# Conservation Voltage Reduction (CVR) via Two-Timescale Control in Unbalanced
Power Distribution Systems

Rahul Ranjan Jha, Anamika Dubey, and Kevin P. Schneider R. Jha, and A. Dubey,
are with the School of Electrical Engineering and Computer Science, Washington
State University, Pullman, WA, 99164 e-mail:
rahul.jha<EMAIL_ADDRESS>P. Schneider is with the Pacific
Northwest National Laboratory, Battelle Seattle Research Center, Seattle, WA
98109 USA e-mail<EMAIL_ADDRESS>
###### Abstract

Voltage control devices are employed in power distribution systems to reduce
the power consumption by operating the system closer to the lower acceptable
voltage limits; this technique is called conservation voltage reduction (CVR).
The different modes of operation for system’s legacy devices (with binary
control) and new devices (e.g. smart inverters with continuous control)
coupled with variable photovoltaic (PV) generation results in voltage
fluctuations which makes it challenging to achieve CVR objective. This paper
presents a two-timescale control of feeder’s voltage control devices to
achieve CVR that includes (1) a centralized controller operating in a slower
time-scale to coordinate voltage control devices across the feeder and (2)
local controllers operating in a faster timescale to mitigate voltage
fluctuations due to PV variability. The centralized controller utilizes a
three-phase optimal power flow model to obtain the decision variables for both
legacy devices and smart inverters. The local controllers operate smart
inverters to minimize voltage fluctuations and restore nodal voltages to their
reference values by adjusting the reactive power support. The proposed
approach is validated using the IEEE-123 bus (medium-size) and R3-12.47-2
(large-size) feeders. It is demonstrated that the proposed approach is
effective in achieving the CVR objective for unbalanced distribution systems.

###### Index Terms:

Three-phase optimal power flow, local control, coordinated control, PV
variability, conservation voltage reduction

##  I Introduction

Distribution Driven by the requirement for improved energy efficiency,
conservation voltage reduction (CVR) has emerged as a popular approach to
reduce the power consumption of voltage dependent loads by operating the
feeder at the lower limits of the acceptable voltages. It is shown in [1] that
there is a 3.04% of annual energy reduction by implementing CVR on all the
distribution feeders throughout the United States. Traditionally, voltages in
electric power distribution systems are maintained by controlling the legacy
voltage control devices such as capacitor banks, load tap changers and voltage
regulators. With IEEE 1547-2018 standard [2], the distributed generators (DGs)
equipped with smart inverters can help maintain distribution feeder voltages
by providing reactive power support. To meet CVR objective, a centralized
voltage control system can be envisioned that aims to minimize the feeder
power consumption by coordinating all voltage control devices: capacitor
switches, regulator taps, and DG smart inverters. Note that solar photovoltaic
(PV) systems (both roof-top and utility scale) are the most prominent DG
technology deployed in distribution feeders [3]. Unfortunately, due to
inherent variability of PV generation, the node voltages vary making it
difficult for the centralized controller to maintain the desired nodal
voltages to meet CVR objective. To minimize the impacts of local PV
variability on optimum decisions from centralized controller, there is a
critical need for the local control of PV inverters that is coordinated with
the centralized voltage control system.

The related literature designs a centralized controller to achieve the CVR
objective by solving a three-phase optimal power flow (OPF) problem to obtain
decision variables for both legacy devices and smart inverters. The resulting
OPF is a mixed integer nonlinear problem (MINLP) which is to be solved for
three-phase unbalanced system and includes both discrete and continuous
decision variables [4, 5, 6]. These methods, however, do not scale for a
large-scale three-phase unbalanced power distribution systems. Moreover, the
existing centralized control methods for CVR assume that the local generation
do not vary during the discrete time intervals on which OPF decisions are
implemented (typically 5-min or 15-min) [7]. Unfortunately, this assumption is
flawed as local PV generation is significantly variable within a 5 or 15-min
time-interval. The variation in local generation leads to nodal voltage
fluctuations rendering the centralized decisions sub-optimal. The voltage
fluctuations resulting from variable PV generation are also known to increase
the number of operations for the legacy devices that may reduce their
operating life. A local control of PV smart inverters can help mitigate the
nodal voltage fluctuations caused by rapid changes in PV generation. However,
the local control alone cannot help meet the CVR objectives that require
coordination of all voltage control devices. This calls for the coordination
between the centralized and local controllers. In this paper, we present a
two-timescale control which coordinates the centralized and local controllers
to achieve CVR objective in the presence of variable generation resources.

The existing local voltage control methods are widely based on sensitivity
matrix and volt-var droop characteristics of smart inverters [8, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18]. For the volt-var droop based method, the set
points for the droop curve are predetermined to simply mitigate any over-
voltage or under-voltage concerns [8, 10, 12]. These methods may cause
voltages to oscillate, may lead to a steady state error, and are unable to
optimize for system-wide objectives. The optimality and the stability of the
droop based voltage control methods is discussed in [18]. In order to remove
the oscillations and improve stability, several authors [11, 12, 13] have
proposed methods to dynamically change the droop points based on the local
measurements. Dynamically obtaining droop set-points at shorter time-intervals
(1-minute or less) remain a challenging problem. The sensitivity based
approach which utilizes $\frac{R}{X}$ ratio of the distribution system is used
to obtain the required reactive power to reduce voltage fluctuations [10].
However, here authors modelled the distribution system as an equivalent
balanced single-phase system. In another work, the coefficients of change in
active and reactive power are determined by repeatedly solving the power flow
equations [15]. The obtained coefficients, however, are not accurate for all
operating conditions. In [19], a decentralized approach for the voltage
control of DGs is proposed by generating voltage reference for each inverters
using PI control. In [20], authors propose a method based on feedback
linearization where system voltage is maintained by providing the required
reactive power to the PVs. Unfortunately, none of the above referred work,
provides a mechanism for coordination among the centralized and local
controllers; that is, the local controllers work independent of the
centralized controller.

Recently, a few researchers have proposed methods to coordinate the control of
the centralized and local controllers [21, 22, 23]. In these articles, the Q-V
points for the volt-var droop curve are periodically updated by solving a
centralized control problem. The local controls are used in sub-intervals to
mitigate any sudden voltage violations. Although, these methods coordinate the
centralized and local control, they do not solve the problem for a three-phase
unbalanced system and ignore the effect of control provided by other inverters
in the system. In [24], authors solve a three-phase OPF to obtain the volt-var
curve for the smart inverter which can perform autonomous voltage control;
however, the obtained curve is based on a few specific planned scenario and do
not generalize for all possible operating conditions. Also, the volt-var curve
required to maintain the pre-specified node voltages is generated without
taking the effects of the var support from other inverters. The combined
effect of CVR and power quality is solved in [25] using the legacy devices and
smart inverters. In this work, capacitor banks and voltage regulators are
controlled heuristically to flatten the voltages to lower value while smart
inverters work autonomously; thus, legacy devices and smart inverters are not
really coordinated to meet the system-wide objectives. Thus, in the existing
literature, the centralized and local control are not properly coordinated to
meet CVR objective when local generation can fluctuate.

Recently, several distributed algorithms have been proposed for the voltage
and reactive power optimization in a power distribution system [26, 27, 28,
29]. For example, in [26, 27], a distributed voltage control approach based on
dual decomposition is proposed for a balanced single-phase system. Here, the
peer-to-peer communication is required to update the Lagrangian multipliers.
The peer-to-peer communication increases the requirements for the
communication infrastructure. In [28], the distribution system is separated
into different areas to reduce the communication requirements compared to
peer-to-peer communication. Here, the alternating direction method of
multipliers (ADMM) is used to solve the distributed optimization problem for
both an equivalent single-phase and three-phase unbalanced distribution
system. The distributed algorithm still takes 100s of iterations and thus
requires 100s rounds of communication among distributed agents to converge for
a single step of the optimization problem. The implementation of the existing
distributed algorithms for the real-time voltage control will require a high-
bandwidth and low-latency communication infrastructure. Thus, there is a
critical need for low-compute and low-communication algorithms to optimize
grid’s voltage control devices while incorporating the effects of local
generation variability. This paper presents a two-timescale coordinated
centralized and local control approach for CVR. The CVR objective is achieved
by optimally controlling feeder’s legacy devices and smart inverters. First, a
centralized controller is proposed that solves an OPF and obtains control set-
points for feeder’s voltage control devices for a three-phase unbalanced
radial distribution system at discrete time intervals (every 5 to 15-min).
Next, an adaptive local controller is proposed that cancels the effect of
local generation variability on nodal voltage fluctuations and thus help
maintain the same nodal voltages as obtained from the centralized controller
based on the forecasted values of local generation. The overall contributions
of this work are threefold: (1) we propose low-compute algorithms for the
coordinated control of system’s legacy devices and smart inverters for CVR;
(2) we propose fast local control methods for smart inverters to mitigate the
effects of local generation/load variability on nodal voltage fluctuations;
(3) we propose low-compute and low-communication algorithms to coordinate the
central and local controllers to achieve the CVR objectives for a system with
significant levels of local generation variability. Note that the central
controller needs to communicate with the local controllers at every 15-min
time-interval, while the local controllers are operating at much faster time-
intervals (real-time to every 1-min depending upon the granularity of the
local measurements).

