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Price-Taker Offering Strategy in Electricity Pay-as-Bid Markets


Abstract and Figures

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.
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Price-Taker Offering Strategy in Electricity
Pay-as-Bid Markets
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
Index Terms—Pay-as-bid, offering strategy, stochastic pro-
gramming, mixed-integer linear program, thermal unit.
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
Feasible region of the unit’s operation
ik Day-ahead market price [e/MWh]
ijk Balancing market price [e/MWh]
jj 0kAcceptance matrix for up-regulation offers
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:
J. Kazempour and P. Pinson are with the Department of Electrical Engi-
neering, Technical University of Denmark, Kgs. Lyngby, Denmark (e-mail:; 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-
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]
iProbability of day-ahead price scenario i
ij Probability of balancing price scenario j, pro-
vided that day-ahead price scenario irealizes
ik Quantity offer at day-ahead market [MWh]
ijk , oDW
ijk Up/down regulation incremental offer [MWh]
ijk , qDW
ijk Up/down regulation quantity offer [MWh]
ijk Total production quantity [MWh]
cijk Operational cost [e/MWh]
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
g(·), h(·)Operational cost functions
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
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.
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
k=αk+δ eDA,d
where eDA,d
kis the amount of energy demand at price pDA,d
The parameters αkand δcontrol the shape of the demand
curve. For the same interval k, the supply curve is
0,if DA,s
,otherwise (2)
where DA,s
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
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
where µγand σ2
γare the mean value and variance of γk,
respectively. In the balancing market, the supply curve at time
interval kis
λ0,if BA,s
where BA,s
k. The variables pBA,s
kand eBA,s
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.,
Parameters βBA
kand e0
kare evaluated by imposing that pBA,s
kwhen eBA,s
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 :iI, k K}is generated
while fixing the exponential parameter νto 5. Then, a set of
market price trajectories {λDA
ik :iI, k K}is generated,
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 :jJ, 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 :iI , j J, k K}is
generated by clearing the balancing market model.
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[λp] = Z
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|λp] = p. (8)
The expected return ρof the producer, following [18], is
E[ρ] = P[λp]E[p|λp]q. (9)
By replacing (7) and (8) in (9) we obtain
E[ρ] = q p Z
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.,
s.t. E[ρb] = pbqbZpb+1
fλ(l)dl, b(11b)
E[cb] = h(qb)Zpb+1
fλ(l)dl, b(11c)
EqbE, 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]:
qiqi0if λiλi0,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
E[ρ] = X
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
where the acceptance matrix Mis defined as
Mii0=(1,if λiλi0
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
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[λλi0]E[p|λλi0]oi0,(17)
where the acceptance probability of each block offer is
P[λλi0] = X
and the expected remuneration price is
E[p|λλi0] = λi0.(19)
Substituting (18) and (19) in (17) renders
E[ρi0] = oi0λi0X
The total expected return can thus be computed as
E[ρ] = X
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
Given the linear formulation in (22), we rewrite the generic
nonlinear model (11) in a linear manner, i.e.,
s.t. qi=X
EqiE, 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.
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.,
ik ). Similarly, the real-time production adjustments
ik and qDW
ik are made scenario-dependent (i.e., qUP
ik qUP
ijk ,
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
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)
ijk =X
ijj 0koUP
ij0k,i, j, k(24c)
ijk =X
ijj 0koDW
ij0k,i, j, k(24d)
ijk =X
ijj 0koUP
ij0k,i, j, k(24e)
ijk =X
ijj 0koDW
ij0k,i, j, k(24f)
ik , oUP
ijk , oDW
ijk Π,i, j, k(24g)
ijk , uijk , yijk , zij k ,i, j, k(24h)
cijk =g(qA
ijk , uijk , yijk , zij k),i, j, k(24i)
ik , oUP
ijk , oDW
ijk 0,i, j, k(24j)
uijk , yijk , zijk ∈ {0,1},i, j, k(24k)
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}.
