1 A General Stochastic Optimization Framework for Convergence Bidding

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A General Stochastic Optimization Framework for
Convergence Bidding
Letif Mones and Sean Lovett
Abstract—Convergence (virtual) bidding is an important part
of two-settlement electric power markets as it can effectively
reduce discrepancies between the day-ahead and real-time mar-
kets. Consequently, there is extensive research into the bidding
strategies of virtual participants aiming to obtain optimal bids
to submit to the day-ahead market. In this paper, we introduce a
price-based general stochastic optimization framework to obtain
optimal convergence bid curves. Within this framework, we
develop a computationally tractable linear programming-based
optimization model, which produces bid prices and volumes
simultaneously. We also show that different approximations and
simplifications in the general model lead naturally to state-of-the-
art convergence bidding approaches, such as self-scheduling and
opportunistic approaches. Our general framework also provides
a straightforward way to compare the performance of these
models, which is demonstrated by numerical experiments on the
California (CAISO) market.
Index Terms—Convergence bidding, virtual bidding, electricity
market, stochastic optimization, portfolio optimization.
NOMENCLATURE
Functions and functionals
ESα
Expected shortfall (conditional value at risk) with
quantile parameter α.
EExpected value.
F,GAbstract functionals used in stochastic optimizations.
VVariance.
L,D
Abstract deterministic functions to compute day-ahead
and delta prices.
ϕ, ψ
Abstract utility functions used in stochastic optimiza-
tions.
Sets
B
Convex set of bids (supply and demand prices and
volumes).
PConvex set of supply and demand prices.
WConvex set of supply and demand volumes.
Variables
n
t
Delta (spread) LMP at node
n
at index
t
in the training
data.
λn
t, πn
t
Day-ahead and real-time LMPs at node
n
at index
t
in the training data.
µ, Σ
Mean vector and covariance matrix of predicted delta
price distribution.
All authors are with Invenia Labs, 95 Regent Street, Cambridge, CB2 1AW,
United Kingdom (e-mails: {firstname.lastname}@invenialabs.co.uk).
d, d
Minimum distance in prices between supply and
demand bid segments.
P , P
Number of potential bid prices for supply and demand
positions.
p, p Supply and demand price variables.
v, v Cleared supply and demand volume variables.
W , W
Upper and lower bounds on net volume in stochastic
optimization problem.
w, w Supply and demand volume variables.
ρ, ˜ρ
Risk and volume-normalized risk upper bound param-
eters.
τ, z
Variables for computing sample-based expected short-
fall.
θParameter vector of a probabilistic model.
bSubmitted bid including (w, w, p, p).
F
Number of physical participants in the day-ahead
market.
K
Number of samples used to compute expected shortfall.
NNumber of biddable nodes in the target period.
r, ˜r
Hourly revenue and volume-normalized revenue vari-
ables.
SNumber of maximum allowed segments in bid curve.
TNumber of training samples.
V
Number of all other virtual participants in the day-
ahead market.
W
Total volume distributed among nodes and positions
in stochastic optimization problem.
X, x
Training feature matrix and test feature vector for a
probabilistic model.
YTraining price matrix for a probabilistic model.
zλ,zπ
Parameters of the day-ahead and real-time OPF prob-
lems.
I. INTRODUCTION
C
ONVERGENCE or virtual bidding plays a critical role
in most electric power markets in the United States.
Independent System Operators (ISOs) use a two-settlement
approach to aid price convergence (i.e. reduce the price gap)
between the day-ahead and real-time (spot) markets. Under
this mechanism, participants can submit convergence supply
(increment) and demand (decrement) bids at each target hour
to the day-ahead market. Convergence bids are input into the
market clearing calculations in the same way as physical bids,
but unlike physical bids, they are settled financially and carry
no obligation to supply or consume physical power in real
time. The revenue of the participant depends on the difference
between day-ahead and real-time Locational Marginal Prices
(LMPs) [
1
]. There is a wide consensus that virtual bidding
arXiv:2210.06543v4 [math.OC] 7 Feb 2023
2
improves the efficiency of electricity market operation by
converging prices [
2
], [
3
], [
4
], [
5
], [
6
], [
7
], [
8
], [
9
] and
increasing day-ahead unit commitment efficiency [
10
], [
11
],
hedges financial risks [
3
], [
8
], increases market liquidity [
8
],
reduces price volatility [
12
], [
6
], and helps prevent the exercise
of market power [2], [8].
