# Distributed energy control in electric energy systems

Rupamathi Jaddivada<EMAIL_ADDRESS>Marija D. Ilic<EMAIL_ADDRESS>Research
Laboratory of Electronics, 10-034, 222 Memorial Drive, Cambridge, MA-02139,
USA  Laboratory for Information and Decision Systems, 32-D726, 32 Vassar
Street, Cambridge, MA-02139, USA

###### Abstract

The power interactions of any component in electric energy systems with the
rest of the system happen naturally, as governed by the energy conservation
principles. There may, however, occur instances when the rate at which power
gets generated by one component through local energy conversion is not exactly
the same as that absorbed by rest of the system. This is when instabilities
get induced. To model and control such instabilities, this paper generalizes
the notion of interaction variable used to characterize diverse system
components in a unified manner. The same variable captures aggregate system-
wide effects and sets reference points for multi-layered distributed output
feedback control. It has a physical interpretation of instantaneous power and
generalized reactive power. The higher layer design utilizes the interactive
energy state-space model to derive intermediate reactive power control, which
becomes a control command to the lower layer physical model. This command is
implemented using either Feedback Linearizing Control (FBLC) or Sliding Mode
Control (SMC), for which sufficient stability conditions are stated. This
paper claims that the proposed design is fundamental to aligning dynamic
interactions between components for stability and feasibility. Without loss of
generality, we utilize a simple RLC circuit with a controllable voltage source
for illustrations, which is a simplified representation of any controllable
component in microgrids.

###### keywords:

Energy dynamics, Generalized reactive power, Interaction variables,
Distributed control, Feedback linearizing control, Sliding mode control

††thanks: This paper was not presented at any IFAC meeting. Corresponding
author Rupamathi Jaddivada. Tel. +1-412-378-1479

, ,

##  1 Introduction

Table 1: Commonly used notation Variables
---
$x$ |  State variable |  $u$ |  Control |  $r$ |  port input
$m$ |  Exogenous disturbance |  $y$ |  Output of interest |  $\tilde{x}$ |  Extended state $\left[x,r,m\right]^{T}$
$w$ | Dual variable associated with control($w^{u}$), disturbance ($w^{m}$) or port input ($w^{r}$)
Superscripts
$r$ |  Interaction port |  $u$ |  Control port |  $m$ |  Disturbance port
out |  Outgoing |  in |  Incoming |  ref |  Setpoint for tracking
Subscripts
$i$ |  General component index |  $j$ |  Neighboring component index |  $z$ |  Variables in energy space
Energy space variables (Definitions available in Appendix A)
$P$ | Instantaneous power (Definition 3) Eg: $P^{u}=uw^{u}$
$\dot{Q}$ | Rate of change of reactive power (Definition 4) Eg: $\dot{Q}^{u}=u\dot{w}^{u}-\dot{u}w^{u}$
$E$ | Stored energy (Definition 5) | $p$ | Time derivative of stored energy ($\dot{E}$)
$E_{t}$ | Stored energy in tangent space (Definition 6) |  $\tau$ | Time constant (Definition 7)
$x_{z}$ | State variable in energy space $\left[E,p\right]^{T}$ | $z$ | Interaction variable $\left[P,Q\right]^{T}$ (Definition 3)

This paper considers the problem of distributed modeling and control design
for the changing electric energy systems. These are large-scale complex
systems for which control is needed to manage solar power, wind power and load
disturbances. To pursue distributed control, we characterize components by
their internal dynamics as well as by their interactions with the neighboring
components. This is done by establishing physics-based structured modeling to
manage complexity. We formalize the inherent multi-layered structures
mathematically and build on these structures for deriving the multi-layered
control design.

From the component’s standpoint, the control objective is to deliver the
required power. However, as system conditions vary, this requires adjusting
voltages in response to current deviations and vice-versa. Such adjustments
lead to persistent time-varying disturbances, which appear as an incremental
negative resistance characteristic undesirable for stable operation. Such
disturbances have been a cause of perpetual concern for system stability [1,
2, 3]. These problems are growing due to high penetration of intermittent
power plants with low inertia.

One of the several approaches to control energy systems subject to time-
varying disturbances is to design robust centralized controllers immune to any
disturbance in a pre-characterized set of disturbances. However, in this
paper, we seek distributed controllers due to privacy restrictions and their
implementation without requiring fast communications. Most existing
distributed control design methods rely on time-scale separation between the
models and control design objectives utilized at multiple control layers [4,
5]. When the assumptions on time-scale separation fail, each component gets
affected by the net power disturbance resulting from interaction with the rest
of the system. Such disturbances can not be treated as independent inputs
since they are fast state-dependent variations.

We propose in this paper a novel modeling approach that lends itself to
distributed control design through the exploitation of inherent structures
resulting from energy conservation principles. The proposed method relies on
information exchange between neighboring components. This provides the
information about the interactive dynamics as power is exchanged back and
forth between the components and its environment. Notably, this is a new area
of research since the derived models are not in standard state-space form. We
provide sufficient mathematical conditions for feasibility and stability in
terms of physically intuitive energy and power flow variables.

The typical design proposed in the literature is based on either hard-to-
implement centralized controllers or on entirely local disturbance rejection-
based controllers, leading to increased control effort and lack of robustness
[6, 7]. While facilitating coordinated control, the existing distributed
controllers require restrictive conditions on internal stability, such as
those of zero-state detectability [8, 9]. It is therefore, hard to obtain
general feasibility, stability, and robustness conditions due to the non-
linearity of the resulting interconnected systems governed by general energy
conservation principles.

As a solution to the limitations mentioned above, [10, 11] proposed a modeling
and control that exploits the inherent structural properties of physical
systems. In particular, this work showed that there exists a transformation of
internal variables into an interaction variable, which can be leveraged for
the local control design to have provable distributed stabilization of the
interconnected system. Interaction variable is defined irrespective of the
complex internal energy conversion processes by utilizing the inherent
structure of interactions across components in the system, resulting from the
first energy conservation law. Over the last two decades, the interaction
variable-based theory has been extended and utilized to design other types of
nonlinear controllers, including control of power-electronics-based inverters,
induction machines, generators, and residential demand-side technologies [12,
13, 14]. In this paper, we extend the previously proposed interaction
variable-based approach to accommodate the interactive time-varying
disturbances.

Contributions and outline: In Section 2, we formally pose the distributed
control problem as a tracking and output-disturbance decoupling problem from
the component’s perspective. We present a general multi-layered modeling
framework that helps re-pose the distributed control problem drawing on the
inherent intuitive structure resulting from the underlying physics, which is
summarized in Section 3. To do so, we extend the previously proposed
interaction variable-based approach introduced in [10, 15] by defining an
intuitive two-dimensional common output variable of interest and its dynamics
based on fundamental energy conservation principles. We thereby obviate the
need to perform complex coordinate transformations searching for common output
variables that would otherwise be needed [16, 17]. Utilizing this modeling, we
present an overview of the interactive control in Section 4. In particular, we
show how the selection of common variables in energy space assists in
designing a control cognizant of both component-level stability conditions
resulting from the internal dynamics and system-level feasibility conditions
pertaining to the interconnection dynamics, derived in Section 5 and 6
respectively. While these are only sufficient conditions, they have far
reaching implications since they offer intuitive understanding in terms of
energy and power flows and are modular. In Section 5, we extend the control
design to an easy-to-implement switching control robust to modeling and
parameter uncertainties. This approach notably supports data-enabled decision-
making due to its minimal dependence on internal physics-based models. We take
an example of an RLC circuit in Section 7 and illustrate the performance and
robustness of our proposed energy control. Finally, we provide some concluding
thoughts in Section 8.

##  2 Distributed control problem formulation

Without loss of generality, consider a simple electrical energy system
comprising two components as shown in Fig. 1. In order to formally pose the
distributed control problem we first devise the component models. One approach
to obtaining these models is tearing of the interconnected system model by
defining additional variables at the intersection of two components, which are
also referred to as port inputs, denoted as $r_{i}$. These inputs dictate the
interaction of the component $i$ with the rest of the system [18].

Figure 1: Interconnected system comprising two components $\Sigma_{1}$ and
$\Sigma_{2}$ with local controllable input $u_{1},u_{2}$ and exogenous
disturbances $m_{2}$

Algebraic relations such as Kirchoff’s current and voltage laws dictate the
relationship between the port inputs and state variables. This relation is
abstracted through a map $L$ as shown in Eqn. (1) [19, 20].

$0=L(x_{1},x_{2},r_{1},r_{2})$ (1)

A schematic of component models comprising the interconnected system is shown
in Figure 1 and the corresponding dynamical models are given in Eqn. (2).

$\displaystyle\text{State Dynamics:}\qquad\qquad\qquad\qquad x_{i}(0)=x_{i,0}$
$\displaystyle\dot{x}_{i}=f_{x,i}(x_{i})+g_{i}^{r}(x_{i})r_{i}+g_{i}^{m}(x_{i})m_{i}+g_{i}^{u}(x_{i})u_{i}$
(2a) $\displaystyle\text{Outputs of
interest}:y_{i}=f_{y,i}(x_{i},u_{i},r_{i},m_{i})$ (2b)
$\displaystyle\text{Common outputs}:z_{i}=f_{z,i}(x_{i},u_{i},r_{i},m_{i})$
(2c)

The notation used here is summarized in Table 1.

The major difficulty in distributed control problem is to find the common
output variable $z_{i}$. Assuming such common output variable can be found,
the control problem can be posed as follows:
(P1):The objective is to design a smooth feedback control
$u_{i}=k_{i}(y_{i},z_{j})~{}\forall i\in\\{1,2\\},j\neq i$ utilizing local
information ($y_{i}$) and minimal information sent by neighboring component
$\Sigma_{j}$ ($z_{j}$), to satisfy following objectives:

  1. 1.

Dissipativity: Each component $\Sigma_{i}$ in closed loop is dissipative [18]
w.r.t. a supply function $w_{i}\left(\left[m_{i},r_{i}\right],y_{i}\right)$.