The acceptance matrices MUP
jj 0kand MDW
jj 0kare defined as
ijj 0k=(1,if λBA
ijk λBA
ij0kand λBA
ijk > λDA
ijj 0k=(1,if λBA
ijk λBA
ij0kand λBA
ijk < λDA
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
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 =
0i, 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
ik E, i, k(26a)
ik qDA
i0kif λDA
ik λDA
i0k,i, i0,k(26b)
ik =qDA
i0kif λDA
ik =λDA
i0k,i, i0,k(26c)
ij0k= 0 if λBA
ik ,i, j0,k(26d)
ij0k= 0 if λBA
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
Constraints (24h) represent the feasible operating region of
the producer and can be replaced by
ijk uij kE , i, j, k(27a)
ijk uij kE , i, j, k(27b)
ijk qA
ij(k1) RUP ,i, j, k(27c)
ij(k1) qA
ijk RDW,i, j, k(27d)
uijk uij (k1) yijk ,i, j, k(27e)
uij(k1) 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
Csxijks +
+CUPyij k +CDWzijk ,i, j, k
0xijks Es,i, j, k , s(28b)
Euij k +X
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.
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)
where E1and E2are 30 MW, while C1=e35/MWh and C2
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
(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
(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
δ β µγσ2
(e/MWh2) (e/MWh2) (e/MWh3) (e/MWh3) (e/MWh)
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
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
(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:
0,if λDA
40,if 39.7λDA
80,if 42.0λDA
120,if λDA
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
39.7 while desires to operate at its full capacity if λDA
51.7. However, when 42.0 λDA
748.2, 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
735.6, or to increase it to 120 MW
in case λBA
755.7. 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
756.0 while that increase is even more (80
MW) in case λBA
758.7. 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.
(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).
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
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
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.
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.
[1] F. Sensfuss, M. Ragwitz, and M. Genoese, “The merit-order effect: A
detailed analysis of the price effect of renewable electricity generation
on spot market prices in Germany,Energy Policy, vol. 36, no. 8, pp.
3086–3094, 2008.
[2] S. Cl`
o, A. Cataldi, and P. Zoppoli, “The merit-order effect in the Italian
power market: The impact of solar and wind generation on national
wholesale electricity prices,” Energy Policy, vol. 77, pp. 79–88, 2015.
[3] J. M. Arroyo and A. J. Conejo, “Optimal response of a thermal unit to
an electricity spot market,” IEEE Trans. Power Syst., vol. 15, no. 3, pp.
1098–1104, 2000.
[4] ——, “Modeling of start-up and shut-down power trajectories of thermal
units,” IEEE Trans. Power Syst., vol. 19, no. 3, pp. 1562–1568, 2004.
[5] A. J. Conejo, F. J. Nogales, J. M. Arroyo, and R. Garc´
ıa-Bertrand, “Risk-
constrained self-scheduling of a thermal power producer,IEEE Trans.
Power Syst., vol. 19, no. 3, pp. 1569–1574, 2004.
[6] R. A. Jabr, “Robust self-scheduling under price uncertainty using
conditional value-at-risk,IEEE Trans. Power Syst., vol. 20, no. 4, pp.
1852–1858, 2005.
[7] A. J. Conejo, F. J. Nogales, and J. M. Arroyo, “Price-taker bidding
strategy under price uncertainty,IEEE Trans. Power Syst., vol. 17, no. 4,
pp. 1081–1088, 2002.
[8] E. Ni, P. B. Luh, and S. Rourke, “Optimal integrated generation
bidding and scheduling with risk management under a deregulated power
market,” IEEE Trans. Power Syst., vol. 19, no. 1, pp. 600–609, 2004.
[9] M. Maenhoudt and G. Deconinck, “Strategic offering to maximize day-
ahead profit by hedging against an infeasible market clearing result,”
IEEE Trans. Power Syst., vol. 29, no. 2, pp. 854–862, 2014.
[10] A. Baillo, M. Ventosa, M. Rivier, and A. Ramos, “Optimal offering
strategies for generation companies operating in electricity spot mar-
kets,” IEEE Trans. Power Syst., vol. 19, no. 2, pp. 745–753, 2004.
[11] V. P. Gountis and A. G. Bakirtzis, “Bidding strategies for electricity
producers in a competitive electricity marketplace,IEEE Trans. Power
Syst., vol. 19, no. 1, pp. 356–365, 2004.
[12] A. G. Bakirtzis, N. P. Ziogos, A. C. Tellidou, and G. A. Bakirtzis,
“Electricity producer offering strategies in day-ahead energy market with
step-wise offers,IEEE Trans. Power Syst., vol. 22, no. 4, pp. 1804–
1818, 2007.
[13] A. J. Conejo, M. Carri´
on, and J. M. Morales, Decision Making Under
Uncertainty in Electricity Markets. Springer, 2010.
[14] Q. Wang, C. Zhang, Y. Ding, G. Xydis, J. Wang, and J. Østergaard,
“Review of real-time electricity markets for integrating distributed
energy resources and demand response,” Applied Energy, vol. 138, pp.