The rest of the paper is organized as follows. In Section II, linear three-
phase unbalanced power flow is presented. The mathematical models for the
distribution system components are discussed in Section III. Section IV
presents the proposed approach for the coordinated voltage control to achive
the CVR objectives. Section V provides the mathematical formulation for the
proposed local controllers. The results and discussions are detailed in
Section VI, followed by conclusion in Section VII.

##  II Three Phase Unbalanced Power Flow

In this section, we introduce the three-phase branch flow equations for an
unbalanced power distribution system. Next, we detail a linearized approximate
three-phase linear power flow model that is known to accurately approximate
the nodal voltages. Note that achieving CVR objective requires a power flow
model that accurately represents nodal voltages. For completeness, in the
results section, we thoroughly validate the accuracy of the detailed three-
phase linearized power flow model.

###  II-A Branch Flow Equations

The three-phase power flow equations as detailed in [30] are described in
(1)-(4). Let there be directed graph as shown in Fig.1,
$\mathcal{G}=(\mathcal{N},\mathcal{E})$ where $\mathcal{N}$ denotes set of
buses and $\mathcal{E}$ denotes set of lines. Each line connects ordered pair
of buses $(i,j)$ between two adjacent nodes $i$ and $j$. Let $\\{a,b,c\\}$
denotes the three phases of the system and $\Phi_{i}$ denotes set of phases on
bus $i$. For each bus $i\in\mathcal{N}$, let phase $p$ complex voltage is
given by $V_{i}^{p}$ and phase $p$ complex power demand is $s_{L,i}^{p}$. Let
$V_{i}:=[V_{i}^{p}]_{p\in\Phi_{i}}$ and
$s_{L,i}:=[s_{L,i}^{p}]_{p\in\Phi_{i}}$. For each line, let $p$ phase current
be $I_{ij}^{p}$ and define, $I_{ij}:=[I_{ij}^{p}]_{p\in\Phi_{i}}$. Let
$z_{ij}$ be the phase impedance matrix.

$\displaystyle
v_{j}=v_{i}-(S_{ij}z_{ij}^{H}+z_{ij}S_{ij}^{H})+z_{ij}l_{ij}z_{ij}^{H}$ (1)
$\displaystyle\text{diag}(S_{ij}-z_{ij}l_{ij})=\sum_{k:j\rightarrow
k}{\text{diag}(S_{jk})}+s_{L,j}$ (2)
$\displaystyle\left[\begin{array}[]{cc}v_{i}&S_{ij}\\\ S_{ij}^{H}&l_{ij}\\\
\end{array}\right]=\left[\begin{array}[]{c}V_{i}\\\ I_{ij}\\\
\end{array}\right]\left[\begin{array}[]{cc}V_{i}\\\ I_{ij}\\\
\end{array}\right]^{H}$ (9)
$\displaystyle\left[\begin{array}[]{cc}v_{i}&S_{ij}\\\ S_{ij}^{H}&l_{ij}\\\
\end{array}\right]:-\text{Rank-1 PSD Matrix}$ (12)

Here, (1) represents voltage drop equation, (2) corresponds to power balance
equation, (3) are variable definitions for power flow quantities, and (4) is a
Rank-1 constraint that makes the associated optimization problem non-convex.

Figure 1: Distribution system branch flow model.

Although in the literature, researchers have proposed methods to to obtain a
convex power flow formulation by either relaxing or approximating the power
flow equations [31, 32, 30], there remain additional non-linearities due to
load models and the nature of control decisions that require attention. For
example, the commonly used ZIP load model that combines constant impedance
($Z$), constant current ($I$) and constant power ($P$) load models leads to
additional non-linearities in the power flow model. Similarly, the decision
variables for capacitor banks’ switching and voltage regulators’ taps, are
discrete leading to a binary/mixed-integer problem. The resulting CVR
optimization problem is thus a large-scale mixed integer nonlinear program
(MINLP) that is computationally intensive to solve and hence not suitable for
real-time control. These challenges led to the development of linearized
three-phase power flow models that can realistically solve large-scale
nonconvex OPF problems for unbalanced distribution systems with integer
decision variables.

###  II-B Linear Three-Phase Power Flow Model

This section describes a three-phase linear power flow model that is an
extension of single-phase distflow equations [33]. The three-phase unbalanced
power flow model involves nonlinear relationship between power, voltage and
current in power balance and voltage equations as observed in equations (1)
and (2). The additional nonlinearity results from the mutual coupling among
the three phases. In linearized three-phase model, it is assumed that the
power loss in a branch is relatively smaller as compared to the power flow in
the branch. Due to this assumption, the nonlinearity due to power loss and
voltage drop are ignored. Further, the nonlinearity due to mutual coupling is
removed by approximating the voltage phase angle difference among the phases.
It is assumed that the phase voltages are $\frac{2\pi}{3}$ apart [30].
Mathematically, the approximation is represented as:

$\frac{V_{i}^{a}}{V_{i}^{b}}\simeq\frac{V_{i}^{b}}{V_{i}^{c}}\simeq\frac{V_{i}^{c}}{V_{i}^{a}}=e^{j2\pi/3}=a^{2}$
(13)

Note that the apparent power can be represented using (6). We use the
approximation (5) to reduce the number of variables in the optimization
problem. Due to (5), the off diagonal terms in the power balance equation can
be represented as a function of diagonal terms as shown in (7) [30].

$\displaystyle S^{aa}_{ij}=V^{a}_{i}*I^{a}_{ij}\hskip
5.69046pt\text{and}\hskip 5.69046ptS^{ab}_{ij}=V^{a}_{i}*I^{b}_{ij}$ (14)
$\displaystyle
S^{ab}_{ij}=V^{a}_{i}*I^{b}_{ij}=\frac{V^{a}_{i}}{V^{b}_{i}}*V^{b}_{i}*I^{b}_{ij}=a^{2}*S^{bb}$
(15)

For the linear AC power flow, it is assumed that the branch power loss are
relatively smaller as compared to the respective branch power flow. Hence, by
ignoring the power losses and using (1), (2) and (7), the linear AC branch
flow equations are obtained (8)-(10), where, $v_{i}^{p}=(V_{i}^{p})^{2}$ and
$v_{j}^{p}=(V_{j}^{p})^{2}$.

$\displaystyle P^{pp}_{ij}$ $\displaystyle=$
$\displaystyle\sum_{k:j\rightarrow k}P^{pp}_{jk}+p^{p}_{Lj}\hskip
19.91684ptp\in{a,b,c}$ (16) $\displaystyle Q^{pp}_{ij}$ $\displaystyle=$
$\displaystyle\sum_{k:j\rightarrow k}Q^{pp}_{jk}+q^{p}_{Lj}\hskip
19.91684ptp\in{a,b,c}$ (17) $\displaystyle v_{i}^{p}-v_{j}^{p}$
$\displaystyle=$
$\displaystyle\sum_{q\in\phi_{j}}2\Re[S_{ij}^{pq}(z_{ij}^{pq})^{*}]\hskip
11.38092ptp,q\in{a,b,c}$ (18)

Since the linear power flow equations do not include the components of power
loss in active and reactive power flow equations, i.e (8) and (9), the line
flows are not well approximated. Note that the voltage equation in (10) takes
the effect of active and reactive power flows along the line on the feeder
voltage drop; it however, does not include the effect of power losses along
the lines on voltage drop. Since, the line losses are significantly less than
the line flow due to load demand, the obtained voltage drops are well
approximated. We further verify in Section 6 that the node voltages obtained
using the linearized power flow model are well approximated for an unbalanced
power distribution system. Since the CVR objective depends on nodal voltages,
the AC linearized power flow equations are accurate for optimizing the CVR
objective.

##  III Distribution System Equipment Models

This section details the mathematical models for voltage dependent components
of the distribution systems. As shown in Section 2, the approximate power flow
equations are a function of the square of nodal voltage magnitude. Therefore,
the distribution system components such as voltage regulators, capacitor
banks, and customers loads are modeled as a function of the square of voltage
magnitude. This ensures that the constraints for CVR optimization problem
remain mixed-integer linear in the problem decision variables. These models
were introduced in our prior work as well [7].

###  III-A Voltage Regulators

The voltage regulation range of a voltage regulator is assumed to be $\pm
10\%$, which is divided into 32-steps. The series and shunt impedance of the
voltage regulator have very small value [34] and hence these are ignored. Let,
for a line $(i,j)$ the voltage regulator is connected to phase $p$ with the
turn ratio of $a^{p}$. The $a^{p}$ has step change of 0.00625 pu for the
values between 0.9 to 1.1. An additional node $i^{\prime}$ is introduced to
model the current equations. Let for $u_{tap,i}^{p}\in\\{0,1\\}$ be a binary
variable defined for each regulator step position i.e. $i\in(1,2,...,32)$.
Also, define a vector $b_{i}\in\\{0.9,0.90625,...,1.1\\}$. Then $V_{i}^{p}$,
$V_{j}^{p}$ where $p\in\Phi_{i}\cap\Phi_{j}$ are given as follows:

$\displaystyle V_{j}^{p}=V_{i^{\prime}}^{p}=a^{p}V_{i}^{p}$ (19)

where, $a^{p}=\sum\limits_{i=1}^{32}b_{i}u_{tap,i}^{p}$ and
$\sum\limits_{i=1}^{32}u_{tap,i}^{p}=1$.

