# Tractable Data Enriched Distributionally Robust Chance-Constrained CVR

Qianzhi Zhang, Fankun Bu,
Yi Guo, and Zhaoyu Wang  This work was supported in part by the U.S.
Department of Energy Wind Energy Technologies Office under Grant DE-EE0008956,
and in part by the National Science Foundation under ECCS 1929975. This work
was partially supported by an ETH Zürich Postdoctoral Fellowship.
(_Corresponding author: Zhaoyu Wang_). Qianzhi Zhang, Fankun Bu, and Zhaoyu
Wang are with the Department of Electrical and Computer Engineering, Iowa
State University, Ames, IA 50011 USA (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>wzy@iastate.edu). Yi Guo is with Power Systems Laboratory at
ETH Zürich, Zürich, 8092, Switzerland (e-mail: guo@eeh.ee.ethz.ch).

###### Abstract

This paper proposes a tractable distributionally robust chance-constrained
conservation voltage reduction (DRCC-CVR) method with enriched data-based
ambiguity set in unbalanced three-phase distribution systems. The increasing
penetration of distributed renewable energy not only brings clean power but
also challenges the voltage regulation and energy-saving performance of CVR by
introducing high uncertainties to distribution systems. In most cases, the
conventional robust optimization methods for CVR only provide conservative
solutions. To better consider the impacts of load and PV generation
uncertainties on CVR implementation in distribution systems and provide less
conservative solutions, this paper develops a data-based DRCC-CVR model with
tractable reformulation and data enrichment method. Even though the
uncertainties of load and photovoltaic (PV) can be captured by data, the
availability of smart meters (SMs) and micro-phasor measurement units (PMUs)
is restricted by cost budget. The limited data access may hinder the
performance of the proposed DRCC-CVR. Thus, we further present a data
enrichment method to statistically recover the high-resolution load and PV
generation data from low-resolution data with Gaussian Process Regression
(GPR) and Markov Chain (MC) models, which can be used to construct a data-
based moment ambiguity set of uncertainty distributions for the proposed DRCC-
CVR. Finally, the nonlinear power flow and voltage dependant load models and
DRCC with moment-based ambiguity set are reformulated to be computationally
tractable and tested on a real distribution feeder in Midwest U. S. to
validate the effectiveness and robustness of the proposed method.

###### Index Terms:

Conservation voltage reduction, data enhancement, distributionally robust
chance-constrained optimization, distribution systems, tractable
reformulation.

##  I Introduction

Conservation voltage reduction (CVR) can reduce the voltage for peak load
shaving and long-term energy-saving [1]. To achieve system-wide optimal
performance, voltage/var optimization-based CVR (VVO-CVR) is previously
studied [1, 2, 3, 4], which can be cast into an optimal power flow program.
While the previous works have contributed valuable insights to VVO-CVR, there
are problems remaining open, summarized as follows:

(1) The impact of load and renewable uncertainties on VVO-CVR: In [2], a
linear least-squares centralized optimization model is developed for
coordinating combinations of voltage regulating devices and PVs to implement
CVR in distribution systems. In [3] and [4], several alternating direction
method of multipliers (ADMM)-based distributed optimization algorithms are
developed for CVR implementation, which can decompose a large-scale VVO-CVR
problem into several small-scale problems with improved scalability. However,
the above centralized or distributed VVO-CVR works [2, 3, 4] are developed
based on deterministic optimization methods, which assume the perfect
forecasts of load and renewable power. Neglecting those prediction errors
could result in potential violations of operational constraints, such as bus
voltage constraints in VVO-CVR. To consider the impacts of load and renewable
uncertainties on voltage regulation for CVR implementation, stochastic
programming (SP) and robust optimization (RO) are applied in some existing
works [5, 6, 7, 8]. In [5] and [6], the scenario-based SP methods aim at
optimizing reactive power dispatch for voltage regulation with expected
performance and an accurate probability distribution model. While the SP
methods need an accurate probability distribution model and a large sampling
number of scenarios, it requires heavy computational efforts. In [7] and [8],
the RO methods are developed to handle the uncertain load and renewable power
production. However, the RO methods can only give a feasible solution for the
worst-case scenario, which is too conservative and hinders the performance of
CVR.

Recently, the distributionally robust optimization (DRO) has been considered a
more effective way to handle uncertainty in power systems, such as economic
and power dispatch [9] and [10]. The DRO can construct an ambiguity set of
probability distributions based on historical data, including all possible
uncertainty distributions. Thus, the DRO can ensure constraints are satisfied
for any distributions in the ambiguity set built upon distribution moments or
structure information. Also, the distributionally robust chance-constrained
(DRCC) models [11] and [12] are developed in power systems, which integrate
chance constraints to enforce certain events within a probability threshold.
Even though DRO and DRCC models have some advantages over conventional
deterministic and stochastic approaches, there are still challenges to use the
DRCC method for CVR implementation, including how to formulate DRCC in a
tractable way for CVR implementation and how to construct an ambiguity set
with limited access to historical and real-time load and PV generation data.

(2) Intractable DRCC and nonlinear voltage-dependent load models: The VVO-CVR
problem is nonlinear and intractable, which makes a distributionally robust
stochastic reformulation of the VVO-CVR problem even more challenging. In
[13], an approximation method is proposed for conventional DRO problems, while
the approximation for DRCC problems is not considered. Also, the loads are
considered as voltage-independent models in above reference of DRO and DRCC
[9, 10, 11, 12]. In practice, the nature of CVR is lowering network voltages
to reduce the voltage, and the literature on the CVR problem validates the
necessity of voltage-dependent loads, such as ZIP load and exponential load
models [3]. In existing research works [14] and [15], some approximation
methods are applied to linearize ZIP load model in deterministic optimization.
However, the linearization and convexification for ZIP load model in DRO or
DRCC with uncertainties of load are not considered in previous works [13, 9,
10, 11, 12, 14, 15].

(3) Availability of data for constructing ambiguity set of load and renewable
uncertainties: Even though the ambiguity set of uncertainty in DRCC can
characterize a group of possibility distributions, defining a high-quality
ambiguity set is non-trivial, as one needs to decide the trade-off between the
conservativeness of decisions and the operational efficiency, while
considering the mathematical tractability. To construct an ambiguity set of
uncertainties in DRCC, the conventional way is using statistical inference and
data analysis methods with the historical data and system feedback
measurements. For example, the existing DRO works [16] and [17] use data-based
method to construct ambiguity set. However, the aforementioned data-based
ambiguity set relies heavily on either sufficiently high-resolution data or
complex machine learning methods.

To capture the uncertainties of load and PV generation through data-based
methods in modern distribution networks, micro-phasor measurement units (PMUs)
and smart meters (SMs) are implemented to record load and PV generation data,
where micro-PMUs111Micro-PMUs are synchrophasor devices that high-speed record
real-time stamped data measurement of power and energy consumption. Micro-PMUs
have a high sampling rate, e.g., one sample per second or higher [18] can
record high-resolution data (1-second resolution or higher), and SMs222SMs are
electronic devices that record power and energy consumption and can
communicate remotely with utility. SMs have a relatively low sampling rate
compared to micro-PMUs, e.g., one sample per hour [18]. can record low-
resolution data (typically 1-hour resolution). However, due to the cost issue,
micro-PMUs are only installed at limited locations in real distribution
networks, while SMs are widely installed in real distribution networks.
Therefore, the access to available data for constructing an ambiguity set is
limited by the number of micro-PMUs and SMs. If we only use limited data to
construct an ambiguity set, it may hinder the performance of the DRCC program.
The data from SMs and micro-PMUs are necessary but need modification to
support our approaches. Therefore, we need to enrich the load and PV
generation data, then construct the data-based ambiguity set for uncertainties
of load and PV generation.

