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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.