In order to express (11) as a function of $v_{i}^{p}=(V_{i}^{p})^{2}$ and
$v_{j}^{p}=(V_{j}^{p})^{2}$, we take square of (11) and define
$a_{p}^{2}=A_{p}$ and $b_{i}^{2}=B_{i}$. Further realizing that
$(u_{tap,i}^{p})^{2}=u_{tap,i}^{p}$, (11) can be reformulated as (12).

$\displaystyle v_{j}^{p}=A^{p}\times v_{i}^{p}$ (20)

###  III-B Capacitor Banks

The per-phase model for capacitor banks is developed. The reactive power
generated by a capacitor bank, $q_{cap,i}^{p}$, is defined as a function of
binary control variable $u_{cap,i}^{p}\in\\{0,1\\}$ indicating the status
(On/Off) of the capacitor bank, its rated per-phase reactive power
$q_{cap,i}^{rated,p}$, and the square of the bus voltage at bus $i$ for phase
$p$, $v_{i}^{p}$.

$q_{cap,i}^{p}=u_{cap,i}^{p}q_{cap,i}^{rated,p}v_{i}^{p}$ (21)

The capacitor bank model is assumed to be voltage dependent and provides
reactive power as a function of $v_{i}^{p}$ when connected, i.e.
$u_{cap,i}=1$. For a three-phase capacitor bank, a common control variable,
$u_{cap,i}^{p}$, is defined for each phase.

###  III-C Distributed Generation with Smart Inverters

The DGs are modeled as negative loads with a known active power generation
equal to the forecasted value. The reactive power support from the DGs depends
on the rating of the smart inverter. In this work, the smart inverter is rated
at $15\%$ higher than the maximum active power rating of the DGs. Let, the
apparent power rating of the DG connected to phase $p$ of node $i$ be
$s_{DG,i}^{rated,p}$ and the forecasted active power generation be
$p_{DG,i}^{p}$. The available reactive power $q_{DG,i}^{p}$ support from the
smart inverters are given by (14). Since $s_{DG,i}^{rated,p}$ and
$p_{DG,i}^{p}$ is known, (14) is simply a box constraint.

$\small-\sqrt{(s_{DG,i}^{rated,p})^{2}-(p_{DG,i}^{p})^{2}}\leq
q_{DG,i}^{p}\leq\sqrt{(s_{DG,i}^{rated,p})^{2}-(p_{DG,i}^{p})^{2}}$ (22)

###  III-D Voltage-Dependent Load Models

The most widely acceptable load model is the ZIP model which is a combination
of constant impedance (Z), constant current (I) and constant power (P))
characteristics of the load [34]. The mathematical representation of the ZIP
model for the load connected at phase $p$ of bus $i$ is given by (15)-(16).

$\displaystyle
p_{L,i}^{p}=p_{i,0}^{p}\left[k_{p,1}\left(\dfrac{V_{i}^{p}}{V_{0}}\right)^{2}+k_{p,2}\left(\dfrac{V_{i}^{p}}{V_{0}}\right)+k_{p,3}\right]$
(23) $\displaystyle
q_{L,i}^{p}=q_{i,0}^{p}\left[k_{q,1}\left(\dfrac{V_{i}^{p}}{V_{0}}\right)^{2}+k_{q,2}\left(\dfrac{V_{i}^{p}}{V_{0}}\right)+k_{q,3}\right]$
(24)

where, $k_{p,1}+k_{p,2}+k_{p,3}=1$, $k_{q,1}+k_{q,2}+k_{q,3}=1$, $p_{i,0}^{p}$
and $q_{i,0}^{p}$ are per-phase load consumption at nominal voltage $V_{0}$.

The ZIP load model represented in (15)-(16) are a function of both $V_{i}^{p}$
and $v_{i}^{p}=(V_{i}^{p})^{2}$. Including (15) and (16) to OPF formulation
will make (8) and (9), earlier linear in $v_{i}^{p}$, nonlinear. Here, we
develop an equivalent load model for voltage-dependent loads using the
definition of CVR factor. Next, an equivalence between ZIP parameters and
proposed CVR factors is obtained.

CVR factor is defined as the ratio of percentage reduction in active or
reactive power to the percentage reduction in bus voltage. Let CVR factor for
active and reactive power reduction be $CVR_{p}$, and $CVR_{q}$, respectively
defined in (17).

$\displaystyle
CVR_{p}=\dfrac{dp_{L,i}^{p}}{p_{i,0}^{p}}\dfrac{V_{0}}{dV_{i}^{p}}\hskip
5.69046pt\text{and}\hskip
5.69046ptCVR_{q}=\dfrac{dq_{L,i}^{p}}{q_{i,0}^{p}}\dfrac{V_{0}}{dV_{i}^{p}}$
(25)

where, $p_{L,i}^{p}=p_{i,0}^{p}+dp_{i}^{p}$ and
$q_{L,i}^{p}=q_{i,0}^{p}+dq_{i}^{p}$. Furthermore,
$v_{i}^{p}=(V_{i}^{p})^{2}$. Therefore, $dv_{i}^{p}=2V_{i}^{p}dV_{i}^{p}$.
Assuming $V_{i}^{p}\approx V_{0}$ and $dv_{i}^{p}=v_{i}^{p}-(V_{0})^{2}$, we
obtain:

$\displaystyle
p_{L,i}^{p}=p_{i,0}^{p}+CVR_{p}\dfrac{p_{i,0}^{p}}{2}\left(\dfrac{v_{i}^{p}}{V_{0}^{2}}-1\right)$
(26) $\displaystyle
q_{L,i}^{p}=q_{i,0}^{p}+CVR_{q}\dfrac{q_{i,0}^{p}}{2}\left(\dfrac{v_{i}^{p}}{V_{0}^{2}}-1\right)$
(27)

Note that the CVR based load model detailed in (18) and (19) is linear in
$v_{i}^{p}$, thus can be easily included in approximate power flow equations
(8) and (9). The CVR factors, $CVR_{p}$ and $CVR_{q}$ are estimated from the
ZIP coefficients of the load. On differentiating the ZIP model detailed in
(15) and (16) and assuming $V_{0}=1$ p.u., we obtain:

$\dfrac{dp_{L,i}^{p}}{dV_{i}^{p}}=p_{i,0}^{p}\left(2k_{p,1}V_{i}^{p}+k_{p,2}\right)\text{,}\dfrac{dq_{L,i}^{p}}{dV_{i}^{p}}=q_{i,0}^{p}\left(2k_{q,1}V_{i}^{p}+k_{q,2}\right)$
(28)

Using (17) and (20) and assuming $V_{i}^{p}\approx V_{0}$, we obtain (21).
Using (21), the CVR factors for customer loads can be obtained from the ZIP
coefficients.

$CVR_{p}=2k_{p,1}+k_{p,2}\hskip 2.84544pt\text{and}\hskip
2.84544ptCVR_{q}=2k_{q,1}+k_{q,2}$ (29)

The proposed load model should accurately represent the ZIP load model. It is
thoroughly validated in authors’ prior work detailed in [7] that the
characteristics of the CVR based load model (detailed in this section) closely
matches the ZIP model for the allowable ANSI voltage ranges i.e. $0.95-1.05$
pu . This proves that the proposed load model in (18)-(19), which is linear in
$v_{i}^{p}$, can be used instead of the ZIP load model.

Figure 2: Integrating Centralized and Local Control.

##  IV Two-Timescale Coordinated Centralized and Local Voltage Control
Framework

The proposed two-timescale approach to coordinate the centralized and local
controls is shown in Fig. 2. The objective is to achieve CVR objective by
optimally controlling voltage control devices to reduce nodal voltages and
consequently help reduce feeder power consumption due to voltage-dependent
loads. Note that the nodal voltages should be maintained within the
recommended ANSI limits (American National Standard for Electric Power Systems
and Equipment). The problem is formulated as a mixed integer linear program
(MILP) where integer decision variables are introduced due to voltage
regulators taps positions, and capacitor banks switches. The desired controls
obtained after solving MILP are implemented to reduce feeder voltages. The
centralized control decisions generate a reference voltage for each feeder
node that, if maintained, will help optimally minimize the power consumption
from voltage dependent loads thus help achieve the CVR objective. In order to
include the effects of variations in PV generation, the centralized control is
supplemented with the local control of smart inverters. The local control help
to maintain the nodal voltages close to the reference voltages obtained using
the centralized controller.

###  IV-A Centralized Control

The objective of the centralized control is to minimize the active power
demand from the substation by controlling system’s voltage control devices
including voltage regulators, capacitor banks and smart inverters. The control
of legacy devices will introduce integer variables in the optimization
problem. We use the three-phase linear power flow model detailed in Section 2.
This results in a MILP problem as described below. Note that the centralized
control is executed at every 15-min interval.

