Electrical Power and Energy Systems 48 (2013) 83–92
Contents lists available at SciVerse ScienceDirect
Electrical Power and Energy Systems
journal homepage: www.elsevier.com/locate/ijepes
Risk-based optimal bidding strategy of generation company in day-ahead
electricity market using information gap decision theory
Sayyad Nojavan, Kazem Zare ⇑
University of Tabriz, P.O. Box 51666-15813, Tabriz, Iran
a r t i c l e
i n f o
Article history:
Received 9 August 2012
Received in revised form 11 November 2012
Accepted 25 November 2012
Available online 11 January 2013
Keywords:
Optimal bidding strategy
Information gap decision theory
Risk management
a b s t r a c t
This paper considers a price-taker generation company to participate in day-ahead electricity energy
market. While making optimal bidding strategy for producer, factors such as the characteristics of generator and the market price uncertainty need to be considered because of having direct impact on the
expected profit and bidding curve. The market price considered an uncertain variable and it is assumed
that the generation company forecasted the market prices. In this study, the uncertainty model of market
price is considered based on the concept of weighted average squared error using a variance–covariance
matrix. Information gap decision theory is used to develop the bidding strategy of a generation company.
It assesses the robustness/opportunity of optimal bidding strategy in the face of the market price uncertainty while producer considers whether a decision risk-averse or risk-taking. It is shown that risk-averse
or risk-taking decisions might affect the expected profit and bidding curve to day-ahead electricity market. A case study is used to illustrate the proposed approach.
2012 Elsevier Ltd. All rights reserved.
1. Introduction
In a day-ahead energy auction, generation companies submit
supply bids and energy service companies submit hourly demand
offers for the next day to market operator. Supply bids and demand
offers can be submitted in the form of a set of price-quantity
($/MW h MW) pairs. The market operator processes the supply
bids and the demand offers and computes the price that clears
the market as well as the trading volume.
The complexity of the problem that a generation company faces
in preparing its optimal bidding strategy arises from the fact that it
must take a decision based on imperfect information on the market
prices. In imperfect electricity market, a generation company could
develop bidding strategies to maximize its own expected profits. A
generation company has to make a decision based on limited information. For example, a generation company does not know the
actual system market clearing price (MCP) beforehand since it
depends on bidding behavior of other participants in the market.
Thus, an optimal bidding strategy is a challenging task for generation companies. This uncertainty implies that the market price is
unknown for generation companies. Therefore, generation companies should forecast the market prices.
Information gap modeling is a stark theory of uncertainty, motivated by severe lack of information. It does, however, have its own
⇑ Corresponding author. Tel./fax: +98 411 3300829.
E-mail addresses: [email protected] (S. Nojavan), kazem.zare@
tabrizu.ac.ir (K. Zare).
0142-0615/$ - see front matter 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ijepes.2012.11.028
particular subtlety. It is facile enough to express the idea that
uncertainty may be either pernicious or propitious. That is, uncertain variations may be either adverse or favorable. Adversity is the
possibility of failure while favorability is the possibility of sweeping success [1].
In this paper, the optimal bidding strategy problem affected by
the uncertainty of market price is formulated and solved using the
information gap decision theory (IGDT). IGDT modeling is put forth
as a basic approach for enhancing decision making under uncertainty, especially when there is a high level of uncertainty and little
information is available.
1.1. Literature review of optimal bidding strategy
The optimal bidding strategy problem optimization models
include many mathematical programming methods. As reviewed
in [2], a method for simultaneous bidding is proposed, where the
bidding prices and quantities on the spot and reserve markets
are calculated by maximizing a stochastic nonlinear objective function of expected profit. A new method to obtain the equilibrium
points for reliability and price bidding strategy of units is proposed
in [3] when the unit reliability is considered in the scheduling
problem. In [4], the linear supply function model is applied so as
to find the supply function equilibrium analytically. It also proposed a new and efficient approach to find SFEs for the network
constrained electricity markets by finding the best slope of the
supply function with the help of changing the intercept. A new
method that uses the combination of particle swarm optimization
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S. Nojavan, K. Zare / Electrical Power and Energy Systems 48 (2013) 83–92
Nomenclature
Indices
t
Vectors
e
P
k
k
time (h) index
vector of market price errors for day-ahead electricity
market, e = [e1, . . . ,eT] ($/MW h)
vector of hourly produced power for day-ahead electricity market, P = [P1, . . . ,PT] (MW)
vector of market price bids for day-ahead electricity
market, k = [k1, . . . ,kT] ($/MW h)
vector of market price forecasts for day-ahead electricity
market,
k ¼ ½k1 ; . . . ; kT ($/MW h)
Constants
a
no-load cost coefficient of generation facility ($)
b
linear cost coefficient of generation facility ($/MW h)
c
quadratic cost coefficient of generation facility ($/
(MW)2 h)
Cstartup
constant startup cost of generation facility ($)
Cshutdown constant shutdown cost of generation facility ($)
Fk
profit target for the robustness function ($)
Fw
profit target for the opportunity function ($)
(PSO) and simulated annealing (SA) to predict the bidding strategy
of generating companies in an electricity market is proposed in [5].