695–706, 2015.
[15] Y. Ren and F. D. Galiana, “Pay-as-bid versus marginal pricing - part I:
Strategic generator offers,IEEE Trans. Power Syst., vol. 19, no. 4, pp.
1771–1776, 2004.
[16] ——, “Pay-as-bid versus marginal pricing - part II: Market behavior
under strategic generator offers,IEEE Trans. Power Syst., vol. 19, no. 4,
pp. 1777–1783, 2004.
[17] D. J. Swider and C. Weber, “Bidding under price uncertainty in
multi-unit pay-as-bid procurement auctions for power systems reserve,
European Journal of Operational Research, vol. 181, no. 3, pp. 1297–
1308, 2007.
[18] D. J. Swider, “Simultaneous bidding in day-ahead auctions for spot
energy and power systems reserve,International Journal of Electrical
Power & Energy Systems, vol. 29, no. 6, pp. 470–479, 2007.
[19] J. Khorasani and H. R. Mashhadi, “Bidding analysis in joint energy and
spinning reserve markets based on pay-as-bid pricing,” IET Generation,
Transmission & Distribution, vol. 6, no. 1, pp. 79–87, 2012.
[20] J. Sadeh, H. R. Mashhadi, and M. A. Latifi, “A risk-based approach
for bidding strategy in an electricity pay-as-bid auction,” European
Transactions on Electrical Power, vol. 19, no. 1, pp. 39–55, 2009.
[21] P. Pinson, H. Madsen, H. A. Nielsen, G. Papaefthymiou, and B. Kl¨
“From probabilistic forecasts to statistical scenarios of short-term wind
power production,Wind Energy, vol. 12, no. 1, pp. 51–62, 2009.
[22] P. Pinson and R. Girard, “Evaluating the quality of scenarios of short-
term wind power generation,Applied Energy, vol. 96, pp. 12–20, 2012.
[23] P. H. D. Andersen, “Optimal trading strategies for a wind-storage power
system under market conditions,” Master’s thesis, Technical University
of Denmark, DTU, DK-2800 Kgs. Lyngby, Denmark, 2009.
[24] N. Growe-Kuska, H. Heitsch, and W. Romisch, “Scenario reduction and
scenario tree construction for power management problems,” in IEEE
Power Tech Conference Proceedings, 2003.
[25] Gurobi Optimization, “Gurobi optimizer reference manual,” 2016.
[Online]. Available:
[26] E. Jones, T. Oliphant, P. Peterson et al., “SciPy: Open source scientific
tools for Python,” 2001. [Online]. Available:
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
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.
... In [25], an optimal bidding strategy is proposed for a producer in a pool-based market and self-scheduling to maximize profits. In [26], a PT bidding strategy is suggested for a thermal unit in a two-stage market in the balancing stage according to pay-as-bid. In [27], a PT bidding strategy is proposed for a storage system in ancillary services for frequency regulation. ...
... In some papers, such as [21] and [22]- [26], the optimal operation condition is determined without the game theory method, while this method is an effective method for finding the NE point in the optimal operation condition [41]. Some papers neglect the optimal operation indices for simplicity [23], and others do not consider the MG's operation formulation to focus on other concepts [24]- [26]. ...
... In some papers, such as [21] and [22]- [26], the optimal operation condition is determined without the game theory method, while this method is an effective method for finding the NE point in the optimal operation condition [41]. Some papers neglect the optimal operation indices for simplicity [23], and others do not consider the MG's operation formulation to focus on other concepts [24]- [26]. In [28]- [30], the optimal operation management of an MG is modeled without DR and the electricity market. ...
Full-text available
This paper proposes a new framework for the optimal operation of a microgrid aggregator (MGA) that participates in an oligopoly electricity market. This aggregator obtains an optimal bidding (power and price) strategy for a multigrid (MG) system, i.e., a community MG. Consequently, the granted quantity, i.e., power in the electricity market, is deployed to optimally schedule the MG’s resources to meet demand. As such, as per three-level optimization, the independent system operator (ISO) clears the market with the goal of maximizing the social welfare in the first stage and determining the hourly market price as well as players’ credited power. In the second-level optimization process, the players select the optimal coefficient supply function equilibrium according to the power granted from the market. In third-level optimization, an optimal scheduling for MGs’ resources and demand would be obtained according to the won power in the market to maximize the aggregator profit. In addition, a price-taker MGA is simulated for comparison with the price–maker MGA to highlight the advantage of the proposed technique. Furthermore, a bidding strategy based on game theory is proposed to obtain the optimal price and power of the oligopoly market players and maximize all players’ profits. Finally, a test system including three generators is created to evaluate the performance of the devised bidding strategy. The results show that the proposed bidding strategy can optimally calculate the focal point of the Nash equilibrium (NE) in the oligopoly electricity market.
