# An Information Theoretic approach to identify Dominant Voltage Influencers
for Unbalanced Distribution Systems

Sai Munikoti, Student Member, IEEE, Mohammad Abujubbeh, Student Member, IEEE,
Kumarsinh Jhala, Member, IEEE, Balasubramaniam Natarajan, Senior Member, IEEE
K. Jhala is with the Center for Energy, Environmental, and Economic Systems
Analysis in the Energy Systems Division at Argonne National Laboratory
(e-mail: kjhala@anl.gov).S. Munikoti, M. Abujubbeh and B. Natarajan are with
Electrical and Computer Engineering, Kansas State University, Manhattan,
KS-66506, USA, (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>bala@ksu.edu)This work has been submitted to the IEEE for possible
publication. Copyright may be transferred without notice, after which this
version may no longer be accessible.

###### Abstract

Smart distribution grid with multiple renewable energy sources can experience
random voltage fluctuations due to variable generation, which may result in
voltage violations. Traditional voltage control algorithms are inadequate to
handle fast voltage variations. Therefore, new dynamic control methods are
being developed that can significantly benefit from the knowledge of dominant
voltage influencer (DVI) nodes. DVI nodes for a particular node of interest
refer to nodes that have a relatively high impact on the voltage fluctuations
at that node. Conventional power flow-based algorithms to identify DVI nodes
are computationally complex, which limits their use in real-time applications.
This paper proposes a novel information theoretic voltage influencing score
(VIS) that quantifies the voltage influencing capacity of nodes with
DERs/active loads in a three phase unbalanced distribution system. VIS is then
employed to rank the nodes and identify the DVI set. VIS is derived
analytically in a computationally efficient manner and its efficacy to
identify DVI nodes is validated using the IEEE 37-node test system. It is
shown through experiments that KL divergence and Bhattacharyya distance are
effective indicators of DVI nodes with an identifying accuracy of more than
$90$%. The computation burden is also reduced by an order of $5$, thus
providing the foundation for efficient voltage control.

###### Index Terms:

voltage influencing score (VIS), voltage influencer nodes, DERs, rank,
analytical.

##  I Introduction

The power grid is evolving with the increasing integration of renewable
distributed energy resources (DERs). While offering the benefits of reduced
carbon footprint, DERs also impose new technical challenges. Specifically,
active consumers with rooftop photovoltaics and distributed generation are
expected to alter their generation and usage patterns to follow the trends of
time-varying electricity prices [1]. This in turn induces frequent power
variations. The uncontrolled operations of DERs under this condition leads to
voltage fluctuations in the distribution system. Grid operators have to manage
DERs’ operations to assure the reliability of the grid. Therefore, various
control algorithms have been considered for regulating power
injection/consumption across various buses of the network to mitigate voltage
fluctuations [2, 3, 4]. However, traditional methods of voltage control
involving voltage regulators, on-load tap changing transformers are slow and
inadequate to deal with bi-directional power flows and fast dynamics. This
necessitates the development of computationally efficient and fast dynamic
voltage control algorithms that can handle the dynamics of power/voltage
variation [5, 6, 7]. The speed of response to impending voltage issues and the
computational efficiency of these dynamic algorithms rely on our ability to
select the optimal set of nodes for control that have the highest influence on
the system voltage profile.

To this end, we introduce the notion of dominant voltage influencer (DVI)
nodes. Throughout this paper, we refer to actor nodes as the nodes where power
changes (due to DER unit and/or load dynamics) and observation nodes represent
the set of nodes where voltage state is monitored. Thus, DVI nodes for a
particular observation node denote all those actor nodes that have a
relatively high impact on the voltage state of that observation node compared
to the rest of the actor nodes. Hence, altering the actions of DVI nodes
result in the highest reduction in voltage issues at the observation node. The
nodes could be DVI because of their size (generation/load capacity) and/or
locations in the distribution network. Conventional methods of identifying
such DVI nodes involve Monte-Carlo simulations using load flow algorithms.
These classical methods possess various drawbacks including (1) high
computational complexity, (2) numerical results with no analytical insights,
and (3) scenario dependent results with no generalization. These factors limit
the applicability of conventional approaches in modern distribution systems.
Thus, there is a need to develop an analytical and computationally efficient
algorithm for identifying DVI nodes in three phase unbalanced distribution
systems, which in turn can be used to develop effective voltage control
algorithms.

For identifying DVI nodes, we need to quantify the impact of actor nodes on
other nodes of the network in terms of voltage. Therefore, first, we develop a
novel metric namely voltage influence score (VIS) for quantifying the voltage
influencing capacity of an actor node on any arbitrary observation node. The
VIS is based on an analytical method of voltage sensitivity. Thereafter, the
VIS is employed to rank the actor nodes for any arbitrary observation node and
identify the most influential ones, i.e., DVI nodes. The proposed approach
allows us to identify DVI nodes without relying on computationally expensive
Monte-Carlo simulations thereby, significantly reducing the computation time.

Related work:  Various control algorithms are proposed in the literature for
regulating voltage fluctuations in the distribution system. Primarily, the
control algorithms can be broadly categorized into two main categories: (1)
centralized [3, 8, 2, 9], and (2) decentralized [4, 10, 11] control
approaches. Centralized voltage control algorithms are generally based on
classical power flow solution, which can be computationally expensive for real
time applications. On the other hand, decentralized voltage control methods
trade-off performance for reduced computational complexity. The speed,
complexity and performance of both centralized and decentralized algorithms
rely on the size and space of control actions. In modern distribution systems,
the change in power consumption or injections due to multiple active consumers
necessitates the need for analyzing a large number of scenarios, expanding the
size and space for determining optimal control actions. This may further
increase the computational complexity of traditional voltage control
algorithms. Other approaches to bring voltage to safe operational limits in
distribution systems involve the installation of additional control devices
such as, static-var compensators [12, 13], dynamic voltage restorers [14], and
grid-edge volt-var controllers [15]. Despite their effectiveness in voltage
control, the installation of additional devices in the system can be
expensive. Therefore, a practical and economical solution to regulate voltage
fluctuations is to utilize smart grid edge devices like PV inverters. To this
end, multiple dynamic algorithms have been proposed to control inverters and
loads to regulate voltages [5, 6, 7]. In [5], authors randomly select and shut
down the thermostatically controlled loads during the time of high voltage
unbalance. Similarly, [7] proposes a data-driven voltage control algorithm to
regulate voltage violations by minimizing power injections at nodes with over
voltage violations. In a nutshell, all these prior works use heuristic based
methods to identify nodes whose generations need to be curtailed. However,
such irrational selection is not desirable in terms of yielding optimal
solution and providing fairness to all generators participating in a
transactive energy market.