In [6], the optimal bidding strategy of wind power producer in
adjustment spot markets is addresses, in order to increase the
wind producer revenues through a stochastic optimization process
which considers the uncertainty of the random variables involved,
namely short-term wind power prediction, intra-day market price
prediction and imbalance price prediction. The self-scheduling
problem of the power company is a profit maximization problem,
which is formulated and solved as a mixed integer linear program
(MILP) in [7], considering the co-optimization of energy as well as
primary (up/down), secondary (up/down) and tertiary (spinning/
non-spinning) reserve markets in a pool-based framework similar
to the Greek day-ahead market structure. In [8], a probabilistic
Price Based Unit Commitment (PBUC) approach using Point Estimate Method (PEM) is employed to model the uncertainty of market price and generation sources for optimal bidding of a virtual
power plant (VPP) in a day-ahead electricity market. In [9], a stochastic mixed-integer linear programming model for determining
the optimal bidding strategy considering the uncertain market
prices is proposed. A methodology for profit maximizing bidding
under price uncertainty in a day-ahead is proposed in [10] and
multi-unit and pay-as-bid procurement auction for power systems
reserve is proposed and market price uncertainty is modeled using
the probability distribution. The bidding decision making problem
in electricity pay-as-bid auction is studied in [11] from a supplier’s
point of view. The market clearing price (MCP) is considered as a
continuous random variable with a known probability distribution
function (PDF), and an analytic solution is proposed. In [12], a stochastic multi-objective model for self-scheduling of a power producer is presented and the price forecasting inaccuracies are
modeled as scenarios tree using a combined fuzzy c-mean/
Monte-Carlo Simulation (FCM/MCS) method. An evolutionary
imperfect information game approach to analyze the bidding strategy in electricity markets with price-elastic demand is proposed in
[13]. A new method for determination of the optimal bidding strategies among generating companies in the electricity markets using
agent-based approach and numerical sensitivity analysis is presented in [14]. An optimal risky bidding strategy for a generating
company by self-organizing hierarchical particle swarm optimiza-
PG,max
PG,min
Rdown
Rup
T
Tup
Tdown
kt
maximum power output of generation unit (MW)
minimum power output of generation unit (MW)
ramping down (shutdown) limit of generation unit
(MW/h)
ramping up (startup) limit of generation unit (MW/h)
number of time periods (h)
minimum up time (h)
minimum down time (h)
forecasting the market price at time t ($/MW h)
Variables
a
Pt
Ut
kt
X on
t
X off
t
uncertainty variable
produced power from generation unit at time t (MW)
binary variable, which is equal to 1 if the generation unit
is selected at time t; otherwise, it is 0
market price at time t ($/MW h)
time duration for which the unit is ON at time t (h)
time duration for which the unit is OFF at time t (h)
Functions
a^ ðP; F k Þ ‘‘robustness’’ function
^
bðP;
F w Þ ‘‘opportunity’’ function
tion with time-varying acceleration coefficients (SPSO–TVACs) is
discussed in [15]. In [16], a new genetic algorithm is proposed
for determining the optimal bidding strategy of generating company in the day-ahead electricity market. This approach is developed from viewpoint of a generating company that participates
in a market and wish to maximize its own profit. In [17], a
mixed-integer nonlinear programming approach is considered to
enable optimal hydro scheduling for the short-term time horizon,
including the effect of head on power production, start-up costs related to the units, multiple regions of operation, and constraints on
discharge variation. In [18], a new stochastic framework for clearing of day-ahead reactive power market is presented. The uncertainty of generating units in the form of system contingencies
are considered in the reactive power market-clearing procedure
by the stochastic model in two steps. The Monte-Carlo Simulation
(MCS) is first used to generate random scenarios. Then, in the second step, the stochastic market-clearing procedure is implemented
as a series of deterministic optimization problems (scenarios)
including non-contingent scenario and different post-contingency
states. In [19], the coordinated interaction between units’ output
and electricity market prices, the benefit/risk/emission comprehensive generation optimization model with objectives of maximal
profit and minimal bidding risk and emissions is established.
1.2. Literature review of market price modeling
In the technical literature, several techniques to forecast shortterm electricity prices have been reported, namely hard and soft
computing techniques.
The hard computing techniques include autoregressive integrated moving average (ARIMA) [20], wavelet-ARIMA [21], time
series model [22], and mixed model [23] approaches. The soft computing techniques include neural networks (NNs) [24], fuzzy neural networks (FNNs) [25], weighted nearest neighbors (WNNs)
[26], adaptive wavelet neural network (AWNN) [27], hybrid intelligent system (HIS) [28], neuro-fuzzy models [29], and cascaded
neuro-evolutionary algorithm (CNEA) [30]. A combination of neural networks with wavelet transform (NNWT) has also been recently proposed [31], presenting a good trade-off between
forecasting accuracy and computation time.
S. Nojavan, K. Zare / Electrical Power and Energy Systems 48 (2013) 83–92
IGDT, which makes minor assumptions on the structure of
uncertainty, is developed in [1] as an alternative approach for decision making under uncertainty. The IGDT models do not use the
measure functions such as probabilistic density, or fuzzy membership functions. In other words, it models the error between the actual and the forecasted variable. In this paper, IGDT is used as a tool
to model the uncertainty of market price of optimal bidding strategy for a generating company.