  2. 2.

Internal stability: It is further desired of the supply function to satisfy
$w_{i}\left(\left[m_{i},r_{i}\right],y_{i}\right)\leq 0\quad\forall y_{i}$ and
admissible component inputs $r_{i}$ and $m_{i}$.

  3. 3.

Output regulation: Local outputs of interest $y_{i}(t)$ must be regulated to a
fixed reference value $y_{i}^{ref}$

  4. 4.

Feasibility: The common output variables track a consistent reference given by
smooth map $\psi_{i}$ as
$z_{i}^{ref}(t)=\psi_{i}(x_{i}(t),z_{j}(t))~{}~{}\forall j\in\mathcal{C}_{i}$
where $\mathcal{C}_{i}$ represents the set of component indexes corresponding
to the neighbors.

The objectives stated in (P1) are multi-fold including stabilization of
internal dynamics, tracking of common output variables and the robustness to
local exogenous disturbances. It is apparent that the control objectives above
can be contradicting each other. This problem only gets more complex with the
different types of disturbances entering the system.

##  3 Interactive energy-based modeling

Next, we introduce a common output variable that helps solve the different
control objectives posed in (P1). We refer to this common output variable as
an interaction variable. Its physical meaning is the energy and reactive power
absorbed by the component by virtue of its own energy conversion processes.
More fundamentally, it is defined as follows:

######  Definition 1.

(Interaction Variable) [10, 21] 111This definition was provided in particular
for electric power systems under an assumption of real-reactive power
decoupling [10, 11]. We now further extend this notion by relaxing the
decoupling assumption.
The interaction variable $z_{i}^{r,out}$ is defined as a function of local
variables that satisfies the property

$z_{i}^{r,out}=\text{constant}$ (3a) when all interconnections are removed.
Let $E_{i}$, $p_{i}$, $P_{i}^{u}$, $P_{i}^{m}$, $\dot{Q}_{i}^{u}$,
$\dot{Q}_{i}^{m}$ represent the stored energy, rate of change of stored
energy, instantaneous power at control terminal and disturbance terminal and
generalized rate of reactive power at control and disturbance terminals
respectively of component $i$. Each of these variables is defined as a
function of local state variables and state derivatives in Appendix A.
Mathematically, the interaction variable is then defined as
$\begin{array}[]{*{20}{l}}{z_{i}^{r,{\rm{out}}}=\left[{\begin{array}[]{*{20}{l}}{\int\limits_{0}^{t}{\left({{p_{i}}(s)+\frac{{{E_{i}}(s)}}{{{\tau_{i}}}}-P_{i}^{u}(s)-P_{i}^{m}(s)}\right)ds}}\\\
{\int\limits_{0}^{t}{\left({-{{\dot{p}}_{i}}(s)+4{E_{t,i}}(s)-\dot{Q}_{i}^{u}(s)-\dot{Q}_{i}^{m}(s)}\right)ds}}\end{array}}\right]}\end{array}$
(3b)

The existence and the defining structural properties of the interaction
variable were proved in [21]. In order to differentiate the interactions
resulting from internal energy conversion processes as per the Definition 3,
and the ones obtained as a result of interconnection, we utilize the
superscripts ‘$\rm{out}$’ and ‘$\rm{in}$’ respectively. The incoming
interaction variable is a result of interconnection, as shown in the zoomed-
out representation of the interconnected system in Fig. 2.

Figure 2: Zoomed-out representation for the interconnected system: Incoming
interactions ($z_{i}^{r,in}$) are shown with blue arrows, while the outgoing
ones ($z_{i}^{r,out}$) by virtue of local energy conversion dynamics are shown
in brown for each of the components in the closed-loop. After interconnection,
the incoming interaction variable is equal to the negative of outgoing
interaction variables of its neighbors.

We next introduce the zoomed-in component models. The physical models
introduced in Eqn. (2) are related to the interaction variable and its
dynamics, through a bi-directional dynamic mapping as shown in Fig. 3.

Figure 3: Interactive stand-alone model of an open-loop component $i$: The
lower layer models are utilized to compute the outgoing interaction variable
$\dot{z}_{i}^{r,\rm{out}}$, which drive the higher-layer energy dynamics of
the component. The incoming interaction variable from the grid
$\dot{z}_{i}^{r,\rm{in}}$, is utilized by the lower-layer models to evaluate
the extended state trajectories $\tilde{x}_{i}=\left[x_{i},m_{,}r_{i}\right]$
given their initial conditions.

From this figure, it is important to note that the physical lower layer model
is characterized now using extended state space
$\tilde{x}_{i}=\left[x_{i},m_{i},r_{i}\right]^{T}$. Here, we characterize the
persistent disturbances and incoming interactions at the ports through the
instantaneous power and generalized reactive power at ports by the vector
$\dot{z}_{i}^{m}$ and $\dot{z}_{i}^{r,in}$ respectively. In addition to the
state dynamics, it now becomes imperative to model the dynamics of the
disturbance and port inputs in order to include the dependence of
instantaneous and reactive power entering the respective terminals
($\dot{z}_{i}^{m},\dot{z}_{i}^{r,in}$). Characterization of these power
variables, in turn, helps retain the linear structure of the higher layer
energy space model in Fig. 3, also elucidated in Eqn. (5). The dynamics of
extended state variables are expressed in Eqn. (4).

$\begin{array}[]{l}{\rm{Extended~{}state~{}space~{}model}}:~{}{{\tilde{x}}_{i}}(0)={\left[{\begin{array}[]{*{20}{c}}{{x_{i0}}}&{{m_{i0}}}&{{r_{i0}}}\end{array}}\right]^{T}}\\\
\underbrace{\left[{\begin{array}[]{*{20}{l}}\begin{array}[]{l}{{\dot{x}}_{i}}\\\
{{\dot{m}}_{i}}\end{array}\\\
{{{\dot{r}}_{i}}}\end{array}}\right]}_{{{\dot{\tilde{x}}}_{i}}}=\underbrace{\begin{array}[]{l}\left[{\begin{array}[]{*{20}{l}}\begin{array}[]{l}{f_{x,i}}({x_{i}})+{g_{i}}^{m}({x_{i}}){m_{i}}+{g_{i}}^{r}({x_{i}}){r_{i}}\\\
{f_{m,i}}({{\tilde{x}}_{i}})+g_{m,i}^{m}({{\tilde{x}}_{i}}){m_{i}}+{g_{m,i}}^{r}({{\tilde{x}}_{i}}){r_{i}}\end{array}\\\
{{f_{r,i}}({{\tilde{x}}_{i}})+g_{r,i}^{m}({{\tilde{x}}_{i}}){m_{i}}+{g_{r,i}}^{r}({{\tilde{x}}_{i}}){r_{i}}}\end{array}}\right]+\\\
\left[{\begin{array}[]{*{20}{c}}0&0\\\ {g_{m,i}^{zm}({{\tilde{x}}_{i}})}&0\\\
0&{g_{r,i}^{zr}({{\tilde{x}}_{i}})}\end{array}}\right]\left[\begin{array}[]{l}{{\dot{z}}_{i}}^{m}\\\
{{\dot{z}}_{i}}^{r,in}\end{array}\right]+\left[{\begin{array}[]{*{20}{l}}\begin{array}[]{l}{g_{i}}^{u}({x_{i}})\\\
g_{m,i}^{u}({{\tilde{x}}_{i}})\end{array}\\\
{g_{r,i}^{u}({{\tilde{x}}_{i}})}\end{array}}\right]{u_{i}}\end{array}}_{{{\tilde{f}}_{{{\tilde{x}}_{i}}}}\left({{{\tilde{x}}_{i}},{{\dot{z}}_{i}}^{m},{{\dot{z}}_{i}}^{r,in},{u_{i}}}\right)}\end{array}$
(4)

Here $f_{x,i},g_{i}^{r},g_{i}^{m}$ are the same functions defined in Eqn.
(2a). The rest of the functions are a result of expressions for reactive power
at respective ports according to Definition. 4.

###  3.1 Distributed control problem posing in energy space

We propose to characterize the output variables $y_{i}$ and $z_{i}$ in energy
space for obtaining an intuitive map $\psi_{i}$ in the fourth objective of
tracking in problem (P1) for re-interpreting the distributed control problem
posed for a feasible and stable interconnected system. We henceforth use the
notation $y_{z,i}$ and $\dot{z}_{i}^{r,out}$ characterized in energy space to
represent the output variables $y_{i}$ and $z_{i}$ respectively in the problem
formulation (P1). The model for use with the problem formulation (P1) in
transformed state space is shown in Eqn. (5). Here, the state variables are
aggregate dynamical energy variables denoted as
$x_{z,i}=\left[E_{i},p_{i}\right]^{T}$.

$\displaystyle\text{Energy space state dynamics:}\qquad x_{z,i}(0)=x_{z,i0}$
(5a)
$\displaystyle\dot{x}_{z,i}=A_{z,i}x_{z,i}+B_{t}E_{t,i}\left(\dot{x}_{i}\right)+B_{z}\left(\dot{z}_{i}^{r,out}+\dot{z}_{i}^{u}+\dot{z}_{i}^{m}\right)$
$\displaystyle\text{Rate of change of common outputs:
}~{}~{}z_{i}^{r,out}(0)=z^{r,out}_{i0}$
$\displaystyle\dot{z}_{i}^{r,out}=\phi_{z,i}(x_{i},r_{i},u_{i},m_{i},\dot{z}_{i}^{r,in})$
(5b) Outputs of interest:
$\displaystyle{y_{z,i}}=\underbrace{\left[{\begin{array}[]{*{20}{c}}{\frac{1}{{{\tau_{i}}}}}&0\end{array}}\right]}_{{C_{z,i}}}{x_{z,i}}+\underbrace{\left[{-1\;\;0}\right]}_{{D_{z}}}\left({{{\dot{z}}_{i}}^{u}+{{\dot{z}}_{i}}^{m}}\right)$
(5d)

The energy state space evolution is given in Eqn. (5a), which was first
postulated in [22] as a generalization of energy conservation principles. In
Eqn. (5), the constant matrices and vectors used in the model are:
$B_{t}=\left[0,4\right]^{T},B_{z}=\left[1~{}~{}-1\right]^{T}$ for any
component and matrix
$A_{z,i}=\left[{\begin{array}[]{*{20}{c}}-1/{\tau_{i}}&0\\\
0&0\end{array}}\right]$ depends only on the time constant $\tau_{i}$ defined
in Definition 7.