The problem of how to select optimal convergence bids
is important both to virtual participants and to the efficient
operation of electricity markets. While this problem bears
some relationship to classical problems of risk-averse portfolio
optimization, it has several unique aspects. The most important
is that the clearance of a convergence bid is not based on
a simple bilateral auction, but rather on the solution of the
day-ahead unit commitment and optimal power flow (OPF)
problems. A convergence bid on a given node at a given target
hour is submitted as a volume-price pair (i.e. bid volume
of power in unit of MWh and bid price in unit of
$
/MWh).
In the day-ahead market, a convergence bid with a supply
position is cleared if the (day-ahead) LMP at the specific
hour and node is greater or equal than the bid price and
similarly, a convergence bid with demand position is cleared if
the corresponding LMP is less or equal than the bid price. As
part of the optimization problem, the cleared convergence bid
will alter the day-ahead market solution and therefore the day-
ahead LMPs. Accordingly, a convergence bidding approach
can be modelled by a bilevel optimization problem, where the
upper level is a financial problem subject to lower level market
clearance problems [13], [14], [15].
Ideally, virtual participants want to obtain nodal bid curves
including segments with optimal bid volumes and prices. This
requires solving stochastic optimization problems that treat both
bid volumes and prices as decision variables at the same time.
However, even without the bilevel consideration, co-optimizing
both prices and volumes results in large mixed-integer optimiza-
tion problems, which are typically computationally infeasible
for realistic data. This leads to the consideration of simplified,
computationally feasible versions of this problem. A typical
approach is to focus only on a subset of the original variables
in the decision making process, while using fixed values or
simple heuristics to obtain the rest. For example, optimal bid
volumes can be derived with fixed prices [
16
], [
17
], or optimal
prices with fixed quantity of power [18], [19], [20].
In this paper, we consider a general framework for financially-
based convergence bidding and derive an optimization model
that provides optimal virtual bid curves. More specifically, our
contributions are as follows:
We introduce a fully price-based stochastic optimization
framework for obtaining optimal bid curves.
The general framework is primarily based on some
predictive joint distribution of the day-ahead and delta
prices at the target hours and includes both bid volumes
and bid prices as decision variables (referred to as a VP-
model).
We provide a detailed discussion on approaches for
approximating the joint distribution of the day-ahead and
delta prices.
We show that the general framework encompasses many
state-of-the-art approaches as special cases. More precisely,
omitting bid prices from the decision making model and
focusing only on bid volumes leads to self-scheduling
approaches or V-models, while omitting bid volumes
and focusing only on bid prices leads to opportunistic
approaches or P-models.
Within the general framework, we introduce a new
optimization model that 1. provides optimal bid curves
(i.e. with multiple bid segments) 2. for both supply and
demand positions 3. by finding optimal bid prices and
volumes simultaneously 4. using a linear programming
problem. The model avoids integer variables and is
therefore computationally affordable for large numbers of
nodes and training data.
The general framework also provides a straightforward
way to carry out a fair comparison among different (VP,
V and P) models. We demonstrate it by comparing state-
of-the-art (V and P) approaches to our model (VP) on the
California (CAISO) market.
