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1
Price-Taker Offering Strategy in Electricity
Pay-as-Bid Markets
Nicol`
o Mazzi, Jalal Kazempour, Member, IEEE, and Pierre Pinson, Senior Member, IEEE
Abstract—The recent increase in the deployment of renewable
energy sources may affect the offering strategy of conventional
producers, mainly in the balancing market. The topics of optimal
offering strategy and self-scheduling of thermal units have been
extensively addressed in the literature. The feasible operating
region of such units can be modeled using a mixed-integer linear
programming approach, and the trading problem as a linear
programming problem. However, the existing models mostly
assume a uniform pricing scheme in all market stages, while
several European balancing markets (e.g., in Germany and Italy)
are settled under a pay-as-bid pricing scheme. The existing
tools for solving the trading problem in pay-as-bid electricity
markets rely on non-linear optimization models, which, combined
with the unit commitment constraints, result in a mixed-integer
non-linear programming problem. In contrast, we provide a
linear formulation for that trading problem. Then, we extend
the proposed approach by formulating a two-stage stochastic
problem for optimal offering in a two-settlement electricity
market with a pay-as-bid pricing scheme at the balancing stage.
The resulting model is mixed-integer and linear. The proposed
model is tested on a realistic case study against a sequential
offering approach, showing the capability of increasing profits in
expectation.
Index Terms—Pay-as-bid, offering strategy, stochastic pro-
gramming, mixed-integer linear program, thermal unit.
NOMENCLATURE
Indices and Sets
i, i0(I)Indices of day-ahead market price scenarios
j, j0(J)Indices of balancing market price scenarios
s(S)Index of generation blocks
k(K)Index of time intervals
ΠFeasible region of the unit’s offer curves
ΩFeasible region of the unit’s operation
Parameters
λDA
ik Day-ahead market price [e/MWh]
λBA
ijk Balancing market price [e/MWh]
MUP
jj 0kAcceptance matrix for up-regulation offers
MDW
jj 0kAcceptance matrix for down-regulation offers
ECapacity of the unit [MW]
EMinimum power limit of the unit [MW]
RUP, RDW Ramp-up and down limits [MW/h]
N. Mazzi is with the Department of Industrial Engineering, University of
Padova, Padova, Italy (e-mail: nicolo.mazzi@dii.unipd.it).
J. Kazempour and P. Pinson are with the Department of Electrical Engi-
neering, Technical University of Denmark, Kgs. Lyngby, Denmark (e-mail:
seykaz@elektro.dtu.dk; ppin@elektro.dtu.dk). J. Kazempour and P. Pinson
are partly supported by the Danish Innovation Fund through the projects
5s - Future Electricity Markets (12-132636/DSF) and CITIES (DSF-1305-
00027B).
EsSize of generation block s[MW]
CUP,CDW Start-up and shut-down costs [e]
C0Cost at the minimum production level [e]
CsMarginal cost of generation block s[e/MWh]
πDA
iProbability of day-ahead price scenario i
πBA
ij Probability of balancing price scenario j, pro-
vided that day-ahead price scenario irealizes
Variables
qDA
ik Quantity offer at day-ahead market [MWh]
oUP
ijk , oDW
ijk Up/down regulation incremental offer [MWh]
qUP
ijk , qDW
ijk Up/down regulation quantity offer [MWh]
qA
ijk Total production quantity [MWh]
cijk Operational cost [e/MWh]
ρUP
ijk , ρDW
ijk Up/down regulation profit [e]
yijk , zijk Start-up/shut-down (binary) status of the unit
uijk Commitment (binary) status of the unit
Functions
g(·), h(·)Operational cost functions
I. INTRODUCTION
IN recent years the power sector has experienced a sig-
nificant increase in the deployment of renewable energy
sources, such as wind and solar power. These sources are usu-
ally traded at zero marginal cost and their growing penetration
is leading to a decrease of the prices in the day-ahead market
[1]-[2]. Moreover, they can only be predicted with a limited
accuracy, thus leading to real-time imbalances and increasing
the need for balancing energy. These changes may affect the
strategy of the conventional producers in both day-ahead and
balancing markets.
A. Literature Review
The optimal offering strategy and self-scheduling of con-
ventional thermal units have already been widely studied in
the literature. Ref. [3] addresses the optimal response of a
thermal generator to a given set of electricity market prices
in terms of both energy and reserve. A mixed-integer linear
programming (MILP) problem is developed considering a non-
convex cost function, as well as its start-up costs, ramp rates
and minimum-up and -down constraints. The same authors
in [4] propose a detailed formulation to model start-up and
shut-down characteristics of a thermal generator. Other works,
such as [5] and [6], include risk measures while optimizing
2
the self-scheduling problem of thermal units. References [3]-
[6] demonstrate that a detailed modeling of the generator’s
feasibility region and its production cost function may be
essential for deriving its optimal self-scheduling. Indeed, the
inter-temporal constraints (e.g., ramping constraints) and non-
convex costs (e.g., start-up and shut-down costs) may affect
the optimal solution. In this context, the pioneering paper
[7] presents an offering strategy for a price-taker producer
under price uncertainty. It develops a set of rules that aim to
translate the results of a self-scheduling problem into market
offers. Ref. [8] presents an algorithm for offering and self-
scheduling of a unit including risk management. Ref. [9]
presents an offering strategy for a price-taker power producer
that aims to maximize profit expectation while hedging against
possible infeasible schedules. Other works relax the price-taker
assumption and develop tools for strategic offering considering
the impact of power producer’s decisions on market prices.
