Ratha et al. Moving from Linear to Conic Markets for Electricity Moving from
Linear to Conic Markets for Electricity Anubhav Ratha Technical University of
Denmark, Kgs. Lyngby, Denmark<EMAIL_ADDRESS>Flemish Institute for
Technological Research (VITO), Mol, Belgium<EMAIL_ADDRESS>Pierre
Pinson Technical University of Denmark, Kgs. Lyngby, Denmark<EMAIL_ADDRESS>Hélène Le Cadre, Ana Virag Flemish Institute for Technological Research
(VITO), Mol, Belgium<EMAIL_ADDRESS>Jalal Kazempour
Technical University of Denmark, Kgs. Lyngby, Denmark<EMAIL_ADDRESS>
We propose a new forward electricity market framework that admits
heterogeneous market participants with second-order cone strategy sets, who
accurately express the nonlinearities in their costs and constraints through
conic bids, and a network operator facing conic operational constraints. In
contrast to the prevalent linear-programming-based electricity markets, we
highlight how the inclusion of second-order cone constraints enables
uncertainty-, asset- and network-awareness of the market, which is key to the
successful transition towards an electricity system based on weather-dependent
renewable energy sources. We analyze our general market-clearing proposal
using conic duality theory to derive efficient spatially-differentiated prices
for the multiple commodities, comprising of energy and flexibility services.
Under the assumption of perfect competition, we prove the equivalence of the
centrally-solved market-clearing optimization problem to a competitive spatial
price equilibrium involving a set of rational and self-interested participants
and a price setter. Finally, under common assumptions, we prove that moving
towards conic markets does not incur the loss of desirable economic properties
of markets, namely market efficiency, cost recovery and revenue adequacy. Our
numerical studies focus on the specific use case of uncertainty-aware market
design and demonstrate that the proposed conic market brings advantages over
existing alternatives within the linear programming market framework. spatial
equilibrium; mechanism design; pricing; electricity markets; conic economics;
second-order cone programming

##  1 Introduction

The spatial price equilibrium problem, as first analyzed by Enke (1951) and
Samuelson (1952), computes commodity prices and trade flows that satisfy
partial equilibrium conditions over a network. In a two-sided auction
framework, this problem involves price-quantity supply offers and demand bids
matched by an auctioneer to maximize the social welfare, contingent on the
spatial constraints. Historically, spatial price equilibrium problems rely on
linear programming (LP) theory, originally proposed in 1939 by Kantorovich
(1960), to derive the market equilibrium prices from marginal equalities.
Despite the success of LP in achieving optimal market-clearing outcomes and
efficient prices with satisfactory computational effort, it is potentially
limiting in physical systems as it may fail to accurately represent the
nonlinear operational characteristics of assets and the network. Examples of
such physical systems include electricity (Bohn et al. 1984), natural gas (De
Wolf and Smeers 2000), water (Cai et al. 2001), heat (Mitridati et al. 2020),
telecommunication networks (Courcoubetis and Weber 2003) and supply chains
(Snyder et al. 2014), where simplifying assumptions become necessary to remain
within the LP framework. Even in non-physical systems operating under
uncertainty, the use of LP compels a linear modeling of uncertainty and
potential risk preferences, which is limiting at times.

A natural question then arises, why should the market-clearing problem be
confined to the LP framework? This question is pertinent since LP is the
simplest mathematical framework in the convex optimization theory, while more
general convex frameworks such as conic programming are available, which
contain LP as a special case. If such frameworks enable a more accurate
representation of physical assets and networks as well as uncertainty, while
retaining the advantages of LP in terms of optimality, pricing, and
computational ease, it is then, indeed, appealing to adopt them. Leveraging
the recent mathematical and computational advances in conic programming
(Alizadeh and Goldfarb 2003, Mosek ApS 2021), in this work, we introduce and
analyze spatial price equilibrium in a market-clearing context based on the
second-order cone programming (SOCP) framework, and demonstrate how it
outperforms the LP-based markets. Although our theoretical results are
generalizable to any market-clearing context, we choose the domain of
electricity markets for exposition as it is rich in examples that worsen the
adverse impacts of the limitations imposed by LP-based markets, which we
discuss in the following.

###  1.1 Limitations of LP-based Electricity Markets

In electricity markets, the nonlinearities arise from the costs (utilities)
and constraints of producers (consumers) and the physics of power flow in the
network. Currently, real-world electricity markets follow the original
proposal by Bohn et al. (1984) to solve the spatial price equilibrium problem
as an LP to obtain optimal production and consumption quantities and the
spatially-differentiated nodal electricity prices, commonly referred to as
locational marginal prices (LMPs) in the industry (Kirschen and Strbac 2018).
To reach climate change mitigation goals, electricity systems are
transitioning towards a more sustainable future (Chu and Majumdar 2012), by
integrating larger shares of weather-dependent renewable energy sources such
as wind and solar. In Denmark, for instance, 50% of annual national
electricity consumption in 2020 was supplied by such renewable energy sources,
and by 2030, this share is envisioned to reach 100% (The Danish Government’s
Climate Partnership 2020). Similar, if not as ambitious, targets have been
adopted by countries across the world to reduce electricity-related CO2
emissions (IRENA 2015). This trend challenges LP-based markets on three
accounts.

First, with large shares of variable and unpredictable renewable energy,
electricity markets are exposed to significant uncertainty, which needs to be
accounted for. Classical methods within the LP framework, such as scenario-
based stochastic programs (Pritchard et al. 2010, Zavala et al. 2017) and
robust optimization techniques (Bertsimas et al. 2013) are unsuitable in a
practical setting as they suffer from computational intractability and
solution conservativism, respectively. Going beyond LP-based markets is
beneficial from an uncertainty modeling perspective as it allows alternative
approaches that may outperform the classical methods. An example is chance-
constrained programming (Kuang et al. 2018), which admits nonlinear yet convex
computationally tractable and analytically expressable uncertainty models,
paving the way for uncertainty-aware electricity markets in practice.

Second, accommodating the uncertainty requires resources that provide
flexibility services by adapting their operational status in response to the
uncertain supply. Such flexible resources include, among others, energy
storage (Kim and Powell 2011), flexible consumers (Anjos and Gómez 2017), as
well as the coordination with natural gas (Thompson 2013) and district heating
(Mitridati et al. 2020) sectors. LP-based markets impose limitations because
operational characteristics of flexibility providers, which are typically
nonlinear, when approximated by linear constraints shrinks their actual
feasibility set, thereby undermining the amount of flexibility that can be
harnessed. At times, ignoring the characteristics induces operational and
reliability risks for the flexibility provider and the electricity system, as
witnessed during the 2014 polar vortex event in Northeastern United States
(PJM Interconnection 2014) and more recently, during the cold weather event in
Texas which is still under investigation (Bushnell 2021). The need for an
asset-aware electricity market-clearing remains unfulfilled within the LP
framework.

Third, accurate operational modeling of flexible resources is valuable only if
the network constraints are represented in sufficient detail, which is crucial
as flexible resources are dispersed across the electricity network.
Conventionally, a linear approximation of electricity network constraints is
adopted to retain an LP-based market-clearing problem. With this
approximation, the market-clearing problem may result in procurement of
flexibility services from resources in a way that the simplified network
constraints do not allow the flexibility to be delivered when needed
(Papavasiliou 2018). This is critical in the case of electricity markets as
the loss of real-time balance between the production and consumption of
electricity in the system may lead to minor, localized supply disruptions at
best and a large-scale cascading blackout at worst (Daqing et al. 2014). A
network-aware procurement of flexibility services is vital for maintaining the
real-time balance.

On the above three accounts, the LP framework falls short in meeting the
challenges of electricity markets of the future, necessitating a more advanced
yet practical alternative.

###  1.2 Towards Conic Economics

Augmenting the LP-based market-clearing problem with conic constraints
alleviates the previously-discussed limitations to a great extent.
Specifically, in electricity markets, admitting conic constraints incorporates
the nonlinearities arising from modeling uncertainty, operational constraints
of flexible resources, and physically-accurate representation of power flows.
For instance, conic constraints resulting from the convex reformulation of
chance constraints not only enable a computationally tractable model of the
uncertainty from renewable energy but also an analytical characterization of
the risk faced by the electricity markets while mitigating it (Mieth et al.
2020). Towards an asset-aware market, conic constraints model the
nonlinearities associated with the flow of natural gas in pipelines that are
fuel conduits for gas-fired power producers, which are currently the primary
flexibility providers in electricity markets (Byeon and Van Hentenryck 2020).
Lastly, in the context of network-awareness, the foreseen evolution of
decentralized market structures closer to consumption (Parag and Sovacool
2016), and the flexibility available therein, is unlocked through the adoption
of conic constraints, which are necessary to accurately model the networks.

A market-clearing problem with conic constraints fosters the so-called conic
economics. Coined by Raissi (2016), conic economics was introduced in the
context of general equilibrium theory, focusing on mitigation of financial
risk. In a broader sense, in this work, we argue that the inclusion of second-
order cone (SOC) constraints based on the Lorentz cones into the spatial price
equilibrium problem alleviates the limitations of the prevalent LP-based
framework. Enabling convexity-preserving modeling of nonlinearities, the
resulting SOCP-based market-clearing problems are efficiently solved in
polynomial time using interior-point methods (Alizadeh and Goldfarb 2003) by
several off-the-shelf commercial solvers such as MOSEK, Gurobi, and CPLEX. Our
proposed conic market is uncertainty-, asset- and network-aware, which is
crucial for many physical and non-physical systems, apart from the electricity
system. Indeed, advances towards including SOC constraints in electricity
market-clearing problems have been recently made, in particular by Kuang et
al. (2018), Papavasiliou (2018), Dvorkin (2020) and Mieth et al. (2020).
However, these works address one of three challenges discussed, i.e., related
to uncertainty-awareness, asset-awareness or network-awareness; thereby,
lacking a general appeal, and are limited in their practical applicability.
Moreover, the non-generality of the bids implies that flexible resources,
which are critical in the future electricity markets, are excluded.

###  1.3 Contributions

As a broad contribution, our work generalizes the prevalent LP-based market-
clearing problem to the SOCP framework and applies conic duality to analyze
the market equilibrium prices and the economic properties of the underlying
market-clearing problem. Beyond the SOCP framework, semidefinite programming
(SDP) which operates on the cone of semi-definite matrices instead of the
Lorentz cone, allows for further broadening of the scope of the market-
clearing, albeit at the cost of a higher computational burden. Our theoretical
results and their proofs build on Lagrangian duality involving generalized
inequalities (Boyd and Vandenberghe 2004), which lay the foundation for an
SDP-based market-clearing framework to be developed in the future. In the
following, we discuss the specific contributions of this work from three
perspectives, i.e., from a market design perspective, theoretical perspective,
and finally from a practitioner’s perspective.

From a market design perspective, our primary contribution is an original
proposal for a general conic electricity market. By general, we imply a market
framework that is uncertainty-, asset- and network-aware. Our proposed market
framework is uncertainty-aware by design, as it admits a chance-constrained
market-clearing formulation. Towards an asset-aware electricity market, we
enable heterogeneous market participants to accurately express SOC-
representable nonlinearities in their cost (or utility) functions and
constraints. Nonlinear network flow models underlying the physical delivery
associated with the trades are also included in our proposal, leading to
network-aware electricity markets. With the needs of future electricity
markets in view, flexibility providers take a central role in our market
design.

From a theoretical perspective, we first formulate the market-clearing problem
as a centrally-solved optimization problem and address the challenge of robust
solvability of SOCP problems. Theorem 3.2 provides the necessary and
sufficient conditions for optimality and robustness of the market-clearing
outcomes, while Theorem 3.3 gives an analytical expression for conic spatial
prices for the traded commodities. Connecting the centrally-solved
optimization problem to a spatial equilibrium problem involving rational and
self-interested actors, Theorem 3.4 leverages conic duality to prove the
equivalence of the optimization to a competitive equilibrium. Under common
assumptions, Theorem 3.6 proves the satisfaction of economic properties,
namely efficiency, cost recovery and revenue adequacy (Schweppe et al. 1988),
in the proposed market-clearing. This analytically supports that the move
towards conic markets does not incur the loss of any economic properties
compared to the prevalent LP-based markets.

From a practitioner’s perspective, we illustrate the generality of our
proposed market-clearing framework by defining a bid format for conic markets,
enabling heterogeneous market participants to express their preferences. Our
numerical studies highlight how the conic market encompasses an uncertainty-
aware electricity market that efficiently remunerates the mitigation of
uncertainty by the market participants, such that the real-time balance in the
electricity system is ensured. We compare the proposed SOCP-based market-
clearing proposal with two LP-based uncertainty-aware benchmarks, highlighting
the advantages of moving towards a conic market framework.

##### Paper organization:

In section 2 we introduce the market setting, illustrate the relevance of SOC
constraints via examples, introduce the bidding format, and present the
general conic market-clearing as an optimization problem. In section 3, we
analyze the spatial equilibrium underlying the optimization problem and
discuss the economic properties constituting the market equilibrium. Next,
section 4 presents numerical results on one of the market-clearing use cases
by comparing an uncertainty-aware conic market with the available alternatives
within the LP domain. Finally, section 5 concludes by highlighting the key
findings of this work and discusses future perspectives. We provide proofs of
our theoretical results and discuss the market-clearing problems employed in
our numerical studies in the e-Companion to the paper.

##### Notation:

The set of natural and real numbers is denoted by $\mathbb{N}$ and
$\mathbb{R}$, respectively, whereas $\mathbb{R}_{+}$ and $\mathbb{R}_{-}$,
respectively, denote the sets of non-negative and non-positive real numbers.
Upper case alphabets with a script typeface, such as $\mathcal{A}$, represent
sets, while vectors are denoted by lower case boldface and matrices by upper
case boldface alphabets. For a vector $\mathbf{x}$, the operator
$\mathbf{x}^{\top}$ denotes its transpose, $\norm{\mathbf{x}}_{2}$ represents
its Euclidean norm and $\text{diag}(\mathbf{x})$ returns a diagonal matrix
with vector $\mathbf{x}$ as the leading diagonal. We retrieve the $k$-th
element of the vector $\mathbf{x}$ as the scalar $x_{k}$ and use the operator
$[\cdot]_{k}$ to retrieve the $k$-th element of a general vector expression.
$\mathbb{0}$ and $\mathbb{1}$ are vectors of zeros and ones; arithmetic
operators $\leq$, $=$, and $\geq$ on vectors are understood element-wise. For
a matrix $\mathbf{M}\in\mathbb{R}^{p\times q}$,
$[\mathbf{M}]_{(:,k)}\in\mathbb{R}^{p}$ retrieves its $k$-th column while
$[\mathbf{M}]_{(k,:)}\in\mathbb{R}^{1\times q}$ retrieves its $k$-th row. The
operator $\text{tr}(\mathbf{M})$ returns the trace of the matrix $\mathbf{M}$,
while the expression $\mathbf{M}\succcurlyeq 0$ indicates its positive-
semidefiniteness. Lastly, the operator $\otimes$ denotes the Kronecker
product.

##  2 A General Conic Market for Electricity

We discuss the setting of our conic market-clearing problem in section 2.1,
followed by introducing SOC constraints in a market-clearing context in
section 2.2. In section 2.3, we discuss the equality constraints and present
the conic market bids in section 2.4. Lastly, we formulate the general market-
clearing problem in section 2.5 as an SOCP problem.

###  2.1 Market Setting

We consider a forward electricity market involving multiple discrete clearing
periods within the finite time horizon of a single day. We focus on hourly
electricity markets prevalent across the world and collect the hours of the
day in a set $\mathcal{T}=\\{1,2,\dots,T\\}$, where $T=24$.