We further define the common output variable as the interaction variable
defined in Eqn. (3b). This variable can only be numerically computed by
utilizing the analytical expressions given for the rate of change of
interaction variable through an abstract map $\phi_{z,i}$ of internal
variables as shown in Eqn. (5b) [21]. Since outgoing interaction variable by
Definition 3 is a function of local states and state derivatives (function of
port input $r_{i}$), the rate of change of outgoing interaction variable
depends on the rate of change of incoming interaction variable. Such
dependence makes the modeling framework inherently interactive. For more
details on the interactive modeling approach, the reader is referred to [22,
23, 21].

We propose to characterize the output variable of interest to have an
interpretation of power produced or absorbed by the component. Over relatively
slower time scales, instantaneous power balance or reactive power balance can
be set up as a control objective since one implies the other over these time
scales. Furthermore, if only one physical control input is available, only one
out of the two balance equations can be set as a control objective. We thus
select the output variable of interest as the instantaneous power absorbed the
component after energy dynamics settle i.e. after $\dot{E}_{i}=0$.

The control problem ${(P1)}$ can better be posed in reduced order linear
energy space. However, the model is dependent on the internal dynamics and
vice-versa as shown in Fig. 3. This notion of interactive modeling is also
inherent in Willems’ seminal paper on behavioral modeling of physical systems
[18]. While standard state space modeling has been central to the development
of control theory for decades, we emphasize that interactive modeling is
instrumental for distributed implementation. Next, utilizing these interactive
models, we propose a multi-layered interactive control design to solve the
problem (P1).

##  4 Multi-layered distributed control design

In this section, we provide an overview of the proposed multi-layered control
architecture, that utilize the inherent structure of linear energy space
models for a provable control design. The proposed method constitutes two
parts of the design summarized as follows:

  * •

Component-level: Given the incoming interaction variable $\dot{z}_{i}^{r,in}$
from the interconnection-level control, map in Eqn. (6) is utilized to obtain
the reference point $y_{z,i}^{ref}$. The control objective of this layer is to
ensure that the outputs of interest $y_{z,i}$ track the feasible reference
point $y_{z,i}^{ref}$. This is achieved by two-layered control at the
component level as follows:

    * –

The linearity of the higher layer energy-space model shown in Fig. 3 is
utilized to design a control in energy space that ensure $y_{z,i}$ tracks
$y_{z,i}^{ref}$,

    * –

The lower layer implementation is then performed through a diffeomorphic map
between the energy space variables and the conventional space variables at the
control ports.

  * •

Interconnection-level: Given the ranges of outgoing interaction variable
bottom-up by the components for a period of time, the zoomed-out
interconnected system model depicted through Fig. 2 is utilized to obtain
optimal values of incoming interaction variable that need to be distributed
among multiple components, so as to guarantee feasibility and efficient
utilization of available controllers. For a two-component interconnection of
interest in this paper, optimal computation of incoming interaction variable
over a period of time is not necessary. We instead propose a simple modular
feasibility conditions that each component in closed-loop can check for before
getting connected to the interconnection with the grid. For a complete optimal
control problem formulation, the interested readers are referred to [24].

The feasibility of a component’s interconnection with the rest of the system
is dictated primarily by the existence of a solution for the interface
variables dictated by Kirchoff’s current or voltage laws. The port current and
voltage variables can enter the component dynamical models non-linearly that
makes the problem of answering the feasibility questions quite involved [23].
It has been shown in [22] that instantaneous power and generalized reactive
power balance equations shown through the equality constraints in Fig. 2 imply
satisfaction of Kirchoff’s current and voltage laws both statically and
dynamically. As a result, we can better answer feasibility questions under
this new modeling framework, especially when systems are subject to persistent
disturbances.

The reference values of the outputs should then be equal to incoming
interaction variable so that instantaneous power balance equation over
relatively slower timescales is satisfied. In terms of the feasibility
objective in problem (P1), an intuitive mapping
$\psi_{i}:(x_{i},\dot{z}_{j}^{r,out})\rightarrow y_{z,i}^{ref}~{}~{}\forall
j\in\mathcal{C}_{i}$ is proposed as:

$y_{z,i}^{ref}=\left[{1\;\;0}\right]\dot{z}_{i}^{r,in}=-\left[{1\;\;0}\right]\sum\limits_{j\in{{\mathcal{C}}_{i}}}{\dot{z}_{j}^{r,out}}={D_{z}}\sum\limits_{j\in{{\mathcal{C}}_{i}}}{\dot{z}_{j}^{r,out}}$
(6)

The second equality is a result of the application of generalized Tellegen’s
theorem [22, 25]. Incorporating such specifications in the control design
enables the component to generate or absorb energy at a rate equal to the one
that the rest of the system injects into the component. Such an objective,
when incorporated by all components in the system, leads to the components
interactively achieving consensus on power-sharing, thereby ensuring a
feasible interconnection. However, further investigation is needed to ensure
the existence and reachability of a feasible equilibrium through proper
control design. In particular, we have only addressed the fourth objective
stated in problem (P1) by defining a consistent map $\psi_{i}$. We focus on
the second and third objectives in Section 5, and the first objective in
Section 6.

Before formalizing the claims in the rest of the paper, we make the following
assumptions that typically hold in electric energy systems.

######  Assumption 1.

The interactive model of the sub-system $\Sigma_{i}$ as defined in Fig. 3 has
sufficient smoothness conditions to make the sub-system well defined. More
precisely, for any $\tilde{x}_{i,0}$ and admissible pair of
$\dot{z}_{i}^{m}(t),\dot{z}_{i}^{r,\rm{in}}(t)$, there exists at least one
solution $\tilde{x}_{i}(t)\forall t\in[0,\infty)$ so that the output of
interest $y_{z,i}$ is locally square integrable.

######  Assumption 2.

The extended state space of $\Sigma_{i}$ is reachable from origin. More
precisely, given any $\tilde{x}_{i}$ and $t_{1}$, there exists a $t_{0}\leq
t_{1}$ and an admissible incoming interaction $\dot{z}_{i}^{r,in}(t)$ and
local disturbances $\dot{z}_{i}^{m}$ such that the extended states can be
driven from $[x_{i}(t_{0}),r_{i}(t_{0})]={0}$ to
$[x_{i}(t_{1}),r_{i}(t_{1})]=[x_{i},r_{i}]$

######  Assumption 3.

The stored energy in tangent space $E_{t,i}$ as stated in Definition 6 is
positive at all times.

######  Assumption 4.

The stored energy function $E_{i}(x_{i})$ is a positive definite function and
the time constant $\tau_{i}$ as defined in Definition 7 is assumed positive at
all times.

In Assumption 1, the admissible inputs refer to the class of inputs belonging
to a set of finite energy signals, i.e. they must remain square integreable.
We can also characterize the set with other additional constraints such as
those corresponding to physical equipment protection limits. Assumptions 3 and
4 mean that the components’ inherent inertia and damping (See Definitions 6
and 7) are positive which is true for most physical systems.

##  5 Interactive component-level control

Given a consistent reference value for the output variable, map based on local
measurements such as the one obtained by the mapping Eqn. (6), we propose a
two-layered control that goes hand-in-hand with the interactive models
proposed in Section 3, to satisfy the control objectives stated in (P1). By
utilizing the energy space model, we first design the control in higher layer
to ensure feasibility by ensuring the tracking error $y_{z,i}-y_{z,i}^{ref}$
is driven to zero. In this linear energy space layer, we treat
$u_{z,i}=\dot{Q}_{i}^{u}$ as the virtual control input.

Figure 4: Block diagram of closed loop module formed by component-level
distributed interactive control design satisfying objectives stated in control
problem (P1).

This design is mapped to physical state space by utilizing the Definition 4.
We can design a dynamic controller to be implemented as in Eqn. (7).
Corresponding dynamic map is abstracted through the function $f_{i}^{u}$ also
utilized in the control implementation block in Fig. 4.

$\dot{u}_{i}=\frac{\dot{w}_{i}^{u}}{w_{i}^{u}}u_{i}-\frac{1}{w_{i}^{u}}\dot{Q}^{u}_{i}=f_{i}^{u}(\tilde{x}_{i},u_{z,i})$
(7)

Note that the map $f_{i}^{u}$ is well defined since $w_{i}^{u}$ is non-zero
for non-zero higher layer design variables
$\dot{z}_{i}^{u}=\left[P_{i}^{u},\dot{Q}_{i}^{u}\right]$. A schematic of the
proposed design in shown in Fig. 4.

The proposed two-layered design can also be interpreted as an input-output
linearization based control. In particular, consider the proposed output
variable in Eqn. (5d). By taking its time derivative and plugging in the
relations given by the interaction model in Eqn. (5a), we have the dynamic
input-output relation between the second component of input $\dot{z}_{i}^{u}$
and the chosen output $y_{z,i}$ as shown in Eqn. (8).