... The integrated model presented here complements the framework proposed by [8] and determines 1) power and reserve schedules of energy resources and, simultaneously, decides on not only 2) the quantities that are traded, but also 3) the price at which they are offered to the market, while 4) also taking into account bid acceptance and rejection. These four components are highly interdependent. ...
... In addition, many practical relevant constraints on three levels-the plant level, the grid level, and the market levelare considered. The modular structure of [8] is preserved by clearly separating physical power balances from economically tradeable energy and reserve quantities. Thus, the framework can be readily extended further, e.g., to more generic aggregators with other types of controllable power plants, such as conventional power plants or dispatchable loads. ...
... • to extend the integrated mixed-integer linear scheduling and bidding framework of [8] to provide the hitherto missing but important guarantee for pay-as-bid remuneration to uphold the non-anticipativity property. The provided guarantee then allows for ...
Participants in energy auction markets must simultaneously decide on the quantity and price of their offers while attempting to ensure bid acceptance through the market clearing process. To participate profitably in these markets, energy resource aggregators require accurate knowledge about the costs and quantity of electricity that can be produced and consumed, i.e., knowledge about scheduling decisions. When scheduling and bidding decisions are combined, complex interdependencies arise. Integrated scheduling and bidding with pay-as-bid remuneration has long been treated as a non-linear problem. However, by sampling stochastic parameter distributions, a practical and useful mixed-integer linear programming formulation for the integrated problem can be derived. Within the sampled distribution, the integrated model produces a priori optimal expected profit decisions. Recently, such a linear formulation for a price-taker with pay-as-bid remuneration was proposed for the first time. Our contribution completes that formulation to guarantee non-anticipativity, standardizes it with uniform remuneration, and extends it to integrated scheduling and bidding of energy and control reserve. An additional focus is placed on reserve provision through storage plants. A novel reserve scheduling model with reserve availability guarantees is introduced. This allows for leveraging the full flexibility of storage plants and, simultaneously, reveals information about the direction of reserve power flows. The integrated model is validated in a case study based on real-world data by investigating trade-offs for scheduling and bidding decisions through analysis of the Pareto front of three competing objectives. The results show that a large proportion of bids are placed at price levels with high acceptance probability, which reduces risk. Decreasing storage utilization rates increase risk and reduce solution robustness because of lost flexibility, and result in increased reserve market participation.
... In addition to its application in power system optimization, the IGDT has also made a beneficial attempt in power trading. Mazzi [29] assumed that the market members are the recipients of the market price and then declared the price based on the marginal operating cost of the unit; Y. Shen [30] and Li [31] et al. determined the declared power based on the operating arrangement of the unit on this basis. Zhao [32] et al. conducted a robust model based on IGDT for power allocation in the spot power market, which provides a risk-averse tool for power generation companies to make power allocation decisions with different expected returns. ...
Full-text available
Distributed energy resources aggregators (DERAs) are permitted to participate in regional wholesale markets in many counties. At present, new market players such as aggregators participate in China’s power market transactions. However, studies related to market trading strategy have mostly focused on centralized wind power and PV generation units. Few studies have been conducted on the decision-making strategies for DERAs in China’s power market. This paper proposes an auxiliary decision-making model for distributed energy systems to participate in the day-ahead market with more reasonable trading strategies. Firstly, the Gaussian mixture model (GMM) is used to deal with the uncertainties of wind power and photovoltaic (PV) output in the distributed energy system. Secondly, the information gap decision theory (IGDT) is used to deal with the uncertainty of price fluctuations in the spot electricity market. Thirdly, according to the different risk preferences of the DERAs facing market price fluctuation, the robust decision model and opportunity decision-making model in the day-ahead market are constructed, respectively. Finally, to deal with the irrational behavior of the DERAs’ perception of “gain” and “loss” with market risks in China’s two-tier market environment, the prospect theory and the marine predator’s algorithm (MPA) are employed to obtain a day-ahead trading decision scheme for DERA. The analyses show that RDES with robust preference can withstand greater price volatility in the day-ahead market; they will reduce the bidding expectations and increase the system operating cost to improve the achievability of the expected revenue. However, DERAs under the opportunity strategy is more inclined to sell electricity to the market and offset system operating costs with revenue. The proposed model can provide strategic reference for DERAs with different risk preferences to bid in day-ahead market and can improve the level of aggregators’ participation in electricity trading.