This can be done through a residual demand model [10] or a
bilevel optimization setup [11]-[12].
By analyzing the optimization models in [3]-[12], we can
identify two different sets of variables and constraints. The
first set defines the feasibility region and the cost function of
the production unit. For instance, references [3]-[6] show how
to successfully model it as a MILP problem. The second set
simulates the trading problem, i.e., how the power producer
participates in the market (e.g., through non-decreasing step-
wise offering curves), while considering the market clearing
mechanism (endogenously or exogenously) and the pricing
scheme (e.g., uniform or pay-as-bid). The trading problem can
be modeled using a linear programming (LP) approach [13],
under price-taker assumptions and uniform pricing scheme.
However, even though European day-ahead electricity markets
are mostly settled under a uniform pricing scheme, several
balancing markets (e.g., in Germany and Italy [14]) are settled
under a pay-as-bid pricing scheme.
The topic of trading under a pay-as-bid scheme and price
uncertainty has not been extensively addressed in the literature.
An analysis on optimal offering under pay-as-bid and uniform
pricing schemes is presented in [15] and [16]. They obtain
profit expectation and variance for both pricing schemes, while
assuming that the market price follows a uniform distribution.
Ref. [17] proposes a methodology that aims to maximize the
profit expectation in a day-ahead pay-as-bid auction for power
system reserve. Offering strategies for a joint energy and spin-
ning reserve market under pay-as-bid pricing are presented in
[18] and [19]. Risk aversion is introduced in [20]. References
[15]-[20] show how to model the trading problem under pay-
as-bid pricing scheme using a non-linear programming (NLP)
approach. However, they do not consider an accurate modeling
of production unit’s operational constraints. Introducing the
feasibility region would result in a mixed-integer non-linear
problem (MINLP), which may have high computational cost
and, generally, do not guarantee the optimality of the solution.
It is worth mentioning that [19] proposes to solve the trading
problem in pay-as-bid markets under price uncertainty with a
two-step approach, obtaining the expected profit as a linear
function of the quantity offer. However, this approach is not
applicable in case of problems with inter-temporal constraints
or with more complex cost functions.
B. Approach and Contributions
Compared to the available literature, this paper provides a
novel approach that allows to cast the optimal price-taker trad-
ing problem in pay-as-bid markets under price uncertainty as
an LP problem. For that purpose, continuous random variables
(i.e., market-clearing prices) are represented as discrete vari-
ables. We test our formulation against the existing continuous
NLP alternative. In a simple setup, these two models bring
similar optimal solutions, while our LP approach drastically
reduces the computational cost. Hence, we demonstrate that
our LP model is a good approximation of the NLP one.
However, the value of our LP formulation, with respect to the
NLP one, arises when including the feasible operating region
of the unit.
Then, we use the proposed LP approach to build a multi-
stage stochastic programming problem with recourse. This
efficient decision-making tool could be used by a price-taker
conventional producer to derive its best day-ahead market
offer curves. In line with current practice in several Euro-
pean electricity markets, we consider a two-settlement market
framework, in which the day-ahead market is cleared based
on a uniform pricing scheme, while a pay-as-bid pricing
scheme is used in the balancing stage. The forecast market
prices in both stages are given but uncertain. This uncertainty
is properly characterized by generating a set of foreseen
scenarios. The resulting model is a stochastic MILP problem,
where non-convexities (i.e., binary variables) arise from the
unit commitment constraints. To the best of our knowledge,
this kind of stochastic MILP optimization model for obtaining
the offering strategy of a price-taker thermal producer in a
two-settlement electricity market with a pay-as-bid pricing
scheme in the balancing stage is not available in the literature.
It is worth mentioning that [12] provides a formulation for
obtaining optimal offering curves in markets settled under a
pay-as-bid pricing scheme for a price-maker producer. How-
ever, market problems with equilibrium constraints may have
high computational cost and rely on strong assumptions on
opponents’ behavior. Hence, when the production unit has a
negligible impact on the market, a price-maker setup may not
be the preferable choice.
C. Paper Organization
The remaining of the paper is organized as follows. Section
II presents the electricity market framework, the modeling
assumptions, and the methodology for generating market price
scenarios. Section III provides an overview of the existing NLP
setup as well as the proposed LP setup for deriving the offering
strategy of a price-taker producer under a pay-as-bid pricing
scheme. Then, Section IV extends the proposed approach and
develops a two-stage model for a price-taker producer to derive
its best offering curves in the day-ahead market, considering
a pay-as-bid pricing scheme in the balancing stage. Section
V presents a verification test to assess the performance of
the proposed LP trading model, as well as an application of
the two-stage model using a realistic test case. Finally, the
conclusions are drawn in Section VI.
3
II. MODELING ASSUMPTIONS AND MARKET PRICE
SCE NARIOS
We consider a single conventional producer that trades in a
two-settlement electricity market framework. The day-ahead
market is cleared once a day, at noon, simultaneously for
the whole 24 hourly trading periods of the following day.