II. METHODOLOGY
A. Setup
Suppose we have an electricity market with
N
nodes
potentially biddable by virtual participants. Our aim is to
derive bids (volumes and prices) for the target period of the
day-ahead market using historical price data. Let
λn
t
,
πn
t
and
n
t=λn
tπn
t
be the corresponding locational marginal
prices (LMPs) of the day-ahead and real-time markets as
well as the delta (spread) prices, where
n= 1, . . . , N
and
t= 1, . . . , T
, with
T
number of training samples. Let
λt
,
πt
and
t
denote the vectors of all corresponding nodal prices at
time
t
. Furthermore, without any indices,
and
λ
represent
vector random variables with dimension
N
, while
n
and
λn
denote their nth components, respectively.
Electric power markets support the submission of conver-
gence bid curves, i.e. multiple bid segments with different bid
prices for the same target hour and node. Therefore, we wish
to derive optimal supply and demand convergence bid curves,
each with a maximum number of
S
segments including supply
volumes
wn
s0
and prices
pn
s
as well as demand volumes
wn
s0
and prices
pn
s
, where
n= 1, . . . , N
and
s= 1, . . . , S
.
In this work, we follow a ‘block’ interpretation of bid curves,
where a bid curve consists simply of several bids, which are
cleared independently (cf. MISO [
21
]). This can be converted
straightforwardly to the ‘tiered‘ interpretation used by some
markets, where a bid curve consists of monotonically increasing
segments, at most one of which is cleared (cf. CAISO [22]).
We emphasize that in the general framework, at the target
hour for a given node
n
we can have the following four possible
bid types, where sand s0denote bid segment indexes:
No bid position: s:wn
s=wn
s= 0.
Supply position only: s:wn
s>0and s0:wn
s0= 0.
Demand position only: s:wn
s= 0 and s0:wn
s0<0.
Both supply and demand positions:
s, s0:wn
s>0
and
wn
s0<0.
3
B. General stochastic optimization problem for convergence
bidding
We suggest the following fully price-based general form of
stochastic optimization problem for convergence bidding, in
order for any specific virtual participant to determine optimal
supply and demand bid curves with corresponding volumes
and prices:
max
w,pRS×N
w,pRS×N
Fp(∆)ϕ(∆, λ, w, w, p, p)
s.t.Gp(∆)ψ(∆, λ, w, w, p, p)0,
(w, w)∈ W,
(p, p)∈ P,
(1)
where
p(∆, λ)
is a given joint distribution of the random
variables representing delta and day-ahead prices.
ϕ
and
ψ
are
some utility functions of these prices (e.g. financial revenue),
as well as volumes (
w
,
w
) and bid prices (
p
,
p
), which are the
decision variables. Clearly,
ϕ
and
ψ
are also random variables
due to
and
λ
. In order to take uncertainties of the outcomes
of the decision into account, statistical functionals
F
(e.g.
expectation value) and
G
(e.g. some risk measure) of the
utility functions are used.
W
and
P
are (convex) sets of the
volumes and bid prices, respectively.
We note that in a more general framing, the delta and day-
ahead prices are deterministic functions of the bids of all
physical and virtual participants based on the corresponding
optimal power flow calculations carried out by the ISOs:
max
b
Fp(b,{bphys
i}F
i=1,{bvirt
j}V
j=1,zλ,zπ)[ϕ(∆, λ, b)]
s.t.Gp(b,{bphys
i}F
i=1,{bvirt
j}V
j=1,zλ,zπ)[ψ(∆, λ, b)] 0,
b∈ B,
with λ=Lb, {bphys
i}F
i=1,{bvirt
j}V
j=1, zλ,
∆ = D(λ, zπ),
(2)
where
b= (w, w, p, p)
is the full set of bids submitted
by our specific virtual participant and
B
is a convex set,
while
bphys
i= (wphys
i, wphys
i, pphys
i, pphys
i)
,
i= 1, . . . , F
and
bvirt
j= (wvirt
j, wvirt
j, pvirt
j, pvirt
j)
,
j= 1, . . . , V
represent bids
from physical and all other virtual participants and
F
and
V
are the number of physical and all other virtual participants,
respectively.