... LSE needs to optimize the bidding strategy in WEM and the IBDR strategy in REM. In WEM, LSE buys electricity through bidding, and the bidding strategy π b (t) can be expressed by a monotonically increasing function [28]: (8).where L t is the electricity that LSE expects to purchase at price p t in time slot t, α t determines the highest purchase price of LSE, and β t ≤ 0 determines the trend of the bidding curve. ...
An incremental incentive mechanism considering consumer differences is proposed. • Excessive consumer surplus is avoided through changes of incremental incentive. • Highly flexible consumers can obtain higher revenue through the redistribution of incentive. • A model-free approach is proposed to solve the asynchronous optimization problem. A R T I C L E I N F O Keywords: Incentive-based demand response Incremental incentive mechanism Consumer surplus Deep reinforcement learning A B S T R A C T Demand response has been proven to be an effective way to improve energy utilization efficiency. It is notable that the diversified characteristics of residential consumers, which many greatly affect its performance in demand response, have not been fully considered in existing incentive mechanisms. In this paper, an incremental incentive mechanism for incentive-based demand response (IBDR) is proposed, in which consumers obtain different incentives according to the increment of response, so that the incentive can follow the change of consumers' marginal cost. We theoretically illustrate that the proposed incremental incentive mechanism can effectively improve the profit of load service entity (LSE), as well as the benefit of highly flexible consumers, compared with other existing incentive mechanism. In practice, LSE's bidding strategy in the day ahead market is affected by the intraday IBDR strategy that cannot be known in advance. In order to solve the bidding problem with incomplete information in the day ahead market, we propose an asynchronous double-interaction deep reinforcement learning (DRL) algorithm to maximize LSE's cumulative profit of multiple time slots throughout the day. Numerical simulation results show that the proposed mechanism can improve the consumers' response depth while reducing the unit incentive cost, and the proposed DRL algorithm has relatively stable and satisfactory performance even in highly uncertain environment.
In pay-as-bid peer-to-peer (P2P) energy trading, various types of prosumers and consumers can participate, regardless of their offers. Thus, various types of participants impact the network differently. However, very few pay-as-bid P2P energy trading studies have specifically discussed appropriate compensation for network usage, although the market is implemented in existing utility-owned grids. Therefore, to improve the performance of pay-as-bid P2P energy trading, it is important to determine the appropriate compensation to utilities for network usage. This study aims to obtain an appropriate network cost allocation method for pay-as-bid P2P energy trading. Hence, the authors present a review of pay-as-bid P2P market mechanisms and various network cost allocation (NCA) methods. Additionally, a comprehensive evaluation framework is proposed to determine the most appropriate NCA method for the pay-as-bid P2P energy trading system. A comparison was made between various NCA methods to investigate the outcomes of the implementation of different NCA methods to various market conditions. The study constructs a case study based on the operator-oriented P2P model to represent the pay-as-bid P2P energy trading system. The simulation of pay-as-bid P2P energy trading with large participant number is applied in the IEEE 69-bus distribution system. The study concluded that applying the appropriate NCA method would improve the performance of pay-as-bid P2P energy trading operation.
This paper addresses the question of how much to bid to maximize the profit when trading in two electricity markets: the hourly Day-Ahead Auction and the quarter-hourly Intraday Auction. For optimal coordinated bidding many price scenarios are examined, the own non-linear market impact is estimated by considering empirical supply and demand curves, and a number of trading strategies is used. Additionally, we provide theoretical results for risk neutral agents. The application study is conducted using the German market data, but the presented methods can be easily utilized with other two consecutive auctions. This paper contributes to the existing literature by evaluating the costs of electricity trading, i.e. the price impact and the transaction costs. The empirical results for the German EPEX market show that it is far more profitable to minimize the price impact rather than maximize the arbitrage.
Italy promoted one of the most generous renewable support schemes worldwide which resulted in a high increase of solar power generation. We analyze the Italian day-ahead wholesale electricity market, finding empirical evidence of the merit-order effect. Over the period 2005–2013 an increase of 1 GWh in the hourly average of daily production from solar and wind sources has, on average, reduced wholesale electricity prices by respectively 2.3€/MWh and 4.2€/MWh and has amplified their volatility. The impact on prices has decreased over time in correspondence with the increase in solar and wind electricity production. We estimate that, over the period 2009–2013, solar production has generated higher monetary savings than wind production, mainly because the former is more prominent than the latter. However, in the solar case, monetary savings are not sufficient to compensate the cost of the related supporting schemes which are entirely internalized within end-user tariffs, causing a reduction of the consumer surplus, while the opposite occurs in the case of wind.