To address the above challenges, this paper proposes a tractable data-based
DRCC-CVR model under the uncertainties of voltage-dependent load and PV
generation. Inspired by our previous work [18], we apply the data enrichment
method to both load and PV generation data from SMs and micro-PMU, then
construct the ambiguity set of uncertainties with the enriched data for the
proposed DRCC-CVR. This data enrichment method enables a strong connection
between the moment-based representation of the limited data set and the
moment-based ambiguity set of the DRCC model, which avoids over- or under-
conservativeness of the decisions. The main contributions of this paper are
three-fold:

  * •

DRCC-CVR model with uncertainties of load and solar PV generation: To consider
the impacts of load and renewable uncertainties on voltage regulation and
energy-saving performance of CVR in the unbalanced three-phase distribution
systems, we present a deterministic VVO-CVR model and extend it by introducing
chance constraints for possible voltage violations due to the load and the
renewable uncertainties. To obtain proper conservative solutions robust to the
high-resolution dataset and ensure a better performance for energy-saving and
voltage regulation of CVR, we further integrate the chance constraints with
DRO techniques to propose a DRCC-CVR model.

  * •

Linearization and tractable reformulation of DRCC-CVR model: To make our
proposed DRCC-CVR model tractable, we present the linearized version of the
power flow model and voltage-dependent ZIP load model. Then, we reformulate
the chance constraints with a moment-based ambiguity set of load and PV
generation uncertainties in a tractable way.

  * •

Data-enriching moment-based ambiguity set with SM and micro-PMU data: To
guarantee the performance of the data-based DRCC-CVR model, we leverage the
data enrichment method to recover high-resolution load and PV generation data
from SMs and micro-PMUs. The moment information of load and PV generation
uncertainties are execrated from the enriched data to construct the moment-
based ambiguity sets for the proposed DRCC-CVR model.

##  II The Problem Formulation and the Proposed Method

###  II-A Solving a VVO-CVR Problem in the Unbalanced Three-phase Distribution
Networks

In this paper, we consider an unbalanced three-phase radial distribution
network that consists of $N$ buses denoted by a set $\mathcal{N}$ and $N-1$
branches denoted by a set $\mathcal{E}$. Let $bp(i)$ denote the bus that
immediately precedes bus $i$ along the radial network headed by the feeder
head bus. The three-phase indices $\phi_{a},\phi_{b},\phi_{c}$ are simplified
as $\phi$. The time instance is represented by $t$. Distributed assets are
located at different buses including voltage dependent ZIP loads and solar PV
distributed generators. We assume that the customers are either equipped with
SMs or micro-PMUs, which monitor the active and reactive load power and active
PV generation power with proper time resolution. For each bus
$i\in\mathcal{N}$, let $p^{\rm ZIP}_{i,\phi,t},q^{\rm
ZIP}_{i,\phi,t}\in\mathbbm{R}^{3\times 1}$ denote the vector of three-phase
active and reactive voltage-dependent ZIP loads at time $t$. For each bus
$i\in\mathcal{G}$, let
$p^{g}_{i,\phi,t},q^{g}_{i,\phi,t}\in\mathbbm{R}^{3\times 1}$ denote the
vector of three-phase active and reactive power outputs of the $i$-th PV
inverter at time $t$; $V_{i,\phi,t}\in\mathbbm{R}^{3\times 1}$ represents the
vector of three-phase voltage magnitude at time $t$,
$v_{i,\phi,t}:=V_{i,\phi,t}\odot V_{i,\phi,t}\in\mathbbm{R}^{3\times 1}$
represents the vector of three-phase squared voltage magnitude at time $t$.
For each branch $({i,j})\in\mathcal{E}$, let $z_{ij}=r_{ij}+{\bf
i}x_{ij}\in\mathbbm{C}^{3\times 3}$ denotes matrix of the three-phase
impedence of line $ij$, where $r_{ij}$ and $x_{ij}$ are the matrices of the
three-phase resistance and reactance, respectively. Let
$S_{ij,\phi,t}=P_{ij,\phi,t}+{\bf i}Q_{ij,\phi,t}\in\mathbbm{C}^{3\times 1}$
denotes the vector of three-phase apparent power, where $P_{ij,\phi,t}$ and
$Q_{ij,\phi,t}$ are the vector of three-phase active and reactive power flow
trough line $ij$ from bus $i$ to bus $j$ at time $t$. Let the line active and
reactive power flows, nodal active and reactive power injections, and squared
voltage magnitudes be denoted by the following column vectors:
$P:=\\{P_{bp(i)i,\phi,t},\forall i,t,\phi\\}$, $Q=\\{Q_{bp(i)i,\phi,t},\forall
i,t,\phi\\}$, $p=\\{p_{i,\phi,t},\forall i,t,\phi\\}$,
$q=\\{q_{i,\phi,t},\forall i,t,\phi\\}$, and $v=\\{v_{i,\phi,t},\forall
i,t,\phi\\}$. $\odot$ and $\oslash$ denote the element-wise multiplication and
division.

The classic VVO-CVR program can be formulated as a deterministic problem (1),
which aims to reduce the total power consumption of the entire distribution
network while maintaining a feasible voltage profile within the predefined
bounds across the distribution network as follows:

$\displaystyle\min_{P,Q,p,q,v}\hskip 5.69054pt$
$\displaystyle\sum_{t\in[t,t+T]}\sum_{j:0\rightarrow j}\sum_{\phi\in\\{\rm
a,b,c\\}}{\rm Re}\\{S_{0j,\phi,t}\\},$ (1a) s.t. $\displaystyle P_{ij,\phi,t}$
$\displaystyle=\sum_{k:j\rightarrow
k}P_{jk,\phi,t}-p^{g}_{j,\phi,t}+p_{j,\phi,t}^{\rm
ZIP}+\varepsilon_{ij,\phi,t}^{p},$ (1b) $\displaystyle Q_{ij,\phi,t}$
$\displaystyle=\sum_{k:j\rightarrow
k}Q_{jk,\phi,t}-q^{g}_{j,\phi,t}+q_{j,\phi,t}^{\rm
ZIP}+\varepsilon_{ij,\phi,t}^{q},$ (1c) $\displaystyle v_{j,\phi,t}$
$\displaystyle=v_{i,\phi,t}-2\big{(}\bar{r}_{ij}\odot
P_{ij,\phi,t}+\bar{x}_{ij}\odot
Q_{ij,\phi,t}\big{)}+\varepsilon_{i,\phi,t}^{v},$ (1d) $\displaystyle p^{\rm
ZIP}_{i,\phi,t}$ $\displaystyle=p^{\rm
L}_{i,\phi,t}\odot\big{(}k_{i,1}^{p}\cdot
v_{i,\phi,t}+k_{i,2}^{p}\cdot\sqrt{v_{i,\phi,t}}+k_{i,3}^{p}\big{)},$ (1e)
$\displaystyle q^{ZIP}_{i,\phi,t}$ $\displaystyle=q^{\rm
L}_{i,\phi,t}\odot\big{(}k_{i,1}^{q}\cdot
v_{i,\phi,t}+k_{i,2}^{q}\cdot\sqrt{v_{i,\phi,t}}+k_{i,3}^{q}\big{)},$ (1f)
$\displaystyle-Q_{i,\phi,t}^{\rm cap}$ $\displaystyle\leq q^{g}_{i,\phi,t}\leq
Q_{i,\phi,t}^{\rm cap},\forall i\in\mathcal{G},$ (1g) $\displaystyle
Q_{i,\phi,t}^{\rm cap}$ $\displaystyle=\sqrt{(S^{\rm
cap}_{i,\phi,t})^{2}-(p^{\rm g}_{i,\phi,t})^{2}},\forall i\in\mathcal{G},$
(1h) $\displaystyle v^{\rm min}$ $\displaystyle\leq v_{i,\phi,t}\leq v^{\rm
max},\forall i\in\mathcal{N}.$ (1i)

