsystems. This paper addressed three problems: 1) distributed analysis which
decomposes a centralized dissipativity condition into dissipativity of
individual subsystems. In comparison with other papers, this article proposes
a distributed dissipative analysis condition that only uses the knowledge of
the subsystem dynamics and information about the dissipativity of its
neighbors. 2) It then formulates a procedure to synthesize local controllers
to guarantee dissipativity. 3) They then formulated a control synthesis
procedure that is compositional. Therefore, when a new subsystem is added, the
design procedure uses only the knowledge of the dynamics of the newly added
subsystem and the dissipativity of its neighboring to synthesize local control
inputs such that the new network is dissipative.

In the conference paper [4], a new distributed passivity-based control
strategy for DC MGs has been proposed using a consensus-based algorithm to
achieve average voltage regulation while proper current sharing. In comparison
with other methods where the Laplacian matrix determined the network, the
design of the communication network is independent of the topology of the
microgrid. The stability analysis is achieved by solving the LMI problem. The
idea of interconnection and damping assignment passivity-based control (IDA-
PBC) has been proposed in [5, 6, 7]. The authors in [5] proposed a novel
damping (suppression of power and current oscillations) control for V/I droop-
controlled DC MGs. The damping controller is designed using the IDA-PBC. This
idea comes from the fact that the conventional passivity-based control (PBC)
is very sensitive to load variations. The paper [6] also used the idea in [5]
where a new decentralized control method based on passivity control is
proposed for two DC distributed energy sources supplying the DC MG. The paper
[7] also used the idea of IDA-PBC to improve the damping of the system. The
application of this paper is battery where despite the variation of
irradiation, the bus voltage and power remain constant by finding the
irradiation levels through simulation for which the battery charges and
discharges and remains idle.

A new passivity-based control approach for DC MGs is proposed in [8]. This
approach is a data-driven control design and thus model-free and bypasses
mathematical models to represent the network and guarantees stability. The
authors in [9, 10, 11] proposed a robust PBC to solve the instability problem
of constant power loads (CPLs) by adding the nonlinear disturbance observer
(NDO). This paper aims to achieve 1) stabilize and regulate the DC-bus voltage
using PBC 2) by combining the NDO with PBC, the steady-state error will be
eliminated through the feedforward channel and improve the control robustness
against CPL variations and 3) eliminate the voltage mismatch between the
parallel connected converters using I-V droop control.

In the paper [12], an adaptive passivity-based controller (PBC) alongside an
adaptive extended Kalman Filter (AEKF) is presented for DC MGs. The proposed
controller estimates the unknown CPL power and prevents the number of
measurement sensors. The main contribution of paper [13] is presenting a
power-voltage droop controller in DC MG under explicit consideration of the
electrical networked dynamics. Inspired by passivity-based control design,
dissipation equality is derived in terms of the shifted Hamiltonian gradient.
In the paper [14], a comprehensive passivity-based stability criterion is
introduced by considering all input perturbations which provides a generic
criterion for each individual interface converter.

###  I-A Main Contribution

In this paper, a passivity-based control (PCB) approach is developed in a
decentralized manner. Additionally, a distributed control algorithm is
utilized within the communication channels to accomplish two main objectives
in DC MGs: voltage restoration and current sharing. main contributions of this
paper can be outlined as follows:

###  I-B Organization

The structure of this paper is as follows: Section II provides an overview of
several essential notations and foundational concepts related to the
configurations of dissipativity and networked systems. Section III discusses
the formulation of DC microgrid modeling, while Section IV delves into the
modeling of networked systems. In Section V, the proposed passivity-based
control is introduced. Finally, Section VI provides a conclusion.

##  II Preliminaries

###  II-A Notations

The notation $\mathbb{R}$ and $\mathbb{N}$ signify the sets of real and
natural numbers, respectively. For every $N\in\mathbb{N}$, we define
$\mathbb{N}_{N}\triangleq\\{1,2,..,N\\}$. A block matrix $A$ with dimensions
$n\times m$ can be denoted as
$A=[A_{ij}]_{i\in\mathbb{N}_{n},j\in\mathbb{N}_{m}}$, where $A_{ij}$
represents the $(i,j)^{\text{th}}$ block of $A$. It is worth noting that
either subscripts or superscripts can be used for indexing, for example,
$A_{ij}=A^{ij}$. $[A_{ij}]_{j\in\mathbb{N}_{m}}$ and
$\text{diag}([A_{ii}])_{i\in\mathbb{N}_{n}}$ represent a block row matrix and
a block diagonal matrix, respectively. $A^{\top}$ represents the transpose of
a matrix $A$, and $(A^{\top})^{-1}=A^{-\top}$. $\mathbf{0}$ and $\mathbf{I}$
stand for the zero and identity matrices, respectively. The representation of
a symmetric positive definite (semi-definite) matrix $A\in\mathbb{R}^{n\times
n}$ is $A=A^{\top}>0\ (A=A^{\top}\geq 0)$.

###  II-B Dissipativity

Consider a dynamic control system

$\displaystyle\dot{x}(t)=f(x(t),u(t)),$ (1) $\displaystyle y(t)=h(x(t),u(t)),$

where $x(t)\in\mathbb{R}^{n}$, $u(t)\in\mathbb{R}^{q}$,
$y(t)\in\mathbb{R}^{m}$, and
$f:\mathbb{R}^{n}\times\mathbb{R}^{q}\rightarrow\mathbb{R}^{n}$ and
$h:\mathbb{R}^{n}\times\mathbb{R}^{q}\rightarrow\mathbb{R}^{m}$ are
continuously differentiable. Eq. (1) has an equilibrium point when there is a
unique $u^{*}\in\mathbb{R}^{q}$ such that $f(x^{*},u^{*})=0$ for all
$x^{*}\in\mathcal{X}$, where $\mathcal{X}\subset\mathbb{R}^{n}$. Both $u^{*}$
and $y^{*}\triangleq h(x^{*},u^{*})$ are implicit functions of $x^{*}$. These
conditions are met for every $x^{*}\in\mathcal{X}$. The equilibrium-
independent-dissipativity (EID) feature [15], which is described below, may be
employed for the dissipativity analysis of (1) without the explicit knowledge
of its equilibrium points.
Definition 1. If there is a continuously differentiable storage function
$V:\mathbb{R}^{n}\times\mathcal{X}\rightarrow\mathbb{R}$, the system is known
as EID under supply rate
$s:\mathbb{R}^{q}\times\mathbb{R}^{m}\rightarrow\mathbb{R}$. For all
$(x,x^{*},u)\in\mathbb{R}^{n}\times\mathcal{X}\times\mathbb{R}^{q}$,
$V(x,x^{*})>0$ with $x\neq x^{*}$, $V(x^{*},x^{*})=0$, and
$\dot{V}(x,x^{*})=\nabla_{x}V(x,x^{*})f(x,u)\leq s(u-u^{*},y-y^{*})$.

Depending on the utilized supply rate $s(.,.)$, this EID property may be
specialized. The concept of the $X-\text{EID}$ property is used in the sequel.
This property is characterized by a quadratic supply rate, which is determined
by the coefficient matrix $X=X^{\top}\in\mathbb{R}^{q+m}$.
Definition 2. If the system (1) is EID under the quadratic supply rate, it is
called $X-\text{EID}$ with $X\triangleq[X^{kl}]_{k,l\in\mathbb{N}_{2}}$:

$s(u-u^{*},y-y^{*})\triangleq\begin{bmatrix}u-u^{*}\\\
y-y^{*}\end{bmatrix}^{\top}\begin{bmatrix}X^{11}&X^{12}\\\
X^{21}&X^{22}\end{bmatrix}\begin{bmatrix}u-u^{*}\\\ y-y^{*}\end{bmatrix}$

######  Remark 1.

If the system (1) is $X-\text{EID}$ with:
1) $\begin{bmatrix}\mathbf{0}&\frac{1}{2}\mathbf{I}\\\
\frac{1}{2}\mathbf{I}&\mathbf{0}\end{bmatrix}$, then it is passive;
2) $\begin{bmatrix}-\nu\mathbf{I}&\frac{1}{2}\mathbf{I}\\\
\frac{1}{2}\mathbf{I}&-\rho\mathbf{I}\end{bmatrix}$, then it is strictly
passive ($\nu$ and $\rho$ are the input feedforward and output feedback
passivity indices, respectively);
3) $\begin{bmatrix}\gamma^{2}\mathbf{I}&\mathbf{0}\\\
\mathbf{0}&-\mathbf{I}\end{bmatrix}$, then it is $L_{2}$-stable ($\gamma$ is
the $L_{2}$-gain, denoted as $L2G(\gamma)$).

If the system (1) is linear time-invariant (LTI), a necessary and sufficient
condition for the system (1) to be X-EID is provided in the following
proposition, which regards it as a linear matrix inequality (LMI) problem.

######  Proposition 1.

[16] The linear time-invariant (LTI) system

$\displaystyle\dot{x}(t)=$ $\displaystyle Ax(t)+Bu(t)$ (2) $\displaystyle
y(t)=$ $\displaystyle Cx(t)+Du(t)$

is $X$-EID if and only if there exists $P>0$ such that

$\begin{bmatrix}-\mathcal{H}(PA)+C^{\top}X^{22}C&-PB+C^{\top}X^{21}+C^{\top}X^{22}D\\\
\star&X^{11}+\mathcal{H}(X^{12}D)+D^{\top}X^{22}D\end{bmatrix}\geq 0$ (3)

The following corollary takes into account a specific LTI system that has a
local controller (a setup that we will see later on) and gives an LMI problem
for synthesizing this local controller to optimize or enforce X-EID.

######  Corollary 1.

[16] The LTI system

$\dot{x}(t)=(A+BL)x(t)+\eta(t),$ (4)

is $X-\text{EID}$ with $X^{22}<0$ from external input $\eta(t)$ to state
$x(t)$ if and only if there exists $P>0$ and $K$ such that

$\begin{bmatrix}-(X^{22})^{-1}&P&0\\\ \star&-\mathcal{H}(AP+BK)&-I+PX^{21}\\\
\star&\star&X^{11}\end{bmatrix}\geq 0$ (5)

and $L=KP^{-1}$.