Day-ahead Market-Clearing Commodity 1: energy Commodity 2: adjustment policy
(flexibility)Real-time Operation $\bullet$ energy physically delivered
$\bullet$ hourly-activated policiesDay: ($D-1$)Day: $D$1tHours24 Figure 1:
Illustration of commodities traded in an uncertainty-aware electricity market

#### Commodities:

Without loss of generality, we assume that the participants in the forward
market compete at the day-ahead stage. The hourly clearing periods at the day-
ahead stage, therefore, correspond to the hours of the next day, which we
collectively refer to as the real-time operation. The two types of commodities
to be traded are (i) energy and (ii) flexibility services. Both types of
commodities are traded hourly in the day-ahead market, which is a purely
financial market in the sense that the physical delivery of these commodities
occurs during the real-time operation. The former commodity, i.e., energy,
represents the quantity in MWh to be exchanged among the market participants
during the real-time operation. The latter, i.e., flexibility services, refers
to the exchanges that contribute to the supply-demand balance during the real-
time operation. For instance, these exchanges could be a result of adjustment
policies of flexible market participants in response to an operational need
that may arise during the real-time operation. A potential occurrence of an
imbalance between the total production and consumption of energy in the system
during the real-time operation is such an operational need. Figure 1
illustrates a two-commodity market, highlighting the day-ahead clearing and
the activities during the real-time operation, which we employ in our
numerical studies in section 4. As another example, flexibility services could
also be traded for mitigation of a foreseeable congestion in parts of the
electricity network. We define a set $\mathcal{P}=\\{1,2,\dots,P\\}$ to denote
the $P$ commodities traded in the market.

#### Market participants:

Our market framework admits heterogeneous competing participants buying or
selling one or more commodities in the market. We introduce the notation and
discuss properties applicable to all participants here, while delegating the
elaboration on various kinds of participants to section 2.2. Let the set
$\mathcal{I}=\\{1,2,\dots,I\\}$ collect $I$ participants, such that $I\geq 2$.
For each participant $i\in\mathcal{I}$, let
$\mathbf{q}_{it}\in\mathbb{R}^{K_{i}}$ denote the $K_{i}\geq P$ number of
decision variables at hour $t$. We retrieve the $k$-th element of the decision
variable at hour $t$ as $q_{itk}\in\mathbb{R},\;\forall
k\in\\{1,2,\dots,K_{i}\\}$. To facilitate a multi-period market-clearing, we
stack $\mathbf{q}_{it}$ for the $T=24$ hours, extending each participant’s
decision vector to $\mathbf{q}_{i}\in\mathbb{R}^{K_{i}T}$. The first $P$
elements of the vector $\mathbf{q}_{it}$ represent contribution towards the
$P$ commodities in hour $t$. For notational compactness, we introduce
$\mathbf{q}_{ip}\in\mathbb{R}^{T},\;\forall p\in\mathcal{P}$ as a subvector of
$\mathbf{q}_{i}$ extracting the hourly contributions by participant $i$
towards the trades of the $p$-th commodity over the $T$ hours. Apart from the
contributions towards the commodity trades, each participant may have
$K_{i}-P$ state variables at each hour $t$, which are involved in the
participant’s operational constraints. Let
$c_{it}(\mathbf{q}_{it}):\mathbb{R}^{K_{i}}\mapsto\mathbb{R}$ denote the
participant’s cost function, such that each $c_{it}(\mathbf{q}_{it})$ is
increasing, convex and twice-differentiable in $\mathbf{q}_{it}$ and satisfies
$c_{it}(\mathbf{0})=0$. We adopt a sign convention that
$c_{it}(\mathbf{q}_{it})>0$ applies for injection into the network and
$c_{it}(\mathbf{q}_{it})<0$ for withdrawal, thereby representing the convex
cost of injection and concave benefit of withdrawal. The temporally-separable
structure of the cost function ensures its convexity while accommodating
participants such as firms owning energy storage units, who toggle between
being producers (discharging) and consumers (charging).

#### Electricity network:

We represent the electricity network as a directed graph
$(\mathcal{N},\mathcal{L})$ formed by a set of nodes
$\mathcal{N}=\\{1,2,\dots,N\\}$, each potentially hosting multiple market
participants, and a set of lines $\mathcal{L}$ comprised of pairs of nodes
$(n,n^{\prime})$ that are connected. We define
$\mathcal{I}_{n}\subseteq\mathcal{I},\;\forall n\in\mathcal{N}$ as the set of
market participants connected to node $n$. The quantity of power flowing over
each line and the direction of the flow is governed by Kirchhoff’s laws during
the real-time operation. Moreover, the flow across the network is limited by a
maximum flow quantity in each line, known as the thermal limit or rated
capacity of a power line.

#### System operator:

A system operator ensures that (i) optimal market-clearing outcomes and
efficient prices are achieved, and (ii) a continuous balance between
consumption and generation is maintained during the real-time operation, while
satisfying the transport limits of the underlying electricity network.
Evidently, these roles correspond to the tasks of operating the market and
operating the network, respectively111Electricity markets across the world
adopt various organizational structures to assign these roles. For instance,
in the United States, independent system operators typically consolidate these
two roles and operate electricity markets wherein the network constraints are
considered at a nodal level, see CAISO (2020) for example. In Europe, the
electricity market operator is not necessarily responsible for the real-time
operation of the system. For example, the market operator Nord Pool clears the
day-ahead (and intra-day) markets in the Nordic and Baltic countries, while
the national transmission system operators are responsible for the real-time
balance within their countries (Nord Pool 2021). In this case, the day-ahead
markets adopt a simplified zonal market framework, implying that the network
constraints are ignored within each bidding zone, which often encompasses a
country or even multiple countries.. In this paper, we assume that the system
operator is responsible for both the day-ahead market-clearing and the real-
time operation, which is consistent with the prevalent organization in the
United States.

#### Competition and timeline:

For the simplicity of exposition, this work assumes that no market participant
acts strategically to exercise market power. Moreover, we overcome the non-
convexity arising from integrality constraints by accepting offers (bids) from
participants already committed to producing (consuming)222This assumption
implies that a unit commitment problem, prevalent in the United States, is
solved a priori. As we discuss in section 2.4, electricity markets in Europe
do not solve a unit commitment problem, while admitting complex bid structure
that allows market participants to internalize their commitment decisions
within the bids. Nevertheless, unit commitment decisions may be incorporated
into an extension of the proposed conic market-clearing problem, rendering it
as a mixed-integer second-order cone program. Commercial conic solvers can
already solve such problems for large dimensions, coupling branch-and-bound
strategies with primal-dual interior-point methods.. In our market setting,
the committed market participants submit their day-ahead supply offers and
demand bids to the central system operator, before a predefined gate closure
time, without any knowledge of the bids and offers from other participants.
Thereafter, the market is cleared by the system operator matching the offers
with the bids, while ensuring the feasibility of network constraints. Finally,
cleared prices and quantities for all the commodities are publicly disclosed
and the acceptance or rejection of bids made by the participants is
communicated to them.

#### Payment mechanism:

We consider a uniform pricing scheme for pricing of energy as in majority of
electricity markets across the world. This implies that all accepted supply
offers and demand bids are cleared with a common price for a given commodity
at a given location and time. Alternative payment mechanisms such as pay-as-
bid and Vickrey-Clarke-Groves pricing in a single commodity setting (Vickrey
1961) are other possibilities and our market-clearing problem is generalizable
to admit them. For instance, electricity markets in Germany, Switzerland and
Italy use pay-as-bid payments for flexibility services needed for real-time
balance in the network (Wang et al. 2015). Moreover, countries such as Denmark
use a two-price scheme for balancing services during real-time operation,
i.e., the remunerations towards mitigating over- and under-supply are priced
distinctly (Nord Pool 2021). However, due to the advantages underlying the
uniform pricing scheme (Federico and Rahman 2003, Cramton and Stoft 2007),
day-ahead electricity markets around the world are moving towards a uniform
pricing scheme, thus justifying our choice.

In what follows, recalling the significance of future electricity markets to
be uncertainty-, asset- and network-aware discussed in section 1, we explain
how SOC constraints enable such markets.

###  2.2 SOC Constraints in a Market-Clearing Problem

Our general market framework admits heterogeneous market participants of four
kinds. First, we consider conventional power producers such as firms that own
coal-fired, gas-fired, nuclear, and hydro power plants. These producers are
dispatchable at the day-ahead stage, i.e., they are able to plan their
production during real-time operation with a high degree of certainty. Some
are flexible, e.g., gas-fired and hydro, meaning that their planned production
quantities can be modified during the real-time operation, if needed. On the
contrary, some producers are relatively less flexible, e.g., coal-fired and
nuclear power producers. Second, we consider firms owning weather-dependent
renewable power production sources such as wind and solar power plants. These
producers are non-dispatchable and inflexible. Third, we consider energy
consumers such as small-scale end-consumers or large industries. Some of these
consumers may be flexible and can be counted as flexible resources. Last, we
consider additional types of flexibility providers, which may not necessarily
be conventional power producers or flexible consumers, for instance, firms
owning energy storage units.

SOC constraints applicable to these market participants are derived from a
second-order cone, which is a convex set and is alternatively referred to as
Lorentz cone or ice-cream cone. For the variable $\mathbf{q}_{i}$ of
participant $i,\;\forall i\in\mathcal{I}$, a SOC constraint in its general
form is given by

$\displaystyle\norm{\mathbf{A}_{i}\mathbf{{q}}_{i}+\mathbf{b}_{i}}_{2}\leq\mathbf{d}^{\top}_{i}\mathbf{{q}}_{i}+e_{i}\quad\Leftrightarrow\quad\begin{bmatrix}\mathbf{A}_{i}\\\
\mathbf{d}_{i}^{\top}\end{bmatrix}\mathbf{q}_{i}+\begin{bmatrix}\mathbf{b}_{i}\\\
{e}_{i}\end{bmatrix}\;\in\;\mathcal{C}_{i}\;\subseteq\;\mathbb{R}^{m_{i}+1}\;,$
(1)

where $\mathcal{C}_{i}$ is a second-order cone of dimension $m_{i}+1$, where
$m_{i}\in\mathbb{N}$. The dimensions $m_{i}+1$ of the cone reflect the
relationship within the decision variables. Considering the heterogeneous mix
of market participants involved, the dimensions of the cone in the SOC
constraints are not necessarily identical among the various participants or
even among the constraints of each participant. Parameters
$\mathbf{A}_{i}\in\mathbb{R}^{m_{i}\times K_{i}T}$,
$\mathbf{b}_{i}\in\mathbb{R}^{m_{i}}$, $\mathbf{d}_{i}\in\mathbb{R}^{K_{i}T}$
and $e_{i}\in\mathbb{R}$ embody the structural and geometrical information for
each constraint. Constraint (1) admits a set of points that lie on or within
the inverse image of the Lorentz cone under an affine transformation. Since
these operations are convexity-preserving, a SOC constraint is a convex
constraint on the variable $\mathbf{q}_{i}$ (Lobo et al. 1998). We use the two
equivalent forms in (1) interchangeably, preferring the form with Euclidean
norm in the following sections that discuss the modeling of participant
constraints and the conic form in the analytical proofs provided in the
Appendix.

#### Linear constraints:

Any single-period or multi-period linear constraint arising from the operation
of physical assets owned by a participant $i$ can be represented by (1).
Examples of such constraints for the power producers are their minimum or
maximum production limits and ramping rate constraints, which limit the change
in production over subsequent hours. Such linear constraints can be
represented by the SOC constraint (1), provided that $\mathbf{A}_{i}$ is a
null matrix or the cone is dimensioned such that $m_{i}=0$. In the former
case, (1) reduces to the linear constraint
$0\leq\mathbf{d}_{i}^{\top}\mathbf{q}_{i}+e_{i}$, which denotes a halfspace.
Whereas in the latter case, (1) reformulates into a linear constraint
$0\leq\mathbf{d}_{i}^{\top}\mathbf{q}_{i}+e^{{}^{\prime}}_{i}$, where
$e^{{}^{\prime}}_{i}=e_{i}-\norm{\mathbf{b}_{i}}_{2}$.

#### Quadratic constraints:

While linear constraints are admissible in LP-based electricity markets,
convex quadratic constraints are not. However, any convex quadratic
operational constraint related to the assets of participant $i$ can be
represented by (1) and therefore, considered within our proposed conic market-
clearing problem. An example of such a constraint stems from quadratic
operational costs faced by the participants. Such quadratic costs arise, for
instance, from frequent ramping needs (Kuang et al. 2019). Although the cost
naturally appears in the objective function of the optimization problem, it is
analytically and computationally appealing to move the quadratic terms to the
constraints, leaving the objective function to be linear (Mosek ApS 2021). We
will later illustrate how such quadratic costs can be recast as SOC
constraints in Example 2.9. More generally, any convex quadratic constraint
can be represented by (1) if $\mathbf{d}_{i}=\mathbf{0}$ and $e_{i}\geq 0$.

#### Other nonlinear constraints:

In a more general sense, beyond linear and convex quadratic constraints, (1)
captures the relationship among the $K_{i}T$ number of decision variables for
each market participant, contributing towards asset-awareness of the market-
clearing problem. For instance, such nonlinear constraints arise in the
coordination between the electricity system and natural gas system, wherein
the operational constraints of the natural gas system become relevant to the
day-ahead electricity market-clearing problem333The interdependence between
electricity and natural gas systems primarily arises due to the significant
role played by gas-fired power plants in providing flexibility to the
electricity system facing uncertainty. Congestion in the gas network during
the real-time operation of the electricity system jeopardizes the fuel-
availability for gas-fired power plants and therefore, adversely impacting the
flexibility provision (Byeon and Van Hentenryck 2020).. One of the crucial
operational constraints of the gas system is the relation between gas flows
and nodal pressures in the network. The steady-state flow of gas in the
pipelines is represented by a non-convex quadratic equality constraint that
relates the squared flow magnitude with the difference in squared pressures at
the terminal nodes. A convex relaxation of this non-convex equality takes the
form of SOC constraints, see Example 2.1.

######  Example 2.1 (Natural gas network)

Let $\varphi\in\mathbb{R}_{+}$ denote the flow of gas along a gas pipeline
connecting two terminal nodes, the sending node $s$ and the receiving node
$r$, each with nodal pressures denoted by $\pi_{s}\in\mathbb{R}_{+}$ and
$\pi_{r}\in\mathbb{R}_{+}$, respectively. As proposed by Borraz-Sánchez et al.
(2016), the convex relaxation for the non-convex equality between the flow
along a pipeline and the pressures at the terminal nodes is expressed as

$\displaystyle\varphi^{2}\;\leq\;\beta^{2}\;(\pi_{s}^{2}-\pi_{r}^{2}),$ (2)

where $\beta\in\mathbb{R}_{+}$ is a constant which encodes the friction
coefficient and geometry of pipelines. Ignoring all other pipelines and
considering the gas network operator as an electricity market participant, we
can denote its decision vector as
$\mathbf{q}_{\text{GN}}=\begin{bmatrix}\varphi\quad\pi_{r}\quad\pi_{s}\end{bmatrix}^{\top}$
and can represent the constraint (2) as a SOC constraint of the form (1) with
parameters

$\displaystyle\mathbf{A}=\begin{bmatrix}\frac{1}{\beta}\quad 0\quad 0\\\
0\quad 1\quad
0\end{bmatrix}\;,\;\;\mathbf{b}=\mathbb{0}_{2}\;,\;\;\mathbf{d}=\begin{bmatrix}0\quad
0\quad 1\end{bmatrix}^{\top}\;\;\text{and}\;\;e=0,$

which is a three-dimensional SOC constraint, i.e., $m_{i}=2$. Similar SOC
constraints are included for other pipelines in the gas network and over
various hours, therefore enabling the consideration of gas network operational
constraints within the electricity market-clearing problem.  $\blacksquare$

Lastly, beyond the asset-awareness discussed so far, SOC constraints in their
general form enable an uncertainty-aware market-clearing problem.
Specifically, chance constraints enable endogenous modeling of uncertainty and
risk faced by market participants and are analytically reformulated as SOC
constraints under some mild conditions (Nemirovski and Shapiro 2007), see
Example 2.2.