Furthermore, there are some efforts where system is partitioned into smaller
clusters and control is implemented for each cluster in a distributed manner
[16, 17, 18, 19]. In [16, 17], for instance, authors propose a new clustering
method based on voltage sensitivity with respect to power fluctuation due to
battery storage systems (BSS). However, this method is scenario specific since
the location of BSSs in the system is fixed. Another sensitivity based zonal
voltage control method is proposed in [18] where all nodes in a particular
cluster are assumed to participate in power curtailment for voltage control.
This is difficult to implement as it is not economical or even feasible for
utilities to have access to control all nodes. There are some approaches which
also include topological data for system clustering[20, 19, 21]. For the
approach in [19] to work, the nodal Q-V sensitivities must be computed to
iteratively evaluate the modality index for each cluster [22]. Nevertheless,
the computational complexity of this method increases with the size of the
system as computing the Q-V based modality index of each cluster is based on
forming the adjacency matrix of the system. Additionally, the entire body of
prior work on identifying dominant nodes of voltage fluctuations does not
consider stochasticity associated with DER power injections or user load
variations.

To address these demerits of existing approaches, analytical and
computationally efficient approaches to identify DVI nodes that have
relatively high influence on the voltage state of critical nodes are needed.
The work in this paper addresses this essential research gap using a new
information theoretic voltage influencing index that is applicable to three-
phase unbalanced systems.
Contributions: This paper proposes a novel approach to identify DVI nodes that
can be used for network clustering and distributed control mechanisms in
unbalanced distribution systems with DERs. The key contributions of this paper
include:

  * •

An analytical and computationally efficient method of identifying dominant
voltage influencer nodes is proposed for three phase unbalanced distribution
system.

  * •

This work introduces a VIS metric that quantifies the voltage influencing
capacity of nodes with DERs/active loads, and a computationally efficient
method to compute it.

  * •

Compared to [23], this work has three major advancements (1) the probabilistic
model of voltage fluctuations is derived for a three phase unbalanced
distribution systems; (2) The voltage influencing capacity is quantified for
each pair of nodes; (3) In addition to identifying the most dominant node,
this work provides ranks of all dominant influencer nodes based on their VIS;

  * •

The effectiveness of proposed method is evaluated on the standard unbalanced
IEEE 37-node test system yielding an accuracy of more than $90$% with the
computation complexity reduced by an order of $5$ compared to a classic load
flow based approach.

The paper is organized as follows. Section II presents a conventional method
to identify DVI nodes followed by an analytical method of voltage sensitivity
analysis. Section III computes probability distributions of voltage change.
Various information theoretic indicators of the DVI nodes are proposed in
Section III, which are used to compute VIS. The performance of the proposed
approach is tested using the IEEE 37-node test system in Section IV.
Conclusions and future-work are presented in Section V.

##  II Background

This section introduces the conventional approach to identify Dominant voltage
influencer (DVI) nodes in a power distribution network. Then, the fundamental
analytical expressions of voltage sensitivity are discussed which are later
used to model voltage fluctuations for identifying DVI nodes.

###  II-A Conventional approach to identify DVI nodes

The DVI nodes for an observation node are the nodes that have high impact on
the voltage fluctuations at the observation node. An actor node can be a DVI
due to association of two factors: (1) location of the actor node, i.e. phase
and bus of the distribution network, and (2) generation/load capacity of
DER/loads connected at the actor node which influences the variance of power
change at that node. Generally, simulation-based scenario analysis is used as
a major planning tool to identify DVI nodes. A typical approach involves
following steps [23]:

####  II-A1 Step 1- Compute variance of voltage change at each phase of
observation node due to all actor nodes

The variance of voltage change at each phase of the observation node is
computed by running multiple power flow based Monte-Carlo simulations with
varying power, which captures temporal variation of generations.

####  II-A2 Step 2- Calculate reduction in variance of voltage change at the
observation node due to each actor node by setting power drawn/injected by the
actor node as zero

This step requires repetition of Step 1 for each actor node after setting the
variance of actor node as zero.

####  II-A3 Step 3- Rank actor nodes based on the reduction in variance caused
by the removal of the corresponding actor node:

Actor nodes are ranked in an ascending order with topmost and bottommost rank
assigned to those actor node that causes maximum and minimum reduction in
variance of voltage change at the observation node, respectively.

The scenario-based simulations incur high computational complexity, which
grows with the size of the network. Therefore, various information theoretic
metrics are explored in the next section that can identify DVI nodes in a
computationally efficient manner, and can be used as a tool in various power
system operations.

Algorithm 1 Proposed method to identify DVI nodes

0: Distribution Network with the target observation node.

0: Ranked actor nodes for the input observation node.

1: Compute distribution of voltage change at the input observation node due to
each actor node at a time.

2: Compute distribution of voltage change at the input observation node due to
aggregate effect of all actor nodes.

3: Find statistical distance between the distributions computed in steps 1 &
2\.

4: Compute VIS for each pair of observation and actor nodes.

5: Rank actor nodes in the ascending order based on their respective VIS.

6: return ranks and VIS of actor nodes for an input observation node.