###  II-C Networked Systems

####  II-C1 Configuration

The networked system $\Sigma$ depicted in Fig. 1 is composed of $N$
independent dynamic subsystems $\\{\Sigma_{i}:i\in\mathbb{N}_{N}\\}$ and a
static interconnection matrix $M$ that characterizes the interconnections
between the subsystems, an exogenous input signal $w(t)\in\mathbb{R}^{r}$
(e.g. disturbance) and an interested output signal $z(t)\in\mathbb{R}^{l}$
(e.g. performance).

Figure 1: A generic networked system $\Sigma$

The dynamics of the subsystem $\\{\Sigma_{i}:i\in\mathbb{N}_{N}\\}$ are
provided by

$\displaystyle\dot{x}_{i}(t)$ $\displaystyle=f_{i}(x_{i}(t),u_{i}(t)),$ (6)
$\displaystyle y_{i}(t)$ $\displaystyle=h_{i}(x_{i}(t),u_{i}(t)),$

where $x_{i}(t)\in\mathbb{R}^{n_{i}}$, $u_{i}(t)\in\mathbb{R}^{q_{i}}$,
$y_{i}(t)\in\mathbb{R}^{m_{i}}$.
On the other hand, the interconnection matrix $M$ and the corresponding
interconnection relationship are given by:

$\begin{bmatrix}u\\\ \bar{u}\\\ z\end{bmatrix}=M\begin{bmatrix}y\\\ \bar{y}\\\
w\end{bmatrix}\equiv\begin{bmatrix}M_{uy}&M_{u\bar{y}}&M_{uw}\\\
M_{\bar{u}y}&M_{\bar{u}\bar{y}}&M_{\bar{u}w}\\\
M_{zy}&M_{z\bar{y}}&M_{zw}\end{bmatrix}\begin{bmatrix}y\\\ \bar{y}\\\
w\end{bmatrix}$ (7)

where $u\triangleq[u_{i}^{\top}]^{\top}_{i\in\mathbb{N}_{N}}$ and
$y\triangleq[y_{i}^{\top}]^{\top}_{i\in\mathbb{N}_{N}}$.

####  II-C2 Dissipativity Analysis

Inspired by [15], the following proposition exploits the $X_{i}$-EID
properties of the individual subsystems $\Sigma_{i},i\in\mathbb{N}_{N}$ to
formulate an LMI problem so as to analyze the Y-EID property of the networked
system $\Sigma$, where
$\text{Y}=\text{Y}^{\top}=[\text{Y}^{kl}],k,l\in\mathbb{N}_{2}$.

######  Remark 2.

A subsystem is required to be finite-gain $\mathcal{L}_{2}$ stable with some
$\mathcal{L}_{2}$-gain $\gamma$ from its disturbance $w(t)$ to performance
$z(t)$. This specification is equivalent to $\mathbf{Y}$-dissipativity with:

$\mathbf{Y}=\begin{bmatrix}\gamma^{2}\mathbf{I}&0\\\
0&-\mathbf{I}\end{bmatrix}=\begin{bmatrix}\tilde{\gamma}\mathbf{I}&0\\\
0&-\mathbf{I}\end{bmatrix}$ (8)

Let $\tilde{\gamma}\triangleq\gamma^{2}$, and we need
$0<\tilde{\gamma}<\bar{\gamma}$, where $\bar{\gamma}\in\mathbb{R}>0$ is a
constant value.

######  Proposition 2.

[16] Let
$\hat{x}^{*}\triangleq[[x_{i}^{*}{{}^{\top}}]_{i\in\mathbb{N}_{N}},[\bar{x}_{i}^{*\top}]_{i\in\mathbb{N}_{\bar{N}}}]^{\top}$
be an equilibrium point of networked (under $w(t)=0$) with
$x_{i}^{*}\in\mathcal{X}_{i},\forall i\in\mathbb{N}_{N}$,
$\bar{x}_{i}^{*}\in\bar{\mathcal{X}}_{i},\forall i\in\mathbb{N}_{\bar{N}}$ and
$z^{*}$ be corresponding output. Then, the network is Y-dissipative if there
exist scalars $p_{i}\geq 0,\forall i\in\mathbb{N}_{N}$ and
$\bar{p}_{i},i\in\mathbb{N}_{\bar{N}}$ such that

$\begin{bmatrix}M_{uy}&M_{u\bar{y}}&M_{uw}\\\
\mathbf{I}&\mathbf{0}&\mathbf{0}\\\
M_{\bar{u}y}&M_{\bar{u}\bar{y}}&M_{\bar{u}w}\\\
\mathbf{0}&\mathbf{I}&\mathbf{0}\\\ \mathbf{0}&\mathbf{0}&\mathbf{I}\\\
M_{zy}&M_{z\bar{y}}&M_{zw}\end{bmatrix}^{\top}\begin{bmatrix}X_{p}&\mathbf{0}&\mathbf{0}\\\
\mathbf{0}&\bar{X}_{\bar{p}}&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&-\mathbb{Y}\end{bmatrix}\star\leq 0,$ (9)

where the components $X_{p}$ and $X_{\bar{p}}$ for $i\in\mathbb{N_{N}}$ are
defined as follows:

$\displaystyle
X_{p}\triangleq\begin{bmatrix}\text{diag}(p_{i}X_{i}^{11})&\text{diag}(p_{i}X_{i}^{12})\\\
\text{diag}(p_{i}X_{i}^{21})&\text{diag}(p_{i}X_{i}^{22})\end{bmatrix}$ (10)
$\displaystyle\bar{X}_{\bar{p}}\triangleq\begin{bmatrix}\text{diag}(\bar{p}_{i}\bar{X}_{i}^{11})&\text{diag}(\bar{p}_{i}\bar{X}_{i}^{12})\\\
\text{diag}(\bar{p}_{i}\bar{X}_{i}^{21})&\text{diag}(\bar{p}_{i}\bar{X}_{i}^{22})\end{bmatrix}$

####  II-C3 Topology Synthesis

The following proposition develops an LMI problem to synthesize the
interconnection matrix $M$ (7) to enforce the $\mathbb{Y}$-EID property for
the networked system $\Sigma$. Nonetheless, similar to [17], we must initially
establish two mild assumptions.

######  Assumption 1.

For the networked system $\Sigma$, the provided Y-EID specification is such
that $\textbf{Y}^{22}<0$.

######  Remark 3.

According to Rm. 1, As. 1 holds if the networked system $\Sigma$ must be
either: (1) L2G($\gamma$) or (2) OFP($\rho$) (or IF-OFP($\nu,\rho$)) with some
$\rho>0$, i.e., $L_{2}$-stable or passive, respectively. Therefore, As. 1 is
mild since it is usually preferable to make the networked system $\Sigma$
either $L_{2}$-stable or passive.

######  Assumption 2.

Every subsystem $\Sigma_{i}$ in the networked system $\Sigma$ is $X_{i}$-EID
with $X_{i}^{11}>0$ or $X_{i}^{11}<0,\forall i\in\mathbb{N}_{N}$.

######  Remark 4.

Based on Rm. 1, As. 2 holds if the networked subsystem
$\Sigma_{i},i\in\mathbb{N}_{N}$ must be either: (1) L2G($\gamma_{i}$) (as
$X_{i}^{11}=\gamma_{i}^{2}\mathbf{I}>0$), or (2) IFP($\nu_{i}$) (or IF-
OFP($\nu_{i},\rho_{i}$)) with $\nu_{i}<0$ (i.e., non-passive), or (3)
IFP($\nu_{i}$) (or IF-OFP($\nu_{i},\rho_{i}$)) with $\nu_{i}>0$ (i.e.,
passive). Thus, As. 2 is also mild.

######  Proposition 3.

Under Assumptions 1 and 2, the network system NSC 4 can be Y-EID
($\mathbf{Y}$-dissipative from $w(t)$ to output $z(t)$) by synthesizing the
interconnection matrix $M$ by solving LMI problem:

Find: $\displaystyle
L_{uy},L_{u\bar{y}},L_{uw},L_{\bar{u}y},L_{\bar{u}\bar{y}},L_{\bar{u}w},M_{zy},M_{z\bar{y}},M_{zw},$
(11)
$\displaystyle\\{p_{i}:i\in\mathbb{N}_{N}\\},\\{\bar{p}_{i}:i\in\mathbb{N}_{\bar{N}}\\}$
Sub. to: $\displaystyle p_{i}\geq 0,\forall i\in\mathbb{N}_{N},\ \
\bar{p}_{i}\geq 0,\forall i\in\mathbb{N}_{\bar{N}},\ \text{and}\
\eqref{NSC4YEID},$

with $\scriptsize\begin{bmatrix}M_{uy}&M_{u\bar{y}}&M_{uw}\\\
M_{\bar{u}y}&M_{\bar{u}\bar{y}}&M_{\bar{u}w}\end{bmatrix}=\begin{bmatrix}\textbf{X}_{p}^{11}&\mathbf{0}\\\
\mathbf{0}&\bar{\textbf{X}}_{\bar{p}}^{11}\end{bmatrix}^{-1}\begin{bmatrix}L_{uy}&L_{u\bar{y}}&L_{uw}\\\
L_{\bar{u}y}&L_{\bar{u}\bar{y}}&L_{\bar{u}w}\end{bmatrix}$.