In objective function (1a), the three-phase active power supplied from the
substation of the feeders ${\rm Re}\\{S_{0j,\phi,t}\\}$ is minimized over a
moving finite horizon $[t,t+T]$ for energy-saving with CVR implementation.
Constraints (1b)-(1d) are defined by the unbalanced three-phase version of
DistFlow model [3]. Constraints (1b) and (1c) guarantee the nodal active and
reactive power balance. Constraint (1d) calculates the voltage difference
between bus $i$ and bus $j$. The detailed formulations of nonlinear terms
$\varepsilon_{ij,\phi,t}^{p}$, $\varepsilon_{ij,\phi,t}^{q}$ and
$\varepsilon_{i,\phi,t}^{v}$ can be found in [4]. If the network is not too
severely unbalanced, the voltage magnitudes between the phases are similar and
the relative phase unbalance $\alpha$ is small [3], then we can use unbalanced
three-phase resistance matrix $\bar{r}_{ij}$ and reactance matrix
$\bar{x}_{ij}$ in constraint (1d). More details about $\bar{r}_{ij}$ and
$\bar{x}_{ij}$ for unbalanced three-phase distribution systems can be referred
to [3]. The implementation of CVR requires the modeling of voltage-dependent
ZIP active and reactive loads $p^{\rm ZIP}_{i,\phi,t}$ and $q^{\rm
ZIP}_{i,\phi,t}$, as shown in (1e) and (1f). $p^{\rm L}_{i,\phi,t},q^{\rm
L}_{i,\phi,t}\in\mathbbm{R}^{3\times 1}$ are the vectors of three-phase active
and reactive load time-series multipliers on bus $i$ at time $t$,
respectively. $k_{i,1}^{p}$, $k_{i,2}^{p}$, $k_{i,3}^{p}$ and $k_{i,1}^{q}$,
$k_{i,2}^{q}$, $k_{i,3}^{q}$ are constant-impedance (Z), constant-current (I)
and constant-power (P) coefficients for active and reactive ZIP loads on bus
$i$. Constraint (1g) limits the reactive power output $q^{g}_{i,\phi,t}$ of
the PV inverters by the available reactive power capacity $Q_{i,\phi,t}^{\rm
cap}$. Constraint (1h) calculates $Q_{i,\phi,t}^{\rm cap}$ with the total
capacity of the PV inverter $S^{\rm cap}_{i,\phi,t}$ and the active power
output of PV inverter $p^{g}_{i,\phi,t}$. Based on IEEE 1547-2018 Standard
[19], the PV inverters can provide reactive power injection or absorption
$q^{g}_{i,\phi,t}$ to achieve fast voltage regulation. In this work, we focus
on proposing a CVR model by optimally controlling the injection or absorption
of reactive power in the PV inverters against the uncertainties of loads and
renewable powers. While the dispatches of the on-load tap changers (OLTCs) and
the capacitor banks (CBs) are slow and limited by a certain number of
switching operation, which cannot response the uncertainties of loads and
renewable powers. To consider the impacts of those conventional voltage
regulation devices on CVR performance, a hierarchical control method [4] and
[20] can be easily implemented to coordinate PV inverters, OLTC and CBs from
different control stages. Note that the coordination between PV inverters,
OLTC and CBs is out of the scope of this paper. In constraint (1i), the
squared bus voltage magnitude $v_{i,\phi,t}$ is limited by $v_{\rm min}$ and
$v_{\rm max}$, which are typically $[0.95^{2},1.05^{2}]$ p.u., respectively.

The deterministic VVO-CVR problem (1) has an underlying assumption that the
load and PV generation predictions are perfect, which means $p^{\rm
L}_{i,\phi,t}$, $q^{\rm L}_{i,\phi,t}$ in constraints (1e) and (1f), and
$p^{\rm g}_{i,\phi,t}$ in constraint (1h) are predefined constant parameters.
The more realistic setting is to take the load and PV generation prediction
errors into account. To do this, we can replace the deterministic parameters
$p^{\rm L}_{i,\phi,t}$, $q^{\rm L}_{i,\phi,t},p^{\rm g}_{i,\phi,t}$ by
uncertainty variables. Particularly, we introduce the regularized uncertainty
variables $p^{\rm L,\xi}_{i,\phi,t}\in[0,1]$, $q^{\rm
L,\xi}_{i,\phi,t}\in[0,1]$ to replace deterministic load multipliers $p^{\rm
L}_{i,\phi,t}$, $q^{\rm L}_{i,\phi,t}$, as shown in (2a) and (2b). We reserve
the super-script $\xi$ to define the random variables, which also apply to the
rest of the definition blow. Then, we introduce an auxiliary variables
$\alpha_{i,\phi,t}^{\rm q}\in[-1,1]$, which represents the ratio between
reactive power output $q^{\rm g}_{i,\phi,t}$ and reactive power capacity.
Here, we use $Q_{i,\phi,t}^{\rm cap,\xi}$ to represent the square root term
$\sqrt{(S^{\rm cap}_{i,\phi,t})^{2}-(p^{\rm g,\xi}_{i,\phi,t})^{2}}$, so that
the constraint (1h) can be reformulated as constraint (2c).

$\displaystyle p^{\rm ZIP}_{i,\phi,t}$ $\displaystyle=p^{\rm
L,\xi}_{i,\phi,t}\odot\big{(}k_{i,1}^{p}\cdot
v_{i,\phi,t}+k_{i,2}^{p}\cdot\sqrt{v_{i,\phi,t}}+k_{i,3}^{p}\big{)},$ (2a)
$\displaystyle q^{\rm ZIP}_{i,\phi,t}$ $\displaystyle=q^{\rm
L,\xi}_{i,\phi,t}\odot\big{(}k_{i,1}^{q}\cdot
v_{i,\phi,t}+k_{i,2}^{q}\cdot\sqrt{v_{i,\phi,t}}+k_{i,3}^{q}\big{)},$ (2b)
$\displaystyle q^{\rm g}_{i,\phi,t}$ $\displaystyle=\alpha_{i,\phi,t}^{\rm
q}Q_{i,\phi,t}^{\rm cap,\xi},\forall i\in\mathcal{G}.$ (2c)

We can define a uncertainty variable vector
$\xi_{i,\phi,t}=[(p_{i,\phi,t}^{L,\xi})^{\top},(q_{i,\phi,t}^{L,\xi})^{\top},(p^{\rm
g,\xi}_{i,\phi,t})^{\top},(Q_{i,\phi,t}^{\rm cap,\xi})^{\top}]^{\top}$ to
include all the uncertainty variables. To be simplified, we avoid the indices
of ${i,\phi,t}$ in vector $\xi$. To consider the impacts of uncertainty $\xi$
on voltage regulation performance, we can extend the deterministic
maximum/minimum voltage constraint (1i) to two chance constraints (3a) and
(3b) as follows:

$\displaystyle\mathbb{P}\\{v-v_{\textrm{max}}\leq 0\\}$ $\displaystyle\geq
1-\epsilon,$ (3a) $\displaystyle\mathbb{P}\\{-v+v_{\textrm{min}}\leq 0\\}$
$\displaystyle\geq 1-\epsilon.$ (3b)

where $\epsilon$ is a pre-defined risk level of failing to satisfy bus voltage
constraint against uncertainties in $\xi$. To further make the solution robust
to a group of probability distributions with controllable conservativeness, we
introduce DRO and an ambiguity set of uncertainty to chance-constrained CVR
and compactly formulate a DRCC-CVR problem (4a)-(4c) as follows:

$\displaystyle\underset{x}{\min}\hskip
5.69054pt\underset{\xi\sim\mathbb{P}\in\mathcal{P}}{\max}E_{\mathbb{P}}\\{f(x,\xi)\\},$
(4a) s.t. $\displaystyle g_{1}(x)\leq 0,$ (4b)
$\displaystyle\mathbb{P}\left\\{g_{2}(x,\xi)\leq 0\right\\}\geq 1-\epsilon,$
(4c)

where $x$ represents the decision variable vector (i.e. reactive power
dispatch of PV inverter) and $\xi\sim\mathbb{P}\in\mathcal{P}$ in objective
(4a) means that the uncertainty variable vector $\xi$ following the
distribution $\mathbb{P}$ within an ambiguity set of distributions
$\mathcal{P}$. Constraints (1b)-(1h) can be represented by the compact
constraint (4b) and the chance constraints (3a) and (3b) can be represented by
the compact constraint (4c).

###  II-B Our Proposed Method

The challenges of solving this DRCC-CVR problem (4a)-(4c) can be summarized as
follows: (i) the nonlinear power flow model and voltage-dependent ZIP load
model, and the chance constraints with the random variables make the DRCC-CVR
problem (4a)-(4c) intractable to be solved; and (ii) even though we can
reformulate the DRCC problem in a tractable way, the limited access to the
high-resolution load and PV data will lead to an ill-posed DRCC-CVR problem
and hinder the performance of CVR implementation. To address those challenges,
we propose a solution to address these challenges, as shown in Fig. 1, which
includes the following two-fold:

Figure 1: Overall framework of the tractable DRCC-CVR with data enrichment
method and enriched data-based moment-based ambiguity set.

