t1t1footnotetext: This material is based upon work partially supported by the
MIT Energy Initiative, MathWorks Mechanical Engineering Fellowship, and the
U.S. Department of Energy under Award Number DE-IA0000025. The views and
opinions of authors expressed herein do not necessarily state or reflect those
of the United States Government or any agency thereof.

# A Reactive Power Market for the Future Grid

Adam Potter<EMAIL_ADDRESS>Rabab Haider<EMAIL_ADDRESS>Giulio Ferro
<EMAIL_ADDRESS>Michela Robba<EMAIL_ADDRESS>Anuradha M.
Annaswamy<EMAIL_ADDRESS>Department of Mechanical Engineering, Massachusetts
Institute of Technology, Cambridge, MA, USA  Department of Informatics,
Bioengineering, Robotics and Systems Engineering, University of Genoa, Genoa,
Italy

###### Abstract

As pressures to decarbonize the electricity grid increase, the grid edge is
witnessing a rapid adoption of distributed and renewable generation. As a
result, traditional methods for reactive power management and compensation may
become ineffective. Current state of art for reactive power compensation,
which rely primarily on capacity payments, exclude distributed generation
(DG). We propose an alternative: a reactive power market at the distribution
level designed to meet the needs of decentralized and decarbonized grids. The
proposed market uses variable payments to compensate DGs equipped with smart
inverters, at an increased spatial and temporal granularity, through a
distribution-level Locational Marginal Price (d-LMP). We validate our proposed
market with a case study of the US New England grid on a modified IEEE-123
bus, while varying DG penetration from 5% to 160%. Results show that our
market can accommodate such a large penetration, with stable reactive power
revenue streams. The market can leverage the considerable flexibility afforded
by inverter-based resources to meet over 40% of reactive power load when
operating in a power factor range of 0.6 to 1.0. DGs participating in the
market can earn up to 11% of their total revenue from reactive power payments.
Finally, the corresponding daily d-LMPs determined from the proposed market
were observed to exhibit limited volatility.

###### keywords:

reactive power , power distribution economics, distribution level market ,
optimal power flow, inverter based resources , power factor , distributed
optimization

##  1 Introduction

With a growing demand for zero-carbon generation in the electricity sector
comes an increase in grid-edge distributed energy resources (DERs), and in
particular, a subset of DERs referred to as distributed generation (DGs). The
increasing penetration of these small-scale resources disrupts the existing
industry practice for power dispatch, device control, and market compensation
mechanisms. Traditionally, large controllable generators at the transmission
level have been responsible for maintaining grid stability and power quality;
however new strategies are required for the emerging grid. While recent
efforts such as FERC Order 2222 [1] in the US indicate a move towards
accommodation of DERs, new electricity market structures are needed to fully
utilize the flexibility they offer at high penetration and lead to a feasible
framework for their integration while preserving grid stability and power
quality. This paper identifies the requirements for such a market structure,
and proposes one with a focus on a reactive power market at the distribution
level.

Reactive power is intimately connected with power quality, and its regulation
is achieved in the transmission grid by altering the power factors (PFs) of
large synchronous generators. The outputs of these generators are easily
controllable thus permitting an efficient regulation of PF, and have therefore
been sufficient for ensuring power quality and voltage stability at the
transmission level. PF can be regulated at the requisite level even with a few
large synchronous generators as the target PF value rarely drops below 0.95
[2]. In contrast to this practice at the transmission level, power quality in
the distribution grid is typically achieved through devices such as on-load
tap changers, capacitor banks, and voltage regulators, which locally inject
reactive power. Such a solution however becomes quickly inadequate in a
distribution grid with an increased penetration of DERs. Emerging solutions
that allow regulation in a wide range of PF correspond to inverter-based
resources, such as DGs which are often equipped with smart inverters, and can
adjust their PF using suitable power electronics control, quickly and at low
costs. Further, since reactive power travels poorly, the spatially distributed
support offered by these DERs offers a unique grid flexibility option if their
reactive power capabilities can be coordinated. This in turn implies that DGs
that enable such an additional flexibility must be compensated for through an
appropriate market mechanism. A market structure that compensates DGs for
their services, particularly for reactive power, is structurally different
from the current practice [3]. The development of such a market is the topic
of this paper, wherein we propose an energy market for the distribution grid
which prices reactive power. We build on the authors’ earlier work in [4, 5]
which proposed an energy market that enables fine granularity, in both
location and in time, of power injections and corresponding energy prices.
Unlike those works, here we focus primarily on reactive power, including the
deployment of DER resources to provide reactive power, the financial
remuneration for the corresponding grid services, and the role of the market
in an unbalanced distribution grid with increasing DER penetration.

Currently, reactive power markets do not exist. Any compensation for
generators that provide reactive power are static rates, designed to recover
the capital costs of installing the equipment or based on the LOC (lost
opportunity cost) of not selling real power. These payments can consist of
capacity, or fixed payments, based on generator’s reactive power capabilities
and utilization, or variable, payments based on the delivered kVARs of
reactive power. These static rates do not reflect the temporal shifts in
supply and demand, or spatial variation across the grid. Further, these
payments are typically cleared on a case-by-case basis for each generator, a
process which is generally exclusive to larger generators. This approach of
static case-by-case rates will be insufficient in accommodating a deep
penetration of DGs. These DGs offer significant operational flexibility at
little to no cost, and can be leveraged to provide much needed grid services
through the clean energy transition. To do so, a market mechanism which prices
their reactive power capabilities and their spatial and temporal variations is
vital. This is possible with approaches based on optimal power flow (OPF),
which allow the simultaneous determination of prices corresponding to both
real and reactive power injections, enables an increasing penetration of DERs,
and accommodates grid physics. In this paper, we propose a reactive power
market that allows variable payments based on an OPF approach.

The reactive power market proposed in this paper is a part of a distribution-
level market with oversight assumed to be provided by a distribution system
operator (DSO). A DSO-centric market has several advantages, the first of
which is that it enables DERs to directly participate in a local setting, and
have the market location be collocated electrically, such as at a primary
feeder node or a secondary feeder node. It also avoids tier bypassing, which
can arise if DERs are directly dispatched into the wholesale markets in
response to the bulk grid objectives without considering local constraints
[6]. Our proposed reactive power market sits at the primary feeder level (4 to
35 kV level), with any resources and uncontrollable loads residing in the
secondary feeders and below aggregated upward. Each Primary Market Operator
(PMO) is assumed to have oversight over one primary feeder, and determines the
schedule for DERs in terms of both real and reactive power injections as well
as the retail electricity price of both real and reactive power, which we
denote as a distribution-level Locational Marginal Price (d-LMP). Each PMO
pays the participating DGs in that feeder at the d-LMP. The d-LMP also
corresponds to the payment made by the loads to the PMO. As we effectively
split the grid into smaller service regions, the proposed retail market has
the potential to lead to less expensive and more extensive procurement and
payment for reactive power resources. The study in [7] corroborates this
statement as well. A DSO can be tasked with the oversight of these PMOs and
perhaps be commissioned as a non-profit entity, similar to ISO/RTOs, by the
corresponding Public Utility Commission.

The underlying framework utilized to determine the market schedules and prices
is based on constrained optimization, an initial version of which was proposed
in [4, 5]. In the present work, we extend [4, 5] in two ways: (1) we consider
a multi-phase unbalanced distribution grid; and (2) we carry out a detailed
treatment of reactive power deployment and pricing. The case study in [4, 5]
considered a specific level of penetration of DG of approximately 70%,
consisting entirely of real power generation. In this paper, we significantly
extend the scope, by modeling each DG as an inverter-based resource with
reactive power capabilities and variable PF, over a range of DG penetration
from 5% to 160%. We do not consider reactive power support from flexible
consumption, though our market structure can extended to accommodate them as
well. We also consider an unbalanced distribution grid, not considered in [4,
5], using a current injection (CI) model, and include analytical support for
the reactive power price resulting from this model. We place our reactive
power market in context of industry needs. Through numerical case study, we
demonstrate how our market effectively utilizes DERs to provide appropriate
local reactive power support while optimizing network-level objectives subject
to grid constraints. We validate the market’s flexibility and performance over
a range of conditions, including DG penetration and inverter PF and discuss
the implications of our market in the key areas of price volatility and
supporting investment decisions.