$\displaystyle{{{\dot{y}}_{z,i}}=-4{E_{t,i}}+\underbrace{\left({{{\dot{P}}_{i}}^{r,out}+{{\dot{Q}}_{i}}^{r,out}+\dot{Q}_{i}^{m}}\right)}_{{\eta_{i}}({{\tilde{x}}_{i}},{\dot{\tilde{x}}_{i}},\dot{z}_{i}^{m})}+\underbrace{\dot{Q}_{i}^{u}}_{{u_{z,i}}}}$
(8a)
$\displaystyle{{\dot{\tilde{x}}}_{i}}={{\tilde{f}}_{{{\tilde{x}}_{i}}}}({{\tilde{x}}_{i}})+\tilde{g}_{{{\tilde{x}}_{i}}}^{m}({{\tilde{x}}_{i}}){{\dot{z}}_{i}}^{m}+\tilde{g}_{{{\tilde{x}}_{i}}}^{r}({{\tilde{x}}_{i}}){{\dot{z}}_{i}}^{r,in}$
$\displaystyle{\qquad\qquad+\tilde{g}_{{{\tilde{x}}_{i}}}^{u}({{\tilde{x}}_{i}}){U_{i}}({{\tilde{x}}_{i}},{P_{i}}^{m},{y_{z,i}})}$
(8b)

Here, the map $U_{i}(x_{i},P_{i}^{m},y_{z,i})\rightarrow u_{i}$ is defined as
in Eqn. (9) to obtain physical control $u_{i}$ given the output variable in
energy space.

${u_{i}=\frac{1}{{w_{i}^{u}}}\left({\frac{1}{2}{x_{i}}^{T}{B_{i}}({x_{i}}){x_{i}}-{y_{z,i}}-{m_{i}}w_{i}^{m}}\right)}$
(9)

The input-output model in Eqn. (8) has a relative degree equal to one since
the first time derivative of the output is directly dependent on the control
in energy space $u_{z,i}=\dot{Q}_{i}^{u}$. The rest of the state variables
characterizing the internal dynamics, are not directly affected by the virtual
control in energy space $u_{z,i}$. They are only affected by physical control
$u_{i}$, which is re-expressed through the map
$U_{i}(x_{i},P_{i}^{m},y_{z,i})$.

The normal form model is useful in direct control of outputs of interest, the
usage of which is further warranted by the fact that it is diffeomorphic to
the model in Eqn. (2a) and (7) [21]. We next propose two different energy-
space control design methods, with qualitatively different implications on the
performance.

###  5.1 Feedback linearizing control

In order to ensure the tracking objective on $y_{z,i}$ following
$y_{z,i}^{ref}$, we propose control design in energy space as:

$u_{z,i}=-{\hat{\eta}_{i}}-K_{i}(y_{z,i}-y_{z,i}^{ref})+\dot{y}_{z,i}^{ref}$
(10)

Here $K_{i}$ is a positive constant and $\hat{\eta}_{i}$ is an estimate of the
$\eta_{i}$ defined in Eqn. (8a), which undoubtedly is a complex design task.
However, although the definition of terms in use for $\eta_{i}$ involve higher
order terms in conventional state space, its estimate can be obtained using
the linear energy state space model of Eqn. (5). This discussion is however
out of scope of this paper and preliminary work on this topic can be found in
[26]. For the purposes of this paper, we assume that an accurate estimate of
$\eta_{i}$ can be obtained. Substituting the control design of Eqn. (10) into
Eqn. (8a), we obtain in closed loop

$\frac{d}{dt}\left(y_{z,i}-y_{z,i}^{ref}\right)=-K_{i}\left(y_{z,i}-y_{z,i}^{ref}\right)+\left(\tilde{\eta}_{i}-4E_{t,i}\right)$
(11)

Here $\tilde{\eta}_{i}=\eta_{i}-\hat{\eta}_{i}$ is the difference between the
physical and measured nonlinearities that are utilized in the higher layer
design in energy space. From the output response in Eqn. (11), we establish
sufficient stability conditions for the closed loop component models. In order
to do so, we need to introduce the notion of dissipativity in the extended
state space. The definition in [27] has been revised to include the effect of
control inputs in extended state space designed using the interactive energy
space models introduced in Fig. 4.

######  Definition 2.

(Dissipative through feedback) The system in Eqn. (8) is said to be
dissipative through feedback with respect to supply rate $s_{i}$ if there
exists a feedback control $u_{z,i}$ and a storage function $S_{i}$ satisfying

$\begin{array}[]{l}\frac{d{S_{i}}}{dt}({{\tilde{x}}_{i}}(t))\leq{{s_{i}}\left({y_{z,i}}({{\tilde{x}}_{i}}(t)),{\dot{z}_{i}^{r,in}(t)}\right)}\end{array}$
(12)

for all $t\geq 0$ and all
$\left(\tilde{x}_{i},\dot{z}_{i}^{r,in}\right)\in\mathcal{\tilde{X}}_{i}\times\mathcal{Z}_{i}$
where $\mathcal{\tilde{X}}_{i}$ and $\mathcal{Z}_{i}$ represents the extended
state space and the incoming interaction variable manifolds respectively.

######  Theorem 1.

(Performance with FBLC control)

Consider the virtual control $u_{z,i}$ designed as in Eqn. (10). Under
assumptions 1 \- 3, the closed loop model formed by equations (8), (10)
establish following properties:

  1. 1.

It is dissipative through feedback w.r.t a supply rate

${s_{i}}\left({{y_{z,i}},{{\dot{z}}_{i}}^{r,in}}\right)=-K_{i}\left({{y_{z,i}}-y_{z,i}^{ref}}\right)+\dot{P}_{i}^{r,in}+\dot{Q}_{i}^{r,in}$
(13a)

if $\tilde{\eta}_{i}\leq\dot{Q}_{i}^{r,in}$ where $y_{z,i}$ is a function of
extended state space as defined in Eqn. (5d) and $y_{z,i}^{ref}=P_{i}^{r,in}$
which is one of the elements of $\dot{z}_{i}^{r,in}$.

  2. 2.

Assume there exists a non-empty equilibrium set defined as
$\left\\{\tilde{x}_{i}^{*}\in\mathcal{\tilde{X}}_{i}|y_{z,i}\left(\tilde{x}^{*}_{i}\right)=P_{i}^{r,in}\right\\}$
Any point in this set is asymptotically stable if

$\left|\tilde{\eta}_{i}-4E_{t,i}\right|\leq
K_{i}\left|y_{z,i}-y_{z,i}^{ref}\right|$ (13b)

######  Proof.

The proof is detailed in Appendix B.1.

It should be noted that the equilibrium set is unknown and is itself possibly
time-varying. As a result, we do not assume stored energy in tangent space is
zero. The sufficienet stability condition in Eqn. (13b) indicates that the
nonlinearity cancellation error needs to be upper bounded by the rate at which
output tracking is performed. The stored energy in tangent space further aids
in satisfaction of this constraint. Upon satisfaction of such condition, the
internal dynamics of the input-output linearized model in Eqn. (8b) for
$y_{z,i}=y_{z,i}^{ref}$ can be deemed to be stable since bounded state
trajectories are assumed. The overall performance is thereby dominated by the
feedback linearizing output in Eqn. (11). We can conclude that even for time-
varying disturbances, the tracking error decays exponentially.

###  5.2 Sliding mode control

For provable performance, the feedback linearizing control requires exact
cancellation of the nonlinear term $\eta_{i}$. Furthermore, obtaining the
measurements needed for the computation of $\eta_{i}$ can be difficult in
general. We thus propose a sliding mode equivalent control as shown in Eqn.
(14), where only bound $\left|\eta_{i}-4E_{t,i}\right|\leq\overline{L}_{i}$
over a pre-specified time needs to be known.

$\dot{Q}_{i}^{u}=-\underbrace{\left(\overline{L}_{i}+K_{i}\right)}_{\alpha_{i}}sign\left(y_{z,i}-y_{z,i}^{ref}\right)+\dot{y}_{z,i}^{ref}$
(14)

Here, the sliding surface is the tracking error
$\left(y_{z,i}-y_{z,i}^{ref}\right)$ and $K_{i}$ is a positive constant. By
selecting a large enough gain value $\alpha_{i}$, the sliding mode controller
is equivalent to FBLC control [28].

######  Theorem 2.

(Performance with SMC)

Under assumption 1 \- 2 the closed loop model formed by equations (8) and (14)
establish following properties:

  1. 1.

Tracking of output variable $y_{z,i}$ to the reference value
$y_{z,i}^{ref}=P_{i}^{r,in}$ is achieved within a finite reaching time upper
bounded by $t_{r}=\frac{{{2}}}{K_{i}}\left|\sigma_{i}(0)\right|$. where
$\left|\sigma_{i}(0))\right|$ is the distance of the operating point from the
sliding surface and $K_{i}=\alpha_{i}-\overline{L}_{i}$ is a positive constant
where $\overline{L}_{i}$ bounds the nonlinearities of Eqn. (8a) as
$\left|\eta_{i}-4E_{t,i}\right|\leq\overline{L}_{i}$

  2. 2.

Assume there exists a non-empty equilibrium set defined as
$\left\\{\tilde{x}_{i}^{*}\in\mathcal{\tilde{X}}_{i}|y_{z,i}\left(\tilde{x}^{*}_{i}\right)=P_{i}^{r,in}\right\\}$.
Any point in this set is asymptotically stable.

######  Proof.

The proof is detailed in Appendix B.2.

The finite settling time in sharp contrast to the asymptotic stability result
stated earlier is crucial, especially in electric energy systems, since these
systems require components to follow higher-layer supervisory control signals
within pre-specified time. The FBLC and SMC controllers result in
qualitatively similar stability results. However, the characterized bound in
SMC $\left|\eta_{i}-4E_{t,i}\right|\leq\overline{L}_{i}$ is much more
conservative than the differential bound
$\left|\tilde{\eta}_{i}-4E_{t,i}\right|\leq 0$ in use with FBLC, thereby also
leading to increased control effort in the case of former. At the same time,
perfect tracking is achieved with SMC and is robust to model and parameter
uncertainties while FBLC only has exponential convergence and is susceptible
to tracking errors that can be upper bounded by the measurement uncertainty.