Generators are remunerated under a uniform pricing scheme
in the day-ahead market. Then, a balancing market is cleared
separately per each hourly interval, one hour prior to real-time
operation. The provision of balancing energy is remunerated
under a pay-as-bid pricing scheme. The intra-day trading floor
is neglected for the sake of simplicity.
The power producer is assumed to be price-taker in both
day-ahead and balancing markets. Hence, the market prices
within the offering strategy problem of that producer are
exogenous, but still uncertain. We model those uncertainties
using a set of scenarios. Uncertainty characterization is a
critical input to stochastic optimization. The quality of the
solution of a stochastic optimization model is indeed strongly
influenced by the quality of the scenarios provided as input.
Given that the purpose of this paper is to analyze and test
an optimization model, we exploit a fundamental model for
generating market prices, instead of using real market data.
This fundamental model generates a set of electricity market
price forecasts, which is required as an input to our proposed
offering strategy.
In the fundamental market model we assume, for the sake
of simplicity, that the only stochastic generation is wind power
generation. A dataset of wind power forecasts for a wind
farm located in Denmark is used. The wind power forecasts
are re-scaled and assumed representative of the aggregated
wind power production in the market area. At the day-ahead
stage, we assume that the demand curve is linear, known,
and different per each hourly interval. Conversely, the supply
curve of conventional producers is quadratic and uncertain.
To model this uncertainty, we consider the coefficient of
the second-degree term (i.e., γk, where kis the index of
time interval) as a random variable with known marginal
distribution. The methodology for fitting such distribution is
beyond the scope of the paper. The coefficient of the first-
degree term is also considered known to simplify the process
of scenario generation. Then, we assume that the stochastic
generation is offered in the day-ahead market at its mean
forecast and at zero price.
At the balancing stage, the supply curve is assumed known
but different from the day-ahead one. Indeed, the participants
in the balancing market (under pay-as-bid pricing scheme) do
not offer their marginal cost, since they have to internalize
the expected revenues into their market offers [15]. Therefore,
we fix a negative price floor λ0and impose γBA
k=η γk
(η > 1), where γBA
kis referred to the supply curve in the
balancing market. Several factors may cause the real-time
power imbalance in the system, e.g., errors in load and wind
forecasts. For the sake of simplicity, we consider the wind
stochasticity as the only source of uncertainty at the balancing
stage. This simplifies the scenario generation process.
A. Market Model
The demand curve of the day-ahead market at hourly
interval kis
pDA,d
k=αk+δ eDA,d
k,(1)
where eDA,d
kis the amount of energy demand at price pDA,d
k.
The parameters αkand δcontrol the shape of the demand
curve. For the same interval k, the supply curve is
pDA,s
k=
0,if ∆DA,s
k≤0
β∆DA,s
k+γk∆DA,s
k2
,otherwise (2)
where ∆DA,s
k=eDA,s
k−WDA
k. Note that pDA,s
kis the price
for scheduling the quantity eDA,s
k, and WDA
kis the amount of
wind power production offered in the day-ahead market. The
parameters βand γkcontrol the shape of the supply curve.
The value of WDA
kis computed as
WDA
k=E[wk]W , (3)
where wkis the normalized value (wk∈[0,1]) of wind power
production, and Wis the total installed wind capacity. The
uncertain parameter γkfollows a Normal distribution, i.e.,
γk∼ N µγ, σ2
γ,(4)
where µγand σ2
γare the mean value and variance of γk,
respectively. In the balancing market, the supply curve at time
interval kis
pBA,s
k=
λ0,if ∆BA,s
k≤0
βBA
k∆BA,s
k+γBA
k∆BA,s
k2+λ0,otherwise
(5)
where ∆BA,s
k=eBA,s
k−e0
k. The variables pBA,s
kand eBA,s
k
are the price and quantity of the balancing market supply
curve, respectively. The term eBA,s
kis obtained as the difference
between eDA
kand the imbalance generated by the stochastic
generation, i.e.,
eBA,s
k=eDA
k−wkW−WDA
k.(6)
Parameters βBA
kand e0
kare evaluated by imposing that pBA,s
k=
λDA
kwhen eBA,s
k=eDA
k. This ensures that the day-ahead and
the balancing market prices coincide when no balancing power
is required.
B. Scenario Generation
We generate scenarios following a methodology presented
in [21] and [22]. These papers propose a method for generating
trajectories of a stochastic process when predictive distribu-
tions are available. The idea is to convert series of forecast
errors into a multivariate Gaussian random variable and use
a unique covariance matrix to describe its interdependence
structure. This covariance matrix can be modeled through an
exponential covariance function [22], where an exponential
parameter (ν > 1) controls the correlation between different
lead times.
First, a set of scenarios {γik :i∈I, k ∈K}is generated
while fixing the exponential parameter νto 5. Then, a set of
market price trajectories {λDA
ik :i∈I, k ∈K}is generated,
4
where the market-clearing price λDA
ik is obtained from the
intersection between the demand and the supply curves at
interval kunder scenario i.
For the balancing stage, we generate a set of wind power
production trajectories {wjk :j∈J, k ∈K}by fixing the
exponential parameter νto 7 as suggested in [22]. The prob-
abilistic forecasts of wind power production are available in
form of 19 quantiles (from 0.05 to 0.95). To fit the cumulative
distribution function we follow the approach of [23]. Then, a
set of market price scenarios {λBA
ijk :i∈I , j ∈J, k ∈K}is
generated by clearing the balancing market model.