The following are the specific contributions of the paper.

  * a.

We present an analysis of the state of art of reactive power compensation in
theory and in practice. We outline how DERs disrupt traditional reactive power
management, warranting new approaches to reactive power dispatch and pricing.
We identify three priorities for a DER-inclusive reactive power market:

    * (a)

Utilize the significant reactive power capabilities of a large penetration of
DERs and their wider PF operating range

    * (b)

Support efficient grid operation by minimizing grid losses using local DERs

    * (c)

Provide fair remuneration to DERs for their grid services

    * (d)

Incentivize long-term growth of DER adoption through market inclusion and
corresponding reliable revenue streams

These constitute the motivation for our reactive power market.

  * b.

We propose a reactive power market for DGs in the distribution grid which meet
the priorities outlined above. The market dispatch and price clearing are
solutions of a linear CI model of the unbalanced distribution grid. The market
decisions are arrived at using a Proximal Atomic Coordination (PAC) based
distributed optimization algorithm, demonstrated to converge to the optimal
point under suitable conditions in [8, 4]. We further analyze the reactive
power price resulting from the CI model, building the case for minimizing line
losses as a network-wide view of reactive power.

  * c.

Through a numerical case study of the US New England grid modeled as a
modified IEEE-123 bus, we demonstrate the following features:

    * (a)

Our market can accommodate a large amount of DGs: We illustrate how reactive
power support can be realized even with a high penetration of DGs. We show in
particular that the revenue stream for reactive support to these DGs remain
steady even as DG penetration increases from 5% to 160%, which indicates the
feasibility of the proposed market at high penetrations, and even at low
penetrations.

    * (b)

Our market uses all available DGs efficiently: We utilize the full flexibility
of DGs to operate over a range of PF, from 0.6 to 0.95, to meet over 45% of
reactive power load, even with the reactive power price per unit kVAR
remaining the same. The corresponding increase in the requisite reactive power
injection is appropriately compensated by an increasing percentage of the
revenue from reactive power support compared to total revenue from DGs.

    * (c)

Our market supports grid voltages: The reactive power support from DGs
spatially distributed throughout the grid increases the mean voltage across
the network, providing necessary support during periods of high demand.

    * (d)

Our market is stable and provides reliable revenue: A detailed assessment of
the price variations across all DGs in the IEEE 123-bus over a week shows that
the price variations primarily follow load fluctuations and are otherwise
minimal.

The remaining paper is organized as follows. Section 2 presents an analysis of
the state of art in both theory and practice, providing a historical
perspective on our identified priorities for reactive power compensation.
Section 3 introduces our OPF-based reactive power pricing. Section 4 presents
numerical evaluation of the proposed reactive power market on an IEEE-123 bus
network in the context of our market design priorities. Section 5 provides
concluding remarks.

##  2 State of Art of Reactive Power Compensation

In the following sections we provide an analysis of the state of art for
reactive power management and pricing in the US111A review of reactive power
markets in EU markets is presented in [9]. We present an analysis of both
industry practices and technical solutions. We identify the design objectives
for reactive power pricing for traditional power systems, and elucidate how
they are inadequate for the emerging grid with high DER penetration. Finally,
we identify the priorities for the emerging grid, which our proposed market in
Section 3 is shown to satisfy. A summary of the analysis is presented in Table
1.

Current Practice (State-of-Art) | Proposed Practice for Emerging Grid (Proposed)
---|---
Methodology for Q pricing |  Limitations of state-of-art |  Methodology for Q pricing |  Features of proposed method
Fixed pricing |  Emerging grid is more dynamic: fixed prices cannot provide adequate signals for devices to respond to |  Dynamic pricing |  Time-varying and spatially varying Q prices provide signal for DERs to respond to system needs
Cost-based pricing |  Not inclusive to DERs which don’t incur high costs for producing Q |  Service-based pricing |  Q price is determined by an OPF, thereby modeling the real-time value of Q injection to the grid
Case-by-case clearing |  Intractable for a large number of devices (i.e. DER-rich grid) |  Coordinated and distributed clearing |  Maintain tractability of large-scale OPF by using distributed optimization
Price deadbands where no payment is made (0.95 to 1 PF) |  Deadbands are anti-competitive and not recommended for inclusive markets |  No price deadbands |  Provide compensation through the full range of PFs enabled by power electronic control
Table 1: Summary of the state-of-art industry practice on reactive power
management and pricing. We propose a reactive power market for the emerging
grid, with new methodologies and features.

###  2.1 History of reactive power pricing in the US

Since the US energy sector was deregulated in the 1990s, there has been much
debate over the treatment of reactive power in a market setting. Early
proposals for compensating reactive power (Q) aimed to address two problems:

  * 1.

Ensuring sufficient Q resources were available in the market: by focusing on
the recovery of Q-enabling equipment costs;

  * 2.

Maintaining grid stability: by efficiently procuring the requisite reactive
power capacity in the long-term

These problems were inherent to the then centralized grid predominantly
consisting of synchronous generators. Early reactive power compensation
methods proposed in [2] and [10] supplied no payment for generators operating
at PFs in the range of 0.95 leading to 0.95 lagging creating a price
“deadband”, citing that synchronous generators do not incur losses within this
operating range. Reactive power prices were set outside this PF range as it
necessitated additional hardware and therefore recovery of equipment cost. Any
further deviations imply a loss of opportunity of selling real power, and
therefore the price is set based on the lost opportunity cost (LOC) [11]. Much
of current practice has followed this procedure. However, this approach is
highly sensitive to the PF range and becomes inadequate as PF varies. Thus,
this approach cannot be used for the emerging grid, where the penetration of
DERs with variable PFs is rapidly increasing. Further, as clearly mentioned in
[3], pricing deadbands for reactive power are anti-competitive and not
recommended for inclusive markets, such as those encouraged in FERC Order 2222
[1]. On the demand side, some utilities enforce a similarly-structured PF
penalty to customers that operate outside of a permitted PF range to recover
costs from the additional electrical line losses incurred. However, this
approach also uses a pricing deadband, is ineffective at signaling demand
response actions, and can result in inequities between customers [12].

In addition to efforts for recovering equipment costs and lost revenue, much
of industry’s attention was focused on the second problem of reactive power
capacity procurement. As utilities rushed to ensure sufficient capacity to
maintain voltage and power quality under high demand, capacity markets rose as
a reliable way to incentivize reactive power capability investments. These
fixed payments were preferred over variable payments for the VAR-hrs produced.
References [13] and [14] use OPF-based methods with the underlying dual
variables being used to generate reactive power prices, but both noted that
the resulting prices were highly volatile. Study [14] ultimately recommended
capacity payments to drive investment. Reference [15] assessed that before
1998 in the US around 80% of reactive power payout was through fixed, or
capacity, payments while only 20% was through variable payments. It should be
noted that even in this paper, which is more than 20 years old, fixed payments
are recommended to be a transitional process phasing into payment purely on
the utilization of reactive energy. For more than 20 years, researchers have
argued fixed payments are outdated and should not remain the dominant
compensation method for Q-based ancillary services. These fixed payments do
not reflect the true value of services provided. Further, while synchronous
generators are limited in their ability to generate Q, which can result in a
capacity shortage, modern DERs such as PV solar equipped with smart inverters
are not. As the penetration of inverter-based resources capable of generating
a surplus of Q increases, the recruitment of Q-generating devices will no
longer be a primary concern. Rather, the challenge of maintaining grid
stability in an inverter-rich grid comes down to effectively utilizing the
large number of available resources, and designing the corresponding payment
structures required to incentivize Q-generation. Finally, and of critical
importance, capacity markets are not inclusive to DERs. Therefore, as we move
towards a grid with a strong influx of DG resources with much wider PF
operating range, such procurement focused capacity markets will become less
important. Like real power, it is crucial that reactive power compensation
transitions to market-based solutions, and accounts for value added to the
grid, the spatial differences in resources, and are inclusive to DERs. In this
paper, we propose a specific distributed optimization based solution that
allows the determination of a reactive power pricing structure that captures a
fine-grain variation of location and time.