We have so far solved the problem of distributed stabilization. However, the
problem (P1) requires regulation of outputs in conventional space $y_{i}$ that
dictate QoS. In order to equip this objective, we propose to revise the map
$\psi_{i}$. Let us consider QoS dictating variable that appears at the
interaction port as an example. The following remark illustrates how such
regulation objectives can get accommodated while retaining the claims stated
in this section.

######  Remark 1.

(Adopting output regulation objective)
Consider the reference for output variables in energy space to be utilized in
the map $\psi_{i}$ in Eqn. (6) in the component-level control, revised as
follows:

$\displaystyle{y_{z,i}}^{ref}={P_{i}}^{r,in}\left({\frac{{{y_{i}}^{ref}}}{{{y_{i}}}}}\right)$
(16a)
$\displaystyle{{\dot{y}}_{z,i}}^{ref}={{\dot{P}}_{i}}^{r,in}\left({\frac{{{y_{i}}^{ref}}}{{{y_{i}}}}}\right)-{P_{i}}^{r,in}\left({\frac{{{y_{i}}^{ref}}}{{{y_{i}}^{2}}}\frac{{d{y_{i}}}}{{dt}}}\right)$
(16b)

All the results derived in this section still hold while the terminal
variables get regulated to the desired reference value $y_{i}^{ref}$.

In this section, we have explained the different control design methods in
energy space that can be utilized to solve problem (P1) from the perspective
of the stand-alone component. Assuming sufficient internal stability
conditions are satisfied at each of the component, we next investigate the
feasibility of interconnected system shown in Fig. 2.

##  6 Interconnection-level feasibility conditions

In this section, we propose sufficient feasibility conditions that can be
checked for in a feed-forward way to ensure feasible interconnection. In
closed loop, the energy space model representation observes dynamical model as
in Eqn. (17).

Interaction model in closed loop: (17)
$\displaystyle\dot{x}_{z,i}=A_{z,i}x_{x,i}+B_{t}E_{t,i}+B_{z}\dot{z}_{i}^{r,out}~{}~{}x_{z,i}(0)=x_{z,i0}$

The matrices $A_{z,i},B_{t},B_{z}$ are the same as defined for Eqn. (5). It
should be noted that $\dot{z}_{i}^{r,out}$ and $E_{t,i}$ here are functions of
local energy conversion dynamics, the trajectories of which evolve as per the
extended space model considered in Eqn. (4), (7) and (10). However, the
extended state trajectories in the closed loop are dictated by the
instantaneous power and generalized reactive power entering the disturbance,
control, and interaction ports. When two or more components are left to
interact, the incoming interaction variable drives the system dynamics. It
changes the natural equilibrium of the stand-alone component model to that of
the interconnected system models. However, this dynamic adjustment at the
interfaces is contingent upon the existence of interconnected system
equilibrium and the dynamical exchange of power across components. The cause
of the interactions is the incoming interaction variable, while the effect is
the outgoing interacting variable as seen by the component. Intuitively the
set characterizing the effect should remain within the one characterizing the
cause, for convergence.

Let us first characterize the variation of incoming and outgoing interaction
variables over a period of time $t\in\left[(k-1)T,kT\right]$ in the sets
$\mathcal{Z}_{i}^{r,in}[k]$ and $\mathcal{Z}_{i}^{r,out}[k]$ respectively,
where $T>>\delta t$ considered in Theorem 1. At each component, we assume that
the set $\mathcal{Z}_{i}^{r,in}[k]$ is given, for which we characterize
$\mathcal{Z}_{i}^{r,out}[k]$ utilizing the map in Eqn. (5b). These sets are
communicated to neighbors and are utilized as $\mathcal{Z}_{j}^{r,out}[k+1]$
for the next time period. Numerical algorithms to characterize these sets are
a topic of future research. In this paper, we assume these sets can be
computed ahead of time and are available to each component. We first state a
general dissipativity result under certain conditions, that the components can
check ahead of time in a distributed manner.

######  Lemma 1.

(Feasibility of component interconnection)
If the set $\mathcal{Z}_{i}^{r,out}[k]$ as characterized by the closed loop
interactive model of $\Sigma_{i}$ $~{}\forall
z_{i}^{r,in}(t)\in\mathcal{Z}_{i}^{r,in}[k]$ observes the condition
$\mathcal{Z}_{i}^{r,out}[k]\subseteq\mathcal{Z}_{i}^{r,in}[k]$, then,
$\Sigma_{i}$ is dissipative with respect to the supply function
$\left(\dot{P}_{i}^{r,\rm{in}}(t)+\dot{Q}_{i}^{r,\rm{in}}(t)\right)~{}\forall
t\in\left[(k-1)T,kT\right]$.

######  Proof.

The proof is elaborated in Appendix B.3.

Each of the components satisfying Lemma 1 implies that there exists an
interconnected system equilibrium $x^{*}(t)$, which need not be known exactly
and is also possibly time-varying because of the time-varying disturbances
entering the components as shown in Figure 1. The next stated corollary
analyses the stability of this interconnected system created by component
interactions through memory-less junctions.

######  Theorem 3.

(Stability of interconnected system)
Given $\mathcal{Z}_{i}^{r,in}[k]$ at each component $\Sigma_{i}$ in the
system, assume $\Sigma_{i}$ in closed loop at each time $t$ observes the
properties stated in Theorem 1
$\forall{z}_{i}^{r,in}(t)\in\mathcal{Z}_{i}^{r,in}[k]$, and satisfies
sufficient feasibility conditions stated in Lemma 1. Several such components
interacting with each other through memory-less junctions result in an
interconnected system that is stable in the sense of Lyapunov $\forall
t\in\left[(k-1)T,kT\right]$ under additional assumptions 3 and 4.

######  Proof.

The proof is provided in Appendix B.4,

The conditions in Lemma 1 are consistent from the perspective of both
component-level and system-level. They are intended to be checked for in a
feed-forward way, which further can be accommodated by a look-ahead
centralized controller with dynamical energy space constraints [24] for
optimal trajectory planning. However, such an extension is out of the scope of
this paper.

##  7 Example of an RLC circuit supplying given power load

In this section, we take an example of an RLC circuit with a controllable
voltage source supplying a given power value as shown in Fig. 5. This system
can be interpreted as two sub-systems interacting similar to the one shown in
Fig. 1, where $x_{1}=\left[i_{1},v_{1}\right]$ represent the local state
variables of source sub-system $\Sigma_{1}$, $R_{1},L_{1},C_{1}$ respectively
represent the resistance, inductance and capacitance of controllable sub-
system $\Sigma_{1}$. We treat $\Sigma_{2}$ in this exercise as a black box
that is absorbing a given amount of power $P_{l,2}$. Since the dynamics of the
second component are not modeled, the outgoing interaction variable of
$\Sigma_{2}$ is $\dot{z}_{2}^{r,out}=\left[{P_{l,2},\dot{Q}_{l,2}}\right]^{T}$
where $\dot{Q}_{l,2}$ is the reactive power rate absorbed by the load in order
to consume given amount of $P_{l,2}$. The incoming rate of interaction
variable into $\Sigma_{1}$ upon interconnection with $\Sigma_{2}$ is equal to
the negative of the rate of outgoing interaction variables of $\Sigma_{2}$.
Furthermore, as seen by the stand-alone component $\Sigma_{1}$,
$\dot{z}_{1}^{r,in}$ can also be interpreted as the disturbance
$\dot{z}_{1}^{m,u}=\left[P_{1}^{m,u},\dot{Q}_{1}^{m,u}\right]^{T}$ entering
$\Sigma_{1}$.

Here $P_{i}^{m,u}$ is the disturbance that enters $\Sigma_{1}$ or the
interaction as a result of energy conversion dynamics of $\Sigma_{2}$. The
circuit may have additional local disturbances entering near the control port
denoted as $P_{1}^{m,m}$. The suffixes $m$ and $u$ indicate the matched and
unmatched counterpart of disturbances as defined in [29]. Our proposed
controller can stabilize disturbances at both matched and unmatched ports. To
restict the scope of this paper, we have considered only the unmatched
disturbances in this paper.

Figure 5: RLC circuit with local controllable voltage input $u_{1}$
($\Sigma_{1}$) supplying constant power to a black box with unknown internal
dynamics ($\Sigma_{2}$) when subject to its own local disturbance
$P_{1}^{m,m}$, where the power drawn by $\Sigma_{2}$, $P_{l,2}$ is interpreted
as another disturbance $P_{1}^{m,u}$ as seen by $\Sigma_{1}$.

The model in standard state space form is given in Eqn. (2a) for the
$\Sigma_{1}$ is given below:

$\begin{array}[]{l}\frac{d}{{dt}}\underbrace{\left[\begin{array}[]{l}{i_{1}}\\\
{v_{1}}\end{array}\right]}_{{x_{1}}}=\underbrace{\left[{\begin{array}[]{*{20}{c}}{-\frac{{{R_{1}}}}{{{L_{1}}}}}&{-\frac{1}{{{L_{1}}}}}\\\
{\frac{1}{{{C_{1}}}}}&0\end{array}}\right]\left[\begin{array}[]{l}{i_{1}}\\\
{v_{1}}\end{array}\right]}_{{f_{x,1}}({x_{1}})}+\underbrace{\left[\begin{array}[]{l}\frac{1}{{{L_{1}}}}\\\
0\end{array}\right]}_{{g_{1}}^{u}({x_{1}})}{u_{1}}+\\\
\qquad\qquad\qquad\qquad\underbrace{\left[{\begin{array}[]{*{20}{c}}{\frac{1}{{{L_{1}}{i_{1}}}}}&0\\\
0&{\frac{{-1}}{{{C_{1}}{v_{1}}}}}\end{array}}\right]}_{{g_{1}}^{m}({x_{1}})}\underbrace{\left[\begin{array}[]{l}{P_{1}}^{m,m}\\\
{P_{1}}^{m,u}\end{array}\right]}_{{m_{1}}}\end{array}$ (18)

As per the problem posing in (P1) it is desired for the control input $u_{1}$
to be designed using local measurements to achieve interconnected system
stability. Optionally, we required the chosen local output variable
$y_{1}=v_{1}$ to be regulated to a reference value of $80$ V. In this section,
we compare our proposed multi-layered distributed control to several other
linear and nonlinear methods pursued in the literature. Throughout the
analysis, we consider $R_{1}=10m\Omega,L_{1}=1.12mH,C_{1}=6.8mF$.