III. TRADING PROB LE M UNDER A PAY-A S-BID PRICING
SCHEME
When trading in an electricity market, power producers can
usually submit price-quantity offers. The quantity identifies the
amount of energy they are willing to produce, and the price
is the minimum price for which they are willing to produce
that energy. Then, the offer is accepted only when the market
price is higher than or equal to the offered one. Since the given
producer is assumed to be price-taker, the market price in each
future time interval is necessarily treated as an exogenous but
uncertain parameter [4]-[5]. In this paper, the market price
λis considered as a random variable following the density
function fλ:R7→ R+. Given a price-quantity offer of a
producer, denoted as (p, q), the acceptance probability of the
offer is
P[λ≥p] = Z∞
p
fλ(l)dl, (7)
where lis an auxiliary integration variable. Under a pay-as-bid
pricing scheme, the expected remuneration price p∗, providing
that the producer’s offer is being accepted, is computed as
E[p∗|λ≥p] = p. (8)
The expected return ρof the producer, following [18], is
E[ρ] = P[λ≥p]E[p∗|λ≥p]q. (9)
By replacing (7) and (8) in (9) we obtain
E[ρ] = q p Z∞
p
fλ(l)dl. (10)
Notice that the expected return in (10) is non-linear. The offers
in generation-side of real-world electricity market are gener-
ally non-decreasing step-wise functions, which can be modeled
through a set of Bprice-quantity offers {(pb, qb), b = 1, .., B}.
The formulation in (10) is extended to the generic model (11)
below including the multiple offer blocks, i.e.,
Max
qb,pbX
b
E[ρb]−E[cb](11a)
s.t. E[ρb] = pbqbZpb+1
pb
fλ(l)dl, ∀b(11b)
E[cb] = h(qb)Zpb+1
pb
fλ(l)dl, ∀b(11c)
qb≥qb−1,∀b(11d)
pb≥pb−1,∀b(11e)
E≤qb≤E, ∀b(11f)
where pB+1 =∞. The parameter cbis the operational cost for
producing the quantity qb, whose value is computed through
the function h(·). Constraints (11b) and (11c) compute the
expected return ρband the operational cost of the offering
block (pb, qb), respectively. Constraints (11d) and (11e) impose
the non-decreasing condition of the offering curve. Finally,
(11f) imposes the minimum and maximum production levels
of the unit.
One of the main contributions of this work is to derive
an alternative linear formulation to (11), which is non-linear.
First, we discretize the continuous random variables. The
uncertain market price can be represented using a set of
possible scenarios {λi, i = 1, .., N }, where each price scenario
λiis associated with a probability πisuch that Piπi= 1. We
consider each price scenario λias the potential offer price of
the price-taker producer, and obtain the optimal quantity offer
qicorresponding to each offer price λi. A collection of all
price-quantity offers, i.e., (λi, qi), builds the offer curve of the
producer, providing that the following conditions are enforced
to ensure that the offer curve is non-decreasing [13]:
qi≥qi0if λi≥λi0,∀i, ∀i0,(12a)
qi=qi0if λi=λi0,∀i, ∀i0,(12b)
where iand i0are indices of the market price scenarios.
Note that this offer curve is now scenario-independent, i.e.,
it is adapted to all scenarios, though it is built based on
scenario-dependent price-quantity offers. Under a uniform
pricing scheme, the expected market return ρcan be computed
as
E[ρ] = X
i
πiλiqi,(13)
However, the market return formulation (13) needs to be
changed under the pay-as-bid scheme, since each block offer
is remunerated at its corresponding offer price. Therefore, we
introduce variable oi, which represents the additional quantity
offered at price λi. Figure 1 illustrates an example offer curve
with three offer blocks. In this curve, λ1,λ2and λ3are not
only the three price scenarios, but also they are price offers
of the price-taker producer. For example, the producer offers
its q1MWh at price λ1(as the first offer block), and then the
additional o2MWh (i.e., q2-q1) at price λ2(as the second
offer block). The total quantity qicorresponding to price λi
can be computed as
qi=X
i0
Mii0oi0,(14)
where the acceptance matrix Mis defined as
Mii0=(1,if λi≥λi0
0,otherwise.(15)
Matrix Mii0indicates whether the offer block (λi0, oi0)is
accepted in the market, providing that the market price re-
alization is λi. The total expected return is computed as the
sum of the expected returns for each offer block (λi0, oi0),
denoted as ρi0, i.e.,
E[ρ] = X
i0
E[ρi0].(16)
5
Fig. 1. An example offer curve with multiple blocks, which shows how
market price scenarios are used as price offers, building an offer curve.