###  2.2 Implementation practices for reactive power pricing

In this section we focus on the Q-pricing strategies implemented in practice.
Our analysis of these current practices suggests a need for market-driven
variable payments to be inclusive to DGs. Reactive power compensation at the
transmission level is implemented using various methods. Since reactive power
capabilities are deemed critical by FERC, monetary incentives are required to
ensure power is delivered and the grid has sufficient capacity. Tariffs are
designed as either fixed or variable payments to asset owners. However, poorly
designed incentives could result in either a reactive power shortage resulting
in voltage control failures, or surpluses that create monetary losses for
utilities and consumers. A snapshot of these methods at various ISO/RTOs is
delineated in this section.

A US FERC report released in 2014 outlines two common payment methods[3]. The
first is a regulatory mandate on generator reactive power support. Under the
assumption that reactive power is inexpensive, all market participants are
expected to help balance the grid without additional payment. This method is
clearly outdated and unfair, and further, is incapable of incorporating DERs.
The second method, developed by the American Electric Power Service (AEP),
permits generators to report expenses explicitly tied to reactive power
capabilities and settle on an appropriate fixed payment. The settlement
process also gives grid operators control over capacity by recruiting and
settling as necessary. This cost-based compensation is commonly used today,
and is appealing because it avoids a market and the associated risks of market
failures while ensuring the grid has appropriate capacity of reactive power.
However, this method is generally limited to large traditional generators who
can afford the cost analyses and settlement process. Additionally, unlike
traditional generators, most DGs come equipped with extensive reactive power
capabilities so expense-based pricing is less applicable. As mentioned
earlier, the report [3] alludes to the need for retail level accommodation of
DERs as their penetration increases. Without a market, recruiting many small
DERs for reactive power support with the industry standard would be
prohibitive and the resulting prices unsubstantiated. A well designed reactive
power market could facilitate the rapid adoption of small-scale distributed
grid assets.

Another point of interest is the 2020 primer from the Solar Energy Industries
Association (SEIA) [16]. The primer suggests variable payments are currently
inadequate for incentivizing reactive power generation; in practice the
ISO/RTOs rely heavily on fixed payments. Figure 1 outlines reactive power
revenue potential for large solar projects. All US ISO/RTOs include variable,
per kVAR-yr, payment based on loss of potential revenue in the real power
market, but only regions with fixed payments provide high or moderate
potential for a reactive power revenue stream. Currently, if a market
participant, for example DERs, did not have access to fixed payments their
revenue potential would be compromised.

Figure 1: Reactive power compensation for ISO/RTO regions in the US with
corresponding estimates on the reactive power revenue potential for solar and
storage in those regions [16].

The issue of loss of potential revenue in the real power market is further
substantiated by data from the New England Independent System Operator (ISO-
NE) in the US settled cost for yearly reactive power capacity. In 2020, the
costs varied from 1.095 $/kVAR-yr to 1.13 $/kVAR-yr; in comparison, the normal
utilization price is 0.05 $/MWh (0.0057 $/kVAR-yr). As the SEIA primer
suggested, current tariff structures heavily favor standby capacity over power
delivered. The report further notes that while the AEP method for reactive
power capacity pricing has been used for several large solar projects, it is
not inclusive to residential solar. Looking at ISO-NE again, the smallest
asset reported had 587 kVARs of capacity, confirming the standard is non-
inclusive to smaller DGs [17]. This implies that despite fixed payments being
the go-to procedure in industry practice, there is still a significant gap in
the efficient integration of small-scale resources. A pricing scheme centered
around fixed payments would be difficult to scale, improperly value resources,
and result in tier bypassing by separating real and reactive power. In our
work, we propose a reactive power market to provide variable payments for
resources in the distribution grid for services provided at the distribution
level, and argue it is effective and reliable enough to facilitate the
transition to high DER penetration.

###  2.3 Technical Solutions for Reactive-power Pricing

A wide variety of pricing and settlement methods for reactive power have been
proposed in the literature. The survey in [9] reviews various approaches such
as optimization, machine learning, and game theory, and presents a discussion
of different market structures including ancillary provisions. In this paper,
we focus on energy markets, and leave the development of ancillary markets at
the distribution level to future work. The survey in [18] presents a review of
both LOC-based and OPF-based methods for reactive power pricing schemes. These
can loosely be distinguished into three categories as described in [19] based
on the relationship between real and reactive power pricing. The first
category includes the cost-based methods mentioned above that keep reactive
power as an independent ancillary service. The dynamics of real and reactive
power are assumed to be decoupled allowing voltage control to be a secondary
concern to that of balancing power. Control of voltage is assumed to be
possible through regulation of PFs using large generators. This solution is
predicated on transmission-level resources, and the assumptions of decoupled
real and reactive power are violated when PFs dip below 0.95, a plausible
occurrence for DERs. This is further exacerbated by the unbalanced physics of
the distribution grid.

The second category proposed in [19] includes a hierarchical approach where
the first step is the clearing of a real power market followed by a reactive
power market. These solutions leverage the advantages of decoupled dynamics
leading to simple market-based approaches. Reactive power is priced
competitively and supports existing real power markets. One such example can
be found in [20] which is based on LOC of selling real power, while others
such as [21] formulate a separate optimal reactive power flow (OQF) problem to
quantify its marginal value. In both cases, the use of a hierarchical
structure may prevent resources that participated in the real power market
from selling reactive power in the following market. This makes it difficult
to leverage DG’s ability to operate at low PFs as they will be incentivized to
sell real power first, thus limiting reactive power ability.

The third category covers simultaneous real and reactive power deployment.
Typical examples are [22, 23, 24] that extract both real and reactive power
prices using an OPF-formulation. By minimizing an objective function, the
resulting Lagrange multipliers (dual variables) represent the marginal cost of
real and reactive power at each load and resource, leading to a reactive power
price. For example, [22] and [23] use voltage and reactive power constraints
to ensure grid-feasible solutions and find the value of reactive power in
meeting these constraints. By solving for both types of power simultaneously,
these approaches can avoid decoupling grid dynamics and assure there is no
mismatch between cleared real and reactive dispatch. Our proposed market
solution belongs to this third category.

###  2.4 Reactive power needs of the emerging grid

The state-of-art in reactive power management and pricing are insufficient in
accommodating a deep penetration of DERs, which is likely to occur in the
future grid. In order to leverage the flexibility and low cost of DERs, a new
set of requirements and objectives are needed. For the emerging low-carbon
grid with high DER penetration, we propose the following set of new priorities
for industry practice:

  * a.

Utilize the significant reactive power capabilities of a large penetration of
DERs and their wider PF operating range

  * b.

Support efficient grid operation by minimizing grid losses using local DERs

  * c.

Provide fair remuneration to DERs for their grid services

  * d.

Incentivize long-term growth of DER adoption through market inclusion and
corresponding reliable revenue streams

In the next section, we introduce our reactive power market and pricing
mechanism for the emerging grid.

##  3 Reactive Power Pricing

The starting point for the reactive power market is an unbalanced distribution
grid for which we are interested in determining the optimal power dispatch,
for both active and reactive power simultaneously. We utilize an OPF approach
to solve this problem, whose primal and dual variables provide the power
dispatch and the local price of both active and reactive power. These local
prices vary temporally and spatially, providing more accurate economic signals
for DER control and fair service-based remuneration. Within the distribution
grid, many DERs may be connected to only a single phase of the three-phase
network, necessitating a high-fidelity power flow model of the unbalanced
distribution grid. In what follows, we present a detailed discussion of the
current injection model of the unbalanced grid, a distributed optimization
based market mechanism, and the specifics of reactive power pricing.