1) Tractable DRCC-CVR model: To reformulate a tractable DRCC-CVR model, we
leverage the linearized Distflow model [3] and linearized voltage-dependent
ZIP load model with Binominal Approximation method [15]. Then, we reformulate
the chance constraints of voltage by a tractable DRCC model with a moment-
based ambiguity set. Compared to other types of DRO, the DRCC with a moment-
based ambiguity set has higher computational efficiency for tractable
reformulation. We will show the linearized version of the power flow model and
voltage-dependent ZIP load model, as well as the tractable reformulation of
the DRCC-CVR problem in Section III.

2) Data enrichment method and moment-based Ambiguity set: As shown in Fig. 1,
there are a large number of service transformers that can collect low-
resolution load and PV generation data by SMs, while only a few service
transformers are installed with micro-PMUs with access to high-resolution load
and PV generation data. To capture the uncertainty of load and PV generation,
we use SMs and micro-PMUs to collect load and PV generation data. Then we
enrich the load and PV generation data and extract the corresponding moment
information of probability distributions from the enriched load and PV data.
Finally, we can construct the ambiguity set with the first two moment
information i.e., mean and variance, and implement the ambiguity set in our
proposed DRCC-CVR model. The purpose of introducing the data enrichment method
in a data-based ambiguity set is to avoid potential over- or under-
conservativeness. We will show the data-enrichment method and the construction
of a moment-based ambiguity set for DRCC-CVR in Section IV.

##  III Tractable Reformulation of DRCC-CVR Model

This section presents the linearized version of the power flow model and
voltage-dependent ZIP load model, and reformulates a tractable version of the
DDCC-CVR problem.

###  III-A Linearized Reformulation of Power Flow and Voltage-Dependent ZIP
Loads

In power flow constraints (1b)-(1d), the nonlinear terms
$\varepsilon_{ij,\phi,t}^{p}$, $\varepsilon_{ij,\phi,t}^{p}$ and
$\varepsilon_{i,\phi,t}^{v}$ make the optimization problems non-convex and NP
hard. In practice, those nonlinear terms are much smaller than the linear
terms in power flow constraints (1b)-(1d). Therefore, the constraints
(1b)-(1d) can be reformulated as constraints (5a)-(5c) with linearized
Distflow model by neglecting those nonlinear terms.

$\displaystyle P_{ij,\phi,t}$ $\displaystyle=\sum_{k:j\rightarrow
k}P_{jk,\phi,t}-p^{g}_{j,\phi,t}+p_{j,\phi,t}^{\rm ZIP},$ (5a) $\displaystyle
Q_{ij,\phi,t}$ $\displaystyle=\sum_{k:j\rightarrow
k}Q_{jk,\phi,t}-q^{g}_{j,\phi,t}+q_{j,\phi,t}^{\rm ZIP},$ (5b) $\displaystyle
v_{j,\phi,t}$ $\displaystyle=v_{i,\phi,t}-2\big{(}\bar{r}_{ij}\odot
P_{ij,\phi,t}+\bar{x}_{ij}\odot Q_{ij,\phi,t}\big{)}.$ (5c)

This linear form of DistFlow has been verified in many previous studies, such
as [3]. The nonlinear term $\sqrt{v_{i,\phi,t}}$ of ZIP loads also introduces
non-convexity to the problem. Because the voltage magnitudes of all buses in a
distribution network stay close to 1 p.u. under normal operating conditions
[3] and [4], the active and reactive ZIP loads can be linearized by Binominal
Approximation Method [15]. Therefore, the squared deviation of voltage
$(\Delta V)^{2}$ is very small, so it can be neglected. Then we have the
following approximations (III-A) of squared voltage magnitude, as follows:

$\displaystyle v$ $\displaystyle=V\odot V=(1+\Delta V)\odot(1+\Delta V)\approx
1+2\Delta V,$ $\displaystyle v$ $\displaystyle=1+\Delta v\approx 1+2\Delta V,$
$\displaystyle\Delta v$ $\displaystyle\approx 2\Delta V,$ (6)

where $v$ and $\Delta v$ are the vectors of squared voltage magnitude and the
derivation from the nominal value, respectively; $V$ and $\Delta V$ are the
vectors of voltage magnitude and the derivation from the nominal value,
respectively. By introducing equation (III-A) and $\sqrt{v}=V=(1+\Delta V)$ to
equations (2a) and (2b), we have the linear approximation of voltage-dependent
active and reactive ZIP loads as follows:

$\displaystyle p^{\rm ZIP}_{i,\phi,t}$ $\displaystyle\approx p^{\rm
L,\xi}_{i,\phi,t}\odot\big{(}(k_{i,1}^{p}+\frac{k_{i,2}^{p}}{2})v_{i,\phi,t}+(k_{i,3}^{p}+\frac{k_{i,2}^{p}}{2})\big{)},$
(7a) $\displaystyle q^{\rm ZIP}_{i,\phi,t}$ $\displaystyle\approx q^{\rm
L,\xi}_{i,\phi,t}\odot\big{(}(k_{i,1}^{q}+\frac{k_{i,2}^{q}}{2})v_{i,\phi,t}+(k_{i,3}^{q}+\frac{k_{i,2}^{q}}{2})\big{)}.$
(7b)

###  III-B Tractable Reformulation of DRCC-CVR with Load and Renewable
Uncertainties

To achieve the tractable reformulation of DRCC-CVR, the power flow constraints
(1b)-(1d) can be compactly formulated as follows:

$\displaystyle-AP$ $\displaystyle=p,$ (8a) $\displaystyle-AQ$
$\displaystyle=q,$ (8b) $\displaystyle-A_{0}v_{0}-A^{\top}v$
$\displaystyle=-2D_{r}P-2D_{x}Q,$ (8c)

where $A_{0}$ and $A$ are the incidence matrices of unbalanced radial
distribution network, $A_{0}$ represents the connection structure between
substation (the feeder head bus) and each of the line segments in
$\mathcal{E}$, $A$ represents the connection structure between the remaining
buses and each of the line segment in $\mathcal{E}$. $v_{0}$ is vector of
square nominal voltage magnitudes. $D_{r}=blkdiag[R_{bp(1)1},...,R_{bp(N)N}]$
and $D_{x}=blkdiag[X_{bp(1)1},...,X_{bp(N)N}]$ are block diagonal matrices of
line segment resistance and reactance, respectively. In equations (8a) and
(8b), the nodal active and reactive power injections can be calculated based
on ZIP loads and PV generations. Based on the compact power flow formulations
(8a)-(8c), we have the compact formulation (9) to represent the relationship
between bus voltage $v$ and bus power injections $p$ and $q$, as follows:

$v=Rp+Xq+\tilde{v},$ (9)

with

$\displaystyle R$ $\displaystyle=2[A^{\top}]^{-1}D_{r}A^{-1},$ $\displaystyle
X$ $\displaystyle=2[A^{\top}]^{-1}D_{r}A^{-1},$ $\displaystyle\tilde{v}$
$\displaystyle=-[A^{\top}]^{-1}A_{0}v_{0}.$

By introducing PV inverter reactive power output equation (2c), linearized ZIP
load equations (7a) and (7b) as bus power injections $p$ and $q$ into the
compact formulation (9), we have an equation as shown in equation (10), which
represents the relationship between vector of squared voltage $v$ and all the
uncertainty variables in vector $\xi$.