In line with (9), we compute the expectation of ρi0as
E[ρi0] = P[λ≥λi0]E[p∗|λ≥λi0]oi0,(17)
where the acceptance probability of each block offer is
P[λ≥λi0] = X
i
Mii0πi,(18)
and the expected remuneration price is
E[p∗|λ≥λi0] = λi0.(19)
Substituting (18) and (19) in (17) renders
E[ρi0] = oi0λi0X
i
Mii0πi.(20)
The total expected return can thus be computed as
E[ρ] = X
i0
oi0λi0X
i
Mii0πi.(21)
Note that (21) is linear. The expected profit E[ρ]can also
be seen as PiE[ρi], where ρiis the return when scenario i
realizes. Therefore, we rewrite (21) as
E[ρ] = X
i
πiρi=X
i
πiX
i0
Mii0λi0oi0.(22)
Given the linear formulation in (22), we rewrite the generic
nonlinear model (11) in a linear manner, i.e.,
Max
ΘX
i
πihρi−cii(23a)
s.t. qi=X
i0
Mii0oi0,∀i(23b)
ρi=X
i0
Mii0λi0oi0,∀i(23c)
ci=h(qi),∀i(23d)
E≤qi≤E, ∀i(23e)
where Θ = {qi, oi, ci, ρi,∀i}. Constraints (23d) compute the
production cost through the function h(·). It is worth mention-
ing that the non-decreasing conditions are not necessary in (23)
since they are inherently included in the way we compute the
accepted quantity qiin (23b). The performance of both models,
i.e., (11) and (23), is analyzed and compared using a simple
case study in Section V-A.
IV. OFFE RING STRATEG Y AS A TW O-STAG E STOCHASTIC
OPTIMIZATION MODEL
Given the linear offering strategy setup under a pay-as-
bid pricing scheme proposed in Section III, we derive the
optimal offering curves of a conventional producer in the two-
settlement electricity market described in Section II. In this
model, we also consider the unit commitment constraints of
that producer. At noon, the power producer has to submit its
offering curves for the day-ahead market of the following day,
based on the price scenarios for both day-ahead and balancing
markets. In this two-stage setup, the producer maximizes
its expected profit from both markets simultaneously, in the
sense that it endogenously determines its future balancing
actions while solving its offering problem in the day-ahead.
Accordingly, we model the day-ahead production level in time
interval k, i.e., qDA
k, as first-stage (here-and-now) decision,
and the up and down production adjustments in the real-
time stage, i.e., qUP
ik and qDW
ik , as second-stage (wait-and-see)
variables. Following the approach described in Section III
for building the producer’s offer curve, we now relax the
day-ahead production variable to be scenario-dependent (i.e.,
qDA
k→qDA
ik ). Similarly, the real-time production adjustments
qUP
ik and qDW
ik are made scenario-dependent (i.e., qUP
ik →qUP
ijk ,
qDW
ik →qDW
ijk ) in order to build the producer’s offer curve in the
balancing stage. In addition, we add a detailed representation
of the feasible operating region of the thermal unit. The
optimization model that the power producer solves to decide
the day-ahead market offers reads as follows
Max
ΞX
ijk
πDA
iπBA
ij hλDA
ik qDA
ik +ρUP
ijk −ρDW
ijk −cij ki(24a)
s.t. qA
ijk =qDA
ik +qUP
ijk −qDW
ijk ,∀i, ∀j, ∀k(24b)
qUP
ijk =X
j0
MUP
ijj 0koUP
ij0k,∀i, ∀j, ∀k(24c)
qDW
ijk =X
j0
MDW
ijj 0koDW
ij0k,∀i, ∀j, ∀k(24d)
ρUP
ijk =X
j0
λBA
ij0kMUP
ijj 0koUP
ij0k,∀i, ∀j, ∀k(24e)
ρDW
ijk =X
j0
λBA
ij0kMDW
ijj 0koDW
ij0k,∀i, ∀j, ∀k(24f)
qDA
ik , oUP
ijk , oDW
ijk ∈Π,∀i, ∀j, ∀k(24g)
qA
ijk , uijk , yijk , zij k ∈Ω,∀i, ∀j, ∀k(24h)
cijk =g(qA
ijk , uijk , yijk , zij k),∀i, ∀j, ∀k(24i)
qDA
ik , oUP
ijk , oDW
ijk ≥0,∀i, ∀j, ∀k(24j)
uijk , yijk , zijk ∈ {0,1},∀i, ∀j, ∀k(24k)
where
Ξ = {qDA
ik , oUP
ijk , oDW
ijk , qUP
ijk , qDW
ijk , ρUP
ijk , ρDW
ijk , qA
ijk ,
uijk , yijk , zijk , xij ks, cij k :∀i, ∀j, ∀j0,∀s, ∀k}.
6
The acceptance matrices MUP
jj 0kand MDW
jj 0kare defined as
MUP
ijj 0k=(1,if λBA
ijk ≥λBA
ij0kand λBA
ijk > λDA
ik
0,otherwise,(25a)
MDW
ijj 0k=(1,if λBA
ijk ≤λBA
ij0kand λBA
ijk < λDA
ik
0,otherwise. (25b)
The objective function (24a) maximizes the expected profit
of the producer from selling energy in both day-ahead and
balancing markets. Constraints (24b) yield the total power
production qA
ijk when both day-ahead price scenario iand
balancing price scenario jrealize at time interval k. For the
same scenario realization, constraints (24c) compute the level
of up-regulation energy qUP
ijk scheduled. Similarly, constraints
(24d) obtain the level of down-regulation energy qDW
ijk . Con-
straints (24e) and (24f) give the expected revenues from selling
regulation energy in the balancing market under a pay-as-bid
pricing scheme. Constraints (24g) include a set of constraints
associated with the offer curves, which is represented later in
Section IV-A. Constraints (24h) force the power producer to
operate in its feasible operating region, which is provided in
Section IV-B. Constraints (24i) compute the operational costs
for given schedule, whose formulation is provided in Section
IV-C.