$\begin{bmatrix}\textbf{X}_{p}^{11}&\mathbf{0}&\mathbf{0}&L_{uy}&L_{u\bar{y}}&L_{uw}\\\
\mathbf{0}&\bar{\textbf{X}}_{\bar{p}}^{11}&\mathbf{0}&L_{\bar{u}y}&L_{\bar{u}\bar{y}}&L_{\bar{u}w}\\\
\mathbf{0}&\mathbf{0}&-\textbf{Y}^{22}&-\textbf{Y}^{22}M_{zy}&-\textbf{Y}^{22}M_{z\bar{y}}&\textbf{Y}^{22}M_{zw}\\\
L_{uy}^{\top}&L_{\bar{u}y}^{\top}&-M_{zy}^{\top}\textbf{Y}^{22}&-L_{uy}^{\top}\textbf{X}^{12}-\textbf{X}^{21}L_{uy}-\textbf{X}_{p}^{22}&-\textbf{X}^{21}L_{u\bar{y}}-L_{\bar{u}y}^{\top}\bar{\textbf{X}}^{12}&-\textbf{X}^{21}L_{uw}+M_{zy}^{\top}\textbf{Y}^{21}\\\
L_{u\bar{y}}^{\top}&L_{\bar{u}\bar{y}}^{\top}&-M_{z\bar{y}}^{\top}\textbf{Y}^{22}&-L_{u\bar{y}}^{\top}\textbf{X}^{12}-\bar{\textbf{X}}^{21}L_{\bar{u}y}&-(L_{\bar{u}\bar{y}}^{\top}\bar{\textbf{X}}^{12}+\bar{\textbf{X}}^{21}L_{\bar{u}\bar{y}}+\bar{\textbf{X}}_{\bar{p}}^{22})&-\bar{\textbf{X}}^{21}L_{\bar{u}w}+M_{z\bar{y}}^{\top}\textbf{Y}^{21}\\\
L_{uw}^{\top}&L_{\bar{u}w}^{\top}&-M_{zw}^{\top}\textbf{Y}^{22}&-L_{uw}^{\top}\textbf{X}^{12}+\textbf{Y}^{12}M_{zy}&-L_{\bar{u}w}^{\top}\bar{\textbf{X}}^{12}+\textbf{Y}^{12}M_{z\bar{y}}&M_{zw}^{\top}\textbf{Y}^{21}+\textbf{Y}^{12}M_{zw}+\textbf{Y}^{11}\end{bmatrix}>0$
(12)

##  III DC Microgrid Modeling

The initial step involves explaining the electrical configuration of a DC MG,
which consists of multiple DGs interconnected via power transmission lines.
Specifically, we utilize the model proposed in the study by [1], which enables
considering various topologies of DC MGs. The whole microgrid can be
represented by a directed connected graph
$\mathcal{G}=(\mathcal{V},\mathcal{E})$ where the $\mathcal{V}=\\{1,...,N\\}$
represents the set of nodes and
$\mathcal{E}\subseteq(\mathcal{V}\times\mathcal{V})$ represents the set of
edges. For a given node $i\in\mathcal{V}$, the set of out-neighbors is
$\mathcal{N}_{i}^{+}=\\{j\in\mathcal{V}:(i,j)\in\mathcal{E}\\}$, the set of
in-neighbors is
$\mathcal{N}_{i}^{-}=\\{j\in\mathcal{V}:(j,i)\in\mathcal{E}\\}$, and the set
of neighbors is $\mathcal{N}_{i}=\mathcal{N}_{i}^{+}\cup\mathcal{N}_{i}^{-}$.
$\mathcal{V}$ is divided into two parts: $\mathcal{D}=\\{1,..,n\\}$ represents
the set of DGs and $\mathcal{L}=\\{n+1,...,n+m\\}$ represents the set of
transmission lines. The DGs are connected through the communication and
physical interconnection power lines. If the DG $i$ is connected to DG $j$,
then $(i,j)\in\mathcal{E}$. The $\Sigma_{i}^{DG},i\in\mathbb{N}_{N}$
represents an MG with $N$ DGs, and the $\Sigma_{l}^{line},l\in\mathbb{N}_{L}$
an MG with $L$ transmission lines. The interface between each DG and the power
grid is established through a point of common coupling (PCC). The loads are
assumed to be connected to the DG terminals for simplicity. Indeed, the loads
can be moved to PCC using Kron reduction, even if they are in a different
location [18]. Given that every DG is only directly connected to the lines,
every edge in $\mathcal{E}$ has one node in $\mathcal{D}$ and another node in
$\mathcal{L}$. This results in $\mathcal{G}$ being a bipartite graph. The
orientation of each edge reflects an arbitrarily chosen reference direction
for positive currents. A line must have both in-neighbors and out-neighbors
since the current entering a line must exit it. Indeed, each node in
$\mathcal{L}$ is connected via two directed edges to two distinct nodes in
$\mathcal{D}$. The incidence matrix for a subsequent matrix is denoted by
$B\in\mathbb{R}^{n\times m}$, where DGs are arranged in rows and lines are
arranged in columns.

$B_{il}=\begin{cases}+1&l\in\mathcal{N}_{i}^{+},\\\
-1&l\in\mathcal{N}_{i}^{-},\ i\in\mathcal{D},l\in\mathcal{N},\\\
0&\text{otherwise}.\end{cases}$ (13)

###  III-A Dynamic model of a DG

Each DG consists of a DC voltage source, a buck converter, and a series RLC
filter. The $i$-th DG unit supplies power to a specific local load located at
the $\text{PCC}_{i}$. Additionally, it is interconnected with other DG units
via power transmission lines. Fig. 2 illustrates a schematic electric diagram
of the $i$th DG system, including load, connecting lines, loads, and a local
controller. By applying Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage
Law (KVL) on the DG side at the $\text{PCC}_{i}$, the following equations are
derived for $\sum_{i}^{DG}$:

$\displaystyle C_{ti}\frac{dV_{i}}{dt}$
$\displaystyle=I_{ti}-I_{Li}(V_{i})-I_{i}$ (14) $\displaystyle
L_{ti}\frac{dI_{ti}}{dt}$ $\displaystyle=-V_{i}-R_{ti}I_{ti}+V_{ti}$

In (14), the symbol $V_{ti}$ represents the input command signal applied to
the buck converter, while $I_{ti}$ denotes the current flowing through the
filter. The variables $R_{ti}\in\mathbb{R}^{+}$, $L_{ti}\in\mathbb{R}^{+}$,
and $C_{ti}\in\mathbb{R}^{+}$ represent the internal resistance, inductance,
and capacitance of the DG converter, respectively.

###  III-B Dynamic model of power line

Each transmission line is modeled using the $\pi$-equivalent representation.
The assumption is made that the line capacitances are consolidated with the
capacitance of the DG filter. Hence, as seen in Fig. 2, the power line denoted
as $l$ is represented by a RL circuit with a positive resistance $R_{l}$ and
inductance $L_{l}$. By applying Kirchoff’s voltage law (KVL) to analyze the
$l$-th power line denoted $\Sigma_{l}^{line}$, one can get:

$\Sigma_{l}^{line}:\frac{dI_{l}}{dt}=-\frac{R_{l}}{L_{l}}I_{l}+\frac{1}{L_{l}}u_{l}$
(15)

where $u_{l}=V_{i}-V_{j}=\sum_{i\in\mathcal{N}_{l}}B_{il}V_{i}$. The variables
$V_{i}$ and $I_{l}$ denote the voltage at the $\text{PCC}_{i}$ and the current
conducting through the $l$-th power line, respectively.

Figure 2: Electrical Schematic of DG-i, load-$i$, local controller-$i$,
$i\in\mathcal{N}_{N}$ and line-$l$, $l\in\mathcal{N}_{L}$

The line current $I_{i}$ is the current injected into the DC MG and is given
by:

$I_{i}=\sum_{l\in\mathcal{N}_{i}^{+}}B_{il}I_{l}+\sum_{l\in\mathcal{N}_{i}^{-}}B_{il}I_{l}=\sum_{l\in\mathcal{N}_{i}}B_{il}I_{l}.$
(16)

###  III-C Load model

The phrase $I_{Li}(V_{i})$ in Fig. 2 stands for the current flowing through
the $i$-th load. The functional dependence on the PCC voltage varies depending
on the type of load, and the term $I_{Li}(V_{i})$ has distinct expressions.
The following load models are included: (1) loads with constant current:
$I_{Lli}=\bar{I}_{Lli}$, (2) constant impedance loads
$I_{LZi}(V_{i})=Y_{Li}V_{i}$, where the conductance of the $i$th load is
$Y_{Li}=1/R_{Li}>0$, and (3) constant power loads
$I_{LPi}(V_{i})=V_{i}^{-1}P_{Li}^{*}$, where $P_{Li}^{*}>0$ represents the
power demand of the load $i$. Therefore, the ZIP loads are as follows:

$I_{Li}(V_{i})=\bar{I}_{Li}+Y_{Li}V_{i}+V_{i}^{-1}P_{Li}^{*}$ (17)

###  III-D State space model of DC MG

By integrating the DG dynamics in Eq. (14) and the error dynamic in Eq. (25),
the state space representation of the DG will be obtained:

$\sum_{i}^{DG}:\begin{cases}\dot{x}_{i}(t)=A_{i}x_{i}(t)+B_{i}u_{i}(t)+C_{i}d_{i}(t)+\xi_{i}(t)\\\
z_{i}(t)=H_{i}x_{i}(t)\end{cases}$ (18)

where $x_{i}=\begin{bmatrix}V_{i}&I_{ti}&v_{i}\end{bmatrix}^{\top}$ is the
state, $u_{i}=V_{ti}$ is the control input,
$d_{i}=\begin{bmatrix}\bar{I}_{Li}&0&V_{ri}\end{bmatrix}^{\top}$ stands for
the exogenous input or disturbance, and $z_{i}=v_{i}$ is the desired
performance since our goal is $\lim_{t\to\infty}V_{i}(t)=V_{ri}$ The term
$\xi_{i}$ is the transmission line coupling corresponding to DG $i$. Thus, we
have:

$\displaystyle\frac{dV_{i}}{dt}$
$\displaystyle=\frac{1}{C_{ti}}I_{ti}-\frac{1}{C_{ti}}I_{Li}(V_{i})-\frac{1}{C_{ti}}I_{i}$
$\displaystyle\frac{dI_{ti}}{dt}$
$\displaystyle=-\frac{1}{L_{ti}}V_{i}-\frac{R_{ti}}{L_{ti}}I_{ti}+\frac{1}{L_{ti}}u_{i}$
(19) $\displaystyle\frac{dv_{i}}{dt}$ $\displaystyle=V_{i}-V_{ri}$

###  III-E The local and global controller

The primary aim of the local controller is to guarantee that the voltage at
($\text{PCC}_{i}$) closely follows a predetermined reference voltage
($V_{ri}$), often supplied by a higher level controller. In the absence of
voltage stabilization in the DC MG, there is a potential for voltages to
exceed critical thresholds, leading to detrimental consequences for the loads
and the DC MG infrastructure.