$\displaystyle
v=R\left(p^{L,\xi}\big{(}(k_{1}^{p}+\frac{k_{3}^{p}}{2})v+(k_{3}^{p}+\frac{k_{2}^{p}}{2})\big{)}-p^{\rm
g,\xi}\right)+X\left(q^{L,\xi}\big{(}(k_{1}^{q}+\frac{k_{3}^{q}}{2})v+(k_{3}^{q}+\frac{k_{2}^{q}}{2})\big{)}-\alpha^{\rm
q}Q^{\rm cap,\xi}\right)+\tilde{v},$ (10)

If we select an appropriate value as a power base and use per unit to
represent $p^{L,\xi}$ and $q^{L,\xi}$, because the load level of distribution
systems is usually not heavy at kW level, then we have $p^{L,\xi}$ and
$q^{L,\xi}$ are much smaller than identity matrix $I$, the network resistance
and reactance matrices $R,X$ are also small in per unit, thus the equation
(11) is valid.

$I-p^{L,\xi}\left(Rk_{1}^{p}+R\frac{k_{2}^{p}}{2}\right)-q^{L,\xi}\left(Xk_{1}^{q}+X\frac{k_{2}^{q}}{2}\right)>0.$
(11)

After we introduce the equation (10) into the deterministic constraint (1g) on
bus voltages, and because equation (11) is valid, we can obtain equation (12),
which represents the impacts of load and PV generation uncertainties on bus
voltage constraints (1i) with $v_{min}$ and $v_{max}$. For simplicity, we also
avoid the indices of $i,\phi,t$ in equations (10) and (12).

$\displaystyle
v_{\textrm{min}}\leq\frac{p^{L,\xi}\left(Rk_{3}^{p}+R\frac{k_{2}^{p}}{2}\right)+q^{L,\xi}\left(Xk_{3}^{q}+X\frac{k_{2}^{q}}{2}\right)+Rp^{\rm
g,\xi}+X\alpha^{\rm q}Q^{\rm
cap,\xi}+\tilde{v}}{I-p^{L,\xi}\left(Rk_{1}^{p}+R\frac{k_{2}^{p}}{2}\right)-q^{L,\xi}\left(Xk_{1}^{q}+X\frac{k_{2}^{q}}{2}\right)}\leq
v_{\textrm{max}},$ (12)

The compact formulation (4c) can be reformulated in a linear form
$a(x)^{\top}\xi+b(x)\leq 0$. We can obtain the formulations of $a(x)$ and
$b(x)$ for chance constraints (3a) and (3b) by introducing the equation (12)
into chance constraints (3a) and (3b). Therefore, the $a(x)$ and $b(x)$ of
chance constraint (3a) can be formulated as (13a) and (13b), respectively.

$\displaystyle a(x)$
$\displaystyle=\begin{bmatrix}\textrm{diag}(v_{\textrm{max}})\left(Rk_{1}^{p}+R\frac{k_{2}^{p}}{2}\right)+\left(Rk_{3}^{p}+R\frac{k_{2}^{p}}{2}\right)\\\
\textrm{diag}(v_{\textrm{max}})\left(Xk_{1}^{q}+X\frac{k_{2}^{q}}{2}\right)+\left(Xk_{3}^{q}+X\frac{k_{2}^{q}}{2}\right)\\\
-R\\\ -Xa_{j}\end{bmatrix},$ (13a) $\displaystyle b(x)$
$\displaystyle=\tilde{v}-v_{\textrm{max}}.$ (13b)

Similarly, the $a(x)$ and $b(x)$ for another chance constraint (3b) can be
formulated as (14a) and (14b), respectively.

$\displaystyle a(x)$
$\displaystyle=\begin{bmatrix}-\textrm{diag}(v_{\textrm{min}})\left(Rk_{1}^{p}+R\frac{k_{2}^{p}}{2}\right)-\left(Rk_{3}^{p}+R\frac{k_{2}^{p}}{2}\right)\\\
-\textrm{diag}(v_{\textrm{min}})\left(Xk_{1}^{q}+X\frac{k_{2}^{q}}{2}\right)-\left(Xk_{3}^{q}+X\frac{k_{2}^{q}}{2}\right)\\\
R\\\ Xa_{j}\end{bmatrix},$ (14a) $\displaystyle b(x)$
$\displaystyle=-\tilde{v}+v_{\textrm{min}}.$ (14b)

A moment-based ambiguity set of load and PV generation uncertainties can be
constructed in (15), as follows:

$\mathcal{D}_{\xi}=\left\\{\xi\sim\mathbb{P}\in\mathcal{P}:\mathbb{E}_{\mathbb{P}_{\xi}}[\xi]=\mu,\mathbb{E}_{\mathbb{P}_{\xi}}[\xi\xi^{T}]=\Sigma\right\\}$
(15)

where $\mu$ and $\Sigma$ represent the mean and covariance of the uncertain
variables of load and PV generation. Finally, based on equations (13a)-(14b)
for $a(x)$ and $b(x)$, we can obtain a second-order conic reformulation (16)
for the DRCC (3a) and (3b) with moment information, mean $\mu$ and covariance
$\Sigma$ of uncertainty variable vector $\xi$, as follows: [21] and [22]:

$a(x)^{\top}\mu+b(x)+\sqrt{\frac{1-\epsilon}{\epsilon}}||\Sigma^{\frac{1}{2}}a(x)||_{2}\leq
0.$ (16)

Even though the mean $\mu$ and covariance $\Sigma$ of load and PV generation
in (16) can be extracted from recorded data of SMs and micro-PMUs, the reality
is that we only have limited access to high-resolution load and PV generation
data. The limited data will lead to the potential ill-posed of DRCC (16), and
further hinder the performance of DRCC-CVR. Therefore, we introduce a data
enrichment method for high-resolution data recovery and ambiguity set
construction in Section IV.

##  IV Data Enrichment Method And Moment-Based Ambiguity Set

This section presents the data enrichment method to recover high-resolution
data of load and PV generation for those service transformers with only SMs.
Then, the ambiguity set with moment information of probability distributions
is constructed based on the enriched load and PV generation data.

###  IV-A Data Enrichment Method with Micro-PMU and SM Data

As shown in Fig. 1, the majority of service transformers are installed with
SMs to record low-resolution load and PV generation data, and only a few
service transformers are also installed with micro-PMU to record high-
resolution data. To enrich the data of load and PV generation, we consider the
service transformers with micro-PMU data as a teacher repository and the
service transformers with only SM data as a student repository. The teacher
service transformers train two models capturing the statistical relationship
between high-resolution data and low-resolution data. The trained models are
utilized to perform data enrichment for those service transformers with only
SMs. There are four main steps:

Step. I Train the load/PV generation data maximum and minimum boundary
inference models: The first step is to train probabilistic models using known
high-resolution load/PV generation data from teacher service transformers with
micro-PMUs. The Gaussian Process Regression (GPR) technique [18] is used to
capture the relationship between the maximum/minimum bounds and the average
values for load/PV generation. More specifically, two GPR models
$GPR^{*}_{s,1}$ and $GPR^{*}_{s,2}$ are trained for the $s$-th teacher service
transformer:

$\displaystyle GPR^{*}_{s,1}$ $\displaystyle:P_{a}(t)\to\overline{P}(t),$
(17a) $\displaystyle GPR^{*}_{s,2}$
$\displaystyle:P_{a}(t)\to\underline{P}(t),$ (17b)

where $P_{a}(t)$ denotes the average load/PV generation over the $t$-th hour,
$\overline{P}(t)$ and $\underline{P}(t)$ denote the upper and lower bounds of
instantaneous load/PV generation within the $t$-th hour, respectively.

Step. II Train the load/PV generation variability inference models: The second
step is to model the probabilistic transition of instantaneous load/PV
generation within their maximum and minimum bounds using the second-order
Markov Chain (MC) model. Specifically, one MC model $MC^{*}_{s}$ is trained
for each service transformer:

$MC^{*}_{s}:\\{P_{t}(m-2),P_{t}(m-1)\\}\to P_{r}(P_{t}(m)),$ (18)

where $P_{t}(m-2)$, $P_{t}(m-1)$, and $P_{t}(m)$ denote the $(m-2)$-th,
$(m-1)$-th, and $m$-th high-resolution load/PV generation samples within the
$t$-th hour. Formulation (18) outputs the probability of $P_{t}(m)$ based on
$P_{t}(m-2)$ and $P_{t}(m-1)$. Note that Step. I and II build the load/PV
generation data boundary inference models and the load/PV generation data
variability inference models with a small number of high-resolution micro-PMU
data of teacher service transformers. Steps. III and IV will extend the
trained load/PV generation probabilistic models to service transformers with
only SMs, so it recovers the high-resolution load/PV generation data masked by
the low-resolution load/PV generation data.