L
is a deterministic function of all bids and some
parameters
zλ
to provide day-ahead LMP prices, while
D
is
a deterministic function of the day-ahead prices and some
parameters
zπ
associated to the real-time market. In this case,
the uncertainty in the corresponding stochastic problem arises
due to the distribution
p({bphys
i}F
i=1,{bvirt
j}V
j=1, zλ, zπ)
. The
potential advantage of this approach is that the price impact of
bids submitted by our specific virtual participant is explicitly
considered. Also, in principle, this approach could provide
accurate estimates of the dependency between LMPs (both
λ
and
) by incorporating structure from the corresponding OPF
solutions. However, there are several challenges with this most
general model that make it impractical to obtain optimal virtual
bids for a specific virtual participant. First, it would require
models for computing day-ahead and real-time prices that are
adequate approximations of the corresponding OPF and LMP
calculations of the ISO; however in practice, full details and
parameters (
zλ
and
zπ
) of the OPF formulations are typically
unavailable to the participant at time of bidding. Since these
models themselves are optimization problems, the resulting
stochastic optimization problem is rather a complicated one
(i.e. non-linear or even non-convex, mixed-integer and possibly
nested optimization problems). Second, it would also need an
appropriate distribution of bids by the other participants and an
extensive sampling of this distribution that would significantly
increase the size and/or the number of the optimization
problems to solve. One way to approximate the most general
model is to apply bilevel formulations ([
13
], [
14
], [
15
])
that try to address the clearing mechanism by solving some
simplified OPF on the lower level. These problems, among other
extensions, could be considered by using
p(∆, λ |w, w, p, p)
,
that is, accounting only for the impact of actions of the specific
virtual participant on the joint distribution of prices. Problem 1
is a further simplification, where the clearing process is not
modelled explicitly and uncertainty about the behaviour of all
(physical and virtual) participants is incorporated into
p(∆, λ)
.
A typical choice for the functions
ϕ
and
ψ
in problem 1 is
the financial revenue. If cleared, a supply bid segment at node
n
results in a payment of
wn
sn
if
λn
pn
s
and similarly,
a demand bid segment has a payment of
wn
sn
if
λn
pn
s
,
where the subscript
denotes the corresponding target hour.
However, our aim is to take situations into account when the
bid prices are set such that certain bids or bid segments do
not clear (i.e. their contribution is 0). Therefore, we consider
the following expression:
ϕ(∆, λ, w, w, p, p) = ψ(∆, λ, w, w, p, p)
= S
X
s=1
1λpsws+ 1λpsws!T
,
(3)
where
designates the Hadamard (i.e. pointwise) product
between two vectors.
1λps
and
1λps
denote vector-valued
indicator functions, whose
n
th component is 1 if the corre-
sponding position is cleared, i.e. a supply bid segment is cleared
if its bid price is less than or equal to the day-ahead price
(
λnpn
s
) and similarly, a demand bid segment is cleared if
its bid price is greater than or equal to the day-ahead price
(λnpn
s).
Using the above definition of the revenue results in the
following stochastic optimization problem:
max
w,pRS×N
w,pRS×N
Fp(∆)
S
X
s=1
1λpsws+ 1λpsws!T
s.t.Gp(∆)
S
X
s=1
1λpsws+ 1λpsws!T
0,
(w, w)∈ W,
(p, p)∈ P.
(4)
It is common to choose
F
as the expected value of the
revenue according to
p(∆, λ)
. Also,
G
is typically some risk
摘要:

1AGeneralStochasticOptimizationFrameworkforConvergenceBiddingLetifMonesandSeanLovettAbstract—Convergence(virtual)biddingisanimportantpartoftwo-settlementelectricpowermarketsasitcaneffectivelyreducediscrepanciesbetweentheday-aheadandreal-timemar-kets.Consequently,thereisextensiveresearchintothebiddin...

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