The state space representation of DC MG is obtained as follows by using Eq.
(III-D):

$\displaystyle\begin{bmatrix}\frac{dV_{i}}{dt}\\\ \frac{dI_{ti}}{dt}\\\
\frac{dv_{i}}{dt}\end{bmatrix}=$
$\displaystyle\begin{bmatrix}-\frac{Y_{Li}}{C_{ti}}V_{i}-\frac{P_{Li}}{C_{ti}}V_{i}^{-1}+\frac{1}{C_{ti}}I_{ti}\\\
-\frac{1}{L_{ti}}V_{i}-\frac{R_{ti}}{L_{ti}}I_{ti}+\frac{1}{L_{ti}}f_{io}\\\
V_{i}\end{bmatrix}+\begin{bmatrix}-\frac{\bar{I}_{Li}}{C_{ti}}\\\ 0\\\
-V_{ri}\end{bmatrix}$ (20) $\displaystyle+$
$\displaystyle\begin{bmatrix}-\frac{1}{C_{ti}}\sum B_{il}I_{l}\\\ 0\\\
0\end{bmatrix}+\begin{bmatrix}0\\\ \frac{1}{L_{ti}}\sum f_{ij}\\\
0\end{bmatrix}$

where

$A_{i}=\begin{bmatrix}-\frac{Y_{i}}{C_{ti}}&\frac{1}{C_{ti}}&0\\\
-\frac{1}{L_{ti}}&-\frac{R_{ti}}{L_{ti}}&0\\\
1&0&0\end{bmatrix},\quad\text{and}\quad B_{i}=\begin{bmatrix}0\\\
\frac{1}{L_{ti}}\\\ 0\end{bmatrix}$ (21)

and the following is the state space representation of the transmission line:

$\sum_{i}^{line}:\dot{x}_{l}=-\frac{R_{l}}{L_{l}}x_{l}+\frac{1}{L_{l}}u_{l}$
(22)

where $x_{l}=I_{l}$. Therefore, by using Eq. (15):

$\begin{bmatrix}\frac{dI_{l}}{dt}\end{bmatrix}=\begin{bmatrix}-\frac{R_{l}}{L_{l}}I_{l}\end{bmatrix}+\begin{bmatrix}\frac{1}{L_{l}}\sum_{i\in\mathcal{N}_{l}}B_{il}V_{i}\end{bmatrix}$
(23)

It can be seen from Eq. (20) that the control input is combination of two
terms, distributed and local controller. Thus, one can define:

$u_{i}(t)\triangleq
V_{ti}(t)=f_{io}(x_{i}(t))+\sum_{j\in\mathcal{N}_{N}}f_{ij}(x_{i}(t))$ (24)

In the following, we describe the local and distributed controllers in detail.

To effectively track a reference voltage, it is imperative to ensure that the
error $e_{i}(t)\triangleq V_{i}(t)-V_{ri}(t)$ converges to zero. In order to
achieve this objective, following the approach proposed in [2], we include
each DG system $\Sigma_{i}^{DG}$ with an integrator state $v_{i}$ which
follows the dynamics as illustrated in Fig. 2.

$\frac{dv_{i}}{dt}=e_{i}(t)=V_{i}(t)-V_{ri}(t),$ (25)

and subsequently, equip it with a state feedback controller

$f_{io}(V_{i},I_{ti},v_{i})=K_{io}x_{i}(t),$ (26)

where
$x_{i}\triangleq\begin{bmatrix}V_{i}&I_{ti}&v_{i}\end{bmatrix}^{\top}\in\mathbb{R}^{3}$
denotes the augmented state of DG and
$K_{io}=\begin{bmatrix}k_{1o}&k_{2o}&k_{3o}\end{bmatrix}\in\mathbb{R}^{1\times
3}$ is the feedback gain matrix.

However, this local controller does not provide global stability in the
presence of two or more interconnected DGs. To address this issue, as shown in
Fig. 2, the distributed controller is employed to not only restore the voltage
to its nominal value but also ensure that the currents are shared properly
among the DGs. The distributed feedback controller is defined as follows:

$\displaystyle\sum_{j\in\mathcal{N}_{N}}f_{ij}(V_{j},I_{tj},v_{j})=\sum_{j\in\mathcal{N}_{N}}K_{ij}x_{j}(t)$
(27)
$\displaystyle=k_{ij}^{V}V_{j}+k_{ij}^{I}I_{tj}+k_{ij}^{v}v_{j}=\bar{K}_{ij}\begin{bmatrix}V_{j}\\\
I_{tj}\\\ v_{j}\end{bmatrix}$

##  IV Networked System Model

The DC MG may be regarded as a networked system configuration, as represented
in Fig. 3.

Figure 3: DC Microgrid dynamics as a networked system configuration

Let us define $u\triangleq[u_{i}]_{i}$ control input of DG $i$,
$\bar{u}\triangleq[\bar{u_{l}}]_{l}$ control input of transmission line,
$x\triangleq[x_{i}]_{i}$ full state of DG $i$, $w\triangleq[w_{i}]_{i}$
disturbance input, $z\triangleq[z_{i}]_{i}$ output performance, and
$\bar{x}\triangleq[\bar{x}_{l}]_{l}$ full state of line $i$. Assume there is
no constant power load ($P_{Li}=0$) as specified in Eq. (17). As a result, the
closed-loop feedback linearization of the DC MG subsystem will be
accomplished.

$\dot{x_{i}}=(A_{i}+B_{i}K_{io})x_{i}+u_{i}(t)=g_{i}(x_{i})+u_{i}$ (28)

where

$u_{i}=\begin{bmatrix}-\frac{\bar{I}_{Li}}{C_{ti}}\\\ 0\\\
-V_{ri}\end{bmatrix}\\\ +\begin{bmatrix}-\frac{1}{C_{ti}}\sum B_{il}I_{l}\\\
0\\\ 0\end{bmatrix}+\begin{bmatrix}0\\\ \frac{1}{L_{ti}}\sum f_{ij}\\\
0\end{bmatrix}$ (29)

So, we have:

$u_{i}=w_{i}+\sum_{l\in\mathbb{N}_{L}}\bar{C}_{il}\bar{x}_{l}+\sum_{j\in\mathbb{N}_{N}}K_{ij}x_{j}=w+\bar{C}\bar{x}+Kx_{j}$
(30)

where
$\bar{C}_{il}=\begin{bmatrix}-\frac{B_{il}}{C_{ti}}&0&0\end{bmatrix}^{\top}$,
and

$K_{ij}=\frac{1}{L_{ti}}\begin{bmatrix}0&0&0\\\
k_{ij}^{V}&k_{ij}^{I}&k_{ij}^{v}\\\ 0&0&0\end{bmatrix}$ (31)

where $\bar{C}=[\bar{C}_{il}]_{i\in\mathbb{N}_{N},l\in\mathbb{N}_{L}}$ and
$K=[k_{ij}]_{i,j\in\mathbb{N}_{N}}$. On the other hand, the line dynamic of
Eq. (23) can be written as:

$\dot{x_{l}}=(A_{l}+B_{l}K_{lo})x_{l}+u_{l}(t)=\bar{g}_{l}(\bar{x}_{l})+\bar{u}_{l}$
(32)

where $K_{lo}$ is the feedback controller for line $l$ and can be defined as
follows:

$f_{lo}(I_{l})=K_{lo}x_{l}(t),$ (33)

such that

$\bar{u}_{l}=\sum_{i\in\mathbb{N}_{N}}C_{il}x_{i}$ (34)

with $C_{il}\triangleq\begin{bmatrix}\frac{B_{il}}{L_{l}}&0&0\end{bmatrix}$.
So we have:

$\bar{u}=Cx$ (35)

where $C=[C_{il}]_{i\in\mathbb{N}_{N},l\in\mathbb{N}_{L}}$. Then, the output
performance can be written as:

$z_{i}=v_{i}=\begin{bmatrix}0&0&1\end{bmatrix}x_{i}=D$ (36)

So we have:

$z=Dx$ (37)

where $D=\text{diag}(\begin{bmatrix}0&0&1\end{bmatrix})$. Now, using the above
equations, we get:

$\begin{cases}M_{ux}=K,\quad M_{u\bar{x}}=\bar{C},\quad M_{uw}=\mathbf{I}\\\
M_{\bar{u}x}=C,\quad M_{\bar{u}\bar{x}}=\mathbf{0},\quad
M_{\bar{u}w}=\mathbf{0}\\\ M_{zx}=D,\quad M_{z\bar{x}}=\mathbf{0},\quad
M_{zw}=\mathbf{0}\end{cases}$ (38)

##  V Passivity-based Control

The organization of this section is as follows. The initial focus is on
introducing the global controller design challenge for the networked system.
Subsequently, we provide a set of essential prerequisites for the
dissipativity properties of the subsystem in order to accomplish feasible and
efficient global controller design. Next, we introduce the enhanced local
controller design problem that ensures the relevant requirements are met. In
conclusion, we provide a concise overview of the whole process for designing
both local and global controllers.

###  V-A Global Controller Synthesis

A subsystem $\hat{\Sigma}_{i},i\in\mathbb{N}_{N}$ (28) is assumed to be
$X_{i}$-EID (strictly passive; i.e., IF-OFP($\nu_{i},\rho_{i}$)) with

$X_{i}=\begin{bmatrix}X_{i}^{11}&X_{i}^{12}\\\
X_{i}^{21}&X_{i}^{22}\end{bmatrix}\triangleq\begin{bmatrix}-\nu_{i}\mathbf{I}&\frac{1}{2}\mathbf{I}\\\
\frac{1}{2}\mathbf{I}&-\rho_{i}\mathbf{I}\end{bmatrix}$ (39)

where $\rho_{i}$ and $\nu_{i}$ are passivity indices that are fixed and known.
The interconnection matrix $M$ (38) and its block
$M_{ux}=[K_{ij}]_{i,j\in\mathbb{N}_{N}}$ will be synthesized by applying these
subsystem EID properties to Prop. 3. By synthesizing the matrices
$[K_{ij}]_{i,j\in\mathbb{N}_{N}}$, we can uniquely compute the controller
gains $\\{k_{ij}:i,j\in\mathbb{N}_{N}\\}$ (31).
We need the overall DC MG dynamics (i.e., the networked system
$\tilde{\Sigma}$) in Fig. (3) to be $L_{2}$-stable (Y-dissipative) based on
Prop 2. The following theorem formulates the centralized controller synthesis
problem as an LMI problem that can be solved using standard convex
optimization toolboxes.

######  Theorem 1.

The closed-loop dynamics of the DG illustrated in Fig. 3 can be finite-gain
$L_{2}$-stable with some $L_{2}$-gain $\gamma$ (where
$\tilde{\gamma}\triangleq\gamma^{2}<\bar{\gamma}$) from the disturbance input
$w(t)$ to the performance output $z(t)$. The interconnection matrix block
$M_{ux}=[K_{ij}]_{i,j\in\mathbb{N}_{N}}$ (38), which is synthesized via the
solution of the LMI problem, is utilized to achieve this.