$\small\text{Minimize}\sum_{p\in\phi_{s},j:s\rightarrow j}{P^{p}_{sj}(t)}$
(30)

Subject to:

$\displaystyle P_{ij}^{pp}(t)=\sum_{k:j\rightarrow
k}P_{jk}^{pp}(t)+p_{L,j}^{p}(t)-p_{DG,i}^{p}(t)\hskip 2.84544pt\forall
i\in\mathcal{N}$ (31) $\displaystyle Q_{ij}^{pp}(t)=\sum_{k:j\rightarrow
k}Q_{jk}^{pp}(t)+q_{L,j}^{p}(t)-q_{DG,i}^{p}(t)-q_{C,i}^{p}\hskip
2.84544pt\forall i\in\mathcal{N}$ (32) $\displaystyle
v_{j}^{p}(t)=v_{i}^{p}(t)-\sum_{q\in\Phi_{j}}{2\mathbb{Re}\left[S_{ij}^{pq}(t)(z_{ij}^{pq})^{*}\right]}\hskip
2.84544pt\forall j\in\mathcal{N}$ (33) $\displaystyle
p_{L,i}^{p}(t)=p_{i,0}^{p}(t)+CVR_{p}(t)\dfrac{p_{i,0}^{p}(t)}{2}(v_{i}^{p}(t)-1)\forall
i\in\mathcal{N_{L}}$ (34) $\displaystyle
q_{L,i}^{p}(t)=q_{i,0}^{p}(t)+CVR_{q}(t)\dfrac{q_{i,0}^{p}(t)}{2}(v_{i}^{p}(t)-1)\forall
i\in\mathcal{N_{L}}$ (35) $\displaystyle
v_{j}^{p}(t)=A_{i}^{p}(t)v_{i}^{p}(t)\hskip
8.5359pt\forall(i,j)\in\mathcal{E_{T}}$ (36) $\displaystyle
A_{i}^{p}(t)=\sum\limits_{i=1}^{32}B_{i}u_{tap,i}^{p}(t),\sum\limits_{i=1}^{32}u_{tap,i}^{p}(t)=1\forall(i,j)\in\mathcal{E_{T}}$
(37) $\displaystyle
q_{C,i}^{p}(t)=u_{cap,i}^{p}(t)q_{cap,i}^{rated,p}v_{i}^{p}(t)\hskip
5.69046pt\forall(i)\in\mathcal{N_{C}}$ (38) $\displaystyle
q_{DG,i}^{p}(t)\leq\sqrt{(s_{DG,i}^{rated,p})^{2}-(p_{DG,i}^{p})^{2}(t)}\hskip
5.69046pt\forall(i)\in\mathcal{N_{DG}}$ (39) $\displaystyle
q_{DG,i}^{p}(t)\geq-\sqrt{(s_{DG,i}^{rated,p})^{2}-(p_{DG,i}^{p})(t)^{2}}\hskip
5.69046pt\forall(i)\in\mathcal{N_{DG}}$ (40) $\displaystyle(V_{min})^{2}\leq
v_{i}^{p}(t)\leq(V_{max})^{2}\hskip 12.0pt\forall i\in\mathcal{N}$ (41)

The branch capacity for a line is defined as the maximum permissible kVA
capacity calculated based on its ampacity. Thus, the loading of a line should
not exceed its maximum permissible capacity as defined below:

$(P_{ij}^{pp})^{2}(t)+(Q_{ij}^{pp})^{2}(t)\leq((S_{ij}^{pp})^{max})^{2}$ (42)

To avoid the resulting nonlinearity in the optimal power flow formulation, the
quadratic constraints in (34) is approximated as linear constraints using a
polygon based linearization method first proposed in [35]. For each line
$(i,j)$ the linear constraints can be formulated as:

$\begin{array}[]{cc}&-\sqrt{3}(P_{ij}^{pp}(t)+R_{ij}^{pp})\leq
Q_{ij}^{pp}(t)\leq-\sqrt{3}(P_{ij}^{pp}(t)-R_{ij}^{pp})\\\
&\frac{-\sqrt{3}}{2}R_{ij}^{pp}\leq
Q_{ij}^{pp}(t)\leq\frac{\sqrt{3}}{2}R_{ij}^{pp}\\\
&\sqrt{3}(P_{ij}^{pp}(t)-R_{ij}^{pp})\leq
Q_{ij}^{pp}(t)\leq\sqrt{3}(P_{ij}^{pp}(t)+R_{ij}^{pp})\\\ \end{array}$ (43)

where, the radius of a hexagon $R_{ij}^{pp}$ is obtained using

$R_{ij}^{pp}=(S_{ij}^{pp})^{max}\sqrt{\frac{2\pi/6}{sin(2\pi/6)}}\hskip
28.45274pt(i,j)\in\mathcal{E}$ (44)

The problem objective (22) is to minimize the summation of active power demand
from the substation in all the three phases. Constraints (23) - (25) are the
linear AC power flow constraints. Constraints (26)-(27) detail nodal load
demand based on the load model detailed in Section 3.4. Constraints (28)-(29)
represent model for voltage regulator control, and (30) represents capacitor
model. Constraints (31)-(32) are the limits on the reactive power support
provided by the PV smart inverters. Constraint (33) represents the ANSI
voltage limits at each node. Constraint (35)-(36) represents the linear branch
capacity limits.

###  IV-B Local Smart Inverter Control

The objective of the local control is to reduce the voltage deviations at the
nodes wrt. the reference nodal voltages, $V_{i,ref}^{p}$, obtained using the
centralized controller. The objective for the local control is as follows:

$\text{Minimize}\hskip 28.45274pt{|V_{i}^{p}(t)-V_{i,ref}^{p}(t)|}\hskip
28.45274pt\forall i\in\mathcal{N_{DG}}$ (45)

where, $V_{i}^{p}(t)$ is the actual nodal voltage at phase $p$ of node $i$.

In section 5, two local control methods are proposed to achieve the local
control objective. The local control updates the smart inverter’s reactive
power support to mitigate the effects of local PV generation variability on
nodal voltage fluctuation. Note that only local measurements available at the
point of connection for the PV smart inverter are used to obtained the control
decisions.

###  IV-C Infrastructure to Implement the Proposed Framework

The required infrastructure to implement the proposed framework is detailed in
this section. The centralized control is located at the utility control center
that receives measurement information using an advanced distribution
management system (ADMS) also located at the utility control center. ADMS
includes: (1) distribution supervisory control and data acquisition (DSCADA) -
to gather information on network device statuses (capacitor banks, voltage
regulator, switches), (2) advanced meter infrastructure (AMI) - to obtain load
data, (3) distributed energy resource management system (DERMS) or distributed
energy resources (DERs) aggregators - to obtain the information on DERs
(generation and available reactive power). For behind-the-meter resources
(such as roof-top PVs), we assume that an aggregator is available to
communicate and control the individual behind-the-meter smart inverters. The
centralized controller after receiving the information on loads, utility-
scale/community solar, aggregated PVs, performs optimization with the
objective of minimizing the power demand from the substation. It solves for
the optimal capacitor bank switch statuses, voltage regulator tap positions,
and smart inverter reactive power support. The control signals are dispatched
to capacitor banks switch, voltage regulators, and DERs [36]. For aggregated
PVs, the decision from the central controller is distributed among the PVs and
communicated to the individual smart inverters by DER aggregators. It is
assumed that the PVs and smart inverters are controllable and have the
communication and measuring capabilities at the point of connection. Mostly,
the load demand measurements are gathered every 15-min time interval using
AMI. Therefore, we assume that the centralized controller operates every
15-min interval.

The centralized control decisions are communicated to individual grid voltage
control devices at every 15-min interval. Thus, the centralized control
signals are fixed for the specified 15-min time-window. As mentioned before,
the variability in PV power generation will lead to voltage fluctuations
within the 15-min decision interval specified for the centralized controller.
A fast local controller is implemented at each smart inverter to update their
reactive power support in response to local changes in PV generation. The
proposed local controller operates in sub-intervals (from real-time to every
1-min interval depending upon the granularity of the local measurements) and
are designed to reduce the deviations between the actual nodal voltages and
the reference voltages specified by the centralized controller for the given
15-min interval.

##  V Proposed Methods for Local Control

The intermittent nature of PV generators causes voltage fluctuation in the
distribution system. The voltage reference for a node obtained from the
centralized controller must be maintained to realize the CVR objective. In
this section, the factors which causes the voltage fluctuations in the
distribution systems are discussed. Later, two local voltage control methods
are proposed to reduce the nodal voltage deviations.

###  V-A PV Variability

PVs have lately emerged as the most prominent DGs connected at the
distribution level. At the distribution level, both small-scale (roof-top PVs)
and utility-scale PV generation plants are being widely deployed throughout
the nation. The PV power generation is, in general, variable due to the
variations in received solar irradiance. Typically, for a clear sky day, it is
assumed that the PV generation follows a smooth and nearly parabolic profile
(see Fig. 3). However, during a cloudy day, the PV irradiance can rapidly
fluctuate leading to a rapidly varying PV generation profile as shown in Fig.
3. Due to intermittent nature of PV power production, there can be a sudden
increase or decrease in nodal voltages causing voltage fluctuations in the
distribution system. The voltage fluctuations can lead to voltage quality
problems, and/or lead to an increased wear and tear of system’s legacy voltage
devices such as capacitor banks and voltage regulators [37]. Therefore, it is
of interest to minimize the nodal voltage fluctuations resulting from
intermittent PV generation.

In the proposed work, the reactive power support from the smart inverters is
utilized to reduce voltage fluctuations in the distribution system due to
random variations in PV power generation. These are referred to as local
control methods for smart inverter operation. Furthermore, it is ensured that
the proposed local control approach is coordinated with the centralized
voltage control approach as detailed in Section 4.1. Note that we have used
three-sigma rule to generate the variable PV power output [38]. The normalized
PV power production with 30 % and 70 % variability is shown in Fig. 3.