######  Example 2.2 (Chance constraints)

Consider a market-clearing problem wherein, in addition to the nominal
production quantities, a flexibility service is contracted from the
flexibility providers in the market. A traded flexibility service is organized
as an adjustment policy. As mentioned in section 2.1, these adjustment
policies allow mitigation of the uncertainty realized during the real-time
operation, while look-ahead decisions are made by the system operator at the
day-ahead market-clearing stage. Such uncertainty could, for example, arise
from imperfect forecasts for the production from weather-dependent renewable
energy sources or from imperfect load forecasts for consumers. Assuming a
single-period market-clearing problem for notational simplicity, let the set
$\mathcal{W}=\\{1,2,\dots,W\\}$ collect the $W$ independent sources of
uncertainty in the electricity system and vector $\bm{\xi}\in\mathbb{R}^{W}$
denote the random forecast errors representing this uncertainty. Assume that
$\bm{\xi}$ follows a probability distribution $\mathbb{P}_{\xi}$,
parameterized by the moments, mean $\bm{\mu}\in\mathbb{R}^{W}$ and covariance
$\bm{\Sigma}\in\mathbb{R}^{W\times W}$, which are estimated by the system
operator with access to a finite number of historical measurements. Under the
chance-constrained optimization framework, the system operator allocates
adjustment policies to flexible producers while allowing them to violate their
operational constraints with a small probability $\hat{\varepsilon}\in[0,1]$.
Assume again that participant $i$ has $K_{i}=2$ decision variables such that
$\mathbf{q}_{i1}=\begin{bmatrix}\hat{q}_{i1}\;\;\alpha_{i1}\end{bmatrix}^{\top}\in\mathbb{R}^{2}$,
where $\hat{q}_{i1}$ and $\alpha_{i1}$ are, respectively, the nominal
production quantity and the adjustment policy. A chance constraint limiting
the total production, i.e., the sum of nominal and adjustment, of the
participant to its upper limit $\overline{Q}_{i}$ is written as

$\displaystyle\mathbb{P}_{\xi}\left(\begin{bmatrix}1\quad\mathbb{1}^{\top}\bm{\xi}\end{bmatrix}\begin{bmatrix}{\hat{q}}_{i1}\\\
\alpha_{i1}\end{bmatrix}\leq\overline{Q}_{i}\right)\geq(1-\hat{\varepsilon}),$
(3a) where the uncertainty is characterized by the total forecast error
$\mathbb{1}^{\top}\bm{\xi}\in\mathbb{R}$. This probabilistic constraint
reformulates to its analytic equivalent, based on Nemirovski and Shapiro
(2007), as $\displaystyle
r_{\hat{\varepsilon}}\norm{\mathbf{X}\mathbb{1}\;\alpha_{i1}}_{2}\leq\overline{q}_{i}-\hat{q}_{i1}-\mathbb{1}^{\top}\bm{\mu}\;\alpha_{i1},$
(3b)

where all vectors of ones are $\mathbb{1}\in\mathbb{R}^{W}$, and
$\mathbf{X}\in\mathbb{R}^{W\times W}$ denotes a factorization of the
covariance matrix such that $\bm{\Sigma}=\mathbf{X}\mathbf{X}^{\top}$. For
instance, since $\mathbf{\Sigma}\succcurlyeq 0$ from the definition of
covariance matrices, such a factorization can be obtained in a computationally
efficient manner using Cholesky decomposition, resulting in $\mathbf{X}$
having a lower-triangular structure. This decomposition is uniquely determined
for covariance matrices that are full rank, i.e., all uncertainty sources are
linearly independent, as assumed. The parameter
$r_{\hat{\varepsilon}}\in\mathbb{R}_{+}$ is a safety parameter chosen by the
system operator relying on the knowledge of the distribution
$\mathbb{P}_{\xi}$, such that $r_{\hat{\varepsilon}}$ increases as
${\hat{\varepsilon}}$ reduces444When the random forecast errors $\bm{\xi}$ are
assumed to be normally distributed, the safety parameter
$r_{\hat{\varepsilon}}$ is given by the inverse cumulative distribution
function of the standard Gaussian distribution evaluated at
$(1-\hat{\varepsilon})$-quantile (Nemirovski and Shapiro 2007). Dropping the
assumption of normality, a more conservative choice of $r_{\hat{\varepsilon}}$
is obtained based on the so-called moment-based distributionally-robust chance
constraints (Wagner 2008), which considers the distribution $\mathbb{P}_{\xi}$
to lie inside an ambiguity set of all probability distributions characterized
by the empirically-estimated moments $\bm{\mu}$ and $\bm{\Sigma}$. In that
case, the safety parameter
$r_{\hat{\varepsilon}}=\sqrt{\frac{1-\hat{\varepsilon}}{\hat{\varepsilon}}}$.
A further generalization that considers even the moments of distribution
$\mathbb{P}_{\xi}$ to lie in well-defined uncertainty sets such as ellipsoids
(Delage and Ye 2010), lead to a semidefinite constraint instead of the SOC
constraint in (3b), leading to a SDP-based market-clearing problem..
Constraint (3b) is represented by the general SOC constraint (1) with
parameters

$\displaystyle\mathbf{A}=\begin{bmatrix}\mathbb{0}\quad\mathbf{X}\mathbb{1}\end{bmatrix}\;,\;\;\mathbf{b}=\mathbb{0}\;,\;\;\mathbf{d}=-1/r_{\hat{\varepsilon}}\begin{bmatrix}1\quad\mathbb{1}^{\top}\bm{\mu}\end{bmatrix}^{\top}\;\text{and}\;\;e=\overline{Q}_{i}/r_{\hat{\varepsilon}}\;,$

thereby resulting in a SOC constraint of dimension $W+1$. Further modeling
details and additional use cases showcasing the analytical reformulation of
more complicated constraints, such as inter-temporal chance constraints of
market participants, are covered in the e-Companion.  $\blacksquare$

Finally, accounting for the potentially multiple SOC constraints faced by
participant $i$, we introduce a set $\mathcal{J}_{i}=\\{1,2,\dots,J_{i}\\}$
collecting the $J_{i}$ constraints that extend (1) as

$\displaystyle\norm{\mathbf{A}_{ij}\mathbf{{q}}_{i}+\mathbf{b}_{ij}}_{2}\leq\mathbf{d}^{\top}_{ij}\mathbf{{q}}_{i}+e_{ij},\;\forall
j\in\mathcal{J}_{i},$ (4)

where each constraint with parameters
$\mathbf{A}_{ij}\in\mathbb{R}^{m_{ij}\times K_{i}T}$,
$\mathbf{b}_{ij}\in\mathbb{R}^{m_{ij}}$,
$\mathbf{d}_{ij}\in\mathbb{R}^{K_{i}T}$ and $e_{ij}\in\mathbb{R}$ corresponds
to a second-order cone $\mathcal{C}\subseteq\mathbb{R}^{m_{ij}+1}$. The
feasibility region for (4) is formed by the Cartesian product of $J_{i}$
second-order cones
$\mathcal{C}_{i}=\prod_{j\in\mathcal{J}_{i}}\mathcal{C}_{ij}=\mathcal{C}_{i1}\times\cdots\times\mathcal{C}_{iJ_{i}}$,
which is again convex.

###  2.3 Equality Constraints

Each participant may be involved in trades corresponding to the $P$
commodities, subject to equality constraints that arise while considering
temporal and spatial dynamics underlying their asset models. We model the
equality constraints on the decision variable $\mathbf{q}_{i}$ of participant
$i$ as

$\displaystyle\mathbf{F}_{i}\mathbf{q}_{i}=\mathbf{h}_{i},$ (5)

where $\mathbf{F}_{i}\in\mathbb{R}^{R_{i}\times K_{i}T}$ and
$\mathbf{h}_{i}\in\mathbb{R}^{R_{i}}$ are parameters encoding the $R_{i}$
equality constraints on $\mathbf{q}_{i}$, such that $R_{i}\leq K_{i}T$ and
$\mathbf{F}_{i}$ has full row rank.

The market-clearing conditions are also modeled as marginal equalities
coupling the decisions of the participants such that supply-demand balance is
ensured for each of the $P$ commodities traded in the market. Modeling these
conditions, given the heterogeneous mix of market participants in our
framework, necessitates the following definition.

######  Definition 2.3 (Physical Fulfillment of Commodity Trades)

For participant $i\in\mathcal{I}$, the hourly injection or withdrawal towards
commodity $p$ is given by $\mathbf{G}_{ip}\;\mathbf{q}_{ip}\in\mathbb{R}^{T}$,
where $\mathbf{G}_{ip}\in\mathbb{R}^{T\times T}$ is a coupling matrix formed
by elements encoding the injection or withdrawal coefficients.  $\blacksquare$

We now elaborate on this definition. The contribution by a dispatchable
producer (either flexible or inflexible) towards the commodity representing
energy is $\mathbf{q}_{ip}\in\mathbb{R}^{T}_{+}$, whereas for inflexible
consumers, the contribution is $\mathbf{q}_{ip}\in\mathbb{R}^{T}_{-}$. For
both kinds of market participants, the coupling matrices $\mathbf{G}_{ip}$ are
identity matrices, i.e., $\mathbf{G}_{ip}=\text{diag}(\mathbb{1})$. Meanwhile,
the contribution from energy storage units $\mathbf{q}_{ip}$ to the commodity
corresponding to energy adopts different signs in the various hours depending
on whether the storage unit is discharging (injection) or charging
(withdrawal). Information on the conversion factors for participants from
other sectors such as natural gas or district heating is also encoded within
the entries of the matrix $\mathbf{G}_{ip}$. Finally, in addition to those
traded system-wide, the market-clearing problem may entail commodities traded
among a subset of all participants and thus, reflect the agreements among that
subset. Therefore, $\mathbf{G}_{ip}$ may be null matrices for some commodities
and for some participants.

###  2.4 Bid Format in the Conic Electricity Market

For the sake of generality, we adopt a common bid format for the supply offers
and demand bids, referring to them as supply and demand bids, respectively and
define them in the following.

######  Definition 2.4 (Conic Market Bids)

Let $\mathcal{B}_{i}$ denote a bid submitted by the market participant $i$ to
the system operator. The bid $\mathcal{B}_{i}$ is a tuple defined as

$\displaystyle\mathcal{B}_{i}:=\Big{(}n_{i},\;\\{\mathbf{A}_{ij},\mathbf{b}_{ij},\mathbf{d}_{ij},e_{ij}\\}_{j\in\mathcal{J}_{i}},\;\mathbf{F}_{i},\mathbf{h}_{i},\;\\{\mathbf{G}_{ip}\\}_{p\in\mathcal{P}},\;\\{\mathbf{c}^{\text{Q}}_{it},\mathbf{c}^{\text{L}}_{it}\\}_{t\in\mathcal{T}}\Big{)},$

where $n_{i}\in\mathcal{N}$ is the electricity network node at which the
participant $i$ is located. Parameters
$\mathbf{A}_{ij},\mathbf{b}_{ij},\mathbf{d}_{ij},e_{ij},\;\forall
j\in\mathcal{J}_{i}$ are linked to the $J_{i}$ SOC constraints;
$\mathbf{G}_{ip},\;\forall p\in\mathcal{P}$ correspond to the coupling
matrices for the $P$ commodities; $\mathbf{F}_{i}$ and $\mathbf{h}_{i}$
correspond to the $R_{i}$ equality constraints; and lastly,
$\mathbf{c}_{it}^{\text{Q}},\mathbf{c}_{it}^{\text{L}},\;\forall
t\in\mathcal{T}$ denote the temporally-separated quadratic and linear bid
prices.  $\blacksquare$

######  Remark 2.5 (Conic Market Bids vs. Price-Quantity Bids)

The bids $\mathcal{B}_{i}$ are a generalization of the so-called price-
quantity bids that form the backbone of various orders placed in currently-
operational LP-based electricity markets, see Nord Pool (2021) for example. As
the name indicates, in addition to the participant’s locational information, a
price-quantity bid comprises a bid price representing the bidder’s
willingness-to-pay or willingness-to-receive for the associated quantity. In
the proposed conic market bids, the quantities are formed by the parameters of
the $j$-th SOC constraint
$\\{\mathbf{A}_{ij},\mathbf{b}_{ij},\mathbf{d}_{ij},e_{ij}\\}$ admitting the
simple linear characterization of quantities $\mathbf{q}_{i}$ when
$\mathbf{A}_{ij}$ is a null matrix or when $m_{ij}=0$, while more intricate
representations of the operational constraints are covered by a suitable
choice of the SOC constraint parameters, as discussed in section 2.2. In an
attempt to accommodate the operational constraints of market participants,
prevalent LP-based markets allow specialized order types, such as block orders
(Nord Pool 2021). These block orders are derived by logically-linking several
price-quantity bids over multiple periods and to some extent, enable market
participants to reflect their operational constraints on their market
participation strategies. Besides the inability to model physically-realistic,
nonlinear operational constraints of the participants, such complex orders
mandate the use of integrality constraints which leads to the loss of
guarantees on the satisfaction of the desired economic properties of the
market.  $\blacksquare$

###  2.5 Market-Clearing as an SOCP Problem

#### Network constraints:

An accurate modeling of the physics of power flows in the electricity network
introduces nonlinearities and non-convexities. Convexification of these power
flow equations have utilized SOCP (Kocuk et al. 2016) and SDP (Lavaei and Low
2012) relaxations to ensure optimality while solving electricity
system–related optimization problems555The authors in Coffrin et al. (2021)
comprehensively benchmark the options available for convex relaxation of power
flow equations in terms of computational requirements and the quality of
relaxation achieved, for test networks of various sizes.. However, electricity
market-clearing problems across the world adopt a linearized approximation of
the power flow equations, relying on a number of assumptions (Cain et al.
2012). Consistent with current practice and for simplicity of exposition, our
formulation adopts a variation of linearized power flow equations that
leverages the so-called Power Transfer Distribution Factor (PTDF) matrix
(Christie et al. 2000). Nevertheless, we demonstrate the network-awareness of
our market-clearing problem by showing how it can be extended to include the
SOCP relaxation of power flow equations in the following example. Known as the
branch flow model (Farivar and Low 2013), this convex relaxation is exact for
the distribution networks that exhibit a radial graph structure under mild
conditions.

######  Example 2.6 (Electricity network)

The power flow equations model the nonlinear relation of active and reactive
power transported along a line with the current and terminal node voltages.
While active power is the actual quantity in MWh of power produced or
consumed, reactive power is a necessary component oscillating within the
network and regulates voltage across the nodes. Consider a power line
$\ell=(n,n^{\prime})\in\mathcal{L}$ connecting adjacent nodes
$n,n^{\prime}\in\mathcal{N}$. Let ${s_{\ell}^{\text{a}}}\in\mathbb{R}$ and
${s_{\ell}^{\text{r}}}\in\mathbb{R}$ denote the active power and reactive
power; $\theta_{\ell}\in\mathbb{R}_{+}$ and $v_{n}\in\mathbb{R}_{+}$ denote
the squared magnitude of current flow in the line $\ell$ and the squared
voltage magnitude at the node $n$, respectively. The SOC relaxation for the
power flow equations (Farivar and Low 2013)
$\forall\ell=(n,n^{\prime})\in\mathcal{L}$ is

$\displaystyle{s_{\ell}^{\text{a}}}^{2}+{s_{\ell}^{\text{r}}}^{2}\leq\theta_{\ell}v_{n}$
(6a)
$\displaystyle{s_{\ell}^{\text{a}}}^{2}+{s_{\ell}^{\text{r}}}^{2}\leq{\overline{s}_{\ell}}^{2},$
(6b)

where $\overline{s}_{\ell}\in\mathbb{R}_{+}$ is a parameter denoting the rated
power transfer capacity of line $\ell$. Constraints (6) can be reformulated as
SOC constraints admitting the variables
$\begin{bmatrix}s_{\ell}^{\text{a}}\quad
s_{\ell}^{\text{r}}\quad\theta_{\ell}\quad v_{n}\end{bmatrix}^{\top}$. For
instance, (6a) is equivalent to a SOC constraint of the general form (1) with
the parameters

$\displaystyle\mathbf{A}=\begin{bmatrix}[l]2\quad 0\quad 0\quad\phantom{-}0\\\
0\quad 2\quad 0\quad\phantom{-}0\\\ 0\quad 0\quad
1\quad-1\end{bmatrix}\;,\;\;\mathbf{b}=\mathbb{0}_{3}\;,\;\;\mathbf{d}=\begin{bmatrix}0\quad
0\quad 1\quad 1\end{bmatrix}^{\top}\;\text{and}\;\;e=0,$

which is a four-dimensional SOC constraint. Similarly, (6b) is a three-
dimensional SOC constraint with a structure similar that in Example 2.
$\blacksquare$

Under the approximate linearization approach, the reactive power
$s^{\text{r}}$ in (6) is ignored and voltage magnitudes across the network are
considered to be 1 per unit, reducing it to

$\displaystyle{{s_{\ell}^{\text{a}}}}^{2}\leq{\overline{s}_{\ell}}^{2}\Leftrightarrow\absolutevalue{s_{\ell}^{\text{a}}}\leq\overline{s}_{\ell}.$
(7)

Following this approach, we now provide an expression for the power flows
using PTDF.