$\displaystyle\min_{Q,\gamma,\\{p_{i}:i\in\mathbb{N}_{N}\\}}\sum_{i,j\in\mathbb{N}_{N}}c_{ij}\|Q_{ij}\|_{1}+c_{0}\tilde{\gamma},\
0<\tilde{\gamma}<\bar{\gamma}$ (40a) $\displaystyle\mbox{Sub. to:
}p_{i}>0,\forall i\in\mathbb{N}_{N},\ \bar{p}_{i}>0,\forall
i\in\mathbb{N}_{L};l\in\mathbb{N}_{L},$ $\displaystyle\mbox{and LMI condition
in Eq.}\ \eqref{globalcontrollertheorem}\ \ $

In this context, the structure of $Q\triangleq[Q_{ij}]_{i,j\in\mathbb{N}_{N}}$
is identical to that of $M_{\eta e}$, where $c_{0}>0$ is a predetermined
constant. Other parameters are defined as
$\textbf{X}^{12}\triangleq\text{diag}(-\frac{1}{2\nu_{i}}\mathbf{I}:i\in\mathbb{N}_{N})$,
$\textbf{X}^{21}\triangleq(\textbf{X}^{12})^{\top}$,
$\bar{\textbf{X}}^{12}\triangleq\text{diag}(-\frac{1}{2\bar{\nu}_{l}}\mathbf{I}:l\in\mathbb{N}_{L})$
$\bar{\textbf{X}}^{21}\triangleq(\bar{\textbf{X}}^{12})^{\top}$,
$\textbf{X}_{p}^{11}\triangleq\text{diag}(-p_{i}\nu_{i}\mathbf{I}:i\in\mathbb{N}_{N})$,
$\textbf{X}_{p}^{22}\triangleq\text{diag}(-p_{i}\rho_{i}\mathbf{I}:i\in\mathbb{N}_{N})$,
$\bar{\textbf{X}}_{\bar{p}}^{11}\triangleq\text{diag}(-\bar{p}_{l}\bar{\nu}_{l}\mathbf{I}:l\in\mathbb{N}_{L})$,
$\bar{\textbf{X}}_{\bar{p}}^{22}\triangleq\text{diag}(-\bar{p}_{l}\bar{\rho}_{l}\mathbf{I}:l\in\mathbb{N}_{L})$,
and $M_{\eta e}\triangleq(\textbf{X}_{p}^{11})^{-1}Q$.

###### Proof.

By applying the networked system dissipativity properties examined in Prop. 2
and the subsystem dissipativity properties assumed in (39) to the
interconnection topology synthesis result given in Prop. 3 yields the proof. ∎

$\begin{bmatrix}\textbf{X}_{p}^{11}&\mathbf{0}&\mathbf{0}&Q&\textbf{X}_{p}^{11}\bar{C}&\textbf{X}_{p}^{11}\\\
\mathbf{0}&\bar{\textbf{X}}_{\bar{p}}^{11}&\mathbf{0}&\bar{\textbf{X}}_{\bar{p}}^{11}C&\mathbf{0}&\mathbf{0}\\\
\mathbf{0}&\mathbf{0}&\mathbf{I}&D&\mathbf{0}&\mathbf{0}\\\
Q^{\top}&C^{\top}\bar{\textbf{X}}_{\bar{p}}^{11}&D^{\top}&-Q^{\top}\textbf{X}^{12}-\textbf{X}^{21}Q-\textbf{X}_{p}^{22}&-\textbf{X}^{21}\textbf{X}_{p}^{11}\bar{C}-C^{\top}\bar{\textbf{X}}_{\bar{p}}^{11}\bar{\textbf{X}}^{12}&-\textbf{X}^{21}\textbf{X}_{p}^{11}\\\
\bar{C}^{\top}\textbf{X}_{p}^{11}&\mathbf{0}&\mathbf{0}&-\bar{C}^{\top}\textbf{X}_{p}^{11}\textbf{X}^{12}-\bar{\textbf{X}}^{21}\bar{\textbf{X}}_{\bar{p}}^{11}C&-\bar{\textbf{X}}_{\bar{p}}^{22}&\mathbf{0}\\\
\textbf{X}_{p}^{11}&\mathbf{0}&\mathbf{0}&-\textbf{X}_{p}^{11}\textbf{X}^{12}&\mathbf{0}&\tilde{\gamma}\mathbf{I}\end{bmatrix}>0$
(41)

###  V-B Necessary Conditions on Subsystem Passivity Indices

######  Lemma 1.

It is necessary for the main LMI condition (41) in Th. (1) to hold that, for
each subsystem $\hat{\Sigma}_{i},i\in\mathbb{N}_{N}$ (28),
$\hat{\Sigma}_{l},l\in\mathbb{N}_{L}$ (32) the passivity indices $\nu_{i}$,
$\nu_{l}$ and $\rho_{i}$, $\rho_{l}$ (39) (let
$\tilde{\rho}_{i}\triangleq\frac{1}{\rho_{i}}$) are such that the LMI problem:

Find: $\displaystyle\nu_{i},\ \bar{\nu}_{l},\ \tilde{\rho}_{i},\
\bar{\rho}_{l},\ \tilde{\gamma}_{i},$ (42a) Sub. to:
$\displaystyle-\frac{\tilde{\gamma}_{i}}{p_{i}}<\nu_{i}<0,$ (42b)
$\displaystyle 0<\tilde{\rho_{i}}<p_{i},$ (42c) $\displaystyle
0<\tilde{\rho_{i}}<\frac{4\tilde{\gamma}_{i}}{p_{i}},$ (42d)
$\displaystyle\bar{\rho}_{l}>\frac{p_{i}\nu_{i}}{\bar{p}_{l}C_{ti}^{2}},$
(42e)
$\displaystyle\bar{\rho}_{l}>\frac{\tilde{\rho}_{i}}{p_{i}\bar{p}_{l}}(\frac{p_{i}}{2C_{ti}}-\frac{\bar{p}_{l}}{2L_{l}})^{2},$
(42f)
$\displaystyle\bar{\nu}_{l}<-\frac{p_{i}\rho_{i}L_{l}^{2}}{\bar{p}_{l}}<0,$
(42g)

is feasible.

###### Proof.

It is expected that for any $i\in\mathbb{N}_{N}$, there will exist scalars
$0<\tilde{\gamma}_{i}<\tilde{\gamma}$ if the global controller theorem (41)
holds. In other words, (41) is true if and only if there exists
$\tilde{\Gamma}\triangleq\text{diag}(\tilde{\gamma}_{i}\mathbf{I}:i\in\mathbb{N}_{N})$
where $0<\tilde{\Gamma}<\tilde{\gamma}\mathbf{I}$.

To ensure the validity of the LMI condition in (41), it is possible to
identify the following set of necessary conditions by examining particular
sub-block matrices of the block matrix in (41):

$\textbf{X}_{p}^{11}>0,\ \bar{\textbf{X}}_{\bar{p}}^{11}>0$ (43a)
$-Q^{\top}\textbf{X}^{12}-\textbf{X}^{21}Q-\textbf{X}_{p}^{22}>0$ (43b)
$\bar{\textbf{X}}_{\bar{p}}^{22}>0,$ (43c)
$\begin{bmatrix}\textbf{X}_{p}^{11}&\textbf{X}_{p}^{11}\\\
\textbf{X}_{p}^{11}&\tilde{\Gamma}\end{bmatrix}>0,$ (43d)
$\begin{bmatrix}\mathbf{I}&D\\\
D^{\top}&-Q^{\top}\textbf{X}^{12}-\textbf{X}^{21}Q-\textbf{X}_{p}^{22}\end{bmatrix}>0,$
(43e)
$\begin{bmatrix}-Q^{\top}\textbf{X}^{12}-\textbf{X}^{21}Q-\textbf{X}_{p}^{22}&-\textbf{X}^{21}\textbf{X}_{p}^{11}\\\
-\textbf{X}_{p}^{11}\textbf{X}^{12}&\tilde{\Gamma}\end{bmatrix}>0,$ (43f)
$\begin{bmatrix}\textbf{X}_{p}^{11}&\textbf{X}_{p}^{11}\bar{C}\\\
\bar{C}^{\top}\textbf{X}_{p}^{11}&-\textbf{X}_{p}^{22}\end{bmatrix}>0,$ (43g)
$\begin{bmatrix}-Q^{\top}\textbf{X}^{12}-\textbf{X}^{21}Q-\textbf{X}_{p}^{22}&-\textbf{X}^{21}\textbf{X}_{p}^{11}\bar{C}-C^{\top}\bar{\textbf{X}}_{\bar{p}}^{11}\bar{\textbf{X}}^{12}\\\
-\bar{C}^{\top}\textbf{X}_{p}^{11}\textbf{X}^{12}-\bar{\textbf{X}}^{12}\bar{\textbf{X}}_{\bar{p}}^{11}C&-\bar{\textbf{X}}_{\bar{p}}^{22}\end{bmatrix}>0,$
(43h)
$\begin{bmatrix}\bar{\textbf{X}}_{\bar{p}}^{11}&\bar{\textbf{X}}_{\bar{p}}^{11}C\\\
C^{\top}\bar{\textbf{X}}_{\bar{p}}^{11}&-Q^{\top}\textbf{X}^{12}-\textbf{X}^{21}Q-\textbf{X}_{p}^{22}\end{bmatrix}>0,$
(43i)

A required condition for (43a) is:

$\displaystyle\mbox{\eqref{Eq:Lm:CodesignConditionsStep2_1}}\iff-
p_{i}\nu_{i}>0\iff\nu_{i}<0,\forall i\in\mathbb{N}_{N}.$ (44)
$\displaystyle\mbox{\eqref{Eq:Lm:CodesignConditionsStep2_1}}\iff-\bar{p}_{l}\bar{\nu}_{l}>0\iff\bar{\nu}_{l}<0,\forall
l\in\mathbb{N}_{L}.$ (45)

A necessary condition for (43b) can be obtained by considering the diagonal
elements of the matrix and using the structure of the matrix $Q$. Note that
each $Q_{ii}$ block has zeros in its diagonal for any $i\in\mathbb{N}_{N}$.