Figure 3: PV generation profile with variability.

###  V-B Local Smart Inverter Control - using Equivalent Thevenin Impedance
Method

The problem objective is to minimize the voltage deviation at distribution
system nodes caused due to the change in active power generation by PVs. We
assume PV system as a current source as shown in Fig. 4. The voltage at a node
$i$ for phase $p$, $V^{p}_{i}$, is given by (46) where, $I^{p}_{inj}$ is the
current injected by the PV system and $Z_{i}$ is the equivalent Thevenin
impedance (positive sequence impedance) at bus $i$.

$V^{p}_{i}=Z_{i}I^{p}_{inj}=(R_{i}+\iota X_{i})I^{p}_{inj}\hskip
14.22636ptp\in\\{a,b,c\\}$ (46)

The current injected at the node $i$, $I^{p}_{inj}$, is obtained using (47),
where $P^{p}_{inj}$ and $Q^{p}_{inj}$ are the per-phase active and reactive
power injected at node $i$ at nominal voltage level, $V^{p}_{nom}$.

$I^{p}_{inj}=\frac{(P^{p}_{inj}+\iota Q^{p}_{inj})^{*}}{V^{p}_{nom}}\hskip
14.22636ptp\in\\{a,b,c\\}$ (47)

Next, we obtain the expression for voltage deviation at a node as a function
of change in the active and reactive power injected at the node in (48). The
deviation in nodal voltage due to the change in active and reactive power is
calculated using (41)-(42).

$\displaystyle dV^{p}_{i}=\frac{\partial V^{p}_{i}}{\partial
P^{p}_{inj}}dP^{p}_{inj}+\frac{\partial V^{p}_{i}}{\partial
Q^{p}_{inj}}dQ^{p}_{inj}\hskip 11.38092ptp\in\\{a,b,c\\}$ (48)
$\displaystyle\frac{\partial V^{p}_{i}}{\partial
P^{p}_{inj}}=\frac{(R_{i}+\iota X_{i})}{V^{p}_{nom}}\hskip
48.36958ptp\in\\{a,b,c\\}$ (49) $\displaystyle\frac{\partial
V^{p}_{i}}{\partial Q^{p}_{inj}}=-\iota\frac{(R_{i}+\iota
X_{i})}{V^{p}_{nom}}\hskip 42.67912ptp\in\\{a,b,c\\}$ (50)

Figure 4: Distribution system connected to PV

Using (40)-(42), we obtain the expression for the voltage deviation as a
function of the equivalent Thevenin impedance and the change in active and
reactive power injected at the node in (51).

$\begin{split}dV^{p}_{i}=\left(\frac{R_{i}}{V^{p}_{nom}}dP^{p}_{inj}+\frac{X_{i}}{V^{p}_{nom}}dQ^{p}_{inj}\right)\\\
+\iota\left(\frac{X_{i}}{V^{p}_{nom}}dP^{p}_{inj}-\frac{R_{i}}{V^{p}_{nom}}dQ^{p}_{inj}\right)\end{split}$
(51)

Figure 5: Phasor diagram for change in voltage

The change in voltage at a node due to the change in current injected at the
node is the vector sum of nodal voltage, $V$, and impedance times injected
current, $ZI$. Since the angle difference between the source node and load
node voltages is small, as shown in Fig. 5, the change in the nodal voltage is
approximately equal to the real part of the product of impedance and current
[34]. Hence, the imaginary part of equation (51) is ignored.

Next, in order to maintain the nodal voltage constant and equal to the value
obtained using the centralized control, the change in voltage $dV^{p}_{i}$
should be made equal to zero. Therefore, equating real part of (51) to zero,
the desired reactive power support required to maintain nodal voltage constant
is obtained in (52). From (52), it can be observed that the change in reactive
power required to maintain the voltage at a node is directly proportional to
the change in active power injected at the node scaled by $R/X$ ratio at the
node.

$dq^{p}_{inj}(t)=-\frac{R_{i}}{X_{i}}dP^{p}_{inj}(t)$ (52)

The local control law stated in (52) requires the Thevenin impedance as seen
from the node and the change in active power generation from PV wrt. the
forecasted value (that was used when solving centralized control problem).The
Thevenin impedance is the positive sequence impedance for each bus in the
distribution system. We use OpenDSS to obtain the positive sequence impedance
at each bus of the given distribution system. Please refer to [34] for a
detailed explanation on the approach used by OpenDSS to calculate the positive
sequence impedance. The change in active power injected by PVs are measured at
each node. Using the positive sequence impedance and change in active power,
the calculated value of $dq^{p}_{inj}$ is used to update the reactive power
support from smart inverter that was previously calculated using the
centralized controller, $q^{p}_{inj}$. The smart inverter reactive power
support at every interval of local control is obtained using
$q^{p}_{inj,new}=q^{p}_{inj}+dq^{p}_{inj}$.

The net reduction in nodal voltage deviations wrt. the voltage reference is
dependent on the Thevenin impedance at the PV location. Here, the Thevenin
impedance is calculated after ignoring the effects of shunt devices in the
distribution system. Hence, the calculated values are not the exact values for
all load and generation conditions. Also, in this method, the compensation
provided by the smart inverters connected at other nodes is ignored.
Therefore, there is no coordination among the smart inverters when trying to
reduce the nodal voltage fluctuations. The local control decisions, therefore,
are approximate.

###  V-C Local Smart Inverter Control - using Power Flow Measurements

In this section, a measurement-based local smart inverter method is proposed
to mitigate the voltage deviations caused by variations in PV power
generation. The proposed method is based on the linear three-phase power flow
equations described in Section 2.2. It should be noted that the proposed
method includes the effects of local control actions from other smart
inverters and results in a better minimization of nodal voltage deviations.

The deviation in nodal voltages due to variation in PV generation can be
obtained by differentiating the three-phase voltage equations detailed in
(10). The differentiated voltage equation is shown in (53), where,
$\theta^{pq}$ is the angle difference between phase voltages which is assumed
to be a constant value, see (5).

$\small\begin{split}dv_{i}^{p}-dv_{j}^{p}=\sum_{q\in\phi_{j}}2\left(r_{ij}^{pq}dP_{ij}^{pq}cos(\theta^{pq})+x_{ij}^{pq}dQ_{ij}^{pq}sin(\theta^{pq})\right)\end{split}$
(53)

Next, same as in the previous section, we equate the changes in nodal voltages
to zero i.e., $dv_{i}^{p}=dv_{j}^{p}=0$. We obtain the mathematical relation
between the change in reactive power flow as a function of change in active
power flow, detailed in (54).

$\small\begin{bmatrix}r_{ij}^{aa}&A_{1}&A_{3}\\\ B_{1}&r_{ij}^{bb}&B_{3}\\\
C_{1}&C_{3}&r_{ij}^{cc}\\\ \end{bmatrix}\begin{bmatrix}dP_{ij}^{a}\\\
dP_{ij}^{b}\\\ dP_{ij}^{c}\\\
\end{bmatrix}+\begin{bmatrix}x_{ij}^{aa}&A_{2}&A_{4}\\\
B_{2}&x_{ij}^{bb}&B_{4}\\\ C_{2}&C_{4}&x_{ij}^{cc}\\\
\end{bmatrix}\begin{bmatrix}dQ_{ij}^{a}\\\ dQ_{ij}^{b}\\\ dQ_{ij}^{c}\\\
\end{bmatrix}=\begin{bmatrix}0\\\ 0\\\ 0\\\ \end{bmatrix}$ (54)

where, $A_{1}=(-\frac{1}{2}r_{ij}^{ab}+\frac{\sqrt{3}}{2}x_{ij}^{ab})$ ,
$A_{3}=(-\frac{1}{2}r_{ij}^{ac}-\frac{\sqrt{3}}{2}x_{ij}^{ac})$
$A_{2}=(-\frac{\sqrt{3}}{2}r_{ij}^{ab}-\frac{1}{2}x_{ij}^{ab})$ ,
$A_{4}=(\frac{\sqrt{3}}{2}r_{ij}^{ac}-\frac{1}{2}x_{ij}^{ac})$
$B_{1}=(-\frac{1}{2}r_{ij}^{ab}-\frac{\sqrt{3}}{2}x_{ij}^{ab})$ ,
$B_{3}=(-\frac{1}{2}r_{ij}^{bc}+\frac{\sqrt{3}}{2}x_{ij}^{bc})$
$B_{2}=(\frac{\sqrt{3}}{2}r_{ij}^{ab}-\frac{1}{2}x_{ij}^{ab})$ ,
$B_{4}=(-\frac{\sqrt{3}}{2}r_{ij}^{bc}-\frac{1}{2}x_{ij}^{bc})$
$C_{1}=(-\frac{1}{2}r_{ij}^{ac}+\frac{\sqrt{3}}{2}x_{ij}^{ac})$ ,
$C_{3}=(-\frac{1}{2}r_{ij}^{bc}-\frac{\sqrt{3}}{2}x_{ij}^{bc})$
$C_{2}=(-\frac{\sqrt{3}}{2}r_{ij}^{ac}-\frac{1}{2}x_{ij}^{ac})$ ,
$C_{4}=(\frac{\sqrt{3}}{2}r_{ij}^{bc}-\frac{1}{2}x_{ij}^{bc})$

Due to the change in active and reactive power generation by the PVs there
will be change in the branch/line power flow. The relationship for the change
in power flow in the line due to change in active and reactive power injected
by the PVs is obtained by differentiating (16) and (9). The change in active
and reactive power flow in the lines for each phase are shown in (55) and
(56), respectively.