Step. III Determine the learning weights of student service transformers with
respect to teacher service transformers: The third step evaluates the low-
resolution data similarity between the teacher with micro-PMUs and student
service transformers with SMs by determining the learning weights as shown in
(19a) and in (19b). The weights $W_{s}$ and $W^{\prime}_{s}$ can represent the
confidence of a student service transformer to learn from multiple teacher
service transformers for enriching the low-resolution load/PV generation data
to high-resolution load/PV generation data.

$\displaystyle W_{s}$
$\displaystyle=\frac{W^{\prime}_{s}}{\sum^{N_{t}}_{s=1}W^{\prime}_{s}},$ (19a)
$\displaystyle W^{\prime}_{s}$
$\displaystyle=\frac{1}{N_{c}N^{s}_{c}}\sum^{N_{c}}_{i=1}\sum^{N^{s}_{c}}_{j=1}||P_{i}-P_{j}^{s}||,s=\\{1,...,N_{t}\\},$
(19b)

where $N_{c}$ and $N_{c}^{s}$ denote the number of customers served by a
student service transformer and the the $s$-th teacher service transformer,
respectively. We can obtain $N_{c}$ daily load/PV generation patterns for that
service transformer, $\\{{P}_{1},\cdots,{P}_{N_{c}}\\}$. Similarly, the
load/PV generation patterns for the $s$-th teacher service transformer are
denoted by $\\{{P}_{1}^{s},\cdots,{P}_{N_{c}^{s}}^{s}\\},s=1,\cdots,N_{t}$.
The weights are used to linearly combining the estimated bounds in (17) and
the probabilistic transition matrices in (18).

Step. IV Extend the trained load/PV generation data probabilistic model: The
fourth step extends the trained probabilistic models of teacher service
transformers in Steps. I and II to student service transformers that only have
SMs for enriching low-resolution load/PV generation data. Specifically, the
$m$-th high-resolution load/PV generation sample is randomly generated based
on the following Bernoulli distribution:

${P}_{t}(m)\sim Be(Pr({P}_{t})),$ (20)

where $Be$ denotes the Bernoulli distribution [18]. Note that for each hourly
load/PV generation sample, (20) can give us $N$’ high-resolution load/PV
generation samples , i.e., $P_{t}(m),m=1,\cdots,N^{\prime}$. Then, the
enriched high-resolution data samples are employed to optimize the mean and
standard deviation of Gaussian distribution using maximum likelihood
estimation:

$(\mu^{*},\Sigma^{*})=\underset{\mu,\Sigma}{argmin}\;\,\prod_{m=1}^{N^{\prime}}f(\mu,\Sigma;P_{t}(m)),$
(21)

where, $f(\cdot)$ denotes the probability density function of Gaussian
distribution. Therefore, the mean and covariance of load and PV generation
uncertainties $\mu$ and $\Sigma$ can be execrated from enriched data in (21)
for the construction of the moment-based ambiguity set (15). i.e., substitute
$\mu^{*}$ and $\Sigma^{*}$ in (21) into (16).

###  IV-B Enriched Data-Based Ambiguity Set and DRCC-CVR Method

The enriched high-resolution data from (20) can recover instantaneous
uncertainties of load and PV generation, which can be further extracted for
the two moment information and construct moment-based ambiguity set in (15).
By considering the data-based moment-based ambiguity set and the reformulation
of tractable DRO and chance constraints, the DRCC-CVR problem can be compactly
reformulated as follows:

$\displaystyle\underset{\xi\in\mathcal{D}_{\xi}}{\text{Objective}}\hskip
5.69054pt(1),$ (22a) s.t. $\displaystyle\text{PV
generations}\left\\{(2c)\right\\},$ (22b) $\displaystyle\text{OPF
constraints}\left\\{\text{(}5)\right\\},$ (22c) $\displaystyle\text{Linearized
ZIP loads}\left\\{(7)\right\\},$ (22d) $\displaystyle\text{Tractable
reformulation of DRCC}\left\\{\text{(}13)-(14),(16)\right\\}.$ (22e)

Note that problem (22a)-(22e) is a tractable linear programming problem, where
the mean values of load and PV are used in compact constraints (22b)-(22d) and
the mean and covariance values of load and PV are used in the compact
constraint (22e) of DRCC reformulation. To summarize the above steps, the
detailed procedure of the proposed DRCC-CVR with an enriched-based ambiguity
set of the uncertainties of load and PV generation is shown in Algorithm 1.

Algorithm 1 DRCC-CVR Model with Enriched-based Ambiguity Set of Uncertainty of
Load and PV Generation

1:Input: High-resolution data from micro-PMUs and low-resolution data from SMs

2:Initialization: Choose hyper-parameters in DRCC-CVR

3:For: $i=1,2,...,N$.

4:Train load/PV generation upper and lower boundary inference models from
teacher service transformers with micro-PMUs by (17a) and (17b).

5:Train load/PV generation variability inference model from teacher service
transformers with micro-PMUs by (18).

6:Determine the learning weights of student service transformers with respect
to teacher service transformers in (19a) and (19b).

7:Extend the trained load/PV generation data to student service transformers
with SMs in (20).

8:End for.

9:Extract the first two moment information of load and PV generation
uncertainties from enriched data in (21)

10:Construct an ambiguity set with the first two moment information in (15).

11:Solve the DRCC-CVR problem (22) with objective (1) and constraints
(2c),(5),(7), (13)-(14), and (16).

12:Output: Reactive power dispatches of PV inverters

##  V Case Studies

This section presents the simulation results, including the enriched data of
load and PV generation for ambiguity set, comparison results of the benchmark
methods and the proposed DRCC-CVR method, and the impacts of hyper-parameter
on the proposed DRCC-CVR method.

###  V-A Simulation Setup

A real-world distribution feeder [23] in Fig. 2 is used to test our proposed
DRCC-CVR method, which is located in Midwest U.S. and shared by our utility
partner. The reason for choosing this real-world distribution feeder as our
test system is that the service transformers of customers in this feeder are
either equipped with SMs or micro-PMUs, which can record low-resolution
(1-hour) data and high-resolution (1-second) data to construct an ambiguity
set of load and PV generation uncertainties for our DRCC-CVR method. More
information on this real-world distribution feeder and the data from SMs and
micro-PMUs can be found in [23]. In Fig. 2, the yellow bot-ted boxes represent
the buses’ service transformers installed with micro-PMUs, and the rest buses’
service transformers are installed with SMs; the blue dots represent the buses
installed with single-phase or three-phase PV generators; the solid, dashed
and dotted lines represent three-phase overhead lines, three-phase underground
cables and single-phase overhead lines, respectively. In this real-world
distribution feeder, the total capacity of PVs can serve around 20% to 30%
load. Adopted from our industrial partner [4], we use the following
coefficients $[k_{1}^{p},k_{2}^{p},k_{3}^{p}]=[0.96,-1.17,1.21]$ and
$[k_{1}^{q},k_{2}^{q},k_{3}^{q}]=[6.28,-10.16,4.88]$ for active and reactive
ZIP loads in our test cases. The base voltage and base power values are 13.8
kV and 100 kVA. The prescribed risk level parameter $\epsilon$ is set to 0.05
for quantifying the 5% violation probability of chance constraints in our
proposed DRCC-CVR model.

Figure 2: A real distribution feeder in Midwest U.S. [23]

We demonstrate the advantages and effectiveness of the enriched data-based
ambiguity set and the proposed DRCC-CVR method through numerical comparisons
of several benchmark methods. The following simulations are built-in MATLAB
R2019b, which integrate YALMIP Toolbox with IBM ILOG CPLEX 12.9 solver for
optimization. All case studies are simulated on a PC with Intel Core i7-4790
3.6 GHz CPU and 16 GB RAM.