###  3.1 Introduction to Current Injection Model

Two distinct types of models have been used to represent the physics of the
distribution grid: modeling branch variables which leads to the Branch Flow
model [25, 26], or modeling nodal variables which leads to the Bus Injection
model [27, 28]. The Branch Flow model based on second order cone programming
(SOCP) has proven to be advantageous in providing tight convex relaxations to
the original AC-OPF problem with exactness under some conditions, and is shown
to be more computationally stable than the Bus Injection model. However, both
Branch Flow and Bus Injection models are typically limited to networks with
radial topologies and balanced networks, and extensions to these models for
unbalanced distribution grids are only valid for a small range of angle
imbalances. This is a limiting assumption for many grids, especially with
increasing penetration of DERs located on single-phase lines. A recent
approach proposed by some of the authors of this paper, denoted as Current
Injection (CI) model [29, 30], avoids this assumption and so is an ideal
candidate for representing unbalanced grids with various single-phase loads
and generation. The CI model uses nodal variables, similar to the Bus
Injection model, but the main idea is to represent all loads and generators as
nodal current injections, with all power, current, and voltage phasors
represented in Cartesian coordinates. The 3-phase impedance matrix is used to
describe the self and mutual inductance between phases to model the coupling
of phases that are common to a distribution grid. More importantly, the key
obstacle of non-convexity of the AC-OPF and the subsequent nonlinearity of
SOCP and SDP convexification strategies are dealt with in the CI approach by
leveraging McCormick Envelope (MCE) based convex relaxation [31] for the
bilinear power relations. The MCE uses the convex hull representation of
bilinear terms to render a linear OPF model. This representation requires
adequate bounds on the nodal voltages and currents to ensure a tight convex
relaxation. To determine these bounds, the CI approach also includes a
carefully designed pre-processing algorithm which uses generation and load
forecasts and grid limits to iteratively calculate tight nodal bounds [29].
The CI model has been shown to perform well on unbalanced networks with local
generation, with maximum 1.2% optimality gap and 0.9% voltage error when
compared to the AC-OPF for a number of use cases [29]. The CI model has also
been used for solving OPF in a distributed way [30] and for voltage support
from inverter-based resources in unbalanced distribution grids [32]. Due to
its overall ability to model unbalanced grids and all of the aforementioned
advantages including the computational simplicity of the linear model, we
adopt the CI approach in this paper.

###  3.2 Problem Formulation

We denote a general distribution network as a graph
$\Gamma(\mathcal{N},\mathcal{E})$, where $\mathcal{N}\coloneqq\\{1,...,N\\}$
denotes the set of nodes, $\mathcal{E}\coloneqq\\{(m,n)\\}$ denotes the set of
edges, and each phase is expressed as
$\phi\in\mathcal{P},\;\mathcal{P}=\\{a,b,c\\}$. The CI-OPF for the network is
then written as:

$\displaystyle\underset{x=\left[{{I}^{R}}\;{{I}^{I}}\;{{V}^{R}}\;{{V}^{I}}\;P\;Q\right]}{\mathop{\min}}\;f$
$\displaystyle\left(x\right)$ (1a) $\displaystyle V$ $\displaystyle=ZI$ (1b)
$\displaystyle P_{j}^{\phi}$
$\displaystyle=V_{j}^{\phi,R}I_{j}^{\phi,R}+V_{j}^{\phi,I}I_{j}^{\phi,I}$ (1c)
$\displaystyle Q_{j}^{\phi}$
$\displaystyle=-V_{j}^{\phi,R}I_{j}^{\phi,I}+V_{j}^{\phi,I}I_{j}^{\phi,R}$
(1d) $\displaystyle P_{j}^{\phi}\tan(\cos^{-1}(-\text{pf}))$
$\displaystyle\leq Q_{j}^{\phi}\leq P_{j}^{\phi}\tan(\cos^{-1}(\text{pf}))$
(1e) $\displaystyle\underline{P}_{j}^{\phi}$ $\displaystyle\leq
P_{j}^{\phi}\leq\overline{P}_{j}^{\phi}$ (1f)
$\displaystyle\underline{Q}_{j}^{\phi}$ $\displaystyle\leq
Q_{j}^{\phi}\leq\overline{Q}_{j}^{\phi}$ (1g)
$\displaystyle\underline{V}_{j}^{\phi,R}$ $\displaystyle\leq
V_{j}^{\phi,R}\leq\overline{V}_{j}^{\phi,R}$ (1h)
$\displaystyle\underline{V}{j}^{\phi,I}$ $\displaystyle\leq
V_{j}^{\phi,I}\leq\overline{V}_{j}^{\phi,I}$ (1i)
$\displaystyle\underline{I}_{j}^{\phi,R}$ $\displaystyle\leq
I_{j}^{\phi,R}\leq\overline{I}_{j}^{\phi,R}$ (1j)
$\displaystyle\underline{I}_{j}^{\phi,I}$ $\displaystyle\leq
I_{j}^{\phi,I}\leq\overline{I}_{j}^{\phi,I}$ (1k)

where $x=\left[{{I}^{R}}\;{{I}^{I}}\;{{V}^{R}}\;{{V}^{I}}\;P\;Q\right]$ is the
decision vector for the CI-OPF problem; $I,V,P,Q$ denote the vector of nodal
current injections, voltages, and real/reactive power injections respectively;
$Z\in\mathbb{R}^{3N\times 3N}$ is the system impedence matrix. We use $x^{R}$
and $x^{I}$ to denote the real and imaginary components of a complex number
$x$; overbar $\overline{{x}}$ and underbar $\underline{{x}}$ denote the upper
and lower limits of a variable ${x}$; $\operatorname{Re}(\cdot)$ and
$\operatorname{Im}(\cdot)$ denote the real and imaginary components of a
complex number. Constraint (1b) describes Ohm’s law, and (1c)-(1d) are the
definitions of real and reactive power. Constraint (1e) models the multiphase
inverter of a DG, where the ratio of P and Q determine the PF setting.
Constraints (1c)-(1k) are for all nodes $j\in\mathcal{N}$ and per each phase
$\phi\in\mathcal{P}$.

Remark 1: The use of current and voltage to model the power physics permits a
simple multi-phase model for the unbalanced distribution grid. The
corresponding definition of real/reactive power as represented in constraint
(1c)-(1d) include bilinear terms which can be nonconvex. To deal with this
nonconvexity, we use a McCormick Envelope based convex relaxation, which
renders a linear model. We then convert the bounds on P and Q to nodal current
limits, $\underline{I}_{j}^{\phi}$ and $\overline{I}_{j}^{\phi}$, for both the
real and reactive components, using a carefully designed pre-processing
algorithm. For brevity, we omit the McCormick Envelope relaxation of the
bilinear terms and the pre-processing. See [30, 29] for details.

Remark 2: Another typical constraint in power systems is a congestion
constraint, which limits the line currents. This constraint can be trivially
included in our CI-OPF formulation by introducing a variable describing the
current flow in the distribution lines, $I_{\text{flow}}$, and rewriting
constraint (1b) as two constraints:

$\displaystyle AV$ $\displaystyle=ZI_{\textrm{flow}}$ (2a) $\displaystyle I$
$\displaystyle=A^{T}I_{\textrm{flow}}$ (2b)

where $A\in\mathbb{R}^{3N\times 3N}$ is the 3-phase graph incidence matrix.

The objective function $f(x)$ in (1a) is chosen to carefully reflect the
market structure. As previously discussed, our proposed market solution covers
simultaneous real and reactive power deployment, and energy price settlements.
To reflect these energy prices, the objective function is chosen to be a
combination of Social Welfare and line losses and is given by:

$\displaystyle f\left(x\right)=$
$\displaystyle\sum_{j\in\mathcal{N}_{G}}\left(a_{j}^{P}P_{j}^{2}+b_{j}^{P}P_{j}+a_{j}^{Q}Q_{j}^{2}+b_{j}^{Q}Q_{j}\right)$
$\displaystyle-\sum_{j\in\mathcal{N}_{L}}\left(\alpha_{j}^{P}(P_{j}-\underline{P_{j}})^{2}+\alpha_{j}^{Q}(Q_{j}-\underline{Q_{j}})^{2}\right)$
$\displaystyle+\zeta\operatorname{Re}\left(I^{H}ZI\right)$ (3)

where the first two terms represents generator cost and customer load
disutility, and the third term represents line losses, in which superscript
$H$ denotes the Hermitian. In (3), $\mathcal{N}_{G}$ and $\mathcal{N}_{L}$ are
the set of generator and load nodes respectively, $\zeta$ describes the
tradeoff between economic and energy efficiency objectives,
$a_{j}^{P},b_{j}^{P},a_{j}^{Q},b_{j}^{Q}$ are generating cost coefficients,
and $\alpha_{j}^{P},\alpha_{j}^{Q}$ are load disutility coefficients. All
nodes $j\in\mathcal{N}_{G}$ and $j\in\mathcal{N}_{L}$ represent active DG and
DR participants in the distribution grid, respectively. Node 1 is treated as
the point of common coupling with the transmission grid, and reflects the
wholesale price of electricity, which is the LMP $\lambda^{P}$ from the WEM.
Thus for $j=1$, $a_{j}^{P}=0$ and $b_{j}^{P}=\lambda^{P}$. The cost
coefficients related to reactive power are chosen as $a_{j}^{Q}=0,\quad
b_{j}^{Q}=0.1b_{j}^{P}$. The motivation for these choices comes from [3] which
cites that reactive power prices are often one-tenth of that of real power. It
should be noted that the weighted combination of the social welfare and line
losses is not standard practice, but is included here as line losses are more
significant for an optimal functioning of a distribution grid compared to a
transmission grid. Further analysis of this objective function as it relates
to reactive power pricing is presented in Section 3.4. The resulting solutions
of the CI-OPF problem forms the backbone of our proposed reactive power
market.

###  3.3 Distributed Market Mechanism

Our proposed retail market mechanism enables market participation of a large
number of DERs spatially distributed throughout the grid, and involves
simultaneous P-Q price and device set point calculations. To solve the
corresponding CI-OPF model in a centralized manner can become quickly
intractable with a large number of DERs participating. An attractive
alternative to centralized perspectives is a distributed one. Herein, each
agent has access to local information and a limited amount of information
shared by its neighbours, which it uses to arrive at the market solution
through iterative computation and communication. Such a distributed approach
leverages the computational and communication capabilities of intelligent grid
edge devices. Further, if well designed, they can be resilient to
communication link and single-point failures, and can preserve the private
information of the DERs while still realizing network-level objectives.
Distributed decision making is an emerging paradigm in the distribution grid
and can enable the decarbonization of future energy systems. Thus, we take a
distributed optimization approach for our retail market. In what follows, we
introduce a Proximal Atomic Coordination (PAC) based algorithm for our retail
market.

The CI-OPF problem (1) can be cast into standard form as:

$\displaystyle\underset{x}{\mathop{\min}}\;\;\frac{1}{2}x^{T}Mx+c^{T}x$
$\displaystyle\text{s. t.}\begin{cases}Gx=b\\\ Hx\leq d\end{cases}$ (4)

where $M$ and $c$ are the quadratic and linear cost matrix and vector
respectively, $G$ and $b$ are the equality constraint matrix and vector (i.e.
representing (1b)-(1d)), and $H$ and $d$ are the inequality constraint matrix
and vector (i.e. representing (1e)-(1k)).

####  3.3.1 Decomposition of CI-OPF

Problem (4) can then be reformulated into $N$ separate coupled optimization
problems, each of the form:

$\displaystyle\underset{a}{\mathop{\min}}\;\;\sum_{j\in\mathcal{N}}\frac{1}{2}a_{j}^{T}M_{j}a_{j}+c_{j}^{T}a_{j}$
$\displaystyle\text{s. t.}\begin{cases}G_{j}a_{j}=b_{j}&(\mu_{j})\\\
H_{j}a_{j}\leq d_{j}&(\lambda_{j})\\\ B_{j}a=0&(\nu_{j})\end{cases}$ (5)

where $M_{j},c_{j},G_{j},b_{j},H_{j},d_{j}$ represent the local cost and
constraint submatrices and subvectors, and $\mu_{j},\lambda_{j},\nu_{j}$
represent the corresponding dual variables for each constraint. In order to
fully distribute the coupled CI-OPF, each agent $j$ will estimate the value of
coupling variables which are owned by a neighbouring agent $i$. A set of
coordination constraints must be added for each subproblem $j\in N$:

$B_{j}{a}=0$ (6)

where $B$ represents the incidence matrix of a directed graph whose nodes are
the owned and copied atomic variables within $a_{1},\ldots,a_{K}$. Hence, we
have for each element of the incidence matrix (where $i$ and $m$ correspond to
indices of atomic variables):

$B_{i}^{m}\triangleq\left\\{\begin{array}[]{rl}-1,&\text{if $i$ is `owned' and
$m$ the related `estimate'}\\\ 1,&\text{if $m$ is `owned' and $i$ a related
`estimate'}\\\ 0,&\text{otherwise}\end{array}\right.$

Submatrices ${B_{j}}$ and $B^{j}$ represent the relevant incoming and out-
going edges of the directed graph for subproblem $j$, respectively. This
coordination constraint (6) can be interpreted as requiring all subproblem
estimated variables in a given ${j}$th subproblem to equal the value of their
corresponding owned in $i$th subproblem, $i\neq j$. In the CI-OPF, the
coupling comes from the power physics of the grid, namely constraint (1b). The
resulting atomic decision variable for agent $j$ is:

$a_{j}=\left[I_{j}^{R};I_{j}^{I};V_{j}^{R};V_{j}^{I};P_{j};Q_{j};\\{\hat{I}_{i}^{R},\hat{I}_{i}^{I}\\}_{\forall
i,\\{(i,j)\\}\in\mathcal{E}}\right]$ (7)

where variable estimates are denoted with a hat.

####  3.3.2 Algorithm statement

The Lagrangian function for each network subproblem is:

$\displaystyle\mathcal{L}_{{j}}\left({a}_{{j}},{\mu}_{{j}},{\nu},\lambda_{j}\right)=\sum_{{j}\in
K}$
$\displaystyle\left[{f}_{{j}}\left({a}_{{j}}\right)+{\mu}_{{j}}^{T}({{G}}_{{j}}{a}_{{j}}-b_{j})\right.$
$\displaystyle\left.+{\nu}^{T}B^{j}{a}_{j}+\lambda_{{j}}^{T}\left({{H}}_{{j}}{a}_{{j}}-{d}_{j}\right)\right]$
(8)

where it can be shown that

$\mathcal{L}\left({a},{\mu},{\nu},\lambda\right)\triangleq\sum_{{j}\in
K}\mathcal{L}_{{j}}\left({a}_{{j}},{\mu}_{{j}},{\nu},\lambda_{j}\right)$

The PAC-based distributed algorithm to solve the CI-OPF is then given by:

$\displaystyle{a}_{{j}}\left[\tau+1\right]$
$\displaystyle=-\left[M_{j}+\rho\gamma\left(G_{j}^{T}G_{j}+B_{j}^{T}B_{j}+\frac{1}{\rho}\right)\right]^{-1}$
$\displaystyle(c_{j}+$
$\displaystyle{\mu}_{j}^{T}G_{j}+{\nu}^{T}B^{j}+\lambda_{j}^{T}H_{j}-\rho\gamma
G_{j}b_{j}-\frac{a_{j}\left[\tau\right]}{\rho})$ (9a)
$\displaystyle\lambda_{j}\left[\tau+1\right]$
$\displaystyle=\mathop{\max}\left[0,\,\lambda_{j}\left[\tau\right]+\rho\gamma\left({{H}}_{{j}}{a}_{{j}}\left[\tau+1\right]-d_{j}\right)\right]$
(9b) $\displaystyle\mu_{j}\left[\tau+1\right]$
$\displaystyle=\mu_{j}\left[\tau\right]+\rho\gamma\left(G_{j}a_{j}\left[\tau+1\right]-b_{j}\right)$
(9c) $\displaystyle\bar{\mu}_{j}\left[\tau+1\right]$
$\displaystyle=\mu_{j}\left[\tau+1\right]+\rho\gamma\left(G_{j}a_{j}\left[\tau+1\right]-b_{j}\right)$
(9d) Commun $\displaystyle\text{icate }\left\\{a_{j}\right\\}_{j\in N}\text{
within network}$ (9e) $\displaystyle\nu_{j}\left[\tau+1\right]$
$\displaystyle=\nu_{j}\left[\tau\right]+\rho\gamma B_{j}a\left[\tau+1\right]$
(9f) $\displaystyle\bar{\nu}_{j}\left[\tau+1\right]$
$\displaystyle=\nu_{j}\left[\tau+1\right]+\rho\gamma{B}_{j}{a}\left[\tau+1\right]$
(9g) Commun $\displaystyle\text{icate }\left\\{\bar{\nu}_{j}\right\\}_{j\in
N}\text{ within network}$ (9h)

Proof and convergence results for the PAC-based distributed algorithm are
detailed in [8].