$\begin{bmatrix}dP_{ij}^{a}\\\ dP_{ij}^{b}\\\ dP_{ij}^{c}\\\
\end{bmatrix}-\begin{bmatrix}dp_{j}^{a}\\\ dp_{j}^{b}\\\ dp_{j}^{c}\\\
\end{bmatrix}=\begin{bmatrix}\sum_{k:j\rightarrow k}dP^{a}_{jk}\\\
\sum_{k:j\rightarrow k}dP^{b}_{jk}\\\ \sum_{k:j\rightarrow k}dP^{c}_{jk}\\\
\end{bmatrix}$ (55) $\begin{bmatrix}dQ_{ij}^{a}\\\ dQ_{ij}^{b}\\\
dQ_{ij}^{c}\\\ \end{bmatrix}-\begin{bmatrix}dq_{j}^{a}\\\ dq_{j}^{b}\\\
dq_{j}^{c}\\\ \end{bmatrix}=\begin{bmatrix}\sum_{k:j\rightarrow
k}dQ^{a}_{jk}\\\ \sum_{k:j\rightarrow k}dQ^{b}_{jk}\\\ \sum_{k:j\rightarrow
k}dQ^{c}_{jk}\\\ \end{bmatrix}$ (56)

Next, in order to reduce the nodal voltage deviations, we use (54)-(56) to
obtain (57) that provides the required reactive power support from each smart
inverter.

$\begin{split}\begin{bmatrix}dq_{j}^{a}\\\ dq_{j}^{b}\\\ dq_{j}^{c}\\\
\end{bmatrix}=\left(-[A_{ij}]^{-1}+\sum_{k:j\rightarrow
k}[A_{jk}]^{-1}\right)\begin{bmatrix}\sum_{k:j\rightarrow k}dP^{a}_{jk}\\\
\sum_{k:j\rightarrow k}dP^{b}_{jk}\\\ \sum_{k:j\rightarrow k}dP^{c}_{jk}\\\
\end{bmatrix}\\\ -[A_{ij}]^{-1}\begin{bmatrix}dp_{j}^{a}\\\ dp_{j}^{b}\\\
dp_{j}^{c}\\\ \end{bmatrix}\\\ \end{split}$ (57)

where,

$\begin{bmatrix}A_{ij}\end{bmatrix}=\begin{bmatrix}x_{ij}^{aa}&A_{2}&A_{4}\\\
B_{2}&x_{ij}^{bb}&B_{4}\\\ C_{2}&C_{4}&x_{ij}^{cc}\\\
\end{bmatrix}^{-1}\begin{bmatrix}r_{ij}^{aa}&A_{1}&A_{3}\\\
B_{1}&r_{ij}^{bb}&B_{3}\\\ C_{1}&C_{3}&r_{ij}^{cc}\\\ \end{bmatrix}$ (58)

Note that the inverters located downstream from a PV node are also providing
the reactive power support and, therefore, affect the power flow along the
line downstream from the node. Therefore, by including downstream line power
flow measurement when calculating the required reactive power support for
smart inverters in (57), the effects of reactive power support provided by
other smart inverters in the system are taken into account. Thus, (57)
provides a better local control to regulate nodal voltages as compared to
Thevenin impedance-based method introduced in the previous section. To
implement this control approach, the local smart inverter requires two local
measurements: (1) local PV generation, and (2) power flow (P, Q) into the
children nodes from the PV nodes.

Figure 6: Modified IEEE 123-bus distribution test feeder. Figure 7: Modified
R3-12.47-2 distribution test feeder.

##  VI Results and Discussions

The proposed control framework is validated using the modified IEEE-123 bus
system and the modified R3-12.47-2 feeder with simulation done on Matlab and
OpenDSS. The interface between Matlab and OpenDSS is established using COM
interface. For the centralized control, the problem is modeled as a MILP which
is solved using CPLEX 12.7. The local control algorithm is executed in Matlab
and the obtained control variables are provided to the distribution model
simulated in OpenDSS. Therefore, although control decisions are based on
linearized power flow model, OpenDSS simulates actual nonlinear three-phase
system and provides a suitable environment for evaluating the performance of
the proposed methods on real-world distribution systems. All the simulations
are done on i7 3.41 GHz processor with 16 GB of RAM. The proposed models are
scalable and suitable for real-time operation. On an average it takes less
than 30 sec. to solve the MILP problem for the centralized controller for both
IEEE 123-bus (with a total of 267 single-phase nodes) and the modified
R3-12.47-2 feeder (with a total of 860 single-phase nodes) test feeders.

The IEEE-123 bus feeder (with a total of 267 single-phase nodes) is a medium
size feeder with several single phase lines and the single phase loads. There
are four capacitor banks and four voltage regulators deployed along the feeder
as shown in Fig. 6. The feeder is modified to include a total of 55 nodes
populated with PVs spread across the feeder; three-phase PVs of 172.5 kVA or
69 kVA ratings and single phase PVs of 23 kVA or 11.5 kVA ratings. The
modified R3-12.47-2 feeder (a total of 860 single-phase nodes) is a large
feeder with single phase and three phase lines and loads. The feeder includes
one substation voltage regulator, one three-phase 600-kVAr capacitor bank and
three 100 kVAr single-phase capacitor banks. In order to implement the
proposed control algorithm, the feeder is modified to include a total of 50
nodes populated with PVs spread across the feeder (see Fig. 7); three-phase
PVs of ratings 690 kVA, 345 kVA, 69 kVA, 172.5 kVA, 138 kVA or 34.5 kVA and
single phase PVs of ratings 23 kVA or 11.5 kVA. The customer loads are assumed
to have a CVR factor of 0.6 for active power and 3 for reactive power [39].
The generation profile for the PV during a clear sky day is used for MILP
problem modeled in centralized control problem. The clear day sky profile
should be thought of as PV forecast and is based on example PV profiles
provided in OpenDSS (see Fig.3). The actual PV profiles to be used in local
control include PV variability that is added to the clear sky PV profile to
generate multiple PV variability cases.

###  VI-A Verification of Approximate Power Flow Formulation

This section validates the three-phase linear AC power flow model detailed in
Section 2.2. The results obtained from the proposed linear power flow model
are compared with the nonlinear power flow solutions obtained using OpenDSS.
Table 1 shows the largest errors in the power flow and bus voltages for the
two test feeders at different loading conditions. It should be noted that
although the error in the power flow obtained using linear power model are
high, nodal voltages are well approximated. The maximum errors in nodal
voltages for the IEEE-123 bus and the modified R3-12.47-2 feeder at peak load
condition are 0.007 pu and 0.002 pu, respectively. The approximated and actual
nodal voltages for 123-bus system are also shown using voltage profile plots
in Fig. 8. Note that approximated voltage closely match the actual nodal
voltages for each phase. Since optimizing for CVR requires a power flow model
that accurately represents nodal voltages so that the voltage can be driven
towards the lower limits specified by ANSI std. (American National Standard
for Electric Power Systems and Equipment) . Since linear power flow model
approximates voltages well, it can be used to model optimization problem to
meet the CVR objective.

Figure 8: IEEE 123-bus: comparing nodal voltages obtained using linearized power flow model and OpenDSS at peak load condition TABLE I: Largest error in linear power flow wrt OpenDSS solutions Test feeder | Loading($\%$) | Error in $S_{flow}$($\%$) | Error in V(pu.)
---|---|---|---
IEEE 123 Bus | 50 $\%$ | 2.76 | 0.001
IEEE 123 Bus | 75 $\%$ | 4.25 | 0.004
IEEE 123 Bus | 100% | 5.76 | 0.007
R3-12.47-2 | 50 $\%$ | 1.1 | 0.0001
R3-12.47-2 | 75 $\%$ | 2.42 | 0.001
R3-12.47-2 | 100 $\%$ | 3.68 | 0.002

###  VI-B Effects of PV Variability on Legacy Device Operations

While implementing the centralized control, the PV generation is assumed to be
constant during the 15-min time interval. However, it is well-known that PV
generation is variable in smaller time intervals. To generate variable PV
profiles, we have used the PV profile for a clear sky day as shown in Fig. 3.
The variable generation is obtained by adding random numbers to the PV profile
for the clear sky day (see Fig. 3). The variability is quantified using the
three-sigma rule.

In this section, we analyze the effects of variable PV generation on the
operation of legacy voltage control devices operating in autonomous control
mode. Note that the proposed local control methods for smart inverter are not
implemented. The simulation is performed using IEEE 123-bus system for every
1-min time interval. The PVs are operating at unity power factor and the
capacitor banks and voltage regulators are used for the voltage regulation.
Table 2 shows the number of switching operations in a day for the four
capacitor banks and five voltage regulators installed in the test feeder. It
can observed from Table 2 that the increase in PV variability causes an
increase in the number of switching operations for the legacy voltage control
devices. The increased switching operations will deteriorate the performance
of the legacy devices. This calls for fast control actions to mitigate the
effects of PV variability on nodal voltage fluctuations. The smart inverters
are a feasible option as they can respond faster than the legacy devices. Note
that the centralized control is not a conducive option for the fast control of
smart inverters in response to rapid changes in PV generation. The centralized
control requires establishing fast communication channels between control
center and smart inverters. The high cost of establishing fast communication
channels, and communication delays or packet drops makes the approach
infeasible for fast control of smart inverters. This makes a strong case for
the local smart inverter control in order to reduce the voltage fluctuations
due to rapidly varying PV generation.