######  Definition 2.7 (Power flows using PTDF)

With the set $\mathcal{I}_{n}\subseteq\mathcal{I}$ collecting market
participants located at node $n$, the active power flow along a line
$\ell=(n,n^{\prime})$ at hour $t$ is given by

$\displaystyle
s_{\ell}^{\text{a}}=\sum_{n\in\mathcal{N}}[\bm{\Psi}]_{(\ell,n)}\left(\sum_{i\in\mathcal{I}_{n}}\sum_{p\in\mathcal{P}}\;[\mathbf{G}_{ip}\mathbf{q}_{ip}]_{t}\right),$

where $\bm{\Psi}\in\mathbb{R}^{L\times N}$ denotes the PTDF matrix of the
electricity network. Derived from the physical parameters of power lines
comprising the network, entries of the PTDF matrix are the sensitivity of
changes in the flow in any line to a unit injection at a given node. The
expression
$\sum_{i\in\mathcal{I}_{n}}\sum_{p\in\mathcal{P}}\;[\mathbf{G}_{ip}\mathbf{q}_{ip}]_{t}\in\mathbb{R}$
computes the algebraic sum of injections or withdrawals of all participants
located at node $n$ towards the $P$ commodities at hour $t$.  $\blacksquare$

###### Remark 2.8

All $P$ commodities share the common physical network for fulfillment of the
trades, which is reflected by the summation in the expression for power flows
in Definition 2.7.  $\blacksquare$

#### Conic market-clearing problem, $\mathcal{M}^{\text{c}}$:

Without loss of generality, we assume that the participants have quadratic
cost functions while satisfying the conditions on the cost functions as
discussed in section 2.1. In the interest of computational and analytical
simplicity, we reformulate the quadratic costs as SOC constraints adopting the
following approach.

######  Example 2.9 (Quadratic cost)

Consider participant $i$ having a quadratic cost (utility) of production
(consumption), such that the cost function at each hour $t$ is given by
$c_{it}(\mathbf{q}_{it})=\mathbf{q}_{it}^{\top}\text{diag}(\mathbf{c}^{\text{Q}}_{it})\mathbf{q}_{it}+{\mathbf{c}^{\text{L}}_{it}}^{\top}\mathbf{q}_{it}$,
where $\mathbf{c}^{\text{Q}}_{it}\in\mathbb{R}^{K_{i}}$ and
$\textbf{c}_{it}^{\text{L}}\in\mathbb{R}^{K_{i}}$ denote the quadratic and
linear cost coefficients, respectively. Let
${\mathbf{C}}_{it}^{\text{Q}}\in\mathbb{R}^{K_{i}\times K_{i}}$ be defined as
a factorization of the quadratic cost matrix such that
$\text{diag}(\mathbf{c}^{\text{Q}}_{it})={{\mathbf{C}}^{\text{Q}}_{it}}^{\top}{{\mathbf{C}}_{it}^{\text{Q}}}$.
The existence of such a factorization is given since
$\text{diag}(\mathbf{c}^{\text{Q}}_{it})\succcurlyeq 0,\;\forall t,\;\forall
i\in\mathcal{I}$. The quadratic objective function of the participant $i$ is
equivalently written as

$\displaystyle\underset{{\mathbf{q}_{it},z_{it}}}{\text{min}}\quad$
$\displaystyle\sum_{t\in\mathcal{T}}\Big{(}{z}_{it}+{\mathbf{c}_{it}^{\text{L}}}^{\top}\mathbf{q}_{it}\Big{)}$
(8a) s.t.
$\displaystyle\norm{{\mathbf{C}}_{it}^{\text{Q}}\mathbf{q}_{it}}_{2}^{2}\leq{z}_{it},\;\forall
t,$ (8b)

where $z_{it}\in\mathbb{R}_{+}$ is an auxiliary variable, resulting in the
linear objective function (8a). The rotated SOC constraint (8b) is in fact a
special form of the general SOC constraint (1), see Mosek ApS (2021). We
illustrate this for single-period market-clearing, i.e., $T=1$, assuming that
participant $i$ has $K_{i}=2$ decisions variables, such that
$\mathbf{q}_{i1}\in\mathbb{R}^{2}$. Extending the decision vector of
participant $i$ to $\mathbf{q}_{i}=\begin{bmatrix}\mathbf{q}_{i1}^{\top}\quad
z_{i1}\end{bmatrix}^{\top}\in\mathbb{R}^{3}$, we have the parameters
$\mathbf{A}_{i}=\begin{bmatrix}\mathbf{C}^{\text{Q}}_{it}\quad\mathbb{0}_{2}\end{bmatrix}$,
$\mathbf{d}_{i}=\begin{bmatrix}0\quad 0\quad 1\end{bmatrix}^{\top}$, while
$\mathbf{b}_{i}=\mathbb{0}_{2}$ and $e_{i}=0$, resulting in a three-
dimensional rotated SOC constraint, i.e., $m_{i}=2$.  $\blacksquare$

The market-clearing problem, hereafter referred to as
$\mathcal{M}^{\text{c}}$, is therefore formulated as

$\displaystyle\underset{{\mathbf{{q}}_{i},\mathbf{z}_{i}}}{\text{min}}\quad$
$\displaystyle\sum_{i\in\mathcal{I}}\;\sum_{t\in\mathcal{T}}\;\Big{(}{z}_{it}+{\mathbf{c}_{it}^{\text{L}}}^{\top}\mathbf{q}_{it}\Big{)}$
(9a) s.t.
$\displaystyle\norm{\mathbf{C}^{\text{Q}}_{it}\mathbf{q}_{it}}_{2}^{2}\leq{z}_{it},\;\forall
t,\;\forall i$ $\displaystyle:$
$\displaystyle(\bm{\mu}^{\text{Q}}_{it},\kappa^{\text{Q}}_{it},\nu^{\text{Q}}_{it})$
(9b)
$\displaystyle\norm{\mathbf{A}_{ij}\mathbf{{q}}_{i}+\mathbf{b}_{ij}}_{2}\leq\mathbf{d}^{\top}_{ij}\mathbf{{q}}_{i}+e_{ij},\;\forall
j\in\mathcal{J}_{i},\;\forall i$ $\displaystyle\colon$
$\displaystyle(\bm{\mu}_{ij},~{}{\nu}_{ij})$ (9c)
$\displaystyle\mathbf{F}_{i}\mathbf{{q}}_{i}=\mathbf{h}_{i},\;\forall i$
$\displaystyle\colon$ $\displaystyle(\bm{\gamma}_{i})$ (9d)
$\displaystyle\sum_{i\in\mathcal{I}}\mathbf{G}_{ip}\mathbf{q}_{ip}=\mathbf{0}_{T},\;\forall
p,$ $\displaystyle\colon$ $\displaystyle(\bm{\lambda}_{p})$ (9e)
$\displaystyle\absolutevalue{\sum_{n\in\mathcal{N}}[\bm{\Psi}]_{(:,n)}\left(\sum_{i\in\mathcal{I}_{n}}\sum_{p\in\mathcal{P}}\;[\mathbf{G}_{ip}\mathbf{q}_{ip}]_{t}\right)}\leq\overline{\mathbf{s}},\;\forall
t,$ $\displaystyle\colon$
$\displaystyle(\underline{\bm{\varrho}}_{t},~{}\overline{\bm{\varrho}}_{t})$
(9f)

where the objective (9a) minimizes social disutility (or maximizes social
welfare) of the market-clearing problem over the time horizon,
$t\in\mathcal{T}$. The optimization variables are the decision variables of
the market participants $\mathbf{q}_{i}\in\mathbb{R}^{K_{i}T},\forall i$ and
$\mathbf{z}_{i}\in\mathbb{R}^{T},\forall i$. Recall that
$\mathbf{q}_{it}\in\mathbb{R}^{K_{i}}$ and $\mathbf{q}_{ip}\in\mathbb{R}^{T}$
are subsets of the $i$-th participant’s decision vector $\mathbf{q}_{i}$. The
Lagrange multipliers or the dual variables associated with the constraints are
shown in parentheses next to them. The dimensions of the parameters appearing
in the constraints, listed in their order of appearance, are
$\mathbf{A}_{ij}\in\mathbb{R}^{m_{ij}\times K_{i}T}$,
$\mathbf{b}_{ij}\in\mathbb{R}^{m_{ij}}$,
$\mathbf{d}_{ij}\in\mathbb{R}^{K_{i}T}$, $e_{ij}\in\mathbb{R}$,
$\mathbf{F}_{i}\in\mathbb{R}^{R_{i}\times K_{i}T}$,
$\mathbf{h}_{i}\in\mathbb{R}^{R_{i}}$, $\mathbf{G}_{ip}\in\mathbb{R}^{T\times
T}$, $[\bm{\Psi}]_{(:,n)}\in\mathbb{R}^{L}$ and
$\overline{\mathbf{s}}\in\mathbb{R}^{L}$. The index $j\in\mathcal{J}_{i}$
refers to the $J_{i}$ SOC constraints of participant $i$ as discussed in
section 2.2, while the index $p\in\mathcal{P}$ refers to the $P$ commodities
traded among the participants. Constraints (9b)-(9d) are participant-specific,
while (9e)-(9f) are related to the commodities exchanged in the market.

As in Example 2.9, the temporally-separable quadratic and linear bid prices of
participant $i$ are given by
$\mathbf{c}_{it}^{\text{Q}},\mathbf{c}_{it}^{\text{L}}\in\mathbb{R}^{K_{i}}$
and ${\mathbf{C}}_{it}^{\text{Q}}\in\mathbb{R}^{K_{i}\times K_{i}}$ is a
factorization of the quadratic cost matrix such that
$\text{diag}(\mathbf{c}^{\text{Q}}_{it})={{\mathbf{C}}^{\text{Q}}_{it}}^{\top}{{\mathbf{C}}_{it}^{\text{Q}}}$.
Lagrange multipliers
$\bm{\mu}^{\text{Q}}_{it}\in\mathbb{R}^{K_{i}},\;\kappa^{\text{Q}}_{it}\in\mathbb{R}_{+}\;\text{and}\;\nu^{\text{Q}}_{it}\in\mathbb{R}_{+},\;\forall
t,\;\forall i$, are participant-specific dual variables associated with the
rotated SOC constraints. Constraints (9c) model the SOC constraints applicable
to the market participants, for instance, those discussed in Examples 2.1 and
2.2, that enable asset- and uncertainty-awareness of the market-clearing
problem. The tuple of dual variables associated with the SOC constraints (9c)
are $(\bm{\mu}_{ij},\nu_{ij})$, where $\bm{\mu}_{ij}\in\mathbb{R}^{m_{ij}}$
and $\nu_{ij}\in\mathbb{R}_{+}$. Likewise, the dual variable
${\bm{\gamma}}_{i}\in\mathbb{R}^{R_{i}}$ associates with the $R_{i}$ equality
constraints for the participant $i$, see section 2.3.

Constraints (9e) are the system-wide balance constraints that couple the
decisions of the market participants for each of the $P$ commodities traded in
the market, such that the Lagrange multipliers associated with these
constraints $\bm{\lambda}_{p}\in\mathbb{R}^{T}$ are interpretable as commodity
prices. For instance, the Lagrange multiplier associated with the balance
between electricity production and consumption is the nodal price of
electricity. Lastly, constraints (9f), defined at each hour $t$, limit the
magnitude of power flow in the lines to their rated capacity,
$\overline{\mathbf{s}}$, as given by Definition 2.7. Given the element-wise
absolute value operator in this constraint, we associate a pair of non-
negative dual variables
$\underline{\bm{\varrho}}_{t},~{}\overline{\bm{\varrho}}_{t}\in\mathbb{R}_{+}^{L}$
with it. Naturally, constraints (9f) can be altered to include asymmetric
limits, i.e., the limits which depend on the direction of flow in the power
lines, without significant change to the analytical results that follow.
Further, note that instead of the chosen PTDF-based formulation, implementing
an SOCP relaxation for the formulation of power flow equations, for instance,
as highlighted in Example 2.6, affects the constraint (9f). However, the
overall market-clearing problem $\mathcal{M}^{\text{c}}$ remains within the
SOCP framework.

The feasibility set for the optimization problem $\mathcal{M}^{\text{c}}$ is
convex and is formed by the intersection of a number of convex sets, which
include (i) halfspaces (9f), (ii) hyperplanes (9d)-(9e), and lastly, (iii) the
Cartesian product of second-order cones (9c) and rotated second-order cones
(9b). Solving problem $\mathcal{M}^{\text{c}}$ entails minimizing a linear
objective function over this feasibility set and therefore, is a convex
minimization problem.

######  Remark 2.10 (Strict convexity)

Observe that the optimization problem $\mathcal{M}^{\text{c}}$ has a strictly
convex objective function (9a) provided every market participant incurs a non-
zero quadratic cost at all hours for each of its $K_{i}$ variables. To see
that, we recall that every positive semidefinite matrix is the Gram matrix for
some set of vectors. Specifically, for the quadratic cost matrix
$\text{diag}(\mathbf{c}_{it}^{\text{Q}})\succcurlyeq 0$,

$\displaystyle\mathbf{q}_{it}^{\top}\text{diag}(\mathbf{c}_{it}^{\text{Q}})\mathbf{q}_{it}=\mathbf{q}_{it}^{\top}{\mathbf{C}_{it}^{\text{Q}}}^{\top}\mathbf{C}_{it}^{\text{Q}}\mathbf{q}_{it}=\norm{\mathbf{C}_{it}^{\text{Q}}\mathbf{q}_{it}}_{2}^{2},\;\forall
t,\forall i,$

such that strict convexity is guaranteed only if the quadratic cost matrix is
positive definite, i.e., $\text{diag}(\mathbf{c}_{it}^{\text{Q}})\succ 0$,
which requires $c^{\text{Q}}_{itk}\neq 0,\;\forall k=1,2,\dots,K_{i},\;\forall
t,\;\forall i$.  $\blacksquare$

##  3 Economic Interpretation and Equilibrium Analysis

After establishing convexity of the conic market-clearing problem
$\mathcal{M}^{\text{c}}$, we next discuss the price formation process for the
$P$ commodities traded in the market.

###  3.1 Conic Spatial Prices for Commodities

Deriving optimal market-clearing prices for the problem
$\mathcal{M}^{\text{c}}$ is not straightforward, since strong duality is not
trivial to establish for an SOCP problem (Ben-Tal and Nemirovski 2001). Unlike
their LP counterparts where strong duality is guaranteed merely by the
feasibility of the primal and dual problems as formulated in Farkas’ lemma,
strong duality in SOCP problems derives from the existence of strictly
feasible666Conventionally, in a market-clearing context, ensuring the strict
feasibility or even merely the feasibility of market-clearing problem is
considered at a market design stage and is not of high relevance for the
market operator. Nevertheless, in the interest of generality and wider
acceptance of our conic market-clearing proposal, we address this crucial
issue in our analytical results. solutions to both the primal and dual
problems (Alizadeh and Goldfarb 2003, Theorem 13). Following Slater’s
constraint qualification for convex optimization problems, strict feasibility
refers to the existence of points within the feasibility set of the primal
problem where all inequalities - SOC and linear - are strictly satisfied.
Additionally, a refinement to Slater’s constraint qualification yields the so-
called essentially strict feasibility of an SOCP problem, which we define for
problem $\mathcal{M}^{\text{c}}$ in the following.