###  II-B Analytical VSA

In a three phase distribution system, change in power at any one phase of the
node causes change in voltage at all phases of all the other nodes.
Traditional methods of voltage sensitivity analysis (VSA) are computationally
complex and less generic. Therefore, we developed a computationally efficient
method of VSA in [24, 25]. For an unbalanced power distribution system, change
in complex voltage $\Delta V_{O}$ at an observation node ($O$) due to change
in complex power at multiple actor nodes can be approximated as:

$\begin{bmatrix}\Delta V_{O}^{a}\\\\[15.0pt] \Delta V_{O}^{b}\\\\[15.0pt]
\Delta
V_{O}^{c}\end{bmatrix}\approx-\sum_{A\epsilon\tilde{A}}\left(\begin{bmatrix}\frac{\Delta
S_{A}^{a\star}Z_{OA}^{aa}}{V_{A}^{a\star}}+\frac{\Delta
S_{A}^{b\star}Z_{OA}^{ab}}{V_{A}^{b\star}}+\frac{\Delta
S_{A}^{c\star}Z_{OA}^{ac}}{V_{A}^{c\star}}\\\\[8.0pt] \frac{\Delta
S_{A}^{a\star}Z_{OA}^{ba}}{V_{A}^{a\star}}+\frac{\Delta
S_{A}^{b\star}Z_{OA}^{bb}}{V_{A}^{b\star}}+\frac{\Delta
S_{A}^{c\star}Z_{OA}^{bc}}{V_{A}^{c\star}}\\\\[8.0pt] \frac{\Delta
S_{A}^{a\star}Z_{OA}^{ca}}{V_{A}^{a\star}}+\frac{\Delta
S_{A}^{b\star}Z_{OA}^{cb}}{V_{A}^{b\star}}+\frac{\Delta
S_{A}^{c\star}Z_{OA}^{cc}}{V_{A}^{c\star}}\end{bmatrix}\right),$ (1)

where $a,b$ and $c$ represent the three phases. $V_{A}^{a}$ and $\Delta
S_{A}^{a}$ represent complex voltage and power changes at the phase $a$ of the
actor node $A$, respectively; $Z$ denotes the impedance matrix including self
and mutual line impedance of the shared path between observation node and
actor node from the source node. $\tilde{A}$ is the set of all actor nodes.
Eqn. (1) follows superposition law and voltage change is aggregated due to the
power change of each actor nodes. For a single actor node, the voltage change
at any phase (say phase $a$) of an observation node $O$ can be decomposed into
real ($\Delta V_{OA}^{a,r}$) and imaginary components ($\Delta V_{OA}^{a,i}$)
as:

$\begin{split}\boldsymbol{\Delta V_{OA}^{a,r}}\approx\Bigg{[}-\frac{(\Delta
P_{A}^{a}R_{OA}^{aa}+\Delta
Q_{A}^{a}X_{OA}^{aa})(V_{A}^{a,r})}{(V_{A}^{a,r})^{2}+(V_{A}^{a,i})^{2}}+\\\
\frac{(\Delta P_{A}^{a}X_{OA}^{aa}-\Delta
Q_{A}^{a}R_{OA}^{aa})(V_{A}^{a,i})}{(V_{A}^{a,r})^{2}+(V_{A}^{a,i})^{2}}-\dots\Bigg{]},\\\\[6.0pt]
\end{split}$ (2)

$\begin{split}\boldsymbol{\Delta V_{OA}^{a,i}}\approx\Bigg{[}-\frac{(\Delta
P_{A}^{a}X_{OA}^{aa}-\Delta
Q_{A}^{a}R_{OA}^{aa})(V_{A}^{a,r})}{(V_{A}^{a,r})^{2}+(V_{A}^{a,i})^{2}}-\\\
\frac{(\Delta P_{A}^{a}R_{OA}^{aa}+\Delta
Q_{A}^{a}X_{OA}^{aa})(V_{A}^{a,i})}{(V_{A}^{a,r})^{2}+(V_{A}^{a,i})^{2}}-\dots\Bigg{]},\\\\[6.0pt]
\end{split}$ (3)

where $\Delta P_{A}^{a}$ and $\Delta Q_{A}^{a}$ are the real and reactive
power change at phase-$a$ of the actor node $A$, $R_{OA}^{aa}$ and
$X_{OA}^{aa}$ are resistance and reactance of shared line between observation
node $O$ and actor node $A$. This work leverages this analytical formulation
of VSA to derive voltage change distributions and consequently identify the
DVI nodes in a three-phase unbalanced distribution system.

##  III Proposed framework

As can be seen from the previous section, the conventional method of
identifying DVI nodes is computationally complex, which limits its use in
real-time operation of power system. Therefore, we propose the use of
information-theoretic distance metrics as potential indicators of DVI nodes.
Fundamentally, the proposed approach consists of four steps. In the first
step, we obtain probability distributions of voltage change at an observation
node due to each actor node as well as due to the aggregate presence of all
actor nodes. These distributions are derived in a computationally efficient
way by employing the analytical expressions discussed in the background
section. In the second step, we compute distances between the probability
distribution due to each actor node and the distribution due to the aggregate
effect of all actor nodes. The Third step utilizes the distances to compute
voltage influencing score (VIS) for each pair of observation and actor nodes.
Finally, for each observation node, all the actor nodes are ranked based on
the computed VIS. The actor node whose voltage change distribution is nearest
to the aggregate voltage change distribution is deduced as the major
influencer of voltage change for that particular observation node. Likewise,
all actor nodes are ranked based on the ascending order of their distances.
The complete procedure is summarized in the Algorithm $1$.

###  III-A Probabilistic model of voltage fluctuations

This section provides a probabilistic model of voltage change in a three-phase
unbalanced distribution system, which consist of multiple spatially
distributed actor nodes with PVs and active consumers. Random change in power
at actor nodes due to intermittent renewable generation causes random voltage
fluctuations. Therefore, probability distribution is needed to quantify
voltage change under such stochastic scenarios. Here, we implement the first
step of the proposed approach, i.e., derive the probability distributions of
voltage change at any observation node due to random power change at a single
actor node as well as due to the aggregate effect of all actor nodes. Let
$\Delta S_{A}^{a}$ be change in complex power at phase $a$ of the actor node
A. Then, using eqn. (1), the voltage change at phase-$a$ of an observation
node $O$ can be expressed as:

$\Delta V_{OA}^{a}=\Delta V_{OA}^{a,r}+j\Delta V_{OA}^{a,i},$ (4)
$\begin{split}\Delta V_{OA}^{a,r}=\sum_{h,u}\frac{-1}{|V_{A}^{h}|}[\Delta
P_{A}^{h}(R_{OA}^{u}cos(\omega_{A})-X_{OA}^{u}sin(\omega_{A}))\\\ +\Delta
Q_{A}^{h}(R_{OA}^{u}sin(\omega_{A})+X_{OA}^{u}cos(\omega_{A}))],\\\ \Delta
V_{OA}^{a,i}=\sum_{h,u}\frac{-1}{|V_{A}^{h}|}[\Delta
P_{A}^{h}(R_{OA}^{u}sin(\omega_{A})+X_{OA}^{u}cos(\omega_{A}))+\\\ \Delta
Q_{A}^{h}(X_{OA}^{u}sin(\omega_{A})-R_{OA}^{u}cos(\omega_{A}))]\end{split}$
(5)

where $h$ $\epsilon$ $\tilde{H}$ and $u$ $\epsilon$ $\tilde{U}$. The sets
$\tilde{H}$ and $\tilde{U}$ denote different phases (i.e., $a,b,c$) and
different phase sequences (i.e., $aa,ab,ac$), respectively. $\Delta P_{A}^{h}$
and $\Delta Q_{A}^{h}$ are the active and reactive power changes,
respectively. $R_{OA}^{h},X_{OA}^{h}$ are the resistance and reactance of
shared path between the observation node $O$ and actor node $A$ from the
source node. $V_{A}^{h}$ denotes the base voltage of actor node $A$. For
brevity, the derivation is shown for phase $a$. However, the same steps can be
followed with the corresponding phase quantities to derive expressions for
other phases. The real and imaginary parts of voltage change can further be
simplified as,