We also compare the number of legacy device operations after implementing the
proposed coordinated centralized and local controller. It should be noted the
number of operations for legacy device are not affected as the PV variability
increases. This is because the proposed controller is programmed to mitigate
the impacts of PV variability using the local control of smart inverters
instead of legacy devices. Recall that the central controller obtains the
optimal control statuses for the capacitor banks switches and voltage
regulators taps at every 15-min interval. These controls are then frozen for
the 15-min time interval and are recalculated only at the next sampling time
interval. Within the 15-min interval, as the PV generation changes, the local
controller operates smart inverter to compensate the effect of PV generation
variability on nodal voltage variations. Thus, when the local control is
implemented to mitigate the PV variability effects, the number of operation of
the legacy devices do not change as these are operated using centralized
controller that executes at every 15-min interval.

TABLE II: Effect of Local Control on Switching Operation of Legacy Device in a Day * | 0% variability | 30% variability | 70% variability
---|---|---|---
Legacy | without | with | without | with
device | control | control | control | control
cap 1 | 15 | 113 | 15 | 95 | 15
cap 2 | 6 | 134 | 6 | 148 | 6
cap 3 | 0 | 68 | 0 | 116 | 0
cap 4 | 23 | 226 | 23 | 218 | 23
VR 1 | 0 | 87 | 0 | 87 | 0
VR 2 | 23 | 315 | 23 | 292 | 23
VR 3 | 47 | 282 | 47 | 263 | 47
VR 4 | 32 | 356 | 32 | 280 | 32
VR 5 | 30 | 315 | 30 | 290 | 30

###  VI-C Centralized Control to Minimize Substation Power Demand (CVR
Objective)

The proposed centralized control is validated on the IEEE 123 bus and the
modified R3-12.47-2 feeder. The set points for legacy devices and smart
inverters are obtained using the centralized control at every 15-min time-
interval. The results demonstrate that the CVR objective is achieved by
maintaining the voltages at the lower levels without violating the specified
ANSI voltage limits.

First, the centralized control is demonstrated using the IEEE 123-bus system.
The control variables for this feeder are the switch statuses for the three
single-phase capacitor banks and a three-phase capacitor bank, tap position
for all the five voltage regulators, and the reactive power support from the
PV smart inverters. The total optimal power consumption supplied from the
substation after implementing the proposed centralized control framework is
shown in Fig. 9 and compared to the case without the centralized controller.
Note that when centralized control is not implemented, it is assumed that DGs
operate at unity power factor, and capacitor banks and voltage regulators
operate in autonomous control modes. It can be observed from Fig. 9 that there
is a reduction in power supplied by the substation. The control decisions for
the legacy devices and smart inverters vary as the loading condition changes.
It is observed that, at the minimum loading condition, all four capacitor
banks are OFF, whereas at the maximum loading, the three-phase capacitor bank
and one of the single phase capacitor bank (at node 88) is ON. The substation
voltage regulator is at -13 tap at minimum load and -8 tap at maximum load
condition. The other voltage regulators also operates according to the control
signals obtained from the centralized controller. The reactive power
supplied/absorbed by the smart inverter varies in order to adjust the nodal
voltages towards the lower ANSI voltage limits. The average reduction in
active power consumption obtained after implementing the centralized control
for the IEEE-123 node system is 60 kW for the day.

To show the effect of CVR on the MV+LV distribution network, we added a
secondary feeder model representing the low voltage network to the IEEE-123
bus system at node 610 ( shown in Fig. 6). The lateral feeder is connected
between node 610 and 610-1. The length of the lateral feeder is taken as 500
ft. A pole-mounted transformer is added at each phase of the lateral feeder
610-1. We have simulated four houses at each pole-mounted transformer, where
each house has rated load of 6 kW. For the secondary feeder we assume there is
100 % penetration of PV, i.e, all the house has maximum PV rating of 6 kW
connected through 7.5 kVA smart inverter. The data for the LV network is shown
in Table 3.

TABLE III: The parameters for the components in the LV network Components | Parameters
---|---
Transformers | 1-$\Phi$ 75 kVA, 2.4/0.277 kV R = 1.1 % X = 2%
Lines | 1-$\Phi$ ,length = 50 feet, $\frac{X}{R}=1.015$

With the LV network, the centralized control operates based on the aggregated
PV generation at node 610 on the primary side of pole-mounted transformer. The
reactive power control obtained from centralized control is then distributed
among each PV’s smart inverter according their kVA ratings. Given that behind-
the-meter PVs are usually not visible to the utility company, we assume a DER
aggregator at node 610 that provides the aggregated PV generation capacity to
the centralized controller and also executes the control of individual smart
inverters based on the control set points obtained from the centralized
controller. The centralized control calculates the required reactive power
from aggregated PVs (and the control set points for other voltage control
devices) by solving the optimal power flow problem. The required reactive
power is communicated to the aggregator that is located at the primary side of
the distribution transformer. The aggregator then distributes and communicates
the required reactive power support to each PV installed at the low voltage
side of the distribution feeder according to their smart inverter ratings. The
results at the maximum and minimum loading condition for MV+LV distribution
network is shown in Table 4 . Here, the simulation is performed by assuming a
clear sky day. For both maximum and minimum loading condition, there is
reduction in power consumption when CVR based control is implemented in the
distribution system.

The secondary feeder has high $\frac{R}{X}$ ratio, thus it will increase the
requirement for the reactive power support from smart inverters in order to
maintain the desired voltage reference. To demonstrate this case, we present a
comparison of reactive power support required when the PVs are assumed to be
connected at the primary feeder-level vs for distributed PV case at the
secondary feeder level (See Fig.10). It can be observed that reactive power
requirement to achieve same voltage regulation is higher when PVs are
connected at the secondary feeder-level.

Figure 9: IEEE-123 CVR benefits observed using centralized control. Figure 10: Comparison of reactive power requirement at node 610-1. TABLE IV: Centralized control for IEEE-123 bus system with LV Network Loading Condition | without CVR (kW) | with CVR (kW)
---|---|---
minimum | 571.35 | 537.16
maximum | 2158.33 | 2034.87

The centralized control is also implemented for the modified R3-12.47-2
feeder. This test case shows the scalibility of the proposed algorithm for
larger systems. At the minimum loading condition, all the capacitor banks are
OFF and the substation voltage regulator is operating at -6 tap for all the
three phases. At the maximum loading condition, the voltage regulator shifts
to -2 tap and the three-phase capacitor banks is ON. The single-phase
capacitor for phase A and C is ON, however, for phase B is OFF. The reactive
power supplied/absorbed by the smart inverters varies accordingly to maintain
the voltages at the nodes towards the lower ANSI limit. The reduction in the
power consumption from the substation for the modified R3-12.47-2 feeder is
shown in Fig. 11. The average power reduction for the day is approximately 145
kW for the modified R3-12.47-2 system.

Figure 11: Modified R3-12.47-2 CVR benefits observed using centralized
control.

###  VI-D Local Smart Inverter Control to Mitigate Effects of PV Variability
when optimizing for CVR

The objective of the local control is to maintain the voltage at each node in
the distribution system close to the voltages obtained by the centralized
controller. In the next section, the results demonstrate the effectiveness of
the local control in maintaining the desired voltages and in optimally
achieving the CVR objectives while considering the variations in PV
generation. Note that there is no delay in executing the local control action
and the local smart inverter control is operating in conjugation with PV
generation without any delay.

####  VI-D1 Reduction in Voltage Deviations

In the following section, the effect of the proposed local control in reducing
the voltage deviation is discussed. In order to quantify the effect of local
control on nodal voltage fluctuations, a power quality index called system
average voltage fluctuation index (SAVFI) proposed in [25] is used. Let the
nodal voltage magnitude at time t be $V_{i}^{t}$. The voltage fluctuation
$\Delta V_{i}(t)$ is defined as the average of the difference in the voltage
magnitude between the two time steps.

$\Delta V_{i}=|V_{i}^{t+1}-V_{i}^{t}|$ (59)

The SAVFI is defined as the average of voltage fluctuation for a time interval
$T$.

$SAVFI=\frac{1}{T}\sum_{n=1}^{T}\Delta V_{i}(t)$ (60)

The SAVFI value for selected nodes of the IEEE 123-bus system is shown in
Table V for 30 % and 70 % PV output power variability cases. It can be
observed from the table the SAVFI increases as PV variability is increased.
Also, it is observed that as the distance of the nodes from the substation
increases the SAVFI value increases for same levels of PV variability. In
order to reduce the voltage fluctuation within the 15-min time interval, both
the impedance-based and the proposed power flow measurement-based local
control methods are implemented at each PV location. It can be observed from
the SAVFI value (see Table V ), that both the local control are able to reduce
the voltage fluctuations at a node. Also, the proposed power flow measurement-
based local control method is able to reduce the voltage fluctuations more
efficiently as compared to impedance-based method. This is because, the
proposed power flow measurement based method takes the reactive power support
provided by other smart inverters into account.