######  Definition 3.1 (Essentially Strict Feasibility)

The market-clearing problem $\mathcal{M}^{\text{c}}$ is essentially strictly
feasible, if there exists a feasible solution tuple denoted by
$(\mathbf{q}_{i},\mathbf{z}_{i}),\;\forall i$ such that

$\displaystyle\norm{\mathbf{C}^{\text{Q}}_{it}\mathbf{q}_{it}}_{2}^{2}<{z}_{it},\;\forall
t,\;\forall i,$
$\displaystyle\norm{\mathbf{A}_{ij}\mathbf{{q}}_{i}+\mathbf{b}_{ij}}_{2}<\mathbf{d}^{\top}_{ij}\mathbf{{q}}_{i}+e_{ij},\;\forall
j\in\mathcal{J}_{i},\;\forall i.{\quad\quad\penalty
9999\hbox{}\nobreak\hfill\quad\hbox{$\blacksquare$}}$

Observe that essentially strict feasibility is a weaker requirement as
compared to strict feasibility. Nevertheless, due to Ben-Tal and Nemirovski
(2001, Theorem 1.4.4), it is necessary and sufficient for strong duality to
hold for $\mathcal{M}^{\text{c}}$. This stems from the fact that the other
inequalities (9f) are linear in decision variables (Boyd and Vandenberghe
2004, §5.2.3). With this background, we provide the analytical results crucial
to deriving optimal prices from the market-clearing problem
$\mathcal{M}^{\text{c}}$.

######  Theorem 3.2 (Strong Duality)

Let $\mathcal{D}^{\text{c}}$ denote the dual problem to the market-clearing
problem $\mathcal{M}^{\text{c}}$. If the set of feasible solutions to the
primal problem $\mathcal{M}^{\text{c}}$ is non-empty, then both the primal
problem $\mathcal{M}^{\text{c}}$ and dual problem $\mathcal{D}^{\text{c}}$ are
essentially strictly feasible, and consequently, strong duality holds for the
primal-dual pair of problems $\mathcal{M}^{\text{c}}$ and
$\mathcal{D}^{\text{c}}$.

Conditioned on feasibility of the primal market-clearing problem
$\mathcal{M}^{\text{c}}$, Theorem 3.2 enables economic interpretations of the
market-clearing outcomes. Relying on classical Lagrangian duality theory, we
provide an analytical expression for the optimal, spatially-differentiated
nodal prices of the $P$ commodities in the following theorem.

######  Theorem 3.3 (Conic Spatial Prices)

The solution to the conic market-clearing problem $\mathcal{M}^{\text{c}}$
results in optimal contributions $\mathbf{q}_{ip}^{\star},\;\forall
i\in\mathcal{I}$ for the $p\in\mathcal{P}$ commodities and the market clears
with optimal prices $\bm{\Pi}^{\star}_{p}\in\mathbb{R}^{N\times T}$ for the
$p$-th commodity given by

$\displaystyle\bm{\Pi^{\star}}_{p}={\bm{\Lambda}}^{\star}_{p}-\bm{\Psi}^{\top}(\overline{\bm{\rho}}^{\star}-\underline{\bm{\rho}}^{\star}),\enspace\forall
p\in\mathcal{P},$ (10)

where
$\underline{\bm{\rho}}^{\star},~{}\overline{\bm{\rho}}^{\star}\in\mathbb{R}^{L\times
T}$ and ${\bm{\Lambda}}^{\star}_{p}\in\mathbb{R}^{N\times T}$ are variables
with stacked columns of optimal dual variables
$\underline{\bm{\varrho}}^{\star}_{t}$,
$\overline{\bm{\varrho}}^{\star}_{t},\;\forall t$ and
$\bm{\lambda}_{p}^{\star},\;\forall n$, respectively, over the $T$ market-
clearing hours, i.e.,

$\displaystyle\overline{\bm{\rho}}^{\star}=[\overline{\bm{\varrho}}^{\star}_{1}~{}\cdots~{}\overline{\bm{\varrho}}^{\star}_{T}],\;\underline{\bm{\rho}}^{\star}=[\underline{\bm{\varrho}}^{\star}_{1}~{}\cdots~{}\underline{\bm{\varrho}}^{\star}_{T}],\;\text{
and
}\;\bm{\Lambda}_{p}^{\star}\coloneqq\mathbb{1}_{N}^{\top}\otimes\bm{\lambda}^{\star}_{p}.$

The commodity prices given by Theorem 3.3 are analogous in structure to the
LMPs resulting from the prevalent LP-based electricity market-clearing
frameworks. Specifically, the optimal prices $\bm{\Pi}_{p}^{\star}$ comprise
of a nodal price component for each commodity $\bm{\Lambda}_{p}^{\star}$ and a
network price component which is non-zero only if congestion arises in the
network as a consequence of the power flows along the lines to fulfill the
trades.

###  3.2 $\mathcal{M}^{\text{c}}$ as a Spatial Equilibrium Problem

In what follows, we present theoretical results that characterize the spatial
price equilibrium underlying the centrally-solved market-clearing problem
$\mathcal{M}^{\text{c}}$. First, we show the equivalence of the optimization
problem $\mathcal{M}^{\text{c}}$ to a spatial equilibrium problem. Referring
to the two roles played by the system operator discussed in section 1,
consider a virtual separation of the system operator into a market operator
and a network operator. First, the market operator, acting as the price
setter, collects the conic market bids from market participants and is
responsible for clearing the day-ahead market and for the real-time operation
of the system. Second, the network operator is responsible for the physical
fulfillment of the commodities exchanged in the market. In the market context,
the network operator can be seen as a spatial arbitrager who buys electricity
at node $n\in\mathcal{N}$ at a certain nodal price and sells it back at an
adjacent node $n^{\prime}\in\mathcal{N}$ at a price corresponding to that node
(Hobbs 2001). Any congestion in the line connecting the nodes $n$ and
$n^{\prime}$ leads to different nodal prices and thereby a non-zero revenue,
the so-called congestion rent, for the network operator.

Consider an equilibrium problem $\mathcal{E}^{\text{c}}$ comprised of a set of
individual optimization problems of the $i\in\mathcal{I}$ market participants,
the network operator and the market-clearing conditions. For market
participant $i$ located at node $n_{i}\in\mathcal{N}$, let
$\mathbf{W}_{ip}\in\mathbb{R}^{N\times T}$ denote the quantities of the
commodity $p\in\mathcal{P}$ transacted (bought or sold) over the $N$ nodes at
the $T$ hours. Under the perfect competition assumption, the commodity prices
$\bm{\Pi}_{p},\;\forall p$ are exogenous to the participants’ optimization
problem. Therefore, participant $i$ maximizes her profit by deciding the
optimal contributions $\mathbf{q}_{ip},\;\forall p\in\mathcal{P}$ commodities
traded in the market, subject to her operational constraints, by solving the
problem

$\displaystyle\underset{{\mathbf{q}_{i},\;\mathbf{z}_{i},\mathbf{W}_{ip}}}{\text{max}}\quad$
$\displaystyle\sum_{p\in\mathcal{P}}\text{tr}({{\bm{\Pi}_{p}}^{\top}}\mathbf{W}_{ip})-\sum_{t\in\mathcal{T}}\left({z}_{it}+{\mathbf{c}_{it}^{\text{L}}}^{\top}\mathbf{q}_{it}\right)$
$\displaystyle\quad\quad-\sum_{t\in\mathcal{T}}{\bm{\omega}_{t}}^{\top}\Big{(}\sum_{p\in\mathcal{P}}\big{(}[\mathbf{W}_{ip}]_{(:,t)}-\mathbb{I}_{n_{i}}[\mathbf{G}_{ip}\mathbf{q}_{ip}]_{t}\big{)}\Big{)}$
(11a) s.t.
$\displaystyle\norm{\mathbf{C}^{\text{Q}}_{it}\mathbf{q}_{it}}_{2}^{2}\leq{z}_{it},\;\forall
t$ $\displaystyle:$
$\displaystyle(\bm{\mu}^{\text{Q}}_{it},\kappa^{\text{Q}}_{it},\nu^{\text{Q}}_{it})$
(11b)
$\displaystyle\norm{\mathbf{A}_{ij}\mathbf{{q}}_{i}+\mathbf{b}_{ij}}_{2}\leq\mathbf{d}^{\top}_{ij}\mathbf{{q}}_{i}+e_{ij},\;\forall
j\in\mathcal{J}_{i}$ $\displaystyle:$
$\displaystyle(\bm{\mu}_{ij},~{}{\nu}_{ij})$ (11c)
$\displaystyle\mathbf{F}_{i}\mathbf{{q}}_{i}=\mathbf{h}_{i}$ $\displaystyle:$
$\displaystyle(\bm{\gamma}_{i})$ (11d)
$\displaystyle\mathbf{W}_{ip}^{\top}\mathbb{1}=\mathbf{G}_{ip}\;\mathbf{q}_{ip},\;\forall
p\in\mathcal{P},$ $\displaystyle:$ $\displaystyle(\bm{\hat{\lambda}}_{ip})$
(11e)

where the three terms comprising the objective function (11a) are the revenues
generated by the transactions, the cost (utility) of production (consumption)
and the third term is the transport cost for the physical fulfillment of the
transactions at the nodes where they are delivered. Here, the indicator vector
$\mathbb{I}_{n_{i}}\in\mathbb{R}^{N}$ contains the element corresponding to
the location of participant $i$ as $1$, while all other elements are $0$.
Constraints (11e) ensure a balance of the commodity transactions with the
contributions towards the trades as given by Definition 2.3. Variables
$\bm{\hat{\lambda}}_{ip}\in\mathbb{R}^{T}$ are the Lagrange multipliers
associated with these constraints. Note that the nodal commodity prices
$\bm{\Pi}_{p}$ and the price of transmitting power
$\bm{\omega}_{t}\in\mathbb{R}^{N},\;\forall t$ are set by the market operator
and, as such, are considered as fixed by the participant.

Next, acting as a spatial arbitrager, the network operator maximizes
congestion rent, while fulfilling the trades of all the commodities in the
market. Let $\mathbf{y}_{t}\in\mathbb{R}^{N},\;\forall t$ denote the net power
injection comprising of all commodities and all market participants at the $N$
nodes at each hour. The network operator solves the following maximization
problem subject to the network limits

$\displaystyle\underset{{\mathbf{y}_{t}}}{\text{max}}\quad\sum_{t\in\mathcal{T}}\;{\bm{\omega}_{t}}^{\top}\mathbf{y}_{t}\quad\text{s.t.}\quad-\overline{\mathbf{s}}\leq\bm{\Psi}\mathbf{y}_{t}\leq\overline{\mathbf{s}},\quad\colon(\underline{\bm{\varrho}}_{t},~{}\overline{\bm{\varrho}}_{t}),$
(12)

where ${\bm{\omega}}_{t}$ are variables exogenous to the network operator and
the constraints limit the power flows along the lines to their rated capacity
in both directions.

Lastly, the market operator, acting as a price setter, clears the market based
on the following equalities whose shadow prices provide price of the $P$
commodities and price of the transmission service provided by the network
operator

$\displaystyle\sum_{p\in\mathcal{P}}[{\mathbf{Q}_{p}^{\text{inj}}}]_{(:,t)}=\mathbf{y}_{t},\;\forall
t$ $\displaystyle:$ $\displaystyle(\bm{\omega}_{t})$ (13a)
$\displaystyle\sum_{i\in\mathcal{I}}\mathbf{G}_{ip}\mathbf{q}_{ip}=\mathbb{0},\;\forall
p\in\mathcal{P},$ $\displaystyle:$ $\displaystyle(\bm{\lambda}_{p})$ (13b)

where the auxiliary variable
$\mathbf{Q}_{p}^{\text{inj}}\in\mathbb{R}^{N\times T}$ denotes the commodity-
specific net nodal injections, i.e.,

$\displaystyle\mathbf{Q}_{p}^{\text{inj}}$
$\displaystyle\coloneqq\begin{bmatrix}\sum_{i\in\mathcal{I}_{1}}{(\mathbf{G}_{ip}\mathbf{q}_{ip})}^{\top}\\\
\sum_{i\in\mathcal{I}_{2}}{(\mathbf{G}_{ip}\mathbf{q}_{ip})}^{\top}\\\
\vdots\\\
\sum_{i\in\mathcal{I}_{N}}{(\mathbf{G}_{ip}\mathbf{q}_{ip})}^{\top}\end{bmatrix}.$
(14)

The market-clearing equalities (13a) ensure the net injection at each of the
nodes of the network is balanced by the transport service provided by the
network operator. The shadow price of this constraint, $\bm{\omega}_{t}$,
appears as a parameter in the network operator’s maximization (12) and is
interpreted as the price of transmitting power from an arbitrary hub to the
each of the nodes. As discussed previously, (13b) ensure the system-wide
balance of traded commodities.

######  Theorem 3.4 (Competitive Spatial Equilibrium)

The convex market-clearing problem $\mathcal{M}^{\text{c}}$ solved centrally
by the system operator is equivalent to a competitive spatial equilibrium
$\mathcal{E}^{\text{c}}$ comprised of market participants, $i\in\mathcal{I}$
each solving the profit maximization (11), the network operator solving the
congestion rent maximization (12) and the market operator clearing the market
by enforcing the equalities (13).

The proof for Theorem 3.4, provided in the appendix, relies on the equivalence
of the Karush-Kuhn-Tucker (KKT) optimality conditions of the two problems.

######  Corollary 3.5 (Existence and Uniqueness)

The solution to the competitive spatial price equilibrium problem
$\mathcal{E}^{\text{c}}$ exists and is unique in allocations
$\mathbf{q}_{i}^{\star}$, provided all market participants bid with non-zero
quadratic price components, i.e., $c_{itk}^{\text{Q}}\neq 0,\;\forall
k=1,2,\dots,K_{i},\;\forall t,\;\forall i\in\mathcal{I}$.

We next analyze the desirable economic properties underlying the competitive
spatial price equilibrium $\mathcal{E}^{\text{c}}$.

######  Theorem 3.6 (Economic Properties)

The market-clearing problem $\mathcal{M}^{\text{c}}$ and its equivalent
competitive spatial equilibrium $\mathcal{E}^{\text{c}}$ result in optimal
allocations $\mathbf{q}_{i}^{\star},\;\forall i\in\mathcal{I}$ and spatial
commodity prices $\bm{\Pi}_{p}^{\star},\;\forall p\in\mathcal{P}$ such that
the following economic properties are attained at optimality:

  1. (i)

Market efficiency: Under the perfect competition assumption, social welfare is
maximized, such that no participant desires to unilaterally deviate from the
market-clearing outcomes.

  2. (ii)

Cost recovery: Let the market bids $\mathcal{B}_{i}$ for each market
participant $i\in\mathcal{I}$ be such that
$e_{ij}\geq\norm{\mathbf{b}_{ij}}_{2},\;\forall j\in\mathcal{J}_{i}$ and
$\mathbf{h}_{i}=\mathbb{0}$. Then, the optimal allocations
$\mathbf{q}_{i}^{\star},\;\forall i\in\mathcal{I}$ and optimal spatial
commodity prices $\bm{\Pi}_{p}^{\star},\;\forall p\in\mathcal{P}$ ensure cost
recovery for the market participants.

  3. (iii)

Revenue adequacy: The market operator does not incur financial deficit at the
end of the market-clearing horizon, i.e.,

$\displaystyle\sum_{p\in\mathcal{P}}\sum_{i\in\mathcal{I}}\text{tr}({{\bm{\Pi}_{p}^{\star}}}^{\top}\mathbf{W}_{ip}^{\star})-\sum_{t\in\mathcal{T}}{\bm{\omega}_{t}^{\star}}^{\top}\mathbf{y}_{t}^{\star}\geq
0.$

Theorem 3.6 characterizes the economic properties underlying the market-
clearing outcomes from the conic market-clearing $\mathcal{M}^{\text{c}}$.
While market efficiency is guaranteed under the perfect competition
assumption, cost recovery is ensured for all participants under two
conditions, which we elaborate in the following. First, as shown in the
Examples in section 2 and in the e-Companion, the condition
$e_{ij}\geq\norm{\mathbf{b}_{ij}}_{2}$ holds true for all participants in most
practical settings, with the notable exception of market participants having a
non-zero lower bound on their decision variables. The practical issue of non-
guarantee of cost recovery for such market participants also prevails in
currently-operational LP-based electricity markets. The second condition
relates to the homogeneity of the linear equality constraints (9d), i.e.,
$\mathbf{q}_{i}=\mathbb{0},\;\forall i$ is a feasible solution to the market-
clearing problem. Observe that, the feasibility of a zero-allocation solution,
i.e., $\mathbf{q}_{i}=\mathbb{0},\;\forall i$ also satisfies the first
condition, thereby indicating that these two conditions are equivalent.
Lastly, the condition for revenue adequacy for the market operator requires
that the net payments received from the market participants $i\in\mathcal{I}$
less the payments made by the market operator to the network operator towards
transmission service is non-negative.