###  7.1 Constant power load

With the interconnection relations, we have $P_{1}^{m,u}=-P_{l}=-1.2kW$. We
first test a conventional proportional control designed as follows:

$u_{1}={u_{1}^{ref}}-K_{1}i-K_{2}\left(v-v^{ref}\right)$ (19)

Here $u_{1}^{ref}={v_{1}^{ref}+R_{1}\frac{P_{l}}{v_{1}^{ref}}}$ is the
consistent input applied at equilibrium. The gains $K_{1}=0.4512$ and
$K_{2}=0.45$ are the Linear Quadratic Regulator (LQR) tuned gains of the
linearized system matrices around initial conditions $i_{1,0}=1A,v_{1,0}=80V$.
Such control results in a stable response as shown in blue in Fig. 6. Notice
that implementing this control can be difficult because of having to know the
exact reference values and the parameter $R_{1}$.

(a) Current through inductor

(b) Voltage across capacitor

(c) Applied control voltage

Figure 6: Comparison of trajectories obtained by applying constant gain
control (Eqn. (19) in blue) and energy-based control (Equations (7.1) in red
and yellow), for powering constant power load of $1.2kW$ with an additional
objective to regulate terminal voltage $v_{1}$ to $80V$.

Now consider the energy based control with $y_{z,1}$ as defined in Eqn. (5d).
For the system considered in Fig. 5,

$y_{z,1}=R_{1}i_{1}^{2}-u_{1}i_{1}-P_{1}^{m,m}$ (20a) We utilize the higher
layer control design as in Eqn. (10) and the mapping in Eqn. (7), which for
the considered system leads to the following dynamic control
$\displaystyle{\frac{{d{u_{1}}}}{{dt}}=\frac{{{u_{1}}}}{{{i_{1}}}}\frac{{d{i_{1}}}}{{dt}}-\frac{{{u_{z,1}}}}{{{i_{1}}}}}$
(20b) $\displaystyle{{\rm{where}}}$
$\displaystyle{{u_{z,1}}={\eta_{1}}\left({{x_{1}},{{\dot{x}}_{1}}}\right)-{K_{1}}\left({{y_{z,1}}-{y_{z,1}}^{ref}}\right)+{{\dot{y}}_{z,1}}^{ref}}$
(20c)
$\displaystyle{\qquad{\eta_{1}}\left({{x_{1}},{{\dot{x}}_{1}}}\right)=\underbrace{\begin{array}[]{*{20}{l}}{2{L_{1}}\frac{{d{i_{1}}}}{{dt}}+}\\\
{2{C_{1}}\frac{{d{v_{1}}}}{{dt}}}\end{array}}_{4{E_{t}}}+2\underbrace{\left({\begin{array}[]{*{20}{l}}{{v_{1}}\frac{d}{{dt}}\left({{i_{1}}-{i_{2}}}\right)-}\\\
{\left({{i_{1}}-{i_{2}}}\right)\frac{{d{v_{1}}}}{{dt}}}\end{array}}\right)}_{{{\dot{Q}}_{c}}}}$
(20h)
$\displaystyle{\qquad+\underbrace{2{R_{1}}{i_{1}}\frac{{d{i_{1}}}}{{dt}}-2{u_{1}}\frac{{d{i_{1}}}}{{dt}}+4{E_{t}}({x_{1}},{{\dot{x}}_{1}})}_{{{\dot{P}}_{1}}^{r,out}+{{\dot{Q}}_{1}}^{r,out}}}$
(20i)

Figure 7: Conceptual block diagram of distributed energy control
implementation for $\Sigma_{1}$ of Fig. 5

.

It is only for simulation runs that we utilize expressions in Eqn. (20i) and
(20a). A conceptual block diagram of control to be utilized in practice is
shown in Fig. 7. It involves usage of an observer that estimates the values of
$\eta_{1}$ and $y_{z,1}$ that would be needed for energy control in Eqn.
(20c). Notably, both observer and control are designed in linear energy state
space of Eqn. (5). In this paper we have only elaborated on the latter and the
observer design is a topic of future research. Finally, mapping of the
designed energy control $u_{z,1}$ to the physical control $u_{1}$ utilizes
only real time measurements of conventional state variable $i_{1}(t)$ at the
control port. The reference values for use in the control design in Eqn. (10)
for regulation objective requires revision of the output variable references
in energy space revised as shown in Eqn. (16) which translates to the system
under consideration as follows:

$\displaystyle
y_{z,1}^{ref}=P_{1}^{r,in}\frac{v_{1}^{ref}}{v_{1}}=-P_{l,2}\frac{v_{1}^{ref}}{v_{1}}$
(20j)
$\displaystyle{{\dot{y}}_{z,1}}^{ref}={{\dot{P}}_{1}}^{r,in}\left({\frac{{{v_{1}}^{ref}}}{{{v_{1}}}}}\right)-{P_{1}}^{r,in}\left({\frac{{{v_{1}}^{ref}}}{{{v_{1}}^{2}}}\frac{{d{v_{1}}}}{{dt}}}\right)$
$\displaystyle=-{{\dot{P}}_{l,2}}\left({\frac{{{v_{1}}^{ref}}}{{{v_{1}}}}}\right)+{P_{l,2}}\left({\frac{{{v_{1}}^{ref}}}{{{v_{1}}^{2}}}\frac{{d{v_{1}}}}{{dt}}}\right)$
(20k)

The incoming instantaneous power corresponds to non-zero reactive power
absorption characterized in Eqn. (21), which is the prime cause of non-zero
persistent disturbances in conventional state space seen at the interface.

$\dot{Q}_{1}^{r,in}=-v_{2}\frac{d}{dt}\left(\frac{P_{l,2}}{v_{2}}\right)+\left(\frac{P_{l,2}}{v_{2}}\right)\dot{v}_{2}=-\dot{P}_{l,2}+2\frac{P_{l,2}}{v_{2}}\frac{dv_{2}}{dt}$
(21)

The resulting trajectories with energy based control are shown in Fig. 6 in
red and yellow for FBLC and SMC variants respectively. It can be seen that the
trajectories settle much faster energy-based control and also with tolerable
transient overshoots. The gain values can further be adjusted based on the
desired transient response characteristics. Furthermore, notice that the SMC
variant clearly indicates finite settling time, while FBLC variant has
exponential convergence.

###  7.2 Feed-forward checking of feasibility conditions

The sufficient feasibility conditions in Eqn. (28) can be checked in a feed-
forward way through pre-specified ranges. In this example, since $\Sigma_{2}$
is assumed a perfect power-consuming load, it can use its estimated variations
in power to compute ranges of its outgoing interaction variable, which is sent
out as ranges of incoming interaction variable to $\Sigma_{1}$. Given these
ranges, $\Sigma_{1}$ is to compute the ranges of outgoing interaction
variables using the definition in (5b) given its local control saturation and
local predicted disturbance limits. If these ranges are within the range of
the incoming interaction variable sent out by $\Sigma_{1}$, only then is the
system feasible.

Consider the load to be served as shown in Fig. 8.

Figure 8: Non-zero mean unmatched power disturbances under consideration

For this load at every $T_{s}=1s$, we compute the estimated ranges of
instantaneous power and rate of change of instantaneous power for use in Eqn.
(22) to compute range of incoming interaction variable.

$\displaystyle{\rm{Given~{}}}{P_{l,2}}(t)\in\left[{\begin{array}[]{*{20}{c}}{{P_{l,2}}^{\min}[k]}&{{P_{l,2}}^{\max}[k]}\end{array}}\right]\qquad$
(22b) $\displaystyle\Rightarrow
P_{1}^{r,in}(t)\in\left[{\begin{array}[]{*{20}{c}}{-{P_{l,2}}^{\max}[k]}&{-{P_{l,2}}^{\min}[k]}\end{array}}\right]$
(22d)
$\displaystyle{\rm{Given~{}}}{{\dot{P}}_{l,2}}(t)\in\left[{\begin{array}[]{*{20}{c}}{{{\dot{P}}_{l,2}}^{\min}[k]}&{{{\dot{P}}_{l,2}}^{\max}[k]}\end{array}}\right]\qquad$
(22f)
$\displaystyle\Rightarrow{{\dot{Q}}_{1}}^{r,in}(t)\in\left[{\begin{array}[]{*{20}{c}}{-{{\dot{P}}_{l,2}}^{\max}[k]}&{-{{\dot{P}}_{l,2}}^{\min}[k]}\end{array}}\right]$
(22h)
$\displaystyle\quad+\frac{2}{{{\tau^{\prime}}}}\left[{\begin{array}[]{*{20}{c}}{{P_{l,2}}^{\min}[k]}&{{P_{l,2}}^{\max}[k]}\end{array}}\right]\quad\forall
t\in\left[{k{T_{s}},(k+1){T_{s}}}\right]$ (22j)

Here $\tau^{\prime}$ represents the time constant of voltage dynamics to
capture transient overshoots.
$\tau^{\prime}=min\left\\{10\left(\frac{L_{1}}{R_{1}}\right),\frac{1}{K}\right\\}$.
$K$ is the effective control gain corresponding to voltage dynamics. The way
this is computed differs depending on the control design. For the output
feedback control in Eqn. (19), $K=K_{2}$; and for energy-based FBLC control in
Eqn. (10), $K=K_{1}$.

(a) First component (b) Second compnent

Figure 9: Interaction variable for validation of feasibility conditions for
$\Sigma_{1}$ of Fig. 1 when controlled through output feedback control in Eqn.
(19)

The range validation plot for constant gain controllers is shown in Fig. 9.
Notice that the constant gain controller has outgoing reactive power violating
the pre-computed incoming reactive power range at approximately 0.35 seconds,
indicating infeasibility.