$\begin{split}\Delta V_{OA}^{a,r}=\Bigg{[}C_{OA}^{aa,r}\Delta
P_{A}^{a}+D_{OA}^{aa,r}\Delta Q_{A}^{a}+C_{OA}^{ab,r}\Delta P_{b}^{b}+\\\
D_{OA}^{ab,r}\Delta Q_{b}^{b}+C_{OA}^{ac,r}\Delta
P_{c}^{c}+D_{OA}^{ac,r}\Delta Q_{c}^{c}\Bigg{]},\\\ \Delta
V_{OA}^{a,i}=\Bigg{[}C_{OA}^{aa,i}\Delta P_{A}^{a}+D_{OA}^{aa,i}\Delta
Q_{A}^{a}+C_{OA}^{ab,i}\Delta P_{b}^{b}+\\\ D_{OA}^{ab,i}\Delta
Q_{b}^{b}+C_{OA}^{ac,i}\Delta P_{c}^{c}+D_{OA}^{ac,i}\Delta
Q_{c}^{c}\Bigg{]},\end{split}$ (6)

Eqn. (6) is further written in compact form as,

$\begin{split}\Delta V_{OA}^{a,r}=\bm{{C_{OA}^{a,r}}^{\text{T}}\Delta
S_{A}},\\\ \Delta V_{OA}^{a,i}=\bm{{C_{OA}^{a,i}}^{\text{T}}\Delta
S_{A}},\end{split}$ (7)

where,

$\begin{split}\bm{{C_{OA}^{a,r}}^{\text{T}}}=[C_{OA}^{aa,r}\hskip
4.26773ptD_{OA}^{aa,r}\hskip 4.26773ptC_{OA}^{ab,r}\hskip
4.26773ptD_{OA}^{ab,r}\hskip 4.26773ptC_{OA}^{ac,r}\hskip
4.26773ptD_{OA}^{ac,r}]\\\ \bm{{C_{OA}^{a,i}}^{\text{T}}}=[C_{OA}^{aa,i}\hskip
4.26773ptD_{OA}^{aa,i}\hskip 4.26773ptC_{OA}^{ab,i}\hskip
4.26773ptD_{OA}^{ab,i}\hskip 4.26773ptC_{OA}^{ac,i}\hskip
4.26773ptD_{OA}^{ac,i}]\end{split}$

The constants $C$ and $D$ are the functions of the line impedances and base
voltages as explained in eqn. (4). For brevity, the exact expressions are
omitted from here and are provided in the Appendix A. The power change vector
is written as,

$\begin{split}\bm{\Delta S_{A}}=[\Delta P_{A}^{a},\hskip 4.26773pt\Delta
Q_{A}^{a}\hskip 4.26773pt,\Delta P_{A}^{b}\hskip 4.26773pt,\Delta
Q_{A}^{b}\hskip 4.26773pt,\Delta P_{A}^{c}\hskip 4.26773pt,\Delta
Q_{A}^{c}]^{\text{T}}\\\ \end{split}$

Thus, because of random power changes at each actor node, the power change
vector $\bm{\Delta S_{A}}$ is a random vector with mean vector
$\bm{\mu_{\Delta S_{A}}}$ and covariance matrix $\textstyle\sum_{\Delta
S_{A}}$. The covariance matrix $\textstyle\sum_{\Delta S_{A}}$ quantifies the
correlation of power changes among various phases of a particular actor node
$A$. The diagonal elements denote variances of power change at each phase and
off-diagonal elements contain the correlation between the power changes. Thus,
it can be seen from (7) that the voltage change at an observation node $O$ due
to actor node $A$ is the weighted combination of random vector $\bm{\Delta
S_{A}}$, where weights (i.e., $\bm{{C_{OA}^{a,r}}^{\text{T}}}$ and
$\bm{{C_{OA}^{a,i}}^{\text{T}}}$) are constant terms. Invoking the Lindeberg-
Feller central limit theorem, it can be shown that the $\Delta V_{OA}^{a,r}$
and $\Delta V_{OA}^{a,i}$ converges in distribution to a Gaussian random
variables, i.e.,

$\small\begin{split}\Delta
V_{OA}^{a,r}\overset{D}{\rightarrow}\mathcal{N}(\mu_{OA}^{a,r}=\bm{{C_{OA}^{a,r}}^{\text{T}}}\bm{\mu_{\Delta
S_{A}}},{\sigma_{OA}^{a,r}}^{2}=\bm{{C_{OA}^{a,r}}^{\text{T}}}\textstyle\sum_{\Delta
S_{A}}\bm{{C_{OA}^{a,r}}})\\\ \Delta
V_{OA}^{a,i}\overset{D}{\sim}\mathcal{N}(\mu_{OA}^{a,i}=\bm{{C_{OA}^{a,i}}^{\text{T}}}\bm{\mu_{\Delta
S_{A}}},{\sigma_{OA}^{a,i}}^{2}=\bm{{C_{OA}^{a,i}}^{\text{T}}}\textstyle\sum_{\Delta
S_{A}}\bm{{C_{OA}^{a,i}}}),\end{split}$ (8)

where, $\boldsymbol{\mu_{OA}^{a,r}}$ and $\boldsymbol{\mu_{OA}^{a,i}}$ are the
mean vectors of real and imaginary parts of voltage change, respectively.
${\sigma_{OA}^{a,r}}^{2}$ and ${\sigma_{OA}^{a,i}}^{2}$ are the variances of
real and imaginary parts of voltage change, respectively. For investigating
the relationship between the real and imaginary parts of the voltage change, a
new bivariate random vector is defined,