######  Remark 3.7 (Incentive Compatibility)

Roberts and Postlewaite (1976) show that the incentive for actors to deviate
from price-taking, perfectly-competitive behavior decreases to zero as the
number of actors goes to infinity. Thus, assuming a very large number of
market participants, the conic market-clearing proposal tends towards
incentive compatibility, i.e., at the limit participants bid according to
their true preferences. The conditions under which incentive compatibility is
satisfied by $\mathcal{M}^{\text{c}}$ is akin to the prevalent LP-based
electricity markets, thereby preserving this desirable economic property in
the move towards a SOCP-based market-clearing framework.  $\blacksquare$

Beyond the desirable economic properties discussed so-far, we analyze the
robustness of the market-clearing outcomes resulting from
$\mathcal{M}^{\text{c}}$ against small changes in parameters. Strong duality
aside, in the context of a market-clearing problem, this property is crucial
to be studied for SOCP problems. Robust solvability refers to the property of
an SOCP problem such that it remains solvable and obtains the optimal solution
even when the problem parameters are changed by arbitrary small perturbations.
Ben-Tal and Nemirovski (2001) in Proposition 1.4.6 establish that robust
solvability is guaranteed by the strict feasibility of both primal and dual
problems. We provide the following corollary to Theorem 3.2 formalizing robust
solvability of the conic market-clearing problem proposed in this paper.

######  Corollary 3.8 (Robust Solvability of $\mathcal{M}^{\text{c}}$)

The market-clearing problem $\mathcal{M}^{\text{c}}$ is robust solvable, i.e.,
for the $p\in\mathcal{P}$ commodities traded in the market, both the optimal
contributions $\mathbf{q}_{ip}^{\star},\;\forall i$ and the optimal prices
$\bm{\Pi}_{p}^{\star}$ obtained, are robust against small perturbations in the
parameters comprising the bids $\mathcal{B}_{i},\;\forall i\in\mathcal{I}.$

The proof to Corollary 3.8 is a direct consequence of Theorem 3.2. This result
ensures robust outcomes from a large-scale market-clearing problem admitting
hundreds (possibly thousands) of market participants and their SOC
constraints. In that case, robust solvability status of the market-clearing
problem implies that potential numerical pathologies arising from
approximations such as rounding-off errors, estimation of uncertain
parameters, etc. do not cause solvability-related issues.

##  4 Numerical Studies

We perform numerical experiments on a 24-node electricity system wherein the
various market participants, comprising of 6 wind power producers, 9 flexible
and 3 inflexible power producers (PPs), 3 identical energy storage units
(ESUs) owners, and 17 inflexible consumers, are connected at the nodes as
shown in Figure 2(a). The system data is adapted from Conejo et al. (2010) to
include the wind farms and ESUs. We study the market-clearing outcomes under
various renewable energy share (RES) paradigms, ranging from 10% to 60% of the
total energy demand from consumers met by the wind power producers. These
paradigms are derived by suitably varying the installed capacity of the wind
farms while dimensioning the ESUs such that the total available
charging/discharging capacity remains fixed at 12.5% of the total wind farm
installed capacity. Such dynamic dimensioning of storage-related flexibility
is crucial to ensure market-clearing feasibility for the high RES paradigms.
Inspired by the short-term renewable energy targets of around 50% RES share in
various countries (IRENA 2015), we present several key findings for the 50%
RES paradigm. The wind power producers bid at zero prices to ensure acceptance
of bids, and ESU owners bid at prices lower than the cheapest flexible PP. We
assume that the inflexible loads exhibit perfect inelasticity of demand,
rendering the social welfare maximization problem equivalent to finding cost-
minimal dispatch, in expectation, for PPs and ESUs to provide energy and
flexibility services needed to meet the net demand. Here, net demand refers to
the energy demand from inelastic loads reduced by the production from wind
farms during the real-time operation. There is no uncertainty in the inelastic
demand. The expected net demand based on the day-ahead wind power production
forecasts for the 50% RES paradigm is shown in Figure 2(b). Naturally, the
market operator faces high uncertainty in the hours with smaller net demand
due to the high share of weather-dependent renewable energy. To highlight the
impact of network congestion on spatial prices and quantities, we consider two
network configurations: (i) without any network bottlenecks and (ii) with
network bottlenecks induced by reducing capacities of three transmission lines
of the network, shown in blue in Figure 2(a).

$n_{1}$$n_{2}$$n_{3}$$n_{4}$$n_{5}$$n_{6}$$n_{7}$$n_{8}$$n_{9}$$n_{10}$$n_{11}$$n_{12}$$n_{13}$$n_{14}$$n_{15}$$n_{16}$$n_{17}$$n_{18}$$n_{19}$$n_{20}$$n_{21}$$n_{22}$$n_{23}$$n_{24}$250
MW250 MW500
MW$f_{1}$$f_{2}$$f_{3}$$f_{4}$$f_{5/6}$$f_{7}$$f_{8}$$f_{9}$$f_{10}$$f_{11/12}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{1}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{2}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{3}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{4}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{5}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}w_{6}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s_{1}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s_{2}}$${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}s_{3}}$

0.51.01.551015204812162000.81.62.4$\cdot 10^{3}$Net demand [MW] Energy price
[$/MWh] $\cdot 10^{3}$Hours Flexibility payment [$]
(b)(c)(d)$10\%$$20\%$$30\%$$40\%$$50\%$$60\%$

Figure 2: (a) 24-node electricity network showing a visualization of spatial
prices of energy for the network configuration with bottlenecks, (b) expected
net demand for the 50% RES paradigm, (c) system-wide prices for energy and (d)
the total hourly flexibility payments for various RES paradigms

Built on this electricity network backbone, our experiments in section 4.1
demonstrate the uncertainty-awareness of our market-clearing proposal,
followed by an analysis of the market-clearing outcomes and conic spatial
prices constituting the spatial price equilibrium. In section 4.2, we compare
the proposed SOCP market-clearing problem with two uncertainty-aware
alternatives within the LP domain. While we discuss our numerical results
solely in the context of an uncertainty-aware market framework in this
section, the examples provided in section 2.2 can be used to extend the
market-clearing problem to include asset- and/or network-awareness as we
highlight further in section 5.

###  4.1 SOCP-based Uncertainty-aware Energy and Flexibility Market

As discussed in section 2.2, a stochastic market-clearing problem based on
chance-constrained optimization enables endogenous consideration of
uncertainty faced by the electricity markets. Recalling Example 2.2, we
consider a chance-constrained electricity market-clearing problem wherein two
commodities, namely energy and adjustment policies, are cleared by the system
operator. The commodities are traded such that an uncertainty-aware spatial
price equilibrium is achieved. Optimal adjustment policies allocated to
flexibility providers enable the mitigation of uncertainty realized during the
real-time operation from weather-dependent renewable energy sources, while
look-ahead decisions are made by the system operator at the day-ahead market-
clearing stage, as shown in Figure 1. Activated during the real-time
operation, these policies are in per unit and imply the contribution of each
flexibility provider towards mitigating the potential real-time imbalance in
the system. For example, a flexibility provider may, in the day-ahead market,
be allocated a policy corresponding to 10% adjustment, implying that it
contributes to mitigating 10% of any type of imbalance (either over- or under-
supply) during the real-time operation. In addition to payments for energy,
flexibility providers are paid for the flexibility service upfront, i.e., at
the day-ahead market-clearing stage. Considering the novelty of admitting
adjustment policies as opposed to the conventional flexible capacity to meet
the uncertain net demand, we provide insights into the endogenous
consideration of uncertainty and quantify the flexibility payments in the
following. We describe the modeling of market participants, the chance-
constrained market-clearing problem and its SOCP reformulation (which we refer
to hereafter as $\mathcal{M}^{\text{cc}}$) in the e-Companion to the paper.

#### Impact of congestion and uncertainty on prices:

Overlayed on the 24-node electricity network, the density plot in Figure 2(a)
visualizes the impact of network bottlenecks on the day-ahead energy prices
for hour 23 under the 50% renewable energy share paradigm. Figures 2(c) and
2(d) show the commodity prices for the network configuration without
bottlenecks for the various RES paradigms. Observe that with higher shares of
renewable energy, the payment made by the market operator towards flexibility
increases, complementary to the gradual reduction in the energy price due to
wind farms bidding with zero prices. Overall, increasing uncertainty faced at
the day-ahead market-clearing stage leads to lower energy prices while the
payments towards flexibility services increase, thereby resulting in the right
market signals for investments in flexibility over the long run. Note that,
since the adjustment polices are quantified in per unit, the hourly
flexibility payments shown in Figure 2(d) denote total payments made by the
market operator towards flexibility, adopting an allocation determined by the
adjustment policies of individual flexibility providers and as such, following
a differentiated pricing scheme. We now discuss the allocation of adjustment
policies and provide further insights into the pricing of flexibility.

468101200.250.500.751.0Adjustment policies No network bottlenecks4681012With
network bottlenecks468101200.30.60.91.2$\cdot 10^{3}$PP dispatch [MW]
46810124681012-2.0-1.001.02.0$\cdot 10^{2}$ESU dispatch [MW]
4681012468101200.40.81.21.6$\cdot 10^{3}$Flex payment [$]
46810124681012080160240320Hours Flex pay rate [$/MW] 4681012020406080Hours
3(a)(b)(c)(d)(e)(f)(g)(h)(i)(j) Figure 3: Market-clearing outcomes in hours
4-12: (a)-(b) show the adjustment policies, (c)-(f) show the nominal dispatch
of power producers (PPs) and energy storage units (ESUs) for both the network
configurations, and (g)-(h) show the flexibility payments and (i)-(j) show the
flexibility payment rates (FPRs).

#### Flexibility allocation and payments:

For the 50% RES paradigm, Figures 3(a)-3(f) show the optimal allocation of
dispatch and adjustment policies to the PPs ($f1,\;f2,\dots,\;f12$) and to the
ESUs ($s1,\;s2,\;s3$) for selected hours of the day for both network
configurations. First, observe that non-zero adjustment policies are only
allocated to flexibility providers that are also dispatched for the commodity
energy, which is consistent with the requirement that both over- and under-
supply imbalances during the real-time operation are mitigated by the
flexibility delivered. Second, the network configuration with bottlenecks
mandates the allocation of adjustment policies to more number of flexible
power producers, as network congestion is expected to impact the flexibility
delivery during real-time operation. However, note that ESUs are not allocated
adjustment policies in this configuration. This is explained by (i) the
availability of flexible PPs in favorable locations of the network with
respect to congested power lines, and (ii) the inter-temporal constraints and
end-of-day energy balance requirement for ESUs (discussed in the e-Companion).
Indeed, for the paradigm with 60% RES (not shown in the Figure), ESUs in the
case with network bottlenecks are allocated non-zero adjustment policies to
contribute towards flexibility provision, with the market-clearing problem
choosing a more expensive flexibility allocation, offset by the reduced cost
of energy provision. Finally, observe in Figure 3(b) that the adjustment
policies may take negative values, implying that the action
(increasing/decreasing) production by power producers may be in opposition to
the system requirement (under-/over-supply), provided it leads to cost-optimal
flexibility provision under congested network conditions.

Figures 3(g)-3(h) show the flexibility payments made to flexibility providers
under the two network configurations. Observe that flexibility payments are,
in general, higher for the case with network bottlenecks as compared to the
case without any bottlenecks. Moreover, as previously-discussed in reference
to Figure 2(d), flexibility payments are higher for hours with high production
from wind farms. In the short run, this incentivizes market participants to
bid their flexibility in these hours. To further analyze how flexibility is
valued and paid for by the market operator, we introduce an ex-post parameter
called flexibility payment rate (FPR). Defined for each flexibility provider
$i$, FPR is the rate in $\$/\text{MWh}$ at which she is paid for the
flexibility service:

$\displaystyle\text{FPR}_{it}=\frac{[\bm{\Pi}_{p}^{\star}]_{(n_{i},t)}\times\alpha^{\star}_{it}}{\absolutevalue{\hat{q}^{\star}_{it}-\overline{q}_{it}}},$
(15)

where $[\bm{\Pi}_{p}^{\star}]_{(n_{i},t)}$ retrieves the price of the
commodity flexibility service at the node $n_{i}$ where the participant $i$ is
located, $\hat{q}_{it}^{\star}$ and $\alpha^{\star}_{it}$ denote the nominal
dispatch and adjustment policy allocated to the participant $i$, respectively.
The quantity $\overline{q}_{it}$ is the dispatch under the perfect forecast
case, i.e., when the day-ahead market-clearing is deterministic. We obtain
$\overline{q}_{it}$ by solving the market-clearing problem
$\mathcal{M}^{\text{cc}}$ assuming day-ahead forecasts are realized perfectly
during the real-time operation, such that all adjustment policies are set to
zero, i.e., $\alpha_{it}=0,\;\forall t,\;\forall i\in\mathcal{I}$. Figures
3(i) - 3(j) show the FPRs for the various participants for the two network
configurations. Observe that, equal segments within each bar indicate equal
FPRs for all flexibility providers for a given hour. First, we note that, in
general, more flexibility providers being allocated non-zero adjustment
policies in the network configuration with bottlenecks leads to lower FPRs for
the flexibility providers contrary to the one without bottlenecks. A notable
exception is hour 7 in the case without bottlenecks, wherein ESUs are paid for
the flexibility at very high rate. This is a consequence of two contributing
factors: (i) ESUs are dispatched with small nominal quantities in the hours 7
and 8 (see Figure 3(e)) which implies that their ability to provide both
charging and discharging flexibility is valued highly, and (ii) the net demand
follows a steep rise in the hours 6-8 (see Figure 2(b)) and as a result the
market-clearing problem faces a scarcity of not only flexible capacity, but
also the ramping ability needed to meet this change in uncertainty. In
contrast, in hour 8, while the ESUs are still nominally dispatched close to
zero, the flexibility is no longer scarce in the system as the net demand
rises sufficiently enough to lead to the nominal dispatch of power producer
$f1$, thereby homogenizing the FPRs again. Overall, the FPR for a flexibility
provider depends on a number of factors, including the level of uncertainty
perceived by the system (quantified by the forecast error covariance matrix as
well as the day-ahead forecasts), the location of the flexibility provider,
congestion in the network and whether other flexibility providers are
available.

###  4.2 Comparison with LP-based Uncertainty-aware Benchmarks

Next, using numerical simulations corresponding to realizations of the
uncertain wind power production, we compare the market-clearing outcomes and
performance of the proposed SOCP-based uncertainty-aware market-clearing
problem $\mathcal{M}^{\text{cc}}$ to available alternatives within the LP-
domain.

We consider two LP-based market-clearing problems as benchmarks:
$\mathcal{R}1$ and $\mathcal{R}2$. While it is most closely related to the
currently-operational electricity markets, the deterministic market-clearing
problem $\mathcal{R}1$ considers uncertainty during the day-ahead clearing
stage by commissioning flexibility in the form of reserve capacity. The
procured capacity is subsequently activated in real-time electricity markets,
cleared closer to physical delivery. To ensure a cost-optimal allocation of
the reserve capacity, the market operator enforces a exogenously-determined
minimum reserve requirement to procure flexible capacity from flexibility
providers. Market-clearing problem $\mathcal{R}2$ considers uncertainty based
on day-ahead scenarios for realizations of uncertain renewable energy
production. The market operator seeks to maximize the expected social welfare
under uncertainty, resulting in a day-ahead schedule which is then adjusted
during real-time operation for each of the foreseen scenarios. Similarly to
$\mathcal{M}^{\text{cc}}$, this approach considers uncertainty endogenously as
parameterized by the scenarios considered while solving the market-clearing
problem. However, leaving aside the fundamental question of trusting the
scenario-generating agent, this approach mandates a large number of scenarios
to appropriately represent the uncertainty, thereby, limiting its practical
adoption from a computational perspective. Figure 4 illustrates the market-
clearing activities entailed in each of the three problems, starting with pre-
market processing of forecast error samples to generate statistical moments
(mean and covariance), to dimension the reserve requirement and to generate
scenarios, for $\mathcal{M}^{\text{cc}}$, $\mathcal{R}1$ and $\mathcal{R}2$,
respectively. In our numerical studies, we impose a minimum reserve
requirement such that the probability of demand curtailment, due to
unavailability of flexible capacity to be dispatched during the real-time
operation, is less than 5% for the forecast error samples considered.
Moreover, observe that the real-time operation in $\mathcal{M}^{\text{cc}}$
does not involve pricing and can be done without re-optimization provided
uncertainty bounds defined by statistical moments are reliable, whereas both
$\mathcal{R}1$ and $\mathcal{R}2$ involve real-time markets, whose outcomes
are used to price flexibility services. As with $\mathcal{M}^{\text{cc}}$, the
market-clearing formulations and further details on the two reference problems
$\mathcal{R}1$ and $\mathcal{R}2$ are provided in the e-Companion to the
paper.