###  3.4 Reactive Power Pricing

The prices for real and reactive power of a DER at node $j$ are chosen as the
marginal costs of generators/consumers and are therefore determined using the
dual variables of the CI-OPF model. In particular, the Lagrange multipliers
corresponding to (1c) and (1d) are used to determine the price for real and
reactive power respectively. While the discussions below are focused almost
entirely on the reactive power component, the solutions above encompass both
the real and reactive power dispatch simultaneously.

Each market clearing sets the reactive power dispatch, $Q_{j,t}$ for node $j$
and time $t$, and produces the corresponding dual variable $\mu_{j,t}^{Q}$.
This dual variable sets the basis for the time-varying reactive power price
$\bar{\mu}_{j}^{Q}$, which we denote as the d-LMP, determined for node $j$ as:

$\displaystyle\bar{\mu}_{j}^{Q}=\frac{\sum_{t\in
T}\mu_{j,t}^{Q}Q_{j,t}}{\sum_{t\in T}Q_{j,t}}$ (10)

This d-LMP varies daily, calculated as a weighted average of the dual variable
from the OPF problem in (1). Thus for a 5-minute retail market clearing
period, the set $T$ includes 288 points. Such a time-varying price allows DGs
to adjust their generation and/or PF settings and DRs to shift their
consumption behavior, all in a coordinated manner so that the DSO can
accurately recover costs. Rather than exposing end-use customers and DER
owners to the complete dynamics of the electricity system by using
$\mu_{j,t}^{Q}$ as a real-time d-LMP, as wholesale market participation models
(like FERC 2222) would do with a corresponding $\lambda^{Q}$, the averaging
procedure in (10) allows the price volatility to be contained while resulting
in the same payout as the corresponding real-time prices.

####  3.4.1 Reactive price volatility

An important and necessary distinction here is made between the dispatch
decisions and financial transactions within the reactive power market. Each
DER will participate in the 5-minute retail market. For each market period,
DERs will submit iterative bids to neighbouring agents (as per the distributed
PAC-based algorithm), then upon agreement (i.e. convergence of the algorithm),
clear the market at the agreed-upon real-time price $\mu_{j,t}^{Q}$ and commit
to dispatch $Q_{j,t}$. This allows the market to use 5-minute ahead forecasts
for DERs, thereby significantly reducing forecast error as compared to a
24-hour ahead window of a day-ahead market, while creating the requisite
signal for DERs to respond to grid needs at a faster timescale than an hourly
or daily signal. Meanwhile, the financial transaction, i.e. a customer’s bill
or payout to the DER, will reflect the daily d-LMP $\overline{\mu}_{j}^{Q}$ as
calculated in (10). In using a weighted average, the perception of price
volatility is removed, instilling confidence in the retail market. This simple
approach to volatility reduction is shown to be sufficient through simulations
in Section 4, where our reactive power market results in consistent revenue
for DER owners over the week.

####  3.4.2 Market objective function

In this section we present an analysis of how the objective function (3)
affects the individual reactive power prices at the nodes. The first two terms
of the objective function represent the generator cost and customer load
disutility, capturing the P and Q prices at a subset of nodes, i.e. generators
and flexible loads. The third term represents line losses
(${\operatorname{Re}\left(I^{H}ZI\right)}$) and captures a global view of the
network, introducing Q prices at every node. In what follows, we show this
relation between line losses and Q prices.

The losses in an AC network are calculated as:

$\displaystyle S_{l}$ $\displaystyle=\left(I\right)^{H}ZI$
$\displaystyle=\left(I^{I}\right)^{T}RI^{I}+\left(I^{R}\right)^{T}RI^{R}$
$\displaystyle+i\left[\left(I^{I}\right)^{T}XI^{I}+\left(I^{R}\right)^{T}XI^{R}\right]$
(11)

where $i$ is the imaginary unit. The corresponding real and imaginary part of
the complex number $S_{l}$ define the real and reactive power losses, with the
objective function (3) minimizing the real power loss as the third term:

$P_{l}=\left(I^{I}\right)^{T}RI^{I}+\left(I^{R}\right)^{T}RI^{R}$ (12)

Recall that the nodal prices are defined as the Lagrange multipliers
associated with Eqs. (1c)-(1d). These constraints can be re-written in terms
of the nodal currents as:

$\displaystyle Q=$ $\displaystyle-V^{R}I^{I}+V^{I}I^{R}$ $\displaystyle=$
$\displaystyle-(RI^{R}-XI^{I})I^{I}+(RI^{I}+XI^{R})I^{R}$ $\displaystyle=$
$\displaystyle-I^{I}\odot RI^{R}+I^{I}\odot XI^{I}+I^{R}\odot
RI^{I}+I^{R}\odot XI^{R}$ $\displaystyle=$ $\displaystyle I^{I}\odot
XI^{I}+I^{R}\odot XI^{R}$ (13)

where $\odot$ is the Hadamard product, and using the relationship
$V^{R}=RI^{R}-XI^{I}$ and $V^{I}=RI^{I}+XI^{R}$ from (1b). Similarly, we can
show $P=I^{R}\odot RI^{R}+I^{I}\odot RI^{I}$. Comparing (12) and (13), the
reactive power $Q$ is a nonlinear combination of the variables composing the
loss term $P_{l}$. By minimizing $P_{l}$, the objective function then directly
influences the nodal prices of reactive power. Thus, minimizing power loss
serves as a network-wide view of reactive power, and generates an appropriate
price signal.

In the next section we present a numerical case study of the reactive power
market and pricing model.

##  4 Results

Our proposed reactive power market for the distribution grid is evaluated in
this section using an IEEE 123-bus network. The network is modified to include
clusters of rooftop photovoltaic (PV) units at up to 27 of the 123 nodes. To
emulate varying levels of DG penetration, PV clusters are incrementally added
at various nodes. Each PV cluster is chosen to have a capacity of 25-80kW, a
range that represents an addition of 3 to 10 small residential installations
or larger projects such as carports or community solar.

In our simulations, the cluster capacity range was chosen so as to emulate a
range of 5% to 160% DG penetration (ratio of nameplate capacity to average
network load, see (14)), which can reflect the trend that is anticipated over
the coming decade especially in the context of more aggressive DG adoption
scenarios. For New England (US), the current and forecasted DG penetration by
2030 is 4.3% and 14%, respectively [33]. Each DG is assumed to be equipped
with a smart inverter which operates at a flexible PF, with appropriate power
electronic control [34]. Rather than fixing the PF, we enforce a lower limit
for each unit, permitting the market to determine the inverter setting. We
model all inverters as having the same PF limits. The baseline PV generation
data $\alpha^{\text{PV}}(t)$ is taken from the NREL SAM dataset [35], and is
assumed to be the same for each PV cluster. The load profile for each node is
modelled with real-time data from ISO-NE [17], and perturbed for each load at
node $j$ as $\alpha_{j}^{P}(t)=\alpha_{D}(t)\delta^{P}_{j}$ with
$\delta^{P}_{j}\sim\mathcal{N}(0,0.1)$. Wholesale electricity prices are
obtained from ISO-NE’s real-time market to determine $\lambda^{P}$. With these
modifications introduced into the IEEE-123 bus network, the reactive power
price (10) and the reactive power injection $Q_{j}$ are determined using the
PAC-based distributed CI-OPF described by equations (9).

We assess the performance of our proposed retail market by considering the
following scenarios:

  1. a.

with increasing PV penetration; and

  2. b.

with varying lower limit on PF operating range

We also quantify the performance of our retail market by evaluating the
variation in reactive power pricing over a 24-hour period, and over a week.

###  4.1 Metrics for Analysis

To assess the performance of our reactive power market, we introduce the
following metrics.