$\displaystyle\mbox{\eqref{Eq:Lm:CodesignConditionsStep2_2}}\implies$
$\displaystyle\
-Q_{ii}^{\top}(-\frac{1}{2\nu_{i}}\mathbf{I})-(-\frac{1}{2\nu_{i}}\mathbf{I})Q_{ii}-(-p_{i}\rho_{i}\mathbf{I})>0$
$\displaystyle\implies$ $\displaystyle\ p_{i}\rho_{i}>0\iff\rho_{i}>0,\forall
i\in\mathbb{N}_{N}.$ (46)
$\displaystyle\mbox{\eqref{Eq:Lm:CodesignConditionsStep2_3}}\iff$
$\displaystyle-(-\bar{p}_{l}\bar{\rho}_{l}\mathbf{I})>0$
$\displaystyle\implies$
$\displaystyle\bar{p}_{l}\bar{\rho}_{l}>0\iff\bar{\rho}_{l}>0,\forall
l\in\mathbb{N}_{L}.$ (47)

To derive the necessary conditions for (43d)-(43i), we can obtain:

$\displaystyle\mbox{\eqref{Eq:Lm:CodesignConditionsStep2_4}}\iff$
$\displaystyle\ \begin{bmatrix}-p_{i}\nu_{i}&-p_{i}\nu_{i}\\\
-p_{i}\nu_{i}&\tilde{\gamma}_{i}\end{bmatrix}>0\iff-
p_{i}\nu_{i}(\tilde{\gamma}_{i}+p_{i}\nu_{i})>0$ $\displaystyle\iff$
$\displaystyle\ \nu_{i}>-\frac{\tilde{\gamma}_{i}}{p_{i}},\forall
i\in\mathbb{N}_{N};$ (48)
$\displaystyle\mbox{\eqref{Eq:Lm:CodesignConditionsStep2_5}}\implies$
$\displaystyle\ \begin{bmatrix}1&1\\\
1&p_{i}\rho_{i}\end{bmatrix}>0\iff\rho_{i}>\frac{1}{p_{i}},\forall
i\in\mathbb{N}_{N};$ (49)
$\displaystyle\mbox{\eqref{Eq:Lm:CodesignConditionsStep2_6}}\implies$
$\displaystyle\ \begin{bmatrix}p_{i}\rho_{i}&-\frac{1}{2}p_{i}\\\
-\frac{1}{2}p_{i}&\tilde{\gamma}_{i}\end{bmatrix}>0\iff
p_{i}(\rho_{i}\tilde{\gamma}_{i}-\frac{1}{4}p_{i})>0$ $\displaystyle\iff$
$\displaystyle\ \rho_{i}>\frac{p_{i}}{4\tilde{\gamma}_{i}},\forall
i\in\mathbb{N}_{N}.$ (50)
$\displaystyle\mbox{\eqref{Eq:Lm:CodesignConditionsStep2_7}}\implies$
$\displaystyle\
\begin{bmatrix}-p_{i}\nu_{i}&-p_{i}\nu_{i}\frac{B_{il}}{C_{ti}}\\\
-\frac{B_{il}}{C_{ti}}p_{i}\nu_{i}&\bar{p}_{l}\bar{\rho}_{l}\end{bmatrix}>0$
$\displaystyle\iff$
$\displaystyle(-p_{i}\nu_{i}\bar{p}_{l}\bar{\rho}_{l})-(p_{i}^{2}\nu_{i}^{2}\frac{B_{il}^{2}}{C_{ti}^{2}})>0$
$\displaystyle\iff$
$\displaystyle\bar{\rho}_{l}>\frac{p_{i}\nu_{i}}{\bar{p}_{l}C_{ti}^{2}},\forall
i\in\mathbb{N}_{N};l\in\mathbb{N}_{L}.$ (51)
$\displaystyle\mbox{\eqref{Eq:Lm:CodesignConditionsStep2_8}}\implies$
$\displaystyle\
\begin{bmatrix}-p_{i}\rho_{i}&-\frac{p_{i}\nu_{i}B_{il}}{2C_{ti}}-\frac{B_{il}\bar{p}_{l}}{L_{l}}\\\
-\frac{p_{i}\nu_{i}B_{il}}{2C_{ti}}-\frac{B_{il}\bar{p}_{l}}{L_{l}}&\bar{p}_{l}\bar{\rho}_{l}\end{bmatrix}>0$
$\displaystyle\iff$ $\displaystyle
p_{i}\rho_{i}\bar{p}_{l}\bar{\rho}_{l}-(\frac{p_{i}}{2C_{ti}}-\frac{\bar{p}_{l}}{2L_{l}})^{2}>0$
$\displaystyle\iff$
$\displaystyle\bar{\rho}_{l}>\frac{\tilde{\rho}_{i}}{p_{i}\bar{p}_{l}}(\frac{p_{i}}{2C_{ti}}-\frac{\bar{p}_{l}}{2L_{l}})^{2},\forall
i\in\mathbb{N}_{N};l\in\mathbb{N}_{L}.$ (52)
$\displaystyle\mbox{\eqref{Eq:Lm:CodesignConditionsStep2_9}}\implies$
$\displaystyle\
\begin{bmatrix}-\bar{p}_{l}\bar{\nu}_{l}&-\bar{p}_{l}\bar{\nu}_{l}\frac{B_{il}}{L_{l}}\\\
-\bar{p}_{l}\bar{\nu}_{l}\frac{B_{il}}{L_{l}}&-\frac{B_{il}}{L_{l}}\bar{p}_{l}\bar{\nu}_{l}\end{bmatrix}>0$
$\displaystyle\iff$
$\displaystyle(-\bar{p}_{l}\bar{\nu}_{l}p_{i}\rho_{i})-(\frac{\bar{p}_{l}\bar{\nu}_{l}B_{il}}{L_{l}})^{2}>0,$
$\displaystyle\iff$
$\displaystyle\bar{\nu}_{l}<-\frac{p_{i}\rho_{i}L_{l}^{2}}{\bar{p}_{l}},\forall
i\in\mathbb{N}_{N};l\in\mathbb{N}_{L}.$ (53)

In conclusion, we can obtain the set of LMI problems (42) (each in variables
$\nu_{i},\nu_{l},\ \tilde{\rho}_{i},\ \bar{\rho}_{l},\ \tilde{\gamma}_{i}$,
for any $i\in\mathbb{N}_{N};l\in\mathbb{N}_{L}$) by carrying out the
following: (1) compiling the requisite conditions as shown in (44)-(53) and
(2) employing the change of variables
$\tilde{\rho}_{i}\triangleq\frac{1}{\rho_{i}},\forall i\in\mathbb{N}_{N}$, and
(3) recognizing each $p_{i},i\in\mathbb{N}_{N}$ and
$\bar{p}_{l},l\in\mathbb{N}_{L}$ as a predefined positive scalar parameter. ∎

###  V-C Local Controller Synthesis

######  Theorem 2.

The local controller gains $K_{io}$ (26) should be made by solving the LMI
problem for each DG $\Sigma_{i},i\in\mathbb{N}_{N}$ and a certain scalar
parameter $p_{i}>0$. This will create the closed-loop dynamic $\Sigma_{i}$
(28)) IF-OFP($\nu_{i},\rho_{i}$) (as assumed in (39) while also meeting the
conditions set out in Lm. 1.

Find: $\displaystyle\ \tilde{K}_{io},\ P_{i},\ \nu_{i},\ \tilde{\rho}_{i},\
\tilde{\gamma}_{i},$ (54) Sub. to: $\displaystyle\ P_{i}>0,$ $\displaystyle\
\begin{bmatrix}\tilde{\rho}_{i}\mathbf{I}&P_{i}&\mathbf{0}\\\
P_{i}&-\mathcal{H}(AP_{i}+B\tilde{K}_{io})&-I+\frac{1}{2}P_{i}\\\
\mathbf{0}&-I+\frac{1}{2}P_{i}&-\nu_{i}\mathbf{I}\end{bmatrix}>0,$ Sub. to:
$\displaystyle\ -\frac{\tilde{\gamma}_{i}}{p_{i}}<\nu_{i}<0,$ $\displaystyle\
0<\tilde{\rho_{i}}<\min\left\\{p_{i},\frac{4\tilde{\gamma}_{i}}{p_{i}}\right\\}$
$\displaystyle\bar{\rho}_{l}>\frac{p_{i}\nu_{i}}{\bar{p}_{l}C_{ti}^{2}},$
$\displaystyle\bar{\rho}_{l}>\frac{\tilde{\rho}_{i}}{p_{i}\bar{p}_{l}}(\frac{p_{i}}{2C_{ti}}-\frac{\bar{p}_{l}}{2L_{l}})^{2},$
$\displaystyle\bar{\nu}_{l}<-\frac{p_{i}\rho_{i}L_{l}^{2}}{\bar{p}_{l}}<0,$

where $\rho_{i}\triangleq\frac{1}{\tilde{\rho}_{i}}$ and
$K_{io}\triangleq\tilde{K}_{io}P_{i}^{-1}$.

We design the local controller for lines as follows:

######  Theorem 3.

The local controller gains $K_{lo}$ (33) should be made by solving the LMI
problem for each line $\Sigma_{l},l\in\mathbb{N}_{L}$ and a certain scalar
parameter $\bar{p}_{l}>0$. This will create the closed-loop dynamic
$\Sigma_{l}$ (32) IF-OFP($\bar{\nu}_{l},\bar{\rho}_{l}$) while also meeting
the conditions set out in Lm. 1.

Find: $\displaystyle\ \tilde{K}_{lo},\ P_{l},\ \bar{\nu}_{l},\
\bar{\rho}_{l},$ (55) Sub. to: $\displaystyle\ P_{l}>0,$ $\displaystyle\
\begin{bmatrix}\rho_{l}\mathbf{I}&P_{l}&\mathbf{0}\\\
P_{l}&-\mathcal{H}(AP_{l}+B\tilde{K}_{lo})&-I+\frac{1}{2}P_{l}\\\
\mathbf{0}&-I+\frac{1}{2}P_{l}&-\bar{\nu}_{l}\mathbf{I}\end{bmatrix}>0,$ Sub.
to: $\displaystyle\bar{\rho}_{l}>\frac{p_{i}\nu_{i}}{\bar{p}_{l}C_{ti}^{2}},$
$\displaystyle\bar{\rho}_{l}>\frac{\tilde{\rho}_{i}}{p_{i}\bar{p}_{l}}(\frac{p_{i}}{2C_{ti}}-\frac{\bar{p}_{l}}{2L_{l}})^{2},$
$\displaystyle\bar{\nu}_{l}<-\frac{p_{i}\rho_{i}L_{l}^{2}}{\bar{p}_{l}}<0,$

where $K_{io}\triangleq\tilde{K}_{lo}P_{l}^{-1}$.