Owing to the interaction between energy and spinning reserve markets, designing proper bid functions and offering optimal prices to these markets is economically a challenging task from generation companies (GenCos) point of view, especially in the pay-as-bid pricing mechanism. A previously presented only-energy bidding method is generalised in order to model and solve a multimarket bidding problem. Considering a joint probability distribution function for energy and spinning reserve prices, the bidding problem is formulated and solved as a classic optimisation problem. The results show that the contribution of GenCos in each market strongly depends on their production costs, GenCo's risk-aversion degree and the mean values of market prices.
This paper presents a profit-maximizing offering strategy for a price taker power producer participating in the day-ahead spot market as organized under the Power Exchange model. The offering strategy maximizes profit by mitigating profit risk due to price forecast uncertainty leading to an imbalance arising from being accepted by the market operator (MO) to produce an infeasible dispatch schedule. The strategy makes use of a price forecast confidence interval to build hourly offer curves. The offering strategy is applied to realistic price scenarios as obtained from the Iberian MO. The performance is compared to alternative offering strategies using the agent-based OPTIMATE simulator. Results indicate that the offering strategy increases profit, reduces profit risk and reduces balancing costs arising from differences between the accepted and actual dispatch schedule. Consequently, the presented strategy outperforms the literature benchmark strategy in 83% of the cases.
Scenarios of short-term wind power generation are becoming increasingly popular as input to multi-stage decision-making problems e.g.multivariate stochastic optimization and stochastic programming. The quality of these scenarios is intuitively expected to substantially impact the benefits from their use in decision-making. So far however, their verification is almost always focused on their marginal distributions for each individual lead time only, thus overlooking their temporal interdependence structure. The shortcomings of such an approach are discussed. Multivariate verification tools, as well as diagnostic approaches based on event-based verification are then presented. Their application to the evaluation of various sets of scenarios of short-term wind power generation demonstrates them as valuable discrimination tools.
In this paper a novel approach for simultaneous bidding in day-ahead auctions for spot energy and power systems reserve is presented. For the spot market a relatively simple method is considered as a competitive market is assumed. For the reserve market the bidder is considered to behave strategically and the behavior of the competitors is summarized in a joint probability distribution of the market price. This results in a method for simultaneous bidding, where the bidding prices and capacities on the spot and reserve markets are calculated by maximizing a stochastic non-linear objective function of expected profit.
Short-term (up to 2–3 days ahead) probabilistic forecasts of wind power provide forecast users with highly valuable information on the uncertainty of expected wind generation. Whatever the type of these probabilistic forecasts, they are produced on a per horizon basis, and hence do not inform on the development of the forecast uncertainty through forecast series. However, this additional information may be paramount for a large class of time-dependent and multistage decision-making problems, e.g. optimal operation of combined wind-storage systems or multiple-market trading with different gate closures. This issue is addressed here by describing a method that permits the generation of statistical scenarios of short-term wind generation that accounts for both the interdependence structure of prediction errors and the predictive distributions of wind power production. The method is based on the conversion of series of prediction errors to a multivariate Gaussian random variable, the interdependence structure of which can then be summarized by a unique covariance matrix. Such matrix is recursively estimated in order to accommodate long-term variations in the prediction error characteristics. The quality and interest of the methodology are demonstrated with an application to the test case of a multi-MW wind farm over a period of more than 2 years. Copyright © 2008 John Wiley & Sons, Ltd.
The German feed-in support of electricity generation from renewable energy sources has led to high growth rates of the supported technologies. Critics state that the costs for consumers are too high. An important aspect to be considered in the discussion is the price effect created by renewable electricity generation. This paper seeks to analyse the impact of privileged renewable electricity generation on the electricity market in Germany. The central aspect to be analysed is the impact of renewable electricity generation on spot market prices. The results generated by an agent-based simulation platform indicate that the financial volume of the price reduction is considerable. In the short run, this gives rise to a distributional effect which creates savings for the demand side by reducing generator profits. In the case of the year 2006, the volume of the merit-order effect exceeds the volume of the net support payments for renewable electricity generation which have to be paid by consumers.