Forecast error samplesMean,covarianceMinimum reserve requirementScenariosSOCP
chance-constrainedmarket clearingLP deterministicmarket clearingLP scenario-
based market clearingLP deterministicflexibility activationLP deterministic
real-time marketsLP deterministic real-time
markets$\mathcal{M}^{\text{cc}}$$\mathcal{R}1$$\mathcal{R}2$PRE-MARKETDAY-
AHEADREAL-TIME Figure 4: Illustration of the SOCP market-clearing and the LP-
based reference problems No network bottlenecksWith network
bottlenecks1020304050601020304050600200400600Expected renewable energy share
($\%$ of total energy demand)In-sample expected cost
(k$\$$)$\mathcal{M}^{\text{cc}}$$\mathcal{R}1$$\mathcal{R}2$ Figure 5: In-
sample market-clearing cost in the various RES paradigms considered

#### In-sample market-clearing results:

First, we study the in-sample performance of the market-clearing problems. As
shown in Figure 4, in-sample refers to the forecast samples used in the pre-
market stage to obtain the parameters and scenarios for the market-clearing
problems $\mathcal{M}^{\text{cc}}$ and the benchmarks $\mathcal{R}1$ and
$\mathcal{R}2$. For the 6 wind farms in the network, we consider 100 forecast
error samples drawn from a multivariate Gaussian distribution having zero mean
and a standard deviation of 10% of the nominal day-ahead forecast values.
Consequently, the safety parameter $r_{\varepsilon}$ is the inverse cumulative
distribution function of the standard Gaussian distribution evaluated at
$(1-\varepsilon)$–quantile. For $\mathcal{M}^{\text{cc}}$, we fix the system
operator’s constraint violation probability at $\varepsilon=0.05$. Figure 5
shows a comparison of the expected day-ahead market-clearing cost for the
three market-clearing problems under the two network configurations in the
various RES paradigms. Observe that, with increasing share of renewable
energy, due to its exogenous consideration of uncertainty, market-clearing
problem $\mathcal{R}1$ performs increasingly worse compared to the other
problems for both the network configurations, leading to infeasibility of
market-clearing problem at 60% renewable energy share for the case with
network bottlenecks. Furthermore, market-clearing problems
$\mathcal{M}^{\text{cc}}$ and $\mathcal{R}2$ result in comparable in-sample
expected cost. However, it is worth noting that, contrary to
$\mathcal{M}^{\text{cc}}$, $\mathcal{R}2$ does not provide any guarantees on
the feasibility of market-clearing problem when faced with scenarios beyond
those considered as in-sample. In fact, depending on the constraint violation
probability $\varepsilon$ and the number of decision variables, Alamo et al.
(2015, Theorem 4) provides an analytical expression for the number of
scenarios that must be considered while solving $\mathcal{R}2$ to obtain the
same probabilistic feasibility guarantee as $\mathcal{M}^{\text{cc}}$. For the
network considered in this numerical study, this evaluates to $>80,000$
scenarios, thereby rendering problem $\mathcal{R}2$ unable to provide such a
guarantee while being cleared within the desired day-ahead market-clearing
solve times (typically, about 1-2 hours). We discuss the out-of-sample
performance further in the following. The first two rows in Table 1 provide a
comparison for the cost and computation times of the various market-clearing
problems in the 50% RES paradigm.

Parameter | Unit | No network bottlenecks | With network bottlenecks
---|---|---|---
$\mathcal{M}^{\text{cc}}$ | $\mathcal{R}1$ | $\mathcal{R}2$ | $\mathcal{M}^{\text{cc}}$ | $\mathcal{R}1$ | $\mathcal{R}2$
In-sample expected cost | $1000 | 221.85 | 257.85 | 216.49 | 256.62 | 377.56 | 235.16
Market-clearing computation time | s | 2.08 | 0.22 | 178.23 | 2.38 | 0.27 | 215.5
Out-of-sample expected cost | $1000 | 221.76 | 259.57 | 224.29 | 256.51 | 396.86 | 244.28
Out-of-sample infeasibility | % | 0.2 | 3.2 | 0 | 0.2 | 0.6 | 8.8
Load shedding probability | % | 0 | 0 | 0 | 0 | 1.0 | 0.4
Wind curtailment probability | % | 0 | 0 | 46.0 | 0 | 0 | 50.8
Table 1: In-sample and out-of-sample performance of market-clearing problems
for the 50% RES paradigm

#### Out-of-sample performance:

To examine the out-of-sample performance, we perform the deterministic real-
time market clearing, shown as the last stage in Figure 4. Here, the day-ahead
decisions made for each of the market-clearing problems are fixed while
adjustments are made in real-time to meet the uncertainty realization. For the
market-clearing problem $\mathcal{M}^{cc}$, this implies that the flexible
market participants adjust their production in real-time strictly in
accordance to the adjustment policies assigned at the day-ahead market-
clearing stage. On the other hand, for the LP-based reference market-clearing
problems $\mathcal{R}1$ and $\mathcal{R}2$, the adjustments in real-time are
allowed up to the flexible capacity limits obtained at the day-ahead stage.
Consistent with the prevalent practice on handling real-time adjustments in
electricity markets, we introduce a 10% premium over the bid prices in the
day-ahead market-clearing stage to incentivize the market participants to
adjust their schedule during real-time operation. Since some wind forecast
scenarios may lead to infeasibility of the market-clearing problem due to
unavailability of flexibility during the real-time operation, we introduce
contingency actions in the market-clearing that correspond to load shedding
and wind curtailment respectively. Variables corresponding to these
contingency actions are penalized in the market-clearing objective such that
they are the last resort in ensuring supply-demand balance.

200250300350400No bottlenecksWith bottlenecksNetwork configurationOut-of-
sample cost (k$\$$)$\mathcal{M}^{\text{cc}}$$\mathcal{R}1$$\mathcal{R}2$
Figure 6: Out-of-sample market-clearing cost comparison in the 50% RES
paradigm

We prepare a test dataset comprised of 500 wind forecast scenarios, distinct
from those used for in-sample simulations, drawn from the same multivariate
Gaussian distribution. The box and whiskers plots in Figure 6 describe the
distribution of the resulting out-of-sample cost for the various market-
clearing problems under the 50% RES paradigm. For each box, the central line
indicates the median, whereas the ends denote the 25th and 75th percentiles.
The whiskers extend upto 1.5 times the inter-quartile range, while remaining
values are shown as outliers in the form of rings. The second part of Table 1
provides the results from the out-of-sample simulations. First, referring to
Figure 6, we observe that the ordering in terms of expected cost is preserved
from the in-sample simulations, i.e., $\mathcal{R}1$ leads to higher costs. We
note that the market-clearing problem $\mathcal{R}2$ results in out-of-sample
costs which are highly variable compared to those obtained with the SOCP
market-clearing problem $\mathcal{M}^{\text{cc}}$. Second, the out-of-sample
expected cost for $\mathcal{M}^{\text{cc}}$ is close to the in-sample case,
mentioned in Table 1. On the other hand, market-clearing problem
$\mathcal{R}1$ exhibits an increase in cost of 0.75% and 5.34% over their in-
sample values for the network configurations with and without bottlenecks,
respectively. Similarly, problem $\mathcal{R}2$ results in out-of-sample
expected cost which exceeds that in-sample expected cost by 3.54% and 4.1%,
respectively for the network configurations with and without bottlenecks.
Finally, both the LP-based market-clearing problems endure load shedding or
wind curtailment when faced with unknown forecast samples. These results
highlight how the proposed SOCP-based uncertainty-aware market-clearing
problem outperforms the prevalent LP-based alternatives. Next, we provide
comparative insights on the day-ahead market-clearing outcomes for the three
problems.

#### Day ahead-market-clearing outcomes:

0.51.01.551015204812162000.81.62.44122081114Hours Energy price [$/MWh]
(a)$10\%$$20\%$$30\%$$40\%$$50\%$$60\%$$\mathcal{M}^{\text{cc}}$$\mathcal{R}1$$\mathcal{R}2$0.51.01.551015204812162000.81.62.44122081114412200.81.62.4$\cdot
10^{3}$Hours Flexibility payment [$]
(b)$10\%$$20\%$$30\%$$40\%$$50\%$$60\%$$\mathcal{M}^{\text{cc}}$$\mathcal{R}1$$\mathcal{R}2$$\mathcal{M}^{\text{cc}}$$\mathcal{R}1$0.51.01.551015204812162000.81.62.44122081114412200.81.62.4$4$$6$$8$$10$$12$50010001500Hours
PP dispatch [MW]
(c)$10\%$$20\%$$30\%$$40\%$$50\%$$60\%$$\mathcal{M}^{\text{cc}}$$\mathcal{R}1$$\mathcal{R}2$$\mathcal{M}^{\text{cc}}$$\mathcal{R}1$$\mathcal{M}^{\text{cc}}$$\mathcal{R}1$$\mathcal{R}2$0.51.01.551015204812162000.81.62.44122081114412200.81.62.4$4$$6$$8$$10$$12$50010001500$4$$6$$8$$10$$12$-2000200Hours
ESU dispatch [MW]
(d)$10\%$$20\%$$30\%$$40\%$$50\%$$60\%$$\mathcal{M}^{\text{cc}}$$\mathcal{R}1$$\mathcal{R}2$$\mathcal{M}^{\text{cc}}$$\mathcal{R}1$$\mathcal{M}^{\text{cc}}$$\mathcal{R}1$$\mathcal{R}2$$\mathcal{M}^{\text{cc}}$$\mathcal{R}1$$\mathcal{R}2$

Figure 7: Day-ahead market-clearing outcome for various problems: (a) hourly
system-level energy price, (b) hourly flexibility payment, (c) dispatch of
power producers (PPs) and (d) dispatch of energy storage units (ESUs)

For the 50% RES paradigm, Figure 7 compares the day-ahead market-clearing
outcomes among the three problems for the network configuration without any
bottlenecks. Observe that, as expected, the energy price in Figure 7(a)
closely follows the swings in net demand, shown in Figure 2(b). Figure 7(b)
compares the flexibility payment at the day-ahead market-clearing stage for
the problems $\mathcal{M}^{\text{cc}}$ and $\mathcal{R}1$. In contrast to
$\mathcal{M}^{\text{cc}}$ wherein flexibility payments are a result of the
dual solution to the market-clearing problem, flexibility payments in
$\mathcal{R}1$ correspond to the primal solution, i.e., the payments made
towards the reserve capacity to meet the minimum reserve requirement. The
difference in hourly flexibility payments illustrate the challenge in
dimensioning the capacity-based reserves properly. Note that, in absence of
any specific capacity reservation or a meaningful flexibility balance at the
day-ahead clearing stage, $\mathcal{R}2$ mandates the real-time markets for
pricing of flexibility services and is therefore omitted from the
comparison777A real-time balance is indeed enforced in $\mathcal{R}2$, such
that real-time adjustments in each scenario is aligned with the supply-demand
balance. However, the price convergence between day-ahead energy prices and
real-time prices is a desirable property to maximize welfare, as discussed in
Zavala et al. (2017). This price convergence is in fact a characteristic of
our implementation $\mathcal{R}2$, but it limits the ability to explicitly
extract values corresponding to flexibility payment either from the primal or
dual solutions to the market-clearing problem.. Figures 7(c) and 7(d) show the
scheduled dispatch of PPs and ESUs, respectively, for a selection of hours of
the day. On account of the minimum reserve requirement associated with market-
clearing problem $\mathcal{R}1$, in contrast with the other two problems which
consider uncertainty endogenously, the dispatch involves higher generation in
hours 7 through 12. The excess generation, which partly explains the higher
market-clearing cost associated with $\mathcal{R}1$, is used to charge the
ESUs in the network such that sufficient flexible capacity is available to
meet the minimum reserve requirement. This effect further underscores the
importance of endogenous consideration of uncertainty within the market-
clearing framework.

##  5 Concluding Remarks

We proposed and analyzed a new conic formulation for forward markets, which we
applied to the specific case of electricity markets. Our contribution is
threefold. From a market design perspective, we revisited the spatial price
equilibrium problem beyond the LP framework by formulating the market-clearing
problem within an SOCP framework. From a theoretical perspective, we relied on
Lagrangian duality theory for SOCP problems to characterize the solution to
the spatial price equilibrium problem involving rational and self-interested
market participants in terms of optimality and robustness to parameter
perturbation. We derived analytical expressions for conic spatial prices of
the traded commodities and provided analytical proofs to demonstrate that
moving towards conic markets for electricity does not incur the loss of any
desirable economic properties, i.e., the proposed conic market retains market
efficiency, cost recovery and revenue adequacy, under common assumptions also
applicable to LP-based markets. Finally, from a practitioner’s perspective, we
illustrated the generality of our proposed market-clearing framework by
defining a bid format for conic markets.

Our conic market proposal admits SOC-representable nonlinearities underlying
the costs and constraints of market participants and the system operator to
enable an uncertainty-, asset- and network-aware market-clearing problem. This
step in the evolution of electricity markets is crucial for the successful
transition of electricity systems worldwide from dispatchable, fossil
fuel–based system towards a weather-dependent, renewable energy–based system,
supported by a heterogeneous mix of energy and flexibility providers. Our
numerical studies leverage in-sample and out-of-sample simulations to showcase
the benefits of the proposed SOCP-based market-clearing problem over the LP-
based benchmarks in terms of social welfare and its invariability, endogenous
representation of uncertainty, appropriate dimensioning of flexibility need,
and guarantees on the feasibility of the market-clearing problem. We provide
insights into payments associated with the flexibility procurement based on
adjustment policies, which are central to our uncertainty-aware SOCP-based
market-clearing proposal.

This paper opens up a variety of important directions for future work. First,
considering the generality of the proposal, it is potentially interesting to
model market-clearing use cases that involve one or more of the attributes
related to uncertainty-, asset- and network-awareness. For instance, the
examples in section 2 and the theoretical results in section 3 can be used to
study the coordination between the transmission and distribution system
operators in an uncertainty- and network-aware setting. This enables the
flexibility available at the distribution level to be appropriately harnessed
and priced within the market framework while considering a physically-accurate
network constraint representation. Along similar lines is the use case
studying the market-based coupling of the electricity and natural gas sectors.
Here, the focus of the analysis potentially broadens from merely an optimal
network- and asset-aware flexibility procurement to one that designs well-
aligned incentives and risk measures to ensure a reliable and resilient system
operation in both the interdependent sectors. Second, the conic market bids
introduced in section 2.4 merit further evaluation in terms of acceptance by
market participants, alignment with market regulatory agencies as well as
adaptations needed in the prevalent market-clearing algorithms to adopt them.
Intended to accelerate their practical adoption, such evaluations could take
shape as market simulations and participant-specific sensitivity studies to
systematically quantify the costs and benefits of moving towards conic
electricity markets. Third, the theoretical results discussed in the paper can
be extended to include market participants that act strategically. One can
then leverage conic complementarity programs to investigate how exercising
market power influences the market-clearing outcomes, specifically in the
context of the flexibility payments involved in uncertainty-aware use case
analyzed in section 4, and more generally regarding the satisfaction of
desired economic properties. Another important direction to pursue is the
extension of the analytical results we derived on the Lorentz cone to the cone
of semidefinite matrices, which results in an SDP-based market-clearing
framework. Further generalization over the LP framework aside, such an
extension could potentially give rise to markets that consider uncertainty
under the paradigm of distributionally-robust chance-constrained optimization
and that admit an SDP-based convexification approach for non-convex power
flows in the electricity network. Finally, beyond the electricity system, the
theoretical results in this work are of potential interest in competitive
settings involving physical or non-physical systems wherein cost- and
constraint-based nonlinearities currently mandate the use of approximation
techniques via linearization. The work of A. Ratha was supported by a Ph.D.
grant provided by the Flemish Institute for Technological Research (VITO) and
scholarship from Technical University of Denmark (DTU).