The proof follows from: (1) considering the DG dynamic model identified in
(28) and line dynamic model identified in (32) then applying the LMI-based
controller synthesis technique so as to enforce the subsystem passivity
indices assumed in (39), and then (2) enforcing the necessary conditions on
subsystem passivity indices identified in Lm. 1 so as to ensure a feasible and
effective global controller design at (41).

To conclude this section, we summarize the three main steps necessary for
designing local controllers, global controllers and communication topology (in
a centralized manner, for the considered microgrid) as:
Step 1. Choose proper parameters $p_{i}>0,\forall i\in\mathcal{N}_{N}$ and
$\bar{p}_{l}>0,\forall l\in\mathcal{N}_{L}$;
Step 2. Synthesize local controller using Th. (2) and Th. (3).
Step 3. Synthesize global controller using Th. (1).

##  VI Conclusion

In this paper, we propose a passivity-based co-design controller for DC
microgrids where the DGs and lines are considered as two individual subsystems
that are linked through the interconnection matrix. By using a distributed
control approach, the output voltages can be restored to the nominal value
while optimizing the communication topology. Furthermore, the proposed
controller is capable of handling the plug-and-play feature, where the DGs and
the correspondence line can join or leave the microgrid without compromising
the stability of the system since the controller is composed incrementally
without the need to re-design the remaining DGs and lines.

## References

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# Energy-independent optical ${}^{1}S_{0}NN$ potential
from Marchenko equation

N. A. Khokhlov<EMAIL_ADDRESS>Southwest State University, Kursk,
Russia  L. I. Studenikina<EMAIL_ADDRESS>MIREA – Russian Technological
University, Moscow, Russia

###### Abstract

We present a new algebraic method for solving the inverse problem of quantum
scattering theory based on the Marchenko theory. We applied a triangular wave
set for the Marchenko equation kernel expansion in a separable form. The
separable form allows a reduction of the Marchenko equation to a system of
linear equations. For the zero orbital angular momentum, a linear expression
of the kernel expansion coefficients is obtained in terms of the Fourier
series coefficients of a function depending on the momentum $q$ and determined
by the scattering data in the finite range of $q$. It is shown that a Fourier
series on a finite momentum range ($0<q<\pi/h$) of a $q(1-S)$ function ($S$ is
the scattering matrix) defines the potential function of the corresponding
radial Schrödinger equation with $h$-step accuracy. A numerical algorithm is
developed for the reconstruction of the optical potential from scattering
data. The developed procedure is applied to analyze the ${}^{1}S_{0}NN$ data
up to 3 GeV. It is shown that these data are described by optical energy-
independent partial potential.

quantum scattering, Marchenko theory, inverse problem, algebraic method,
numerical solution

###### pacs:

24.10.Ht, 13.75.Cs, 13.75.Gx

##  I INTRODUCTION

The inverse problem of quantum scattering is essential for various physical
applications such as the internuclear potential extraction from scattering
data and similar problems. The main approaches to solving the problem are
Marchenko, Krein, and Gelfand-Levitan theories Gelfand ; Agranovich1963 ;
Marchenko1977 ; Krein1955 ; Levitan1984 ; Newton ; Chadan . The ill-posedness
of the inverse problem complicates its numerical solution. So far, the
development of robust methods for solving the problem remains a fundamental
challenge for applications. This paper considers a new algebraic method for
solving the inverse problem of quantum scattering theory. We derive the method
from the Marchenko theory. To this end, we propose a novel numerical solution
of the Marchenko equation based on the integral kernel approximation as a
separable series in the triangular and rectangular wave sets. Then we show
that the series coefficients may be calculated directly from the scattering
data on the finite segment of the momentum axis. We offer an exact
relationship between the potential function accuracy and the range of known
scattering data.

The concept of optical potential (OP) is a useful tool in many branches of
nuclear physics. There are a number of microscopically derived Timofeyuk2020 ;
Burrows2020 ; Tung2020 ; Dinmore2019 ; Vorabbi2018 ; Wang2018 ; Atkinson2018 ;
Yahya2018 ; Lee2018 ; Vorabbi2017 ; Fernandez-Soler2017 ; Guo2017 ;
Rotureau2017 ; Vorabbi2016  as well as phenomenological Vorabbi2018 ; Guo2017
; Kobos1985 ; Hlophe2013 ; Khokhlov2006 ; Xu2016 ; Ivanov2016 ; Quinonez2020 ;
Jaghoub2018 ; Titus2016  optical potentials used for description of nuclei and
nuclear reactions. Nucleon-nucleon potentials are used as an input for
(semi)microscopic descriptions of nuclei and nuclear reactions Timofeyuk2020 ;
Burrows2020 ; Vorabbi2018 ; Vorabbi2017 ; Guo2017 ; Xu2016 ; Vorabbi2016 . We
analyzed the Marchenko theory Agranovich1963 ; Marchenko1977  and found that
it is applicable not only to unitary $S$-matrices but also to non-unitary
$S$-matrices describing absorption. That is, the Marchenko equation and our
algebraic form of the Marchenko equation allow us to reconstruct local and
energy-independent OP from an absorbing $S$-matrix and characteristics of
corresponding bound states. We applied the developed formalism to analyze the
${}^{1}S_{0}NN$ data up to 3 GeV and showed that these data are described by
energy-independent optical partial potential. Our results contradict
conclusions of Fernandez-Soler2017  where they state that ”… the optical
potential with a repulsive core exhibits a strong energy dependence whereas
the optical potential with the structural core is characterized by a rather
adiabatic energy dependence …” On the contrary, we reconstructed from the
scattering data local and energy-independent $NN$ soft core OP.

##  II Marchenko equation in an algebraic form

We write the radial Schrödinger equation in the form:

$\left(\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}-V(r)+q^{2}\right)\psi(r,q)=0.$
(1)

Initial data for the Marchenko method [1] are:

$\left\\{S(q),(0<q<\infty),\tilde{q}_{j},M_{j},j=1,\ldots,n\right\\},$ (2)

where $S(q)=e^{2\imath\delta(q)}$ is a scattering matrix dependent on the
momentum $q$. The $S$-matrix defines asymptotic behavior at
$r\rightarrow+\infty$ of regular at $r=0$ solutions of Eq. (1) for $q\geq 0;\
\tilde{q}_{j}^{2}=E_{j}\leq 0,E_{j}$ is $j$-th bound state energy
($-\imath\tilde{q}_{j}\geq 0$); $M_{j}$ is $j$-th bound state asymptotic
constant. The Marchenko equation is a Fredholm integral equation of the second
kind:

$F(x,y)+L(x,y)+\int_{x}^{+\infty}L(x,t)F(t,y)dt=0$ (3)

We write the kernel function as

$F(x,y)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}h_{l}^{+}(qx)[1-S(q)]h_{l}^{+}(qy)dq\\\
+\sum_{j=1}^{n_{b}}h_{l}^{+}\left(\tilde{q}_{j}x\right)M_{j}^{2}h_{l}^{+}\left(\tilde{q}_{j}y\right)\\\
=\frac{1}{2\pi}\int_{-\infty}^{+\infty}h_{l}^{+}(qx)Y(q)h_{l}^{+}(qy)dq$ (4)

where

$Y(q)=\left[1-S(q)-i\sum_{j=1}^{n_{b}}M_{j}^{2}\left(q-\tilde{q}_{j}\right)^{-1}\right]$
(5)

Solution of Eq. (3) gives the potential of Eq. (1):

$V(r)=-2\frac{dL(r,r)}{dr}$ (6)

There are many computational approaches for the solution of Fredholm integral
equations of the second kind. Many of the methods use an equation kernel’s
series expansion eprint7 ; eprint8 ; eprint9 ; eprint10 ; eprint11 ; eprint12
; eprint13 ; eprint14 . We also use this technique. Assuming the finite range
$R$ of the bounded potential function, we approximate the kernel function as

$F(x,y)\approx\sum_{k,j=0}^{N}\Delta_{k}(x)F_{k,j}\Delta_{j}(y)$ (7)

where $F_{k,j}\equiv F(kh,jh)$, and the basis functions are

$\left.\begin{array}[]{l}\Delta_{0}(x)=\left\\{\begin{array}[]{lll}0&\text{
for }&|x|>h,\\\ 1+x/h&\text{ for }&-h\leq x\leq 0,\\\ 1-x/h&\text{ for
}&0<x\leq h;\end{array}\right.\\\
\Delta_{n}(x)=\Delta_{0}(x-hn)\end{array}\right\\}$ (8)

where $h$ is some step, and $R=Nh$. Decreasing the step $h$, one can approach
the kernel arbitrarily close at all points. As a result, the kernel is
presented in a separable form. We solve Eq. (3) substituting

$L(x,y)\approx\sum_{j=0}^{N}P_{j}(x)\Delta_{j}(y)$ (9)

Substitution of Eqs. (7) and (9) into Eq. (3), and taking into account the
linear independence of the basis functions, gives

$\sum_{m=0}^{N}\left(\delta_{j\,m}+\sum_{n=0}^{N}\left[\int_{x}^{\infty}\Delta_{m}(t)\Delta_{n}(t)dt\right]F_{n,j}\right)P_{m}(x)=-\sum_{k=0}^{N}\Delta_{k}(x)F_{k,j}$
(10)

We need values of $P_{k}(hp)\equiv P_{p,k}$ $(p,k=0,..,N)$. In this case
integrals in Eq. (10) may be calculated

$\zeta_{n\,m\,p}=\int_{ph}^{\infty}\Delta_{m}(t)\Delta_{n}(t)dt=\frac{h}{6}\left(2\delta_{n\,n}\left(\delta_{n\,p}+2\eta_{n\geq
p+1}\right)+\delta_{n\,(m-1)}\eta_{n\geq p}+\delta_{n\,(m+1)}\eta_{m\geq
p}\right)$ (11)

Here, along with the Kronecker symbols $\delta_{k\,p}$, symbols $\eta_{a}$ are
introduced, which are equal to one if the logical expression $a$ is true, and
are equal to zero otherwise. Considering also that
$\Delta_{k}(hp)\equiv\delta_{k\,p}$, we finally get a system of equations

$\sum_{m=0}^{N}\left(\delta_{j\,m}+\sum_{n=0}^{N}\zeta_{n\,m\,p}F_{n,j}\right)P_{pm}=-F_{p,j}$
(12)

for each $j,p=0,..,N$. Solution of Eq. (12) gives $P_{k}(hp)\equiv P_{p,k}$.
Potential values at points $r=hp$ $(p=0,..,N)$ are determined from Eq. (6) by
some finite difference formula.