$\begin{split}\begin{bmatrix}\Delta V_{OA}^{a,r}\\\ \Delta
V_{OA}^{a,i}\end{bmatrix}\sim\mathcal{N}\Bigg{[}\begin{bmatrix}\mu_{OA}^{a,r}\\\
\mu_{OA}^{a,i}\end{bmatrix},\begin{bmatrix}{\sigma_{OA}^{a,r}}^{2}\hskip
14.22636ptk_{OA}^{a}\\\ k_{OA}^{a}\hskip
14.22636pt{\sigma_{OA}^{a,i}}^{2}\end{bmatrix}\Bigg{]},\end{split}$ (9)

where, $k_{OA}^{a}=\bm{{C_{OA}^{a,r}}^{\text{T}}}\textstyle\sum_{\Delta
S_{A}}\bm{{C_{OA}^{a,i}}}$ is the covariance between the real and imaginary
parts of the voltage change due to single actor node $A$. Eqn. (9) provides
the probability distribution of voltage change at a particular observation
node due to single actor node $A$. A similar approach can be used to compute
individual voltage change distributions due to each actor node in the network.
Now, we need to obtain the voltage change distribution due to the aggregate
effect of all actor nodes. Using the superposition property in (1), the
voltage change due to cumulative effect of power changes at multiple actor
nodes can be expressed as [25],

$\begin{split}\Delta V_{O}^{a,r}=\sum_{A=1}^{L}\Delta
V_{OA}^{a,r},\end{split}$ (10)

where $L$ is the number of actor nodes. By leveraging (7), the net voltage
change can be written as,

$\begin{split}\Delta V_{OA}^{a,r}&=\sum_{A=1}^{N}\underset{1\times
6}{\bm{{C_{OA}^{a,r}}^{\text{T}}}\underset{6\times 1}{\bm{\Delta S_{A}}}}\\\
&=[\bm{{C_{O1}^{a,r}}^{\text{T}}}\bm{{C_{O2}^{a,r}}^{\text{T}}}\ldots\bm{{C_{ON}^{a,r}}^{\text{T}}}][\bm{\Delta
S_{1}}\bm{\Delta S_{2}}\ldots\bm{\Delta S_{N}}]^{\text{T}}\\\
&=\underset{1\times 6N}{\bm{{C_{O}^{a,r}}^{\text{T}}}}\underset{6N\times
1}{\bm{\Delta S}}\end{split}$ (11)

where $\bm{{C_{O}^{a,r}}^{\text{T}}}$ and $\bm{\Delta S}$ are the long
vectors, composed of a constant term and the power change vector corresponding
to each actor node, respectively. Similarly, the imaginary part of voltage
change can be written as,

$\begin{split}\Delta V_{OA}^{a,i}&=\sum_{A=1}^{N}\underset{1\times
6}{\bm{{C_{OA}^{a,i}}^{\text{T}}}\underset{6\times 1}{\bm{\Delta
S_{A}}}}=\underset{1\times
6N}{\bm{{C_{O}^{a,i}}^{\text{T}}}}\underset{6N\times 1}{\bm{\Delta S}}\\\
\end{split}$ (12)

The equations (11) and (12) possess a similar form as that of (7), i.e., the
net voltage change is the weighted combination of power change vector
$\bm{\Delta S}$. Here, weight $\bm{{C_{O}^{a,r}}}$ is a constant vector
comprising of line impedances and node base voltages, whereas, $\bm{\Delta S}$
comprises of power change at all phases of all actor nodes. Now, invoking the
same Lindeberg-Feller central limit theorem, the real $\Delta V_{O}^{a,r}$ and
imaginary $\Delta V_{O}^{a,i}$ part of aggregate voltage change can be shown
to converge in distribution to a Gaussian random variables with the following
parameters:

$\begin{split}\Delta
V_{O}^{a,r}\overset{D}{\rightarrow}\mathcal{N}(\mu_{O}^{a,r}=\bm{{C_{O}^{a,r}}^{\text{T}}}\bm{\mu_{\Delta
S}},{\sigma_{O}^{a,r}}^{2}=\bm{{C_{O}^{a,r}}^{\text{T}}}\textstyle\sum_{\Delta
S}\bm{{C_{O}^{a,r}}})\\\ \Delta
V_{O}^{a,i}\overset{D}{\rightarrow}\mathcal{N}(\mu_{O}^{a,i}=\bm{{C_{O}^{a,i}}^{\text{T}}}\bm{\mu_{\Delta
S}},{\sigma_{O}^{a,i}}^{2}=\bm{{C_{O}^{a,i}}^{\text{T}}}\textstyle\sum_{\Delta
S}\bm{{C_{O}^{a,i}}})\end{split}$ (13)

Similar to the single actor node case, the correlations between the real and
imaginary parts of net voltage change is captured by defining a new random
vector as:

$\begin{split}\begin{bmatrix}\Delta V_{O}^{a,r}\\\ \Delta
V_{O}^{a,i}\end{bmatrix}\sim\mathcal{N}\Bigg{[}\begin{bmatrix}\mu_{O}^{a,r}\\\
\mu_{O}^{a,i}\end{bmatrix},\begin{bmatrix}{\sigma_{O}^{a,r}}^{2}\hskip
14.22636ptk_{O}^{a}\\\ k_{O}^{a}\hskip
14.22636pt{\sigma_{O}^{a,i}}^{2}\end{bmatrix}\Bigg{]}\end{split}$ (14)

where, $k_{O}^{a}=\bm{{C_{O}^{a,r}}^{\text{T}}}\textstyle\sum_{\Delta
S}\bm{{C_{O}^{a,i}}}$ is the covariance between the real and imaginary parts
of net voltage change. Equations (9) and (14) provide the probability
distribution due to single actor node and the aggregation of multiple actor
nodes, respectively. The next sub-section focuses on computing the statistical
distances between these distributions and presents the procedure to rank the
actor nodes.

###  III-B Information theoretic metrics as DVI indicators

This sub-section implements the second step of our proposed approach, i.e.,
calculate statistical distances between the probability distributions (derived
in earlier sub-section), and rank the actor nodes based on the computed
distances. The information theoretic distance metrics which are potential
indicators of DVI nodes are defined first.