####  4.1.1 Scenario metrics

We define DG penetration as

$\displaystyle\text{DG
penetration}=\frac{\sum_{\mathcal{N}_{G}}\textrm{Nameplate
Capacity}}{\sum_{\mathcal{N}_{L}}\textrm{Average Load}}$ (14)

We also introduce a second penetration metric which accounts for the capacity
factor of renewable resources, the DG energy penetration, as

$\displaystyle\text{DG energy penetration}=\frac{\sum_{t\in
T}\sum_{j\in\mathcal{N}_{G}}S_{j,t}}{\sum_{t\in
T}\sum_{j\in\mathcal{N}_{L}}S_{j,t}}$ (15)

where $S$ is apparent power.

####  4.1.2 Economic metrics

The ratio of the reactive power revenue to the total revenue for DGs is
defined as

Q- $\displaystyle\text{Revenue
Ratio}=\frac{1}{|\mathcal{N}_{G}|}\sum_{j\in\mathcal{N}_{G}}\frac{\left(\bar{\mu}_{j}^{Q}\sum_{t\in
T}Q_{j,t}\right)}{\left(\bar{\mu}_{j}^{Q}\sum_{t\in T}Q_{j,t}+\sum_{t\in
T}\mu^{P}_{j,t}P_{j,t}\right)}$ (16)

where $|\cdot|$ denotes the cardinality of set $\mathcal{N}_{G}$, i.e. the
number of generator nodes. That is, the Q-revenue ratio is the average
fraction of the reactive power payout compared to the total payout from real
and reactive power, over the network.

The coefficient of variation $c_{v}$ is defined below as

$\displaystyle
c_{v}=\frac{std\left(\mu_{j}^{Q}\right)}{\overline{\mu_{j}^{Q}}}$ (17)

and is a measure of the reactive price volatility.

####  4.1.3 Grid performance metrics

The DG-Q utilization is defined as

$\displaystyle\text{DG-Q utilization}=\frac{\sum_{t\in
T}\sum_{j\in\mathcal{N}_{G}}Q_{j,t}}{\sum_{t\in
T}\sum_{j\in\mathcal{N}_{L}}Q_{j,t}}$ (18)

That is, the DG-Q utilization is the fraction of reactive power loads that are
served by DGs. Similarly, the DG-P utilization is defined as

$\displaystyle\text{DG-P utilization}=\frac{\sum_{t\in
T}\sum_{j\in\mathcal{N}_{G}}P_{j,t}}{\sum_{t\in
T}\sum_{j\in\mathcal{N}_{L}}P_{j,t}}$ (19)

The remaining load (for both P and Q) is assumed to be served by the
transmission system.

The network line losses are defined as the sum of losses over the grid
calculated as

$\displaystyle\text{Network
Losses}=\sum_{j\in\mathcal{N}_{L}}R_{j}\left(I_{j}^{I^{2}}+I_{j}^{R^{2}}\right)$
(20)

where $R_{j}$ is the resistance of line $j$. The Mean Voltage is the voltage
averaged over nodes and time defined as

$\displaystyle\text{Mean Voltage}=\frac{1}{|\mathcal{N}_{G}|\times
T}\sum_{j\in\mathcal{N}_{G}}\sum_{t\in T}V_{j,t}$ (21)

The PF per DG $k$ is calculated as

$PF_{k}=\cos\left(\arctan\frac{\sum_{\phi\in\mathcal{P}}Q_{k}^{\phi}}{\sum_{\phi\in\mathcal{P}}P_{k}^{\phi}}\right)$

###  4.2 Market Response to Increasing DG Penetration

In this section we investigate the behaviour of the reactive power market for
a range of DG penetration of 5% to 160% obtained from the modified IEEE-123
bus. This corresponds to a DG energy penetration range of 0.5% to 35%. All
inverters were assumed to have a PF limit of 0.9. By monitoring Q prices and
production as DGs are added, we show that our market is capable of utilizing
and incentivizing Q-generation from DGs at a high penetration.

Figure 2 shows the variation in $\bar{\mu}_{j}^{Q}$ and Q revenue, as a
function of the DG penetration. It is clear that $\bar{\mu}_{j}^{Q}$ remains
relatively constant even as the DG penetration increases from 5% to 160%. We
also observe that the Q-revenue ratio remains relatively steady around 5%. As
DG-penetration changes in the range of 5% to 160%, the Q-revenue ratio
declines hovers around 4-5%, before increasing to 6% as DG penetration
increases to 160%. Figure 3 shows the DG-Q and DG-P utilization over the
course of a day, for a DG penetration of 160% (equivalent to a DG energy
penetration of 35%). When solar generation peaks around mid-day, almost 100%
of real power load can be met with local generation. This saturation in real
power means that any additional DG capacity added to the grid will be used for
meeting reactive power load. This shift, wherein DGs move from supplying
primarily real power to reactive power, results in an increase in Q-revenue
ratio. This is seen in Fig. 2, by the Q-revenue ratio inflection point at high
DG penetration. Notably, even at this high penetration, the reactive power
price $\bar{\mu}_{j}^{Q}$ remains steady. Together, these results show that as
more DGs are added, they can support the grid’s real and reactive power needs
without hurting the revenue potential of other DGs. In this way, our market
maintains consistent payouts for reactive power, and can support the grid as
it transitions to high DER penetration.

Figure 2: As the penetration of DGs increases, both the average price and
Q-revenue ratio remain relatively stable. The price of reactive power is not
strongly dependent on the PV penetration, making it more reliable for the
transitioning grid. The revenue ratio shows that even as DGs are added,
reactive power remains a reliable source of revenue.  Figure 3: With a high DG
penetration of 160%, DGs nearly saturate the grid in the middle of the day. At
the same time, DGs cover more than half of the reactive power load; much of
the remaining load is covered by capacitor banks.

Through these new revenue stream for DER owners, the reactive power market
incentivizes DG resources to provide necessary services to the grid, including
improved voltage characteristics. Figure 4 plots the average voltage profile
over the network, when utilizing DGs within the PF of 0.9 to 1, for 1pm on
June 27, 2021. While both the 0% and 120% DG penetration cases (at a PF of
0.9) stay within the specified voltage limits, the 120% case has average nodal
voltages that are 2.2% higher. Even with the limited PF range of just 0.9 to
1, the distributed coordination of DGs enabled by our PAC-based CI-OPF model
leverages the flexibility of the DGs to greatly improve the average voltage
profile, without any overvoltage events which can happen in networks with high
DG penetration. In comparison, the 0% DG Penetration case only receives
voltage support from capacitor banks, which cannot respond quickly to varying
grid conditions.

Figure 4: By adding DGs that provide Q at a PF of 0.9, the average node
voltage of the grid increases by 2.2% while staying within voltage limits. The
DGs are able to provide necessary grid services by supporting voltages during
periods of high demand.

###  4.3 Market Utilization of Variable Power Factors

In this section we investigate the market behaviour under varying PF settings
of the DERs. We vary the PF operating range of all inverters, varying minimum
PF from 0.6 to 0.95 by changing the PF in the inverter P-Q model (1e). This is
a realizable scenario as DGs like PV are being increasingly equipped with
smart inverters with advanced power electronics making them capable of
operating in such a range. Unlike existing methods, our market is able to
utilize DGs at lower PFs to support the grid, while creating a strong economic
incentive for their participation in the reactive power market.

Table 2 and Figure 5 show the corresponding results obtained for June 27,
2021, with a DG Penetration in (14) of 120%. The data in Table 2 illustrates
how the DG-Q utilization varies with PF, and makes it apparent that by
leveraging capabilities of inverters, upwards of 40% of total reactive power
loads can be served by the local DGs over a 24-hour period. As the inverter’s
operating range is increased, which corresponds to a change in minimum PF from
0.95 to 0.6, the DGs are capable of supplying more of the network’s reactive
power load, from 15% to 44%. Note that when operating at a PF of 0.6, the DG
is supplying approximately 30% more reactive power than real power, at no
additional cost to the generator. This is in contrast to synchronous
generators which only operate at minimum PFs of 0.95, beyond which additional
costs are incurred and efficiency degrades. This increased operating range at
low costs means DGs can be an indispensable resource to the grid, when
utilized optimally.