Model (24) can be also used to compute the expected profit
from a sequential offering approach. First, we solve the model
considering the day-ahead scenarios only. To do that, we force
the balancing variables to be null (i.e., qUP
ijk = 0 and qDW
ijk =
0∀i, j, k). The optimal solutions eqDA*
ik represent the optimal
market offers when considering the day-ahead market only.
Then, we solve again the model while imposing qDA
ik =eqDA*
ik .
The optimal solutions eqUP*
ijk and eqDW*
ijk are the balancing market
offers that maximize the expected profit, provided that the day-
ahead offers are eqDA*
ik .
A. Linear Expression of Π
The offer curve constraints (24g), denoted as Π, are
qDA
ik ≤E, ∀i, ∀k(26a)
qDA
ik ≤qDA
i0kif λDA
ik ≤λDA
i0k,∀i, ∀i0,∀k(26b)
qDA
ik =qDA
i0kif λDA
ik =λDA
i0k,∀i, ∀i0,∀k(26c)
oUP
ij0k= 0 if λBA
ij0k≤λDA
ik ,∀i, ∀j0,∀k(26d)
oDW
ij0k= 0 if λBA
ij0k≥λDA
ik ,∀i, ∀j0,∀k(26e)
Constraints (26a) restrict the day-ahead production quantity
of the producer to its capacity. Constraints (26b) and (26c)
enforce the non-decreasing and non-anticipativity conditions
of the producer’s offer curve in the day-ahead, respectively.
These two conditions are required for offer curves to be
submitted to markets settled under a uniform pricing scheme.
Constraints (26d) and (26e) impose that no balancing energy
is contracted when it is not required by the system.
B. Linear Expression of Ω
Constraints (24h) represent the feasible operating region of
the producer and can be replaced by
qA
ijk ≤uij kE , ∀i, ∀j, ∀k(27a)
qA
ijk ≥uij kE , ∀i, ∀j, ∀k(27b)
qA
ijk −qA
ij(k−1) ≤RUP ,∀i, ∀j, ∀k(27c)
qA
ij(k−1) −qA
ijk ≤RDW,∀i, ∀j, ∀k(27d)
uijk −uij (k−1) ≤yijk ,∀i, ∀j, ∀k(27e)
uij(k−1) −uij k ≤zijk ,∀i, ∀j, ∀k(27f)
Constraints (27a) and (27b) impose that the total power pro-
duction lies between its minimum and maximum production
levels. Constraints (27c) and (27d) enforce ramp-up and ramp-
down limits [3]. Constraints (27e) and (27f) determine the
start-up status yijk and the shut-down status zijk . Notice that
constraints (27c)-(27f) require the initial production level and
commitment status qA
0, u0.
C. Linear Expression of Operational Cost
Constraints (24i) define the operational cost of the thermal
unit and can be replaced by
cijk =C0uij k +X
s
Csxijks +
+CUPyij k +CDWzijk ,∀i, ∀j, ∀k
(28a)
0≤xijks ≤Es,∀i, ∀j, ∀k , ∀s(28b)
Euij k +X
s
xijks =qA
ijk ,∀i, ∀j, ∀k. (28c)
Constraints (28a) compute the operational cost of the unit
including its start-up and shut-down costs [4]. Constraints
(28b) and (28c) are auxiliary constraints representing a linear
formulation of the quadratic cost function through a piecewise
approximation [3], where xijks is the amount of energy
produced by the generation block s.
V. CA SE ST UDY
A. LP Trading Model Verification
In this section we test and compare the proposed LP model
(23) and the existing NLP model (11) using a simple test case,
considering a single time period. The market price λfollows
a Normal distribution, i.e.
λ∼ N (50,5) .(29)
The production unit has a capacity Eof 60 MW and a
minimum production level Eof 0 MW. The quadratic cost
function h(·)is approximated with a two-step piecewise linear
cost function, i.e.,
h(q) = C1x1+C2x2,(30a)
q=x1+x2,(30b)
0≤x1≤E1,(30c)
0≤x2≤E2,(30d)
7
where E1and E2are 30 MW, while C1=e35/MWh and C2
=e47/MWh.
The input scenarios for the LP model (23) are selected
following the scenario reduction technique in [24]. We gen-
erate 1000 scenarios from (29) and then keep the 20 most
representative ones. The LP model is implemented using
GUROBI [25] in PYTHON, and it is solved in around 0.001s.
The optimal solution obtained is given in Table I, where the
expected profit E[ρ]is re-computed considering the continuous
distribution of λin (29). The NLP continuous model (11) is
solved using COBYLA [26] algorithm in PYTHON in around
0.115s. Its optimal solution is also reported in Table I. The
two models show similar optimal solutions, while the gap
in the expected profit due to the discretization procedure is
lower than 0.07%. The LP model is solved around 115 times
faster than the NLP one. It is worth mentioning that the
computational time of the NLP model increases to around 3.2s
when the integrals in (11) are numerically computed, instead
of using the cumulative distribution function of (29). However,
the main advantage of the LP formulation is to be more
suitable to be merged with the unit commitment constraints of
the thermal unit. Therefore, a comparison between a MILP and
MINLP offering model would be more appropriate to test the
advantages of our LP formulation. Nevertheless, the solution
of a MINLP problem is out of the scope of this paper.