####  III-B1 Kullback-Liebler distance

Kullback-Liebler (KL) distance quantifies how much one probability
distribution differs from another probability distribution. KL divergence
between two multivariate Gaussian distributions ($\mathcal{N}_{0}$ and
$\mathcal{N}_{1}$) of dimension $k$ with means ($\mu_{0}$ and $\mu_{1}$) and
covariance matrices ($\bm{\textstyle\sum_{0}}$ and $\bm{\textstyle\sum_{1}}$)
can be written as:

$\begin{split}D_{KL}(\mathcal{N}_{0}||\mathcal{N}_{1})=\frac{1}{2}\Bigg{[}tr(\bm{\textstyle\sum_{1}^{-1}}\bm{\textstyle\sum_{0}}\\\
+(\mu_{1}-\mu_{0})^{\text{T}}\bm{\textstyle\sum_{1}^{-1}}(\mu_{1}-\mu_{0})-k+\ln\frac{|\bm{\textstyle\sum_{1}}|}{|\bm{\textstyle\sum_{0}}|})\Bigg{]}\end{split}$
(15)

where $tr(.)$ indicates trace of the matrix. Here, the KL distance between the
distributions of voltage change at the observation node due to change in power
at an actor node $A$ ($\Delta V_{OA}$) and due to change in power at all actor
nodes ($\Delta V_{O}$) is given by

$\begin{split}D_{KL}(\Delta V_{OA}||\Delta
V_{O})=\frac{1}{2}\Bigg{[}tr(\bm{\textstyle\sum_{\Delta
V_{O}^{a}}^{-1}}\bm{\textstyle\sum_{\Delta V_{OA}^{a}}}\\\
+(\mu_{O}^{a}-\mu_{OA}^{a})^{\text{T}}\bm{\textstyle\sum_{1}^{-1}}(\mu_{O}^{a}-\mu_{OA}^{a})-2+\ln\frac{|\bm{\textstyle\sum_{\Delta
V_{O}^{a}}}|}{|\bm{\textstyle\sum_{\Delta V_{OA}^{a}}}|})\Bigg{]},\end{split}$
(16)

where $\bm{\textstyle\sum_{\Delta V_{OA}^{a}}}$ and
$\bm{\textstyle\sum_{\Delta V_{O}^{a}}}$ are the covariances of $\Delta
V_{OA}$ and $\Delta V_{O}$, respectively.

####  III-B2 Bhattacharyya distance

Bhattacharyya (BC) distance measures the similarity of two probability
distributions. It is related to the Bhattacharyya coefficient which is a
measure of the amount of overlap between two statistical samples. BC distance
between the distributions of voltage change at the observation node due to
change in power at an actor node $A$ ($\Delta V_{OA}$) and due to change in
power at all actor nodes ($\Delta V_{O}$) can be expressed as:

$\begin{split}D_{BC}(\Delta V_{OA}||\Delta
V_{O})=\frac{1}{8}(\mu_{O}^{a}-\mu_{OA}^{a})^{\text{T}}\bm{\textstyle\sum}(\mu_{O}^{a}-\mu_{OA}^{a})\\\
+\frac{1}{2}\ln(\frac{|\bm{\textstyle\sum}|}{\sqrt{|\bm{\textstyle\sum_{\Delta
V_{O}^{a}}}||\bm{\textstyle\sum_{\Delta V_{OA}^{a}}}|}})\end{split}$ (17)

where $\bm{\textstyle\sum}=\frac{\bm{\textstyle\sum_{\Delta
V_{O}^{a}}}+\bm{\textstyle\sum_{\Delta V_{OA}^{a}}}}{2}$

###  III-C Voltage Influencing score (VIS)

The influence of an actor node on an observation node needs to be quantified
for identifying its rank among all actor nodes. In this regard, we devise a
novel index to quantify voltage influencing capacity, i.e., Voltage
influencing score (VIS). For a given network scenario, i.e., the location of
actor nodes with the variance of power change from historical data, the
distances between voltage change distributions at an observation node due to
each actor node and due to aggregate effect of all actor nodes are used to
rank the actor nodes. Here, these distances are employed to compute the VIS
between any pair of observation and actor node as:

$VIS(O,A)=\frac{\frac{1}{D(A,O)}-\frac{1}{D(1,O)}}{\frac{1}{D(A^{\prime},O)}-\frac{1}{D(1,O)}},$
(18)

where $D(A,O)$ is the statistical distance between the voltage change
distribution at an observation node $O$ due to aggregate effect of all actor
nodes and when actor node $A$ is solely present in the system. The distance
can be computed with any of the metrics described in the earlier subsection.
For an observation node $O$, the lower the distance, the more the actor node
$A$ contributes to aggregate voltage change and consequently the more
influencing the actor node is and vice-versa. Therefore, the VIS is expressed
in terms of inverse of distance. To provide an absolute sense to the score,
VIS is normalized with minimum and maximum values. For a particular
observation node, the ideal location of actor node $A^{\prime}$ for minimum
distance would be the same observation node location. On the other hand, the
maximum distance location would always be the source node as it has minimum
influence on voltage fluctuations of any observation node. These minimum and
maximum distances are used to normalize VIS as shown in eqn. (18).

In the final step of the proposed approach, VIS is leveraged for ranking the
actor nodes. The nodes are ranked in an ascending order based on VIS. The
topmost (Rank 1) and bottom-most (Rank $L$ for the case of $L$ actor nodes)
ranks will be assigned to the actor nodes in the following way:

$\begin{split}\text{Rank}\hskip
4.26773pt1:\operatorname*{arg\,max}_{A}VIS(O,A)\\\ \text{Rank}\hskip
4.26773ptL:\operatorname*{arg\,min}_{A}VIS(O,A)\\\ \end{split}$ (19)

It is worth to note that the proposed approach can work with any other
information theoretic metric (Frechet distance, Jensen-Shannon distance,
etc.), and for illustration purposes, the method is evaluated with two metrics
(KL divergence & Bhattacharyya distance) in the results section.