Next, we consider the case $l=0$, for which $h_{l}^{+}(qx)=e^{\imath qx}$ and

$F(x,y)=F(x+y)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}e^{\imath q(x+y)}Y(q)dq.$
(13)

We approximate the kernel as follows:

$F(x,y)=F(x+y)\approx\sum_{k=-2N}^{2N}F_{0,k}H_{k}(x+y)$ (14)

where $F_{0,k}\equiv F(kh)$ as in Eq. (7) for $l=0$, and the used basis set is

$\left.\begin{array}[]{l}H_{0}(x)=\left\\{\begin{array}[]{lll}0&\text{ for
}&x<0,\\\ 1&\text{ for }&0\leq x\leq h,\\\ 0&\text{ for
}&x>h,\end{array}\right.\\\ H_{n}(x)=H_{0}(x-hn).\end{array}\right\\}$ (15)

The Fourier transform of the basis set Eq. (15) is

$\tilde{\mathrm{H}}_{k}(q)=\int_{-\infty}^{\infty}\mathrm{H}_{k}(x)e^{-\imath
qx}dx=\frac{\imath\left(e^{-iqh}-1\right)}{q}e^{-\imath qhk}.$ (16)

The function $Y(q)$ may be presented as

$Y(q)=\sum_{k=-2N}^{2N}F_{0,k}\tilde{\mathrm{H}}_{k}(q)=\sum_{k=-2N}^{2N}F_{0,k}\frac{\imath\left(e^{-\imath
qh}-1\right)}{q}e^{-\imath qhk}.$ (17)

The last relationship may be rearranged

$qY(q)=\imath\sum_{k=-2N}^{2N}F_{0,k}\left(e^{-\imath qh}-1\right)e^{-\imath
qhk}\\\ =i\sum_{k=-2N+1}^{2N}\left(F_{0,k-1}-F_{0,k}\right)e^{-\imath
qhk}+\imath\left(-F_{0,-2N}\right)e^{\imath
qh2N}+\imath\left(F_{0,2N}\right)e^{-\imath qh(2N+1)}.$ (18)

Thus, the left side of the expression is represented as a Fourier series on
the interval $-\pi/h\leq q\leq\pi/h$. Taking into account that
$Y(-q)=Y^{*}(q)$, we get

$\left.\begin{array}[]{l}-F_{0,-2N}=\frac{h}{\pi}\int_{0}^{\pi/h}\text{Im}\left(Y(q)e^{-\imath
qh2N}\right)qdq;\\\ \\\
F_{0,k-1}-F_{0,k}=\frac{h}{\pi}\int_{0}^{\pi/h}\text{Im}\left(Y(q)e^{\imath
qhk}\right)qdq\text{ for }k=-2N+1,\ldots,2N-1;\\\ \\\
F_{0,2N}=\frac{h}{\pi}\int_{0}^{\pi/h}\text{Im}\left(Y(q)e^{\imath
qh(2N+1)}\right)qdq.\end{array}\right\\}$ (19)

The system (19) is solved recursively from $F_{0,2N}$. Thus, the range of
known scattering data defines the step value $h$ and, therefore, the inversion
accuracy. Calculation results for the potential function $V(r)=-3\exp(-3r/2)$
are presented in Figs. 1, 2, where $h=0.04$, $R=4$. $S$-matrix was calculated
at points shown in Fig. 1 up to $q=8$. The $S$-matrix was interpolated by a
quadratic spline in the range $0<q<8$. For $q>8$ the $S$-matrix was
approximated as asymptotic $S(q)\approx\exp(-2iA/q)$ for $q>8$, where $A$ was
calculated at $q=8$.

Figure 1: Data used to reconstruct $V(r)=V_{0}\exp(-ar)$, where $V_{0}=-3\
fm^{-2}=-124.5\ MeV$, $a=1.5\ fm^{-1}$. Units correspond to the $NN$ system.

Figure 2: Initial and reconstructed potentials: $V(r)=V_{0}\exp(-ar)$, where
$V_{0}=-3\ fm^{-2}=-124.5\ MeV$, $a=1.5\ fm^{-1}$. Units correspond to the
$NN$ system.

##  III Energy-independent optical ${}^{1}S_{0}NN$ potential

Realistic potentials derived unambiguously from inverse theories should
describe scattering data from zero to infinite energy. It seems that it is
only possible if the available scattering data approach the asymptotic region
below the relativistic region. It is unnecessary because relativistic two-
particle potential models may be presented in the non-relativistic form
Keister1991 . Another problem is the presence of closed channels whose
characteristics are not known. It is usually assumed (for example, for an $NN$
system) that below the inelasticity threshold, effects of closed channels can
be neglected, and a real $NN$ potential may describe the interaction of
nucleons. This assumption is a consequence of the ingrained misconception that
a complex potential corresponds to a non-unitary matrix. One can only assert
that the $S$-matrix is unitary for a real potential.

We have carefully analyzed the Marchenko theory Agranovich1963 ; Marchenko1977
and found that it applies not only to unitary $S$-matrices but also to non-
unitary $S$-matrices describing absorption. That is, the Marchenko theory Eqs.
(1-6) and our algebraic form of the Marchenko equation Eqs. (7-18) allow to
reconstruct local, and energy-independent OP from an absorbing $S$-matrix and
corresponding bound states’ characteristics. We present an absorptive single
partial channel $S$-matrix on the $q$-axis as

$S(q)=\left\\{\begin{array}[]{lll}S_{u}(q)+S_{n}(q)&\text{ for }&q>0,\\\
S^{+}_{u}(-q)-S^{+}_{n}(-q)&\text{ for }&q<0,\end{array}\right.$ (20)

where superscript $+$ means hermitian conjugation. For $q>0$ we define

$S_{u}(q)=e^{2\imath\delta(q)},\\\
S_{n}(q)=-\sin^{2}(\rho(q))e^{2\imath\delta(q)},$ (21)

where $\delta(q)$ and $\rho(q)$ are phase shift and inelastisity parameter
correspondingly. In this case we have instead of Eqs. (19) the following
system

$\left.\begin{array}[]{l}-F_{0,-2N}=\frac{h}{\pi}\int_{0}^{\pi/h}q\left[\text{Im}\left(Y_{u}(q)e^{-\imath
qh2N}\right)-\imath\text{Re}\left(S_{n}(q)e^{-\imath qh2N}\right)\right]dq;\\\
\\\
F_{0,k-1}-F_{0,k}=\frac{h}{\pi}\int_{0}^{\pi/h}q\left[\text{Im}\left(Y_{u}(q)e^{\imath
qhk}\right)-\imath\text{Re}\left(S_{n}(q)e^{\imath qhk}\right)\right]dq\text{
for }k=-2N+1,\ldots,2N-1;\\\ \\\
F_{0,2N}=\frac{h}{\pi}\int_{0}^{\pi/h}q\left[\text{Im}\left(Y_{u}(q)e^{\imath
qh(2N+1)}\right)-\imath\text{Re}\left(S_{n}(q)e^{\imath
qh(2N+1)}\right)\right]dq,\end{array}\right\\}$ (22)

where

$Y_{u}(q)=\left[1-S_{u}(q)-i\sum_{j=1}^{n_{b}}M_{j}^{2}\left(q-\tilde{q}_{j}\right)^{-1}\right].$
(23)

##  IV Results and Conclusions

We applied the developed formalism to analyze the ${}^{1}S_{0}$ $NN$ data. As
input data for the reconstruction, we used modern phase shift analysis data
(single-energy solutions) up to 3 GeV DataScat ; cite . We smoothed phase
shift and inelasticity parameter data for $q>3$ fm-1 by the following
functions:

$\delta(q)\sim-54.56822/q^{3}+57.55296/q^{2}-15.36687/q,\\\ \rho(q)\sim
101.89881/q^{3}-80.13493/q^{2}+15.88984/q,$ (24)

where we fitted the coefficients by the least-squares method. Asymptotics (24)
were used to calculate coefficients of Eqs. (19) with $h=0.0125$ fm
corresponding to $q_{max}\approx 251.3$ fm-1.

Results of our calculations show that these data are described by energy-
independent optical partial potential (Figs. 3,4).

Figure 3: Data used to reconstruct ${}^{1}S_{0}NN$ potential.

Figure 4: Reconstructed ${}^{1}S_{0}NN$ potential.

Thus, we presented a solution of the quantum scattering inverse problem for
the zero orbital angular momentum, the algorithm of which is as follows. We
set the step value $h$, which determines a required accuracy of the potential.
From the scattering data, we determine $F_{0,k}$ from Eqs. (19) for unitary
$S$-matrix or from Eqs. (22) for non-unitary $S$-matrix. Solution of Eqs. (12)
gives values of $P_{k}(hp)$ ($p=0,..,N$). Further, the values of the potential
function (6) are determined by some finite difference formula. Expressions
(7-14) give a method for the Marchenko equation’s numerical solution for an
arbitrary orbital angular momentum $l$, and may be generalized for a case of
coupled channels.

Our results contradict conclusions of Fernandez-Soler2017  claiming that an OP
with a repulsive core exhibits a strong energy dependence. On the contrary, we
reconstructed local and energy-independent $NN$ soft core OP with
$Re(V(0))\approx 14\ GeV$ and $Im(V(0))\approx 19\ GeV$ .

It may be that some local OPs lead to unsatisfactory description of nuclear
reactions Titus2016  ($(d,p)$ transfer reactions). Our approach assumes
inverse scattering reconstruction of local and energy-independent OP
describing all two-particle scattering data (including high energy
asymptotics) and bound states. Such OPs have not been used in nuclear
calculations, though they may give an adequate description of the off-shell
behaviour of the nucleon-nucleon interaction.

The reconstructed $1S0$ $NN$ optical potential may be requested from the
author in the Fortran code.

## References

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