This test case shows that the LP formulation, when pro-
vided with an accurate sampling of input scenarios, brings an
optimal solution close to the continuous NLP alternative. In
the following section, we extend the LP model by including
the unit commitment constraints of the thermal unit in a multi-
time period and multi-stage stochastic optimization problem.
B. Day-ahead Offering Model Test Case
We test the two-stage stochastic optimization model (24)
on a realistic case study. We generate market price scenarios
according to the methodology presented in Section II. The
input parameters are shown in Tables II and III. First, we
generate 300 scenarios for λDA
ik and we select the 20 most
representative ones. Then, for each scenario λDA
ik , we generate
300 scenarios of λBA
ijk and keep the 20 most representative
ones. This procedure results in a scenario tree with 400
branches. We repeat this sampling procedure for different
values of W, i.e. 10, 20 and 30 GW. Figure 2 shows the
20 scenarios of λDA
ik (in blue) and the 20 scenarios of λBA
ijk
(in green) for a given realization of λDA
k(in red), when W
= 20 GW. We consider a thermal unit with a capacity of
E= 120 MW and a minimum production level of E= 40
TABLE I
THE P ROD UCE R’S O PT IMA L MA RK ET OFF ER S OB TAIN ED F ROM T HE LP
AND NLP MODELS
qbλbE[ρ]
(MWh) (e/MWh) (e)
NLP b=1 30 46.6 313.4
b=2 60 51.7
LP b=1 30 46.4 313.2
b=2 60 51.8
TABLE II
PARAMETERS OF THE MARKET PRICE GENERATION MODEL
δ β µγσ2
γλ0
(e/MWh2) (e/MWh2) (e/MWh3) (e/MWh3) (e/MWh)
-6.67×10−31×10−42×10−83×10−9-20
TABLE III
VALUE S OF PAR AM ET ER αk
k12345678
αk(e/MWh) 322 312 315 317 340 349 353 369
k9 10 11 12 13 14 15 16
αk(e/MWh) 394 424 444 445 440 429 437 458
k17 18 19 20 21 22 23 24
αk(e/MWh) 446 423 408 383 373 346 331 332
Fig. 2. Day-ahead and balancing market price scenarios.
MW. Ramping limits are 40 MW for both RUP and RDW. The
quadratic cost function is approximated by a piecewise linear
function of four generation blocks of equal size, i.e. Es= 20
MW ∀s. Table IV shows the marginal cost Csof each block,
the cost C0, the start-up cost CUP and the shut-down cost
CDW.
The optimization model is implemented using GUROBI
in PYTHON environment. We compare the two-stage co-
optimization model with a sequential offering approach. In-
deed, by modeling balancing market variables as recourse
decisions at the day-ahead stage, we co-optimize the offering
strategy for the two markets.
As an example, Table V reports the optimal value of the day-
ahead production variable qDA
ik in time interval k= 7 obtained
from both the co-optimized and sequential approaches. Note
that λDA
i7,i= 1, ..., 20, is the set of the day-ahead price
scenarios, and each member of this set is viewed as a potential
price offer. For the co-optimized approach, the results given
8
TABLE IV
PARAMETERS OF THE COST FUNCTION
C0Cs1Cs2Cs3Cs4CUP CDW
(e) (e/MWh) (e/MWh) (e/MWh) (e/MWh) (e) (e)
2860 23.5 31.5 45.6 72.3 800 100
in Table V can be summarized as given in (31) below:
qDA
7=
0,if λDA
7<39.7
40,if 39.7≤λDA
7<42.0
80,if 42.0≤λDA
7<51.7
120,if λDA
7≥51.7,
(31)
where qDA
7is expressed in MWh and λDA
7in e/MWh. Ac-
cording to (31), a scenario-independent offer curve in day-
ahead can be built using three price-quantity offer points,
i.e., (e39.7/MWh, 40 MW), (e42.0/MWh, 80 MW) and
(e51.7/MWh, 120 MW). A graphic representation of this
curve is provided in Figure 3a, blue curve. Similarly, the day-
ahead offer curve at time interval k= 7 for the sequential
approach can be obtained (red curve). Note that in both
approaches, the producer is not willing to produce if λDA
7≤
39.7 while desires to operate at its full capacity if λDA
7≥
51.7. However, when 42.0 ≤λDA
7≤48.2, the co-optimized
approach suggests to produce 80 MW, while the sequential
approach does 40 MW only. In addition, Figure 3b shows
the offering curve of the producer in the balancing market
at time interval k= 7, provided that the realization of day-
ahead price λDA
7is e44.1/MWh. Based on the co-optimized
approach, the producer is scheduled to produce 80 MW and
then to reduce its production level in the balancing stage
to 40 MW if λBA
7≤35.6, or to increase it to 120 MW
in case λBA
7≥55.7. Unlike the co-optimized approach, the
sequential one schedules the producer at 40 MW in the day-
ahead market, and then provides the up-regulation service only
in the balancing stage. For instance, its production increases
by 40 MW ifλBA
7≥56.0 while that increase is even more (80
MW) in case λBA
7≥58.7. The expected profits obtained from
these two approaches under different conditions are shown
in Table VI. In a case in which W= 10 GW, the expected
profit loss in the sequential approach is around 2%. The power
producer gains a lower expected profit in the day-ahead market
while earning more in the balancing stage, such that its total
expected profit (including both stages) increases as well. This
behavior is more observable in the cases with a higher value
of installed wind capacity. For instance, the loss of profit is
22% and 91% in cases in which Wis equal to 20 and 30 GW,
respectively. The last case (W=30 GW) gives more insight: in
the sequential approach, the producer does not participate in
the day-ahead market, and earns a low profit in the balancing
stage only. In contrast, the producer gains a significant money
in the co-optimized approach, though it loses money in the
day-ahead stage. In fact, it takes such a loosing position in
day-ahead market to be able to produce profitable regulation
services in the balancing stage.