(a) First component (b) Second compnent

Figure 10: Interaction variable for validation of feasibility conditions for
$\Sigma_{1}$ of Fig. 1 when controlled through design in Eqn. (7.1).

In contrast, energy-based controllers for choice of output variable in Eqn.
(20a) remain stable for the values of load power considered. This is validated
by the fact that the outgoing interaction variable always lies within the pre-
computed incoming interaction variable range shown in Fig. 10. The
corresponding plots of internal variables are shown in Fig. 11 in red.

###  7.3 Performance for continuous time-varying load

Consider the load to be served as shown in Fig. 8. This can be perceived as
non-zero-mean unmatched disturbances entering the source sub-system.

(a) Current through inductor

(b) Voltage across capacitor

(c) Rate of change of control voltage

Figure 11: Comparison of trajectories obtained by applying benchmark nonlinear
control (Eqn. (23)) and energy-based controllers (Eqn. (7.1))

The constant gain controllers fail for such time-varying disturbances. We thus
consider another benchmark nonlinear control proposed in [30] based on Brayton
Moser potential function and is shown in Eqn. (23).

Benchmark non-linear control [30]:
$\displaystyle{u_{1}}={R_{1}}{i_{1}}+{v_{1}}^{ref}-{L_{1}}\left(\frac{\Pi}{{v_{1}^{2}}}+K_{3}\right)\left({{{\dot{v}}_{1}}}\right)$
$\displaystyle\qquad\qquad-\left({{K_{1}}\left({{v_{1}}-{v_{1}}^{ref}}\right)+{K_{2}}\frac{{d{v_{1}}}}{{dt}}}\right)$
(23)

Notice that this control utilizes an adaptive gain
$\frac{\Pi}{{v_{1}}^{2}}+k_{3}$ for positive $k_{3}$, where $\Pi$ is the upper
bound on the load power values being served. It has been shown in [30], the
control in Eqn. (23) results in stable performance for all values of
$v_{1}^{ref}>0$ and $P_{l,2}\leq\Pi$. The corresponding trajectories for the
objective of output voltage regulation are shown in Fig. 11 in blue.

For the same disturbances, the energy-based FBLC with choice of output
variable in Eqn. (20a) results in the response shown in red. Also shown in
yellow is the trajectory obtained by replacing the energy space design
$u_{z,i}$ in Eqn. (7.1) with the one in Eqn. (14). Energy space control
explicitly accounts for the interfaces’ continuously varying reactive power
injections and thus both variants of energy space control result in better
tracking performance in terms of the settling time. All the controllers result
in similar voltage regulation with minimal violations. However, the rate of
change of control of our proposed method is much smaller than the nonlinear
benchmark, leading to much lower wear-and-tear. This happens because the
output variable $y_{z,i}$ to which energy space control responds is an
aggregate variable which evolves much slower than conventional state space
variables to which the benchmark controller responds to.

###  7.4 Robustness to parameter uncertainty

The nonlinear benchmark control is very sensitive to the parameter $R_{1}$.
Shown in Fig. 12(a) are the voltage trajectories obtained when this control is
implemented for 10% error in resistance value measurements. Overlaid on top of
it are also the plots obtained with energy-based control. With both FBLC and
SMC variants, we see that voltage regulation is perfect. This happens because
the resistance value is not directly used in the control design. Instead, the
design utilizes the total damping $\frac{E}{\tau}$ as defined in Eqn. (20a),
which can be estimated from available measurements of energy and power
measurements.

(a) Voltage (b) Control

(c) Rate of change of control

Figure 12: Robustness assessment for a time-varying load for 10% error in
resistance value for benchmark nonlinear control (Eqn. (23)) and energy-based
controllers (Eqn. (7.1))

In presence of such parameter uncertainties, the corresponding rate of change
of control trajectories of energy-based SMC control is shown in yellow in Fig.
12(c). It indicates clear chattering, while the rate of change of control
trajectories are much smoother with energy-based FBLC. Note however that the
actual control as in Fig. 12(b) is smooth for all the controllers considered.
It should be noted that in the FBLC variant, we assumed that the disturbance
cancellation terms $\eta_{i}$ in Eqn. (8a) can be measured perfectly, thereby
leading to $\tilde{\eta_{i}}=0$ in Eqn. (11). Thus the voltage trajectories in
Fig. 12(a) appear to be perfect. From these experiments, we see that there is
a tradeoff between the desired transient response, tolerable wear and tear,
measurement and parameter uncertainties in the system.

##  8 Conclusions and future work

In this paper, we consider the problem of distributed stabilization of
interconnected systems. We propose an interactive multi-layered energy space
modeling, based on inherent mathematical structures stemming from the first
principles of energy conservation laws, offering intuition about energy and
power dynamics. The higher layer models in energy space notably are linear and
thus lend themselves to provable control design. The control in the lower
layer physical model is either FBLC or SMC. We show that this multi-layered
control design leads to sufficient modular conditions for provable feasibility
of the interconnected system and the stability of internal dynamics. Notably,
in the prior work, models have been utilized for designing competitive
disturbance rejection control, which treats the interaction dynamics as
exogenous inputs. However, these interaction dynamics seen by each component
$i$ are state-dependent according to energy conservation principles. To
capture this interdependence, we propose collaborative control design which
instead of cancelling the interactions completely, aligns the references in a
cooperative manner. Reference point of component $i$ $y_{z,i}^{ref}$ is
obtained by exchanging the information $y_{z,j}$ from neighboring component
$j$ resulting in their alignment rather than their cancellation. Such
interactive models open new avenues for research on distributed cooperative
control. Finally, we illustrate the effectiveness of the modeling and control
design on an RLC circuit and compare to other linear and nonlinear controllers
considered in the literature.

The control design approach is modular and thus is scalable to arbitrarily
large systems. However, this paper considers an example of single component
subject to time-varying power disturbances for exposition of concepts. Further
simulation-based evidence of multiple controllable components interacting with
each other is needed. The interactive energy-based models proposed in this
paper pose new challenges in distributed numerical simulations and control
implementation. Further investigation of the interplay between numerical
simulations and the hardware implementation of the control is being pursued.

This paper is a result of financial support by the NSF project Early-Concept
Grants for Exploratory Research (EAGER) “Fundamentals of Modeling and Control
for the Evolving Electric Power System Architectures” project ECCS-2002570.

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##  Appendix A Energy space variables definitions

######  Definition 3.

(Instantaneous power)
The power interaction of the component $i$ with the rest of the system is
given by the mapping
${P}_{i}:\mathcal{E}_{i}\times\mathcal{F}_{i}\rightarrow\mathcal{P}_{i}$ and
is defined as $P_{i}=e_{i}^{T}f_{i}$ where $e_{i}\in\mathcal{E}_{i}$ and
$f_{i}\in\mathcal{F}_{i}$ respectively represent the effort and flow variables
appearing at the ports of interconnection.

######  Definition 4.

(Generalized reactive power dynamics)
Generalized reactive power $Q_{i}$ is a quantity whose time derivative is
given by a mapping
$\dot{Q}_{i}:\mathcal{TE}_{i}\times\mathcal{TF}_{i}\rightarrow\mathcal{TQ}_{i}$
and is defined as
$\dot{Q}_{i}=e_{i}^{T}\frac{df_{i}}{dt}-f_{i}^{T}\frac{de_{i}}{dt}$ where
$\mathcal{TE}_{i}$ and $\mathcal{TF}_{i}$ represents the tangent manifold of
efforts and flow variables respectively.

Let the total stored energy in a generic form for the component defined by the
model in Eqn.(2a) be defined using an inertia matrix $H_{i}(x_{i})$.
Similarly, if these elements are lossy, they are also associated with a
symmetric dissipation matrix $B_{i}(x_{i})$. [22, 31].

######  Definition 5.

(Stored energy)
Stored energy of component $i$ is given by the energy function
${E}_{i}:\mathcal{X}_{i}\rightarrow\mathbb{R}$ defined as
$E_{i}(x_{i})=\frac{1}{2}x_{i}^{T}H_{i}(x_{i})x_{i}$ for an inertia matrix
$H_{i}(x_{i})~{}\forall x_{i}\in\mathcal{X}_{i}$.

######  Definition 6.

(Stored energy in tangent space)
For the same positive definite matrix $H_{i}$ as defined in Definition 5, the
stored energy in tangent space $E_{ti}$ is defined over the tangent bundle
$\mathcal{TX}_{i}:=\cup_{x_{i}\in\mathcal{X}_{i}}{x_{i}}\times
T_{x_{i}}\mathcal{X}_{i}$ where $T_{x_{i}}\mathcal{X}_{i}$ is the tangent
space of $\mathcal{X}_{i}$ at $x_{i}$, through the mapping
${E}_{t,i}:\mathcal{TX}_{i}\rightarrow\mathbb{R}$ as
$E_{t,i}(x_{i},\dot{x}_{i})=\frac{1}{2}\dot{x}_{i}^{T}H_{i}(x_{i})\dot{x}_{i}$.

######  Definition 7.

(Time constant)
Dissipation of a component $i$ is described through a dissipation function in
quadratic form through the mapping
${D}_{i}:\mathcal{X}_{i}\rightarrow\mathbb{R}$ defined as
$D_{i}(x_{i})=x_{i}^{T}B_{i}(x_{i})(x_{i})x_{i}$ for some matrix
$B_{i}(x_{i})\forall x_{i}\in\mathcal{X}_{i}$. We define the time constant of
component $i$ as the ratio of stored energy and the damping of the component,
which by utilizing Eqns. as $\tau_{i}=\frac{E_{i}(x_{i})}{D_{i}(x_{i})}$
Utilizing the Definitions 5 and the expression for damping, this quantity can
be upper bounded by the largest singular value of
$D_{i}^{-1}(x_{i})H_{i}(x_{i})$.

