
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
t−πn
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
s≥0
and prices
pn
s
as well as demand volumes
wn
s≤0
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.