##  IV Results and Discussion

The efficacy of the proposed method in identifying dominant voltage influencer
nodes is evaluated in this section. The baseline method is the conventional
Monte-Carlo simulation-based approach, and both algorithms are implemented in
the standard IEEE 37-node test system. This test network is chosen as it
denotes a typical unbalanced distribution system and is used by many
researchers for illustrating the efficiency of their methods [24, 25]. The
nominal voltage of the test system is $4.8$ kV. A scenario is generated with
$15$ actor nodes distributed randomly in the IEEE 37-node network. Change in
real and reactive power at $15$ actor nodes is modeled as zero-mean Gaussian
random vector. It is worth noting that the proposed approach is generic for
any choice of actor node and power change distributions, and the simulated
case-study is merely a way to illustrate its performance. Three different PV
sizes are considered in this case study. The mean and variance of real and
reactive power change of all three kinds of PV capacities along with their
location and phase information are as follows:

$\begin{split}\bm{\Delta S_{A}}\sim\mathcal{N}\Bigg{(}\begin{bmatrix}0\\\
0\end{bmatrix},\begin{bmatrix}1.5\hskip 11.38092pt-0.05\\\ -0.05\hskip
11.38092pt0.25\end{bmatrix}\Bigg{)},\\\ A\hskip 2.84544pt\epsilon\hskip
2.84544pt\\{7^{c},8^{a},12^{c},27^{b},28^{c}\\}\\\ \bm{\Delta
S_{A}}\sim\mathcal{N}\Bigg{(}\begin{bmatrix}0\\\
0\end{bmatrix},\begin{bmatrix}3\hskip 11.38092pt-0.1\\\ -0.1\hskip
11.38092pt0.5\end{bmatrix}\Bigg{)},\\\ A\hskip 2.84544pt\epsilon\hskip
2.84544pt\\{7^{b},18^{a},22^{c},34^{b},36^{b}\\}\\\ \bm{\Delta
S_{A}}\sim\mathcal{N}\Bigg{(}\begin{bmatrix}0\\\
0\end{bmatrix},\begin{bmatrix}4.5\hskip 11.38092pt-0.2\\\ -0.2\hskip
11.38092pt0.75\end{bmatrix}\Bigg{)},\\\ A\hskip 2.84544pt\epsilon\hskip
2.84544pt\\{9^{c},14^{c},26^{c},30^{a},31^{a}\\}\end{split}$ (20)

where superscript over actor nodes, i.e., $\\{a,b,c\\}$ represent respective
phases of actor nodes at which power is varying. The change in power across
different actor nodes can be correlated because of environmental factors. The
DERs such as PVs and wind turbines are expected to exhibit a similar
generation profile due to their geographical proximity. Further, the real and
reactive power of inverter-based DERs is negatively correlated. The underlying
covariance structure $\textstyle\sum_{\Delta S}$ can be estimated based on
historical or irradiance-related data. The base loads on the test network are
the same as mentioned in the IEEE PES Distribution system analysis
subcommittee report. For actor nodes, the variance of change in real power and
reactive power are present in the diagonal elements of covariance matrices as
shown in eqn. (20). The off-diagonal elements of the covariance matrices
capture the covariance between real and reactive power. These covariance of
each actor nodes are combined to form the covariance matrix of power change
vector $\textstyle\sum_{\Delta S}$ which corresponds to all actor nodes. Here,
the correlation coefficient between $\Delta P$’s and $\Delta Q$’s for
different actor nodes within the same phase is kept same and covariance
between cross-phase terms is assumed to be zero. However, the proposed
approach is quite general to accommodate other covariance structures as well.
The variance of nodes other than the actor nodes is set to zero.

TABLE I: Observation node 7 Node | MC | KL | BC
---|---|---|---
7 | 1 | 1 | 1
9 | 0.577 | 0.163 | 0.129
12 | 0.543 | 0.150 | 0.117
22 | 0.530 | 0.127 | 0.096
14 | 0.516 | 0.137 | 0.106
26 | 0.441 | 0.092 | 0.067
28 | 0.350 | 0.084 | 0.060
8 | 0.042 | 0.065 | 0.045
17 | 0.026 | 0.050 | 0.033
27 | 0.001 | 0.043 | 0.029

TABLE II: Observation node 16 Node | MC | KL | BC
---|---|---|---
14 | 1 | 1 | 1
22 | 0.919 | 0.840 | 0.808
18 | 0.782 | 0.170 | 0.133
12 | 0.613 | 0.576 | 0.533
9 | 0.426 | 0.422 | 0.373
7 | 0.209 | 0.290 | 0.242
17 | 0.146 | 0.230 | 0.186
8 | 0.020 | 0.111 | 0.083
26 | 0.001 | 0.176 | 0.138
28 | 0.001 | 0.164 | 0.128

###  IV-A VIS for ranking DVI nodes

To assess the performance of the proposed approach in identifying DVI nodes,
VIS for two arbitrary observation nodes (i.e., $7$, $16$) are computed using
two different distance metrics namely KL divergence (eqn. (15)) and
Bhattacharyya distance (eqn. (16)). The results obtained by using the proposed
approach is validated against ground truth values. The baseline approach to
compute true DVI nodes is based on Monte-Carlo simulations (MCS) of load flow
as explained in Section II-A. Initially, the variance of voltage change
magnitude at a given observation node is computed due to the presence of all
actor nodes. Multiple power change scenarios are simulated by running $100000$
MCS with load flow method. Then, the reduction in the variance of the
magnitude of voltage change is determined by setting the variance to zero for
each actor node sequentially. Again, for each actor node case, different MCS
are executed. Finally, the actor nodes are ranked based on the reduction in
variance they bought when the variance of power change at the corresponding
actor node is set to zero.

Tables II and II tabulates the VIS for observation nodes $7$ and $16$,
respectively. MC, KL and BC refers to baseline MCS approach, proposed KL
divergence and Bhattacharyya distance based methods, respectively. Node in the
tables refers to actor node. It can be inferred that the VIS for the most
dominant actor node is $1$ and it decreases as we move to lower rank nodes.
For observation node-7 with KL distance metric, there is a small difference in
VIS between nodes $22$ and $14$ implying that they exhibit almost equal
voltage influencing capacity. However, there is a considerable difference when
it comes to nodes $14$ and $26$. Thus, VIS allows us to quantitatively
differentiate between dominant influencer nodes.

###  IV-B Ranking of DVI nodes with VIS

DVI nodes are identified by ranking nodes based on VIS. Table III tabulate the
top $10$ dominant voltage influencer nodes for observation nodes $7$ and $16$,
respectively.