Figure 5 shows the variation in reactive power prices and Q-revenue ratio as a
function of PF limit. We observed that $\bar{\mu}_{j}^{Q}$ (left axis of
Figure 5) remained relatively stable, with only an 8% decrease as PF drops
from 0.95 to 0.6. However, the Q-revenue ratio increases from 3% to 10.5%.
This implies that DGs operating at lower PFs can anticipate up to 10% of their
total revenue to come from reactive power payments, which is a significant
amount. Clearly, a retail market for reactive power, such as the one proposed
in this paper, is essential to appropriately compensate DGs for their
services.

Figure 5: Enabling DGs to operate at low PFs increases the fraction of their
revenue from reactive power. DGs that are permitted to have PFs as low as 0.6
sourced over 10% of their revenue from our market’s reactive power payments.

Under this scenario of variable PF over a wide range, the coordination of DGs
by the market results in a notable improvement in the voltage profile. Table 2
outlines the mean voltage over the network and total network losses. As DG
inverters are permitted to operate at lower PFs, the mean voltage across the
network improves as the DG-Q utilization increases. Note that this comes at an
expense of increased network losses: the objective function (3) considers a
trade-off between maximizing the social welfare and minimizing line losses,
capture by parameter $\zeta$. At higher values of $\zeta$, the line losses are
reduced more substantially.

Our results shows that our market will utilize the full flexibility in PF
operating range offered by DGs through the variable payments, something not
possible under current industry practice of fixed pricing.

Power Factor | DG-Q Utilization | Mean Voltage (p.u.) | Network Losses (p.u.)
---|---|---|---
1.0 | 0.000 | 0.990 | 0.0344
0.95 | 0.147 | 0.992 | 0.0353
0.9 | 0.204 | 0.993 | 0.0361
0.8 | 0.284 | 0.996 | 0.0383
0.7 | 0.357 | 0.997 | 0.0399
0.6 | 0.442 | 0.998 | 0.0406
Table 2: DGs are allowed to operate over a range of power factors. As the
power factor deviates from 1.0 the Q utilization increases significantly,
resulting in improved mean voltage with a modest increase in line losses.

###  4.4 Market Stability over a Week

To better understand the market performance over a range of conditions, we
simulate the market over the week of June 27, 2021 to July 3, 2021, using
demand and LMP data from ISO-NE. This week is particularly interesting because
it coincided with the end of a heat wave in New England and thus represents a
larger than average variation in grid demand. For this simulation, a DG
penetration of 120% was used, corresponding to the addition of PV clusters at
20 nodes in the network, each with an inverter with a minimum PF of 0.9.

Figure 6 shows the resulting $\bar{\mu}_{j}^{Q}$ values over the week, for all
20 DGs. The significant price differences between nodes speaks to the
importance of d-LMPs due to their variability in both time and location. The
temporal variation in price exhibited over the week further motivates the need
for time-varying prices in order to reflect the varying grid conditions,
especially with increased DER-penetration.

Figure 6: The daily prices of reactive power vary spatially across nodes and
temporally over a week, but maintain the same order of magnitude. Prices
remain relatively consistent, creating a stable and reliable revenue stream
for DGs. Figure 7: The hourly dual variable $\mu_{j,t}^{Q}$ shows significant
volatility over the course of a single day. The proposed daily d-LMP is
approximately 3.3 less volatile than the hourly price, while both result in
the same payout.

Price volatility is a key concern for reactive power markets [13, 14], and
motivates the weighted average pricing model introduced in (10). The prices in
Fig. 6 are calculated using the dual variables $\mu_{j,t}^{Q}$, which are
shown in Fig. 7 for a single DG over a 24-hr simulation. Compared to the
stable daily prices, the underlying hour-to-hour price exhibits large price
variations, something that is undesirable for a market. The average $c_{v}$
(coefficient of variation, a measure of price volatility) for
$\bar{\mu}_{j}^{Q}$ is 0.42, approximately 3.3 times less volatile than the
hourly $c_{v}$ of 1.39. While the total payout to DERs is the same for the
hourly duals $\mu_{j,t}^{Q}$ or using the daily d-LMPs $\bar{\mu}_{j}^{Q}$,
the daily d-LMPs are significantly less volatile. Averaging over each day,
therefore, is chosen for our reactive power d-LMP, $\bar{\mu}_{j}^{Q}$, in
order to contain the volatility and provide a reliable revenue stream for
DERs. A choice of a larger period such a month or a year would suppress useful
granularity in prices, and render an inefficient pricing mechanism. By
managing the level of volatility the DER owners are exposed to, we instill
confidence in the market while providing reliable revenue streams for DERs,
thereby supporting the long-term development of DGs.

###  4.5 Key Observations

Through these simulations and analyses, the following key observations were
made:

  * a.

Our market allows DGs to contribute a significant amount of grid support by
leveraging a wide range of PFs. In simulation, DGs accounted for 42% of
reactive power when allowed to operate at PFs as low as 0.6 (Table 2).

  * b.

The reactive power market introduced and evaluated in this paper can
incentivize DERs to participate in reactive power support with Q-revenue
streams of up to 10% of total revenue (Fig. 5).

  * c.

The resource coordination enabled by our market structure uses DGs towards
supporting network voltages, with average voltage boosted by up to 2.2% as
compared to zero DG penetration (Fig. 4, Table 2).

  * d.

The limited price volatility of the daily Q d-LMPs (Fig. 6) and the stable
revenue streams under varying DG penetration (Fig. 2) can instill confidence
in the market, through the energy transition. This could encourage additional
DG enrollment into the retail market, and also help drive investment decisions
for additional DER adoption.

##  5 Summary and Concluding Remarks

Recently in the US, the Biden administration has revealed as part of its
climate strategy it aims to reach 40% solar generation by 2035 [36]. Similar
targets have been announced by several countries around the globe. Reaching
such an aggressive goal will require new strategies that do not rely on a
shrinking number of centralized generators for power quality support. DERs
represent a growing resource capable of significant grid support contributions
through their adjustable PFs. However, current market practices are inadequate
to integrate and compensate small grid-edge devices, particularly for reactive
power. Our proposed distribution level retail market achieves this by enabling
DERs to provide grid services and be remunerated at a fair retail rate. In
this paper, we have proposed a reactive power market for DGs in the
distribution grid. The market scheduling and clearing is based on PAC, a
distributed optimization algorithm, which has been shown to converge in other
works. Through a numerical case study of a modified IEEE-123 bus, we
demonstrated that our market accommodates a large amount of DGs with a DG
penetration anywhere between 5% and 160%, with the revenue stream remaining
steady. We also showed that this market utilizes the full flexibility of DGs
with the ability to operate over a range of PF, from 0.6 to 0.95 thereby
meeting over 45% of reactive power load, with the reactive power price per
unit kVAR remaining the same. We also showed through a detailed assessment of
the price variations across all DGs in the IEEE 123-bus over a week shows that
the price variations are non-volatile; they primarily follow load fluctuations
and otherwise exhibit little fluctuations.

The proposed market is structured so as to simultaneously clear real and
reactive power with the resulting solution serving as the DER dispatch, and
the duals providing a basis for real and reactive power pricing. It uses
variable payments and spatially varying prices to better capture the value of
DERs and more effectively control DERs through economic signaling. The
reactive power market incentivizes DERs to enroll and participate in reactive
power support by offering a reliable additional source of income through daily
Q d-LMPs. The results from our numerical study show that this revenue stream
remains dependable even as DG penetration rapidly grows. By building
confidence with stable prices, as shown in Section IV, additional DER
enrollment may perhaps by encouraged into the retail market and help drive
investment decisions for additional DER adoption. Our results appear promising
in this regard and permit better DER integration and utilization to meet
distribution grid objectives, and can enable deeper DER penetration.

Future work will consider how the proposed energy market can interface with
ancillary markets at the distribution level. Our market takes the first step
in enabling resource coordination and results in improved voltage profiles
over the feeder by minimizing line losses. However, as DER penetration
increases, resulting in more variability in load and generation, there may be
a need for additional grid services like voltage control to more closely
manage voltages. Corresponding market derivatives for reactive power may then
be needed in an ancillary market framework.

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