TABLE V: SAVFI for the IEEE 123-bus system at different variability level Node | Without local | Impedance-based | Power flow-based
---|---|---|---
number | control | control | control
30 % PV power variability
7 | 0.5734 | 0.1415 | 0.1335
25 | 1.093 | 0.174 | 0.118
97 | 1.647 | 0.388 | 0.273
114 | 1.738 | 0.369 | 0.30
70 % PV power variability
7 | 1.72 | 0.433 | 0.408
25 | 3.278 | 0.535 | 0.3712
97 | 4.94 | 1.183 | 0.835
114 | 5.21 | 1.126 | 0.915
TABLE VI: SAVFI for the modified R3-12.47-2 system at different variability level Node | Without local | Impedance-based | Power flow-based
---|---|---|---
number | control | control | control
30 % PV output power variability
5 | 0.7132 | 0.1738 | 0.0295
98 | 3.255 | 0.559 | 0.0865
105 | 3.145 | 0.559 | 0.103
249 | 2.987 | 1.43 | 1.12
70 % PV output power variability
5 | 2.65 | 0.549 | 0.068
98 | 11.92 | 2.18 | 0.479
105 | 11.52 | 2.18 | 0.53
249 | 10.93 | 5.31 | 4.19

Similarly, the proposed local control methods are demonstrated to reduce
voltage fluctuation in the modified R3-12.47-2 in Table VI. From Table VI, it
can be observed that the SAVFI is higher for the nodes away from the
substation. It can also be verified from the table that the proposed local
control based on power flow measurements is relatively more effective in
reducing the voltage fluctuations for different levels of PV generation
variability.

Next, the effect of local control in meeting CVR objective while avoiding any
nodal voltage violations regardless of PV variability is examined (see Table
VII). First, we observe the total substation power flow obtained after
implementing the centralized control decisions (at maximum load condition) for
the two cases of PV variability when local control is not implemented. We also
identify the cases of voltage violations resulting in the distribution feeder
due to PV variability when local control is not implemented. For the IEEE-123
node system at the maximum loading condition and at 30% power variability,
three nodes in the distribution system are observed to have voltage violation
if no local control is implemented. However, after implementing local control,
none of the nodes in the system observe voltage violations. Also, the
substation power demand is reduced. Similarly, for 70% PV variability case,
the total number of nodes with voltage violations increases to 13. The local
control ensures no voltage violations and a reduction in substation power
demand. At peak-loading condition, for the modified R3-12.47-2 system, the
number of nodes which observe voltage violation at 30% power variability
without local control is 191 whereas at 70% variability, the number of nodes
with voltage violations increases to 445. After incorporating local control
for the smart inverters, none of the nodes in the system observe voltage
violations for the two cases. For 70% PV variability, there is an increase in
substation power demand. This is due to additional power requirement for the
feeder to supply for losses in order to maintain the feeder voltages within
the required ANSI limits.

TABLE VII: Effect of local control at maximum loading condition PV Variability | Without local control | With local control
---|---|---
Substation | Voltage | Substation | Voltage
Power (kW) | Violations | Power (kW) | Violations
IEEE-123 bus system
30 % | 2216.4 | 3 | 2209.1 | 0
70 % | 2448.6 | 13 | 2440 | 0
Modified R3-12.47-2 system
30 % | 6619.4 | 191 | 6613.3 | 0
70 % | 6978.1 | 445 | 6985.2 | 0
Figure 12: Overvoltage scenario for the IEEE 123-bus system

####  VI-D2 Mitigating Voltage Violations

In this section, the overvoltage and undervoltage cases described in the
previous section are observed in more detail. The overvoltage at nodes due to
variable PV generation are observed when the load demand is less and the
predicted PV power production is lower that the actual PV power production.
The Fig.12, shows the overvoltage scenario in the IEEE-123 node system at the
node 64 and 114. In order to create this case, it is assumed that the
predicted PV power production is 10% of the rated power production in the
15-min time interval and the actual power production for every 1-min (real-
time) interval is around 70-80 % of the rated value. The increased PV power
will increase the voltage at the nodes. It can be observed from the figure
that the voltage at nodes 64 and 114 are above 1.05 pu. This violates the
specified ANSI voltage limits. The local control help to maintain the voltage
close to the voltage obtained from centralized control. Both local control
methods are able to mitigate the overvoltage violation and maintain the nodal
voltages closer to the reference voltage obtained using the centralized
controller.

Similarly, the undervoltage condition at a node is observed when the load
demand is high and the predicted PV power production is larger than the actual
power production. In Fig.13 and Fig.14, the undervoltage scenarios are
simulated for the IEEE 123-bus and the modified R3-12.47-2 test feeders,
respectively. The undervoltage scenario occur when the predicted PV power
production is 90% of the rated power production for the 15-min time interval
but the actual power production (every 1-min interval or real-time) is around
10-20 % of the rated power. For the IEEE-123 node system, node 64 and node 114
observe voltages below 0.95 pu (see Fig. 13). The proposed local control is
able to eliminate the undervoltage conditions at node 64 and 114 and maintain
the voltage closer to the desired voltages obtained from centralized control.
Similarly, the undervoltage scenario is created for the modified R3-12.47-2
system and it can observed from Fig.14 that nodes 147 and 222 observe voltages
less than 0.95 pu. The proposed local control is able to mitigate the
undervoltage and maintain the voltage near to centralized voltage. Thus, local
control methods are effective in mitigating potential voltage violations
resulting from inaccurate forecast of PV generation that is used by the
centralized controller.

Figure 13: Undervoltage scenario for the IEEE 123-bus system Figure 14:
Undervoltage scenario for the modified R3-12.47-2 system

###  VI-E Discussions

In this section, we provide additional discussions regarding the utility of
the proposed two-timescale controller. First, we provide the justification for
adopting a central controller to regulate voltages at the MV level and not
directly at the LV level. Note that to perform OPF, the central controller
requires the knowledge of network connectivity, the locations and ratings of
the legacy devices connected in the system, load demand, and DER generation.
Unfortunately, at present, behind-the-meter resources are not available to the
network operator for the operational decision-making. It requires the
deployment of an extensive communication infrastructure to make the behind-
the-meter data (of individual customers connected to the secondary feeders)
accessible to the operator for operational decision-making. Therefore, we
assume that aggregated load and DER measurements are available at the primary
feeder (MV) nodes that the centralized controller uses for the operational
decision making.

Next, we highlight the reason for having a two-timescale controller. The
primary reason for adopting such architecture is to mitigate the voltage
problems that the central controller is not able to handle. There are two
reasons due to which the decisions of central controller alone may not be
sufficient to mitigate the nodal voltage concerns at the LV level: (1) faster
variability in generation (and/or load), (2) additional voltage drop in
secondary feeders connecting MV and LV level. The local controller operates
purely based on local node measurements i.e. sum of load and generation at a
specific node. Also, the local voltages are affected by the net demand
(generation) at the node. Hence, by acting based on local measurements, the
local controller is able to act for the required voltage levels at the LV
level. Thus, the LV voltage problems that the central controller is unable to
mitigate are resolved by the local controllers.

##  VII Conclusion

This paper presents a two-timescale control approach for system’s voltage
regulation devices to minimize the power consumption from voltage-dependent
customer loads. In first stage, the centralized controller solves a linear
three-phase OPF with mixed-integer/binary decision variables with the
objective of minimizing the active power consumption from the substation. The
centralized controller operates in every 15-min interval and sends the control
signals to both legacy voltage control devices and the smart inverters. To
address the concerns resulting from variable PV generation leading to nodal
voltage fluctuations, at second stage, local control schemes coordinates with
centralized controller is proposed. The local control is implemented for fast
acting voltage control devices such as smart inverters; the control for legacy
devices (capacitor banks and voltage regulators) is fixed for every 15-min
interval based on optimal centralized controller decisions. The local control
schemes are designed to obtain the required reactive power support from smart
inverter to mitigate the voltage deviations resulting from variable PV
generation. First, a Thevenin’s impedance-based method is proposed to minimize
the nodal voltage deviations (due to PV variability) wrt. the reference
voltages obtained after implementing the centralized controller decisions.
This method requires R/X ratio at the point-of-connection (POC) of PV and the
change in local PV power generation wrt. the forecasted PV generation value
used by the centralized controller. This approach, however, does not take the
local decisions of other inverters into account. Therefore, to incorporate the
effects of reactive power support from other smart inverters, a new method
based on local power flow measurements is proposed. In addition to the change
in local PV generation, this method also requires line flow measurements for
the children nodes.

The proposed two-timescale control approach is validated using the IEEE
123-bus and the modified R3-12.47-2 test systems. It is demonstrated that the
proposed control approach is able to help realize CVR benefit for both feeders
even for the cases with high PV variability. The local controls are able to
reduce the voltage deviations caused due to variability in PV generation and
maintain the nodal voltages closer to the reference voltage obtained from the
centralized controller. The power flow measurement-based local control
approach is shown to be relatively more effective compared to the Thevenin’s
impedance-based method. The MILP problem for the centralized controller takes
on an average less than 30 sec. to solve for both the IEEE 123-bus (with a
total of 267 single-phase nodes) and the modified R3-12.47-2 system (with a
total of 860 single-phase nodes) test feeders. The local control is
essentially instantaneous as it requires only simple arithmetic computations.
Therefore, the proposed methods are suitable for the real-time control and
operation of large-scale distribution feeders.

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