TABLE III: DVI nodes for observation nodes 7 and 16 |  Rank $\rightarrow$
---
Metric $\downarrow$
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
Observation node -7
MC | 7 | 9 | 12 | 22 | 14 | 26 | 28 | 8 | 17 | 27
KL | 7 | 9 | 12 | 14 | 22 | 26 | 28 | 8 | 17 | 18
BC | 7 | 9 | 12 | 14 | 22 | 26 | 28 | 8 | 17 | 18
Observation node -16
MC | 14 | 22 | 18 | 12 | 9 | 7 | 17 | 8 | 26 | 28
KL | 14 | 22 | 12 | 9 | 7 | 17 | 26 | 18 | 28 | 8
BC | 14 | 22 | 12 | 9 | 7 | 17 | 26 | 18 | 28 | 8

According to Table III, the MC method indicate that the actor nodes $7$ and
$14$ are Rank-1 nodes for observation node $7$ and $16$, respectively. The
assignment of Rank-1 nodes is carried out correctly by both the metrics of the
proposed approach. The reasons for rank-1 allocation are (1) the high variance
of power change and (2) proximity of actor nodes to the given observation node
in the IEEE 37-node test system.

Apart from rank-1, utilities might be interested in identifying other dominant
nodes which are next to rank-1. In this regard, we use Top-N accuracy, which
can be defined as,

$\small\frac{|\\{\text{Predicted Top-N nodes}\\}\cap\\{\text{True Top-N
nodes\\}}|}{N},$ (21)

where $N$ is the desired number based on the applications. The Top-5 and
Top-10 accuracy results are presented for observation nodes $7$ and $16$ in
Table IV. The last column of the table denotes the mean accuracy when all the
nodes of the IEEE-37 node network act as observation nodes. It can be observed
that distance metrics KL and BC have fairly good identification accuracy, and
the mean accuracy is more than $90\%$. Apart from Rank-1 node, our approach
also provides the correct sequence of actor nodes when compared to baseline MC
approach. However, in some positions, the order of actor nodes is flipped or
offset by one or two units. This is because, the corresponding actor nodes
have an equal influencing capacity for that particular observation node. For
instance, in the case of observation node $7$, the position of actor nodes
$22$ and $14$ are flipped. However, this is not a major concern here, as we
are more interested in correctly identifying nodes lying in a particular band
(i.e., Top-5, Top-10) rather than their exact order. The power at the Top-5 or
Top-10 actor nodes can be efficiently regulated to mitigate voltage
violations.

TABLE IV: Identification accuracy of DVI nodes |  Node $\rightarrow$
---
Accuracy $\downarrow$
| obs. node-7
---
(%)
| obs. node-16
---
(%)
| Mean
---
all nodes (%)
Top-5 | 100 | 80 | 91
Top-10 | 90 | 100 | 93

Furthermore, to get the overall influencing capacity of actor nodes, one can
compute the mean VIS. For each actor node, the mean of VIS is taken across all
observation nodes of the network. It can be observed From Table V that actor
nodes $17$ and $30$ have maximum and minimum influencing capacity among all
the actor nodes present in the network. This assignment is because of their
topological position in the distribution network and the magnitude of power
variance due to associated DER units or load variation.

TABLE V: Average VIS of actor nodes Nodes | VIS | Nodes | VIS | Nodes | VIS
---|---|---|---|---|---
7 | 0.220 | 17 | 0.444 | 28 | 0.156
8 | 0.220 | 18 | 0.304 | 30 | 0.154
9 | 0.212 | 22 | 0.349 | 31 | 0.189
12 | 0.258 | 26 | 0.163 | 34 | 0.378
14 | 0.300 | 27 | 0.166 | 36 | 0.427
TABLE VI: Running time of various approaches | Time
---
Metric
| Running time
---
Top-5 (s)
MC | 12039
KL | 0.92
BC | 0.96

###  IV-C Computational complexity of proposed method

Apart from correctly identifying the dominant voltage influencing nodes, the
proposed approach offers a considerable computational advantage over the
conventional approaches. Table VI reports the execution time of various
approaches to identify Top-5 actor nodes for the observation node $7$. It can
be seen that the proposed approach involving any of the two distance metrics
is multiple order faster than that of the conventional approach, which takes
around $3.2$ hrs. All the experiments are conducted on a machine with Intel i7
processor running at 2.2 GHz. This results demonstrates the computational
advantage of the proposed approach over conventional methods.

The proposed approach of identifying DVI nodes can enhance voltage control
strategies in multiple ways. One of the approaches is based on system
partitioning, where a set of effective actor nodes is used for control
actions. Our approach helps to identify such set of actor nodes in a
computationally efficient manner. Given a particular scenario of power change
in the system, one can determine the set of common observation nodes for which
a particular actor node induces the maximum impact on voltage profiles.
Therefore, the set of common observation nodes can form a cluster and it is
cost effective to control power variations on the most dominant actor node to
quickly restore voltages to their safe operational limits within that cluster.
Along with clustering, our method of ranking nodes can also be used for
optimal allocations of DERs or fault current limiters which involves
sensitivity analysis. These control and asset allocation strategies will be
investigated as part of our future work.

##  V Conclusion and Future work

The conventional methods to identify dominant voltage influencer nodes (DVI)
are Monte-Carlo simulation-based that are computationally expensive.
Therefore, this work focuses on an analytical approach and proposes
statistical distance metrics to devise VIS that quantifies voltage influencing
capacity of actor nodes. The VIS is then leveraged to identify the DVI nodes.
The proposed framework computes VIS not solely on the basis of correlation
between change in power at the actor node and change in voltage at the
observation node, but it also relies on the magnitude of voltage variation
that is being caused due to power change at actor nodes. Effectiveness and
computational efficiency of the proposed method are illustrated by comparing
the results with conventional method of identifying DVI nodes using Monte-
Carlo simulations in IEEE 37-node test system. From the results, it can be
inferred that the proposed statistical distance metrics effectively predict
the DVI nodes while substantially reducing the execution time. This scalable
approach of identifying DVI nodes can find applications in a variety of areas
such as in utilizing DVI nodes for voltage control at strategic locations,
identifying most influencer DERs that are responsible for voltage
fluctuations, among others. Employing DVI nodes for network partitioning and
efficient control applications will be pursued as part of our future work.

## Acknowledgment

This material is based upon work partly supported by the Department of Energy,
Office of Energy Efficiency and Renewable Energy (EERE), Solar Energy
Technologies Office, under Award # DE-EE0008767 and National science
foundation under award # 1855216.

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