###  V-B Original and Enriched Data of Load and PV Generation

As shown in Fig. 2, there are 8 service transformers installed with micro-PMUs
and the rest 34 service transformers installed with SMs, which can record
high-resolution (1-second) and low-resolution (1-hour) load and PV generation
data, respectively. In this sub-section, we use two ways to obtain the mean
and covariance of the uncertainty variables of load and PV generation for
ambiguity sets: (i) we use a statistical method to obtain mean and variance
information of the original load and PV generation data (few high-resolution
data from 8 micro-PMUs and a lot of low-resolution data from 34 SMs). (ii) We
use the data enrichment method to enrich the original load and PV generation
data, then we obtain mean and variance information of the enriched data. To
verify the effeteness of the data enrichment method, we show the empirical
distributions and their according fitted Gaussian distributions of active
load, reactive load, and active power output of PV generation in Fig. 3 and
Fig. 4, respectively. Note that the distributions in the right column and left
column of Fig. 3 and Fig. 4 are obtained from original data and enriched data,
respectively. In Fig. 3 and Fig. 4, each dot denotes the hourly average power
corresponding to the data samples, which means the hourly power is the average
of observations. It can be observed that the distributions obtained from
enriched data have a highly similar fitting to the distributions obtained from
original data. Therefore, the first two moment information extracted from
enriched data can also accurately construct the ambiguity set for the DRCC-CVR
method.

Figure 3: Empirical distribution and fitted Gaussian distribution: (a) active
load (original data); (b) active load (enriched data); (c) reactive load
(original data); (d) reactive load (enriched data). Figure 4: Empirical
distribution and fitted Gaussian distribution: (a) PV generation (original
data); (b) PV generation (enriched data).

###  V-C Voltage Reduction and Power/energy-saving Through CVR Implementation

To serve as a reference to investigate the performance of VVO-CVR, a base case
is firstly built by setting the unity-power factor control model for all PV
generators, which means there is no reactive power support from PV generators.
In this sub-section, we use three ways to implement VVO-CVR: (i) The
deterministic VVO-CVR (Deter-CVR) is solved by deterministic optimization,
where the load and PV prediction errors and uncertainties are neglected. (ii)
The robust VVO-CVR (RO-CVR) is solved by the robust optimization [7] and [8],
where the uncertainties of load and PV generation are considered with 10%
variance from the predictions. (iii) The VVO-CVR is solved by our proposed
DRCC-CVR with an enriched data-based ambiguity set of load and PV generation
uncertainties. The performance through CVR implementation can be evaluated
from three aspects: voltage profile, active power supply from the substation,
and total energy consumption. To show the time-series simulation, the VVO-CVR
is performed in a daily operation of the real-world distribution feeder with
different control strategies. In Fig. 5, the voltage profiles for a selected
bus (bus 23 on phase c) are shown, which are generated from the base case
(without control) and the proposed DRCC-CVR. In Fig. 5, the blue bar and red
bar represent the voltage profiles of the base case and DRCC-CVR,
respectively. It can be observed that all the nodal voltages can maintain
within the predefined range [0.95,1.05] p.u., while the voltage profiles of
DRCC-CVR are overall lower than the voltage profiles of the base case. Because
the DRCC-CVR can optimally dispatch the reactive power from PV inverters to
achieve maximum voltage reduction while still satisfying the voltage
constraints. During the midnight period from 00:00 to 6:00, the voltage
reduction of DRCC-CVR is obviously higher than the voltage reduction of the
base case, as shown in the circled part in Fig. 5. It is because the active
power outputs of PV generators from 00:00 to 6:00 are nearly zero, and
according to the calculations of reactive power output and capacity in
constraints (1g) and equation (1h), the DRCC-CVR has more reactive power
supports to maximize voltage reductions.

Figure 5: Voltage profiles on selected bus 23 on phase c with different
control strategies.

The active power supplies from the substation of the base case, Deter-CVR, RO-
CVR, and DDRC-CVR are shown as different curves in Fig. 6, which represent the
overall active power consumption of the base case, and those VVO-CVR
benchmarks. It can be observed that the proposed DDRC-CVR can effectively
reduce the power supply from the substation compared to the base case and
other methods. In comparison the power saving of RO-CVR is less than the
proposed DDRC-CVR and only slightly better than Deter-CVR. On the one hand,
the proposed DDRC-CVR has an ambiguity set of uncertainties to balance the
trade-off between the conservativeness of decisions and operational
efficiency. On the other hand, RO-CVR is too conservative and hinders the
performance of CVR implementation. The numerical comparisons of total energy
consumption over one day and the energy reduction percentage are presented in
Table I among base case, Deter-CVR, RO-CVR, and DDRC-CVR. The total energy
consumption of base case, Deter-CVR, RO-CVR, and DRCC-CVR are 958.045 kWh,
944.048 kWh, 934.178 kWh, and 898.616 kWh, respectively. Therefore, compared
to the original energy consumption in the base case, the Deter-CVR, RO-CVR,
and DRCC-CVR can achieve 1.461%, 2.491%, and 6.203% of energy savings,
respectively. The computational times of Deter-CVR, RO-CVR, and DRCC-CVR for
this test system are 12.968 seconds, 18.312 seconds, and 21.911 seconds,
respectively. According to the above results in Fig. 5, Fig. 6 and Table I, we
can summarize the differences between Deter-CVR, RO-CVR, and DRCC-CVR: (i)
Among all the methods, DRCC-CVR can achieve the lowest total energy
consumption and highest energy-saving. While the differences between the total
energy consumption and energy-saving results of Deter-CVR and RO-CVR are
small, because Deter-CVR does not consider the errors of load and PV
generation predictions, and RO-CVR provides a conservative solution. Both
Deter-CVR and RO-CVR cannot fully explore the benefit of CVR implementation,
while the proposed DDRC-CVR can better explore the potential of CVR
implementation. (ii) Because of the different ways of handling uncertainties
of load and PV generation, the computation time of DRCC-CVR is slightly slower
than Deter-CVR and RO-CVR. However, their differences in computational time
are very small, which can be neglected for a day-ahead operational
application. Thus the computational efficiency of the DRCC-CVR can be
acceptable.

Figure 6: Substation feed-in active power with different control strategies. TABLE I: Energy Consumption and energy-saving Results With Different Control Strategies | Energy (kWh) | Reduction (%) | Computation (sec)
---|---|---|---
Base Case | 958.045 | - | -
Deter-CVR | 944.048 | 1.461% | 12.968
RO-CVR | 934.178 | 2.491% | 18.312
DRCC-CVR | 898.616 | 6.203% | 21.911

###  V-D Impact of Hyper-Parameters in Performance of DRCC-CVR

It is obvious that more micro-PMUs installed in service transformers can
record more high-resolution data of load and PV generation. Also, more high-
resolution data can be beneficial for recording the uncertainties of load and
PV generation, which is helpful for the data enrichment method and
construction of ambiguity set in DDRC-CVR. To show the impact of micro-PMUs on
the proposed DRCC-CVR, the total energy consumption and energy-saving of DRCC-
CVR with the different numbers of micro-PMUs are presented in Table II. There
are three different tests: the first test only constructs the ambiguity set
based on low-resolution data from SMs without any high-resolution data from
micro-PMU; the second test implements the data enrichment method with high-
resolution data from 4 micro-PMUs and low-resolution data from SMs, then the
ambiguity set is constructed based on the enriched data; the third test has
the high-resolution data from 8 micro-PMUs as the same setting in Section V-B
and Section V-C. It can be observed that a higher number of micro-PMUs can
help to construct a high-quality ambiguity set for DRCC-CVR and achieve better
performance of CVR implementation through DRCC-CVR.