## References

  * Alamo et al. (2015) Alamo T, Tempo R, Luque A, Ramirez DR (2015) Randomized methods for design of uncertain systems: Sample complexity and sequential algorithms. _Automatica_ 52:160–172.
  * Alizadeh and Goldfarb (2003) Alizadeh F, Goldfarb D (2003) Second-order cone programming. _Mathematical Programming_ 95(1):3–51.
  * Anjos and Gómez (2017) Anjos MF, Gómez JA (2017) _Operations Research Approaches for Building Demand Response in a Smart Grid_ , chapter 7, 131–152 (INFORMS TutORials in Operations Research).
  * Ben-Tal and Nemirovski (2001) Ben-Tal A, Nemirovski A (2001) _Lectures on Modern Convex Optimization_ , chapter 1 (Philadelphia: Society for Industrial and Applied Mathematics).
  * Bertsimas et al. (2013) Bertsimas D, Litvinov E, Sun XA, Zhao J, Zheng T (2013) Adaptive robust optimization for the security constrained unit commitment problem. _IEEE Transactions on Power Systems_ 28(1):52–63.
  * Bohn et al. (1984) Bohn RE, Caramanis MC, Schweppe FC (1984) Optimal pricing in electrical networks over space and time. _The RAND Journal of Economics_ 15(3):360–376.
  * Borraz-Sánchez et al. (2016) Borraz-Sánchez C, Bent R, Backhaus S, Hijazi H, Hentenryck PV (2016) Convex relaxations for gas expansion planning. _INFORMS Journal on Computing_ 28(4):645–656.
  * Boyd and Vandenberghe (2004) Boyd S, Vandenberghe L (2004) _Convex Optimization_ (Cambridge university press).
  * Bushnell (2021) Bushnell J (2021) To fix the power market, first fix the natural gas market. Energy Institute Blog, UC Berkeley, March 1, 2021. https://energyathaas.wordpress.com/2021/03/01/to-fix-the-power-market-first-fix-the-natural-gas-market/, Last accessed: 10 March 2021.
  * Byeon and Van Hentenryck (2020) Byeon G, Van Hentenryck P (2020) Unit commitment with gas network awareness. _IEEE Transactions on Power Systems_ 35(2):1327–1339.
  * Cai et al. (2001) Cai X, McKinney DC, Lasdon LS, Watkins DW (2001) Solving large nonconvex water resources management models using generalized benders decomposition. _Operations Research_ 49(2):235–245.
  * Cain et al. (2012) Cain MB, O’Neill RP, Castillo A (2012) _History of Optimal Power Flows and Formulations_ (Federal Energy Regulatory Commission (FERC)).
  * CAISO (2020) CAISO (2020) The role of the California ISO. http://www.caiso.com/about/Pages/OurBusiness/The-role-of-the-California-ISO.aspx, Last accessed: 10 March 2021.
  * Christie et al. (2000) Christie RD, Wollenberg BF, Wangensteen I (2000) Transmission management in the deregulated environment. _Proceedings of the IEEE_ 88(2):170–195.
  * Chu and Majumdar (2012) Chu S, Majumdar A (2012) Opportunities and challenges for a sustainable energy future. _Nature_ 488(7411):294–303.
  * Coffrin et al. (2021) Coffrin C, et al. (2021) The power grid library for benchmarking AC optimal power flow algorithms. arXiv preprint, https://arxiv.org/abs/1908.02788.
  * Conejo et al. (2010) Conejo AJ, Carrión M, Morales JM (2010) _Decision making under uncertainty in electricity markets_ , volume 153, chapter App. B (New York: Springer US).
  * Courcoubetis and Weber (2003) Courcoubetis C, Weber R (2003) _Pricing Communication Networks_ , chapter 9, 219–233 (West Sussex, England: John Wiley & Sons, Ltd).
  * Cramton and Stoft (2007) Cramton P, Stoft S (2007) Why we need to stick with uniform-price auctions in electricity markets. _The Electricity Journal_ 20(1):26 – 37.
  * Daqing et al. (2014) Daqing L, Yinan J, Rui K, Havlin S (2014) Spatial correlation analysis of cascading failures: Congestions and blackouts. _Scientific Reports_ 4(1):5381.
  * De Wolf and Smeers (2000) De Wolf D, Smeers Y (2000) The gas transmission problem solved by an extension of the simplex algorithm. _Management Science_ 46(11):1454–1465.
  * Delage and Ye (2010) Delage E, Ye Y (2010) Distributionally robust optimization under moment uncertainty with application to data-driven problems. _Operations Research_ 58(3):595–612.
  * Dvorkin (2020) Dvorkin Y (2020) A chance-constrained stochastic electricity market. _IEEE Transactions on Power Systems_ 35(4):2993–3003.
  * Enke (1951) Enke S (1951) Equilibrium among spatially separated markets: Solution by electric analogue. _Econometrica_ 19(1):40–47.
  * Farivar and Low (2013) Farivar M, Low SH (2013) Branch flow model: Relaxations and convexification-Part I. _IEEE Transactions on Power Systems_ 28(3):2554–2564.
  * Federico and Rahman (2003) Federico G, Rahman D (2003) Bidding in an electricity pay-as-bid auction. _Journal of Regulatory Economics_ 24(2):175–211.
  * Hobbs (2001) Hobbs BF (2001) Linear complementarity models of nash-cournot competition in bilateral and POOLCO power markets. _IEEE Transactions on Power Systems_ 16(2):194–202.
  * IRENA (2015) IRENA (2015) Renewable energy target setting. https://www.irena.org/publications/2015/Jun/Renewable-Energy-Target-Setting, Last accessed: 10 March 2021.
  * Kantorovich (1960) Kantorovich LV (1960) Mathematical methods of organizing and planning production. _Management Science_ 6(4):366–422.
  * Kim and Powell (2011) Kim JH, Powell WB (2011) Optimal energy commitments with storage and intermittent supply. _Operations Research_ 59(6):1347–1360.
  * Kirschen and Strbac (2018) Kirschen D, Strbac G (2018) _Fundamentals of Power System Economics, 2nd Edition_ , chapter 3, 51–88 (US: John Wiley & Sons, Ltd).
  * Kocuk et al. (2016) Kocuk B, Dey SS, Sun XA (2016) Strong socp relaxations for the optimal power flow problem. _Operations Research_ 64(6):1177–1196.
  * Kuang et al. (2018) Kuang X, Dvorkin Y, Lamadrid AJ, Ortega-Vazquez MA, Zuluaga LF (2018) Pricing chance constraints in electricity markets. _IEEE Transactions on Power Systems_ 33(4):4634–4636.
  * Kuang et al. (2019) Kuang X, Lamadrid AJ, Zuluaga LF (2019) Pricing in non-convex markets with quadratic deliverability costs. _Energy Economics_ 80:123 – 131.
  * Lavaei and Low (2012) Lavaei J, Low SH (2012) Zero duality gap in optimal power flow problem. _IEEE Transactions on Power Systems_ 27(1):92–107.
  * Lobo et al. (1998) Lobo MS, Vandenberghe L, Boyd S, Lebret H (1998) Applications of second-order cone programming. _Linear Algebra and its Applications_ 284(1):193 – 228\.
  * Mieth et al. (2020) Mieth R, Kim J, Dvorkin Y (2020) Risk- and variance-aware electricity pricing. _Electric Power Systems Research_ 189:106804.
  * Mitridati et al. (2020) Mitridati L, Kazempour J, Pinson P (2020) Heat and electricity market coordination: A scalable complementarity approach. _European Journal of Operational Research_ 283(3):1107–1123, ISSN 0377-2217.
  * Mosek ApS (2021) Mosek ApS (2021) MOSEK Modeling Cookbook 3.2.2. https://docs.mosek.com/modeling-cookbook/index.html, Last accessed: 10 March 2021.
  * Nemirovski and Shapiro (2007) Nemirovski A, Shapiro A (2007) Convex approximations of chance constrained programs. _SIAM Journal on Optimization_ 17(4):969–996.
  * Nord Pool (2021) Nord Pool (2021) Nord Pool Day-ahead Trading. https://www.nordpoolgroup.com/trading/Day-ahead-trading/, Last accessed: 10 March 2021.
  * Papavasiliou (2018) Papavasiliou A (2018) Analysis of distribution locational marginal prices. _IEEE Transactions on Smart Grid_ 9(5):4872–4882.
  * Parag and Sovacool (2016) Parag Y, Sovacool BK (2016) Electricity market design for the prosumer era. _Nature Energy_ 1(4):16032.
  * PJM Interconnection (2014) PJM Interconnection (2014) Analysis of operational events and market impacts during the January 2014 cold weather events.
  * Pritchard et al. (2010) Pritchard G, Zakeri G, Philpott A (2010) A single-settlement, energy-only electric power market for unpredictable and intermittent participants. _Operations Research_ 58(4-part-2):1210–1219.
  * Raissi (2016) Raissi M (2016) _Conic Economics_. Ph.D. thesis, University of Maryland, College Park, https://search.proquest.com/docview/1861702044, Last accessed: 10 March 2021.
  * Roberts and Postlewaite (1976) Roberts DJ, Postlewaite A (1976) The incentives for price-taking behavior in large exchange economies. _Econometrica_ 44(1):115–127.
  * Rosen (1965) Rosen JB (1965) Existence and uniqueness of equilibrium points for concave n-person games. _Econometrica_ 33(3):520–534.
  * Samuelson (1952) Samuelson PA (1952) Spatial price equilibrium and linear programming. _The American Economic Review_ 42(3):283–303.
  * Schweppe et al. (1988) Schweppe FC, Caramanis MC, Tabors RD, Bohn RE (1988) _Spot Pricing of Electricity_ , chapter 1 (Boston, MA: Springer US).
  * Snyder et al. (2014) Snyder LV, Scaparra MP, Daskin MS, Church RL (2014) _Planning for Disruptions in Supply Chain Networks_ , chapter 9, 234–257 (INFORMS TutORials in Operations Research).
  * The Danish Government’s Climate Partnership (2020) The Danish Government’s Climate Partnership (2020) Powering Denmark’s Green Transition: Roadmap for a near carbon neutral energy sector to achieve Denmark’s 70% reduction target by 2030. URL https://www.danskenergi.dk/the-climate-partnership-the-energy-and-utilities-sector, Last accessed: 10 March 2021.
  * Thompson (2013) Thompson M (2013) Optimal economic dispatch and risk management of thermal power plants in deregulated markets. _Operations Research_ 61(4):791–809.
  * Vickrey (1961) Vickrey W (1961) Counterspeculation, auctions, and competitive sealed tenders. _The Journal of Finance_ 16(1):8–37.
  * Wagner (2008) Wagner MR (2008) Stochastic 0–1 linear programming under limited distributional information. _Operations Research Letters_ 36(2):150 – 156\.
  * Wang et al. (2015) Wang Q, Zhang C, Ding Y, Xydis G, Wang J, Østergaard J (2015) Review of real-time electricity markets for integrating distributed energy resources and demand response. _Applied Energy_ 138:695–706.
  * Zavala et al. (2017) Zavala VM, Kim K, Anitescu M, Birge J (2017) A stochastic electricity market clearing formulation with consistent pricing properties. _Operations Research_ 65(3):557–576.

Proofs and Market-clearing Problems The e-Companion provides the proofs to the
theoretical results presented in section 3 of the paper and modeling details
and system data for the various market-clearing problems discussed in section
4.

##  6 Lagrangian Duality for SOCP Problems

In section 6.1, we provide a concise theoretical background of SOCP duality,
while directing interested readers to Lobo et al. (1998), Ben-Tal and
Nemirovski (2001) and Alizadeh and Goldfarb (2003) for details. The dual
problem to market-clearing problem $\mathcal{M}^{\text{c}}$ is provided in
section 6.2.

###  6.1 Preliminaries

Consider an arbitrary SOCP problem in the variable
$\mathbf{x}\in\mathbb{R}^{N}$ of the form

$\displaystyle\underset{{\mathbf{x}}}{\text{min}}\quad\mathbf{c}^{\top}\mathbf{x}\quad\text{s.t.}\quad\norm{\mathbf{A}_{j}x+\mathbf{b}_{j}}_{2}\leq\mathbf{d}_{j}^{\top}\mathbf{x}+e_{j},\quad\forall
j=1,2,\dots,J,$ (1)

with parameters $\mathbf{c}\in\mathbb{R}^{N}$,
$\mathbf{A}_{j}\in\mathbb{R}^{k_{j}\times N}$,
$\mathbf{b}_{j}\in\mathbb{R}^{k_{j}}$, $\mathbf{d}_{j}\in\mathbb{R}^{N}$ and
$e_{j}\in\mathbb{R}$. Based on (1), the SOC constraint in the problem (1) may
be expressed as

$\displaystyle\begin{bmatrix}\mathbf{A}_{j}\\\
\mathbf{d}_{j}^{\top}\end{bmatrix}\mathbf{x}+\begin{bmatrix}\mathbf{b}_{j}\\\
e_{j}\end{bmatrix}\in\mathcal{C}_{j}\Leftrightarrow-\begin{bmatrix}\mathbf{A}_{j}\mathbf{x}+\mathbf{b}_{j}\\\
\mathbf{d}_{j}^{\top}\mathbf{x}+e_{j}\end{bmatrix}\leqslant_{\mathcal{C}_{j}}0,$

where $\mathcal{C}_{j}\subseteq\mathbb{R}^{k_{j}+1}$ is the standard second-
order cone and the generalized inequality $\leqslant_{\mathcal{C}_{j}}$
denotes a partial ordering over the cone $\mathcal{C}_{j}$. The Lagrangian
function for (1) writes as

$\displaystyle\Theta(\mathbf{x},\bm{\lambda}_{1},\dots,\bm{\lambda}_{J})=\mathbf{c}^{\top}\mathbf{x}-\sum_{j=1}^{J}\bm{\lambda}_{j}^{\top}\left(\begin{bmatrix}\mathbf{A}_{j}\\\
\mathbf{d}_{j}^{\top}\end{bmatrix}\mathbf{x}+\begin{bmatrix}\mathbf{b}_{j}\\\
e_{j}\end{bmatrix}\right),$

where $\bm{\lambda}_{j}\in\mathbb{R}^{k_{j}+1}$ for $j=1,2,\dots,J$ are
Lagrange multipliers. The dual function is given by

$\displaystyle
g(\bm{\lambda}_{1},\dots,\bm{\lambda}_{J})=\begin{cases}-\sum_{j=1}^{J}\bm{\lambda}_{j}^{\top}\begin{bmatrix}\mathbf{b}_{j}\\\
e_{j}\end{bmatrix},\quad&\mathbf{c}=\sum_{j=1}^{J}\begin{bmatrix}\mathbf{A}_{j}\\\
\mathbf{d}_{j}^{\top}\end{bmatrix}^{\top}\bm{\lambda}_{j},\\\
-\infty,\quad&\text{otherwise}.\end{cases}$

Lastly, the dual problem to (1) after substituting
$\bm{\lambda}_{j}=\begin{bmatrix}\bm{\mu}_{j}^{\top}\quad\nu_{j}\end{bmatrix}^{\top}$,
such that $\bm{\mu}_{j}\in\mathbb{R}^{k_{j}}$ and $\nu_{j}\in\mathbb{R}_{+}$
are auxiliary dual variables, can be written as

$\displaystyle\underset{{\bm{\mu}_{j},\nu_{j}}}{\text{max}}\quad$
$\displaystyle-\sum_{j=1}^{J}(\mathbf{b}_{j}^{\top}\bm{\mu}_{j}+e_{j}\nu_{j})$
(2a) s.t.