TABLE V
OPT IMA L VALU ES OF qDA
i7FO R THE S EQ UEN TI AL A ND CO -OPTIMIZED
AP PROAC HE S
iλDA
i7qDA
i7iλDA
i7qDA
i7
(e/MWh) (MWh) (e/MWh) (MWh)
co-opt seq co-opt seq
1 44.1 80 40 11 36.3 0 0
2 33.0 0 0 12 44.4 80 40
3 45.1 80 40 13 44.6 80 40
4 45.6 80 40 14 43.4 80 40
5 39.7 40 40 15 52.7 120 120
6 37.1 0 0 16 44.7 80 40
7 48.2 80 80 17 42.0 80 40
8 44.9 80 40 18 41.1 40 40
9 51.7 120 120 19 44.3 80 40
10 43.0 80 40 20 46.9 80 40
(a) (b)
Fig. 3. The producer’s optimal offer curve in (a) day-ahead market, (b)
balancing market at time interval k= 7 (W= 20 GW).
VI. CONCLUSIONS AND FU RTH ER WO RKS
This paper presents a novel method for deriving optimal
offering curves of a price-taker conventional producer in an
electricity market under a pay-as-bid pricing scheme. The
importance of this study is that several European balancing
markets in the balancing stage are settled under a pay-as-bid
pricing scheme. The main contribution of this paper is that
we develop an LP approach. In contrast, the existing tools
in the literature are mainly non-linear and less suitable to be
merged to the MILP feasibility region of a thermal unit. Then,
we extend our proposed approach to a two-stage market setup
including day-ahead and balancing stages. In this setup, being
consistent with the structure of several European electricity
markets, the day-ahead market is cleared based on a uniform
TABLE VI
EXP ECT ED P ROFI T OF T HE PR ODU CE R
WModel Profit in DA Profit in BA Total
(GW) (103e) (103e) (103e)
10 co-optimized 16.82 3.25 20.09
sequential 18.08 1.59 19.68
20 co-optimized 4.45 8.27 12.27
sequential 7.68 2.27 9.95
30 co-optimized -8.45 18.44 9.99
sequential 0.00 0.87 0.87
9
pricing scheme, while a pay-as-bid scheme is used in the
balancing stage. To make our setup more realistic, we include
the unit commitment constraints of the thermal units to our
proposed setup, resulting in a MILP model.
We first test our proposed single-stage LP model against the
existing non-linear models. Our LP formulation successes in
well approximating the non-linear one. Then, we compare our
proposed two-stage MILP model against a sequential offering
model, which does not consider the balancing stage while of-
fering in the day-ahead market. Our proposed approach shows
a better performance in terms of expected profit achieved.
In future research it is of interest to test the proposed model
using real market price data. Besides, intra-day markets could
be included in the offering model. These additional trading
floors may bring more business opportunities to the producer.
Moreover, the proposed approach could be extended to derive
the optimal offering strategy of different technologies, such as
energy storage systems. For such facilities, the participation
in the balancing market may be significantly important for
exploiting their flexibility.
ACKNOWLEDGMENT
The authors would like to thank Lesia Mitridati (DTU) for
discussion and feedback on previous versions of the work, and
Stefanos Delikaraoglou (DTU) and Christos Ordoudis (DTU)
for their comments on the final manuscript. The authors would
also like to thank the anonymous reviewers for their valuable
comments and suggestions which have improved the quality
of this paper.
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Nicol`
o Mazzi received the B.Sc. degree in 2011 and the M.Sc. degree
in 2014, both from the University of Padova, Padova, Italy, in energy
engineering. He is currently pursuing the Ph.D. degree at the Department of
Industrial Engineering of University of Padova. His research interests include
energy systems, electricity markets, stochastic optimization and decomposition
techniques.
Jalal Kazempour (M14) is an assistant professor at the Department of
Electrical Engineering, Technical University of Denmark, Kgs. Lyngby, Den-
mark. He received his Ph.D. degree in electrical engineering from University
of Castilla-La Mancha, Ciudad Real, Spain, in 2013. His research interests
include power systems, electricity markets, optimization, and its applications
to energy systems.
Pierre Pinson (M11-SM13) received the M.Sc. degree in applied mathematics
from the National Institute for Applied Sciences (INSA Toulouse, France) and
the Ph.D. degree in energetics from Ecole des Mines de Paris (France). He is a
Professor at the Technical University of Denmark (DTU), Centre for Electric
Power and Energy, Department of Electrical Engineering, also heading a group
focusing on Energy Analytics & Markets. His research interests include among
others forecasting, uncertainty estimation, optimization under uncertainty,
decision sciences, and renewable energies. Prof. Pinson acts as an Editor for
the International Journal of Forecasting, and for Wind Energy.