TABLE II: Energy Consumption and energy-saving Results With Different Number of Micro-PMUs. | Number of
---
micro-PMUs
| Location of
---
micro-PMUs
| Energy
---
(kWh)
| Saving
---
(%)
0 (only SM data) | - | 933.889 | 2.521%
4 |  | $B_{18},B_{25},B_{41},B_{60}$
---
927.817 | 3.155%
8 |  | $B_{18},B_{25},B_{31},B_{41}$
---
$B_{48},B_{52},B_{56},B_{60}$
898.618 | 6.203%

The pre-defined risk level $\epsilon$ is selected as 0.05 in the above
simulation tests. While the different values of $\epsilon$ will influence the
confidence level on chance constraints in the proposed DRCC-CVR and further
influence the benefits of CVR implementation. Therefore, we also test the
DRCC-CVR with three different violation rates (i.e. 0.02, 0.05, 0.1), as shown
in Table III. It can be observed that a larger value of $\epsilon$ leads to
lower energy consumption and a higher energy reduction, which will benefit the
CVR implementation. However, this kind of benefit is achieved by increasing
the violation rate and sacrificing the reliability of operational constraints.
For example, the energy reduction ($6.606\%$) of $\epsilon=0.1$ is only
slightly higher than the energy reduction ($6.203\%$) with $\epsilon=0.1$, but
the risk of constraint violation also become higher when $\epsilon$ increases
from $0.05$ to $0.1$. The results indicate that there is a trade-off between
the maximization of CVR benefit and the reliability of operational constraints
in the proposed DRCC-CVR.

TABLE III: Energy Consumption and energy-saving Results With Different Violation Rates | Energy (kWh) | Reduction (%)
---|---|---
$\epsilon=0.02$ | 904.980 | 4.803%
$\epsilon=0.05$ | 898.616 | 6.203%
$\epsilon=0.1$ | 894.754 | 6.606%

##  VI Conclusion

To better consider the impacts of load and PV generation uncertainties on
voltage regulating performance while implementing CVR in unbalanced three-
phase distribution systems, a DRCC-CVR model with an enriched data-based and
moment-based ambiguity set is proposed to optimally dispatch the reactive
power of PV inverters. The original and intractable DRCC-CVR is approximated
by linearized ZIP load models and reformulated in a tractable way with the
first two moment information of load and PV generation uncertainties. We
further implement a data enrichment method with low-resolution data from SMs
and high-resolution data from micro-PMUs to recover instantaneous
uncertainties of load and PV generation. An ambiguity set is constructed based
on enriched data for DRCC-CVR. Simulation results on a real Midwest U.S.
distribution feeder have validated the effectiveness and robustness of the
proposed DRCC-CVR. According to the case studies, we have shown that: (i) The
data enrichment can construct an accurate ambiguity set of load and PV
generation uncertainties. (ii) Compared to other benchmark methods, the
proposed DRCC-CVR shows a better trade-off between the conservativeness of
decisions and operational efficiency. Thus, the proposed DRCC-CVR can achieve
better CVR performance on voltage reduction and power/energy-saving. (iii)
With a reasonable number of micro-PMUs, a high-quality ambiguity set can be
constructed for DRCC-CVR and better performance of CVR can be achieved. (iv)
To consider the benefits of CVR implementation and the reliability of
operational constraints, a proper risk and confidence level needs to be tuned
in DRCC-CVR.

## References

  * [1] Z. Wang and J. Wang, “Review on implementation and assessment of conservation voltage reduction,” _IEEE Trans. Power Syst._ , vol. 29, no. 3, pp. 1306–1315, 2014.
  * [2] T. V. Dao, S. Chaitusaney, and H. T. N. Nguyen, “Linear least-squares method for conservation voltage reduction in distribution systems with photovoltaic inverters,” _IEEE Trans. Smart Grid_ , vol. 8, no. 3, pp. 1252–1263, 2017\.
  * [3] Q. Zhang, K. Dehghanpour, and Z. Wang, “Distributed CVR in unbalanced distribution systems with PV penetration,” _IEEE Trans. Smart Grid_ , vol. 10, no. 5, pp. 5308–5319, Sept. 2019.
  * [4] Q. Zhang, Y. Guo, Z. Wang, and F. Bu, “Distributed optimal conservation voltage reduction in integrated primary-secondary distribution systems,” _IEEE Trans. Smart Grid_ , vol. 12, no. 5, pp. 3889–3900, 2021.
  * [5] J. A. Momoh, S. S. Reddy, and Y. Baxi, “Stochastic voltage/var control with load variation,” in _2014 IEEE PES General Meeting (PESGM)_ , 2014, pp. 1–5.
  * [6] A. Samimi and A. Kazemi, “Scenario-based stochastic programming for Volt/Var control in distribution systems with renewable energy sources,” _IETE Technical Review_ , vol. 33, no. 6, pp. 638–650, 2016.
  * [7] R. A. Jabr, “Robust volt/var control with photovoltaics,” _IEEE Trans. Power Syst._ , vol. 34, no. 3, pp. 2401–2408, 2019.
  * [8] M. Azarnia and M. Rahimiyan, “Robust volt-var control of a smart distribution system under uncertain voltage-dependent load and renewable production,” _Int. J. Electr. Power Energy Syst._ , vol. 134, p. 107383, 2022.
  * [9] Y. Yang and W. Wu, “A distributionally robust optimization model for real-time power dispatch in distribution networks,” _IEEE Trans. Smart Grid_ , vol. 10, no. 4, pp. 3743–3752, 2019.
  * [10] L. Dong, J. Li, T. Pu, and N. Chen, “Distributionally robust optimization model of active distribution network considering uncertainties of source and load,” _J. Mod. Power Syst. Clean Energy_ , vol. 7, no. 6, pp. 1585–1595, 2019.
  * [11] B. Li, R. Jiang, and J. L. Mathieu, “Distributionally robust chance-constrained optimal power flow assuming unimodal distributions with misspecified modes,” _IEEE Trans. Control. Netw. Syst._ , vol. 6, no. 3, pp. 1223–1234, 2019.
  * [12] L. Yang, Y. Xu, W. Gu, and H. Sun, “Distributionally robust chance-constrained optimal power-gas flow under bidirectional interactions considering uncertain wind power,” _IEEE Trans. Smart Grid_ , vol. 12, no. 2, pp. 1722–1735, 2021\.
  * [13] L. Yang, Y. Xu, H. Sun, and W. Wu, “Tractable convex approximations for distributionally robust joint chance constrained optimal power flow under uncertainties,” _IEEE Trans. Power Syst._ , pp. 1–1, 2021.
  * [14] Z. Shen, Z. Wei, G. Sun, and S. Chen, “Representing ZIP loads in convex relaxations of optimal power flow problems,” _Int. J. Electr. Power Energy Syst._ , vol. 110, pp. 372–385, 2019.
  * [15] F. U. Nazir, B. C. Pal, and R. A. Jabr, “Approximate load models for conic OPF solvers,” _IEEE Trans. Power Syst._ , vol. 36, no. 1, pp. 549–552, 2021.
  * [16] Y. Guo, K. Baker, E. Dall’Anese, Z. Hu, and T. H. Summers, “Data-based distributionally robust stochastic optimal power flow—part I: Methodologies,” _IEEE Trans. Power Syst._ , vol. 34, no. 2, pp. 1483–1492, 2019.
  * [17] ——, “Data-based distributionally robust stochastic optimal power flow—part II: Case studies,” _IEEE Trans. Power Syst._ , vol. 34, no. 2, pp. 1493–1503, 2019.
  * [18] F. Bu, K. Dehghanpour, and Z. Wang, “Enriching load data using micro-pmus and smart meters,” _IEEE Trans. Smart Grid_ , vol. 12, no. 6, pp. 5084–5094, 2021.
  * [19] “IEEE standard for interconnection and interoperability of distributed energy resources with associated electric power systems interfaces,” _IEEE Std 1547-2018 (Revision of IEEE Std 1547-2003)_ , pp. 1–138, 2018.
  * [20] R. Cheng, Z. Wang, Y. Guo, and Q. Zhang, “Online voltage control for unbalanced distribution networks using projected newton method,” _IEEE Trans. Power Syst._ , pp. 1–1, 2022.
  * [21] E. Delage and Y. Ye, “Distributionally robust optimization under moment uncertainty with application to data-driven problems,” _Oper. Res._ , vol. 58, no. 3, pp. 595–612, 2010.
  * [22] G. C. Calafiore and L. El Ghaoui, “On distributionally robust chance-constrained linear programs,” _J Optim Theory Appl._ , vol. 130, no. 1, pp. 1–22, 2006.
  * [23] F. Bu, Y. Yuan, Z. Wang, K. Dehghanpour, and A. Kimber, “A time-series distribution test system based on real utility data,” in _Proc. North American Power Symposium (NAPS)_ , 2019, pp. 1–6.