##  Appendix B Proofs

###  B.1 Theorem 1 of Section 5

######  Proof.

(1)

Consider now the candidate storage function ${S_{i}}=y_{z,i}$ By taking its
time derivative and plugging in the expressions from Eqn. (11), we have
$\displaystyle{\frac{{d{S_{i}}(t)}}{{dt}}=-4{E_{t,i}}+\left({-{K_{i}}\left(\begin{array}[]{l}{y_{z,i}}-\\\
{y_{z,i}}^{ref}\end{array}\right)+{{\tilde{\eta}}_{i}}+{{\dot{y}}_{z,i}}^{ref}}\right)}$
(25c) By Assumption 3, $E_{t,i}\geq 0$. Setting the upper bound on
$\tilde{\eta}_{i}$ and assigning $y_{z,i}^{ref}=P_{i}^{r,in}$, the result
follows.

(2) Select the closed loop potential function as
$V_{i}=\left|y_{z,i}-y_{z,i}^{ref}\right|$. Taking its time derivative and
substituting the derivative expression in Eqn. (11), we obtain

$\begin{array}[]{*{20}{l}}{\frac{{d{V_{i}}(t)}}{{dt}}={\mathop{\rm
s}\nolimits}ign\left(\begin{array}[]{l}{y_{z,i}}-\\\
{y_{z,i}}^{ref}\end{array}\right)\left({-{K_{i}}\left(\begin{array}[]{l}{y_{z,i}}-\\\
{y_{z,i}}^{ref}\end{array}\right)+\left(\begin{array}[]{l}{{\tilde{\eta}}_{i}}-\\\
4{E_{t,i}}\end{array}\right)}\right)}\\\
{\quad\leq-{K_{i}}\left|{{y_{z,i}}-{y_{z,i}}^{ref}}\right|+\left|{{{\tilde{\eta}}_{i}}-4{E_{t,i}}}\right|}\end{array}$
(25d)

The first inequality indicates that for arbitrary $\tilde{\eta}_{i}$ and
$4E_{t,i}$, the set defined by
$\left\\{\tilde{x}_{i}\in\mathcal{\tilde{X}}_{i}|y_{z,i}\left(\tilde{x}_{i}\right)=y_{z,i}^{ref}\right\\}$
is the only possibility for $\frac{dV_{i}}{dt}=0$. From LaSalle’s invariance
principle, we thus have asymptotic stability when condition in Eqn. (13b) is
satisfied.

###  B.2 Theorem 2 of Section 5

######  Proof.

(1) By denoting distance of operating point from the sliding surface as
$\sigma_{i}=y_{z,i}-y_{z,i}^{ref}$, consider
$S_{i}=\frac{1}{2}\sigma_{i}^{2}$.Taking its time derivative and utilizing
Equations (8a) and Eqn. (14), we obtain
$\begin{array}[]{l}{{\dot{S}}_{i}}={\sigma_{i}}{{\dot{\sigma}}_{i}}={\sigma_{i}}\left({\left({-4{E_{t,i}}+{\eta_{i}}+{u_{z,i}}}\right)-{{\dot{y}}_{z,i}}^{ref}}\right)\\\
={\sigma_{i}}\left({-4{E_{t,i}}+{\eta_{i}}}\right)+{\sigma_{i}}\left({-{\alpha_{i}}sign({\sigma_{i}})}\right)\\\
\leq\left|{{\sigma_{i}}}\right|{{\bar{L}}_{i}}-({{\bar{L}}_{i}}+{K_{i}})\left|{{\sigma_{i}}}\right|\\\
\leq-{K_{i}}\left|{{\sigma_{i}}}\right|\leq-
K_{i}{\left({2{S_{i}}}\right)^{\frac{1}{2}}}\end{array}$ (26a) Taking the time
integral on both sides, we can establish
$\begin{array}[]{l}\int\limits_{0}^{t}{{S_{i}}^{-\frac{1}{2}}{{\dot{S}}_{i}}}\leq-\sqrt{2}\int\limits_{0}^{t}{{K_{i}}dt}\\\
\begin{array}[]{*{20}{l}}{\Rightarrow
S_{i}^{\frac{1}{2}}(t)-S_{i}^{\frac{1}{2}}(0)\leq-\frac{K_{i}}{\sqrt{2}}t}\\\
{\Rightarrow\left|{{\sigma_{i}}(t)}\right|\leq\left|{{\sigma_{i}}(0)}\right|-\frac{K_{i}}{\sqrt{2}}t}\end{array}\end{array}$
(26b) Since $|\sigma_{i}|\geq 0$, we have $\sigma_{i}(t)$ approach zero in a
time upper bounded by $t_{r}=\frac{\sqrt{2}}{K_{i}}\left|\sigma_{i}(0)\right|$

(2)

From Eqn. (26b), $\frac{{d{S_{i}}}}{{dt}}\leq-K_{i}\left|\sigma_{i}\right|$.
The right hand sign is negative semi-definite and is zero only when
$\sigma_{i}=0$ for arbitrary operating conditions and time-varying
disturbances. i.e. for when $\tilde{x}_{i}\in
y_{z,i}(\tilde{x}_{i})={P}_{i}^{r,in}$. By LaSalle’s invariance principle, the
equilibrium set thus defined in the theorem statement is asymptotically
stable.

###  B.3 Lemma 1 of Section 6

######  Proof.

At instantaneous time, the sufficient feasibility conditions can also be re-
stated as an inequality condition to be satisfied element-by-element as
follows:

$z_{i}^{r,out}\preceq z_{i}^{r,in}$ (28)

Considering the derivative of the first element of the inequality and the
second element of the inequality and adding them up, we have
${\dot{P}_{i}}^{r,out}+{\dot{Q}_{i}}^{r,out}\leq{\dot{P}_{i}}^{r,in}+{\dot{Q}_{i}}^{r,in}$
(29a) By utilizing the definition of outgoing interaction variable in Eqn.
(17), $\displaystyle{{\dot{P}_{i}}^{r,out}}$
$\displaystyle={\dot{p}_{i}}+\frac{d}{dt}\left({\frac{1}{{{\tau_{i}}}}{E_{i}}}\right)$
(29b) $\displaystyle{\dot{Q}_{i}}^{r,out}$
$\displaystyle={4{E_{t,i}}}-{\dot{p}_{i}}$ (29c) Now combining Eqns. (29a) -
(29c), we have
$\frac{d}{dt}{\underbrace{\left(\int\limits_{0}^{t}{{{{4{E_{t,i}}(s)}}}ds}+{{\frac{1}{\tau}}_{i}}{E_{i}}(t)\right)}_{{S_{i}(t)}}}\leq\underbrace{{\dot{P}_{i}}^{r,in}+{\dot{Q}_{i}}^{r,in}}_{{s_{i}(t)}}$
(29d) The result follows thereby.

###  B.4 Theorem 3 of Section 6

######  Proof.

By adding up the dissipativity conditions in Eqn. (29d) for each of the
components in the network $\mathcal{N}$,

$\begin{array}[]{l}\frac{d}{{dt}}\left({\sum\limits_{i\in{\mathcal{N}}}{{S_{i}}({x_{i}})}}\right)\leq\sum\limits_{i\in{\mathcal{N}}}{\left({\dot{P}_{i}^{r,in}+\dot{Q}_{i}^{r,in}}\right)}\\\
\sum\limits_{i\in{\mathcal{N}}}{\left({\dot{P}_{i}^{r,out}+\dot{Q}_{i}^{r,out}}\right)}\leq\sum\limits_{i\in{\mathcal{N}}}{-\left({\sum\limits_{j\in{{\mathcal{C}}_{i}}}{\left({\dot{P}_{j}^{r,out}+\dot{Q}_{j}^{r,out}}\right)}}\right)}\\\
{\bf{1}}_{|N|\times
1}^{T}\left(\begin{array}[]{l}{{{\bf{\dot{P}}}}^{r,out}}+\\\
{{{\bf{\dot{Q}}}}^{r,out}}\end{array}\right)\leq-{\bf{1}}_{|N|\times
1}^{T}{\bf{L}}_{\left|\mathcal{N}\right|\times\left|\mathcal{N}\right|}\left(\begin{array}[]{l}{{{\bf{\dot{P}}}}^{r,out}}\\\
+{{{\bf{\dot{Q}}}}^{r,out}}\end{array}\right)\\\ {\bf{1}}_{|\mathcal{N}|\times
1}^{T}\left({\bf{I}_{|\mathcal{N}|\times|\mathcal{N}|}+\bf{L}_{\left|\mathcal{N}\right|\times\left|\mathcal{N}\right|}}\right)\left({{{{\bf{\dot{P}}}}^{r,out}}+{{{\bf{\dot{Q}}}}^{r,out}}}\right)\leq
0\\\ \Rightarrow\frac{d}{dt}\left({\bf{1}}_{|\mathcal{N}|\times
1}^{T}\left({\bf{I}_{|\mathcal{N}|\times|\mathcal{N}|}+\bf{L}_{\left|\mathcal{N}\right|\times\left|\mathcal{N}\right|}}\right)\bf{S}\right)\leq
0\end{array}$ (30)

Here, $\bf{P},\bf{Q}$ are the vector form notation of the real and reactive
power of each of the components in the network. $|.|$ operator here represents
the cardinality of the set. $\bf{I}$ and $\bf{1}$ respectively represent the
identity matrix and the column vector comprising element $1$, with its order
in the subscript. $\bf{L}$ represents a symmetric matrix where the element
$L_{ij}=1$ is the components $i$ and $j$ are connected. Finally, $\mathbf{S}$
is the vectorized representation of potential functions considered for each of
the components. Since each element of $\mathbf{S}$ is positive definite due to
the assumptions 3 and 4, we have that the entire bracketed term in the last
equation to be positive definite. This bracketed term can be considered as the
candidate Lyapunov function to thereby prove stabilitiy in the sense of
Lyapunov.