In the current work, IGDT approach addressed the question of
how much uncertainty of market price can affect optimal bidding
strategy of producers. This approach assesses the robustness/
opportunity of optimal bidding strategy in the face of uncertainty
of market price while the producer is considering whether a decision risk-averse or risk-taking. The issue which is addressed here
is to determine an optimal bidding strategy for a producer considering one day planning horizon. The risk associated with this issue
is modeled based on the concepts derived from IGDT approach. For
this purpose, a risk-averse model is proposed to find robust strategies through maximizing the robustness function of the optimal
bidding strategy against the immunity to failure due to low forecasted market prices or profits lower than the expected profit. Furthermore, a risk-taking model is proposed to obtain the benefits of
high market prices through minimizing the opportunity function
of optimal bidding strategy against the immunity to windfall
due to high forecasted market prices or profits higher than the expected profit. It should be noted that IGDT is used to find the optimal procurement strategy of large consumers in [32–35].
The remainder of the paper is organized as follows. The next
section describes the expected profit of a generation facility and
the problem formulation of the risk-based optimal bidding strategy
using IGDT approach. As mentioned, the solution methodology for
price-taker thermal unit is highlighted in this paper. For test results, a numerical example is used to illustrate the bidding approach. Afterwards, the paper is concluded. Finally, an Appendix
provides a background on IGDT.
85
The market price is determined based on the intersection of bidding curves with the required demand. Also it is considered that
the payment will be based on market clearing price (MCP). So,
the generation company must forecast and use it as input variable
to his optimal bidding strategy problem. In this case, the market
uncertainty is fully described by the market price uncertainty,
which affects the bidding strategy of the generation company.
Some backgrounds about price forecasting methods are addressed
in Section 1.2.
The market prices are indicated by
k ¼ ½k1 ; . . . ; kT :
ð1Þ
Since hourly prices are random variables, they can be described
as
kt ¼ kt þ et
8t ¼ 1; . . . ; T
ð2Þ
where et,"t, depicts the error or difference between the actual and
forecasted prices at time t. These errors can be organized as
e ¼ ½e1 ; . . . ; eT :
ð3Þ
2.3. Decision variable
The decision variables in bidding strategy contain the pricequantity pairs. The price will be defined based on (2) after optimizing the objective functions. The quantity is the amount of power
which should be produced from thermal unit at different times.
This variable can be defined as:
P ¼ ½P1 ; . . . ; P T :
ð4Þ
where Pt;"t = 1 . . . T shows the output power at time t. It should be
clarified that the vectors mentioned in (3) and (4) will be determined based on optimization functions based on IGDT.
2.4. Profit function
2. Problem formulation
In this section, the profit function formulation of a generation
company subject to its related constraints is modeled and the
robustness and opportunity functions based on IGDT are derived.
The profit function of a unit for a period of T hours is calculated
by the following formula:
FðP; kÞ ¼ P k0 T
X
fC t ðPt Þ þ C startup U t ð1 U t1 Þ þ C shutdown
t¼1
2.1. Assumptions
In this paper, the following assumptions are considered.
(1) The generating company has a thermal unit and participates
in day-ahead energy market.
(2) The unit participates with its entire available capacity in
day-ahead energy market.
(3) The unit acts as a price-taker, believing that its energy bid
cannot influence the market price.
(4) The unit has a forecasted market price.
(5) The unit has a deterministic estimate of variance–covariance
matrix between hourly market prices.
(6) The unit has perfect information about its technical
constraints.
(7) The price-energy curve is a non-decreasing function.
(8) Minimum up time and minimum down time constraints and
start-up and shutdown costs are considered.
2.2. Market price modeling
In this paper, the generation company acts as a price-taker,
believing that his energy bid cannot influence the market price.
ð1 U t Þ U t1 g
ð5Þ
where superscript ‘‘0 ’’ indicates the transpose function.
The profit is defined as the revenue from the sale of energy
minus the production costs and start up/shutdown cost of the unit.
The operation cost of the available generation thermal unit can be
represented as follows:
C t ðPt Þ ¼ a þ bP t þ cðP t Þ2
ð6Þ
The operating constraints of the generation unit include the following case:
PG;min U t 6 Pt 6 PG;max U t ; 8t ¼ 1; . . . ; T
ð7Þ
Pt Pt1 6 Rup U t ; 8t ¼ 1; . . . ; T
ð8Þ
Pt1 P t 6 Rdown U t1 ; 8t ¼ 1; . . . ; T
on
X t1 T up ½U t1 U t P 0; 8t ¼ 1; . . . ; T
h
i
down
X off
½U t U t1 P 0; 8t ¼ 1; . . . ; T
t1 T
ð9Þ
ð10Þ
ð11Þ
The minimum and maximum outputs of the unit are represented
using (7). Eqs. (8) and (9) describe the ramping rate limits of generation unit. Finally, the minimum up/down time constraints are expressed as (10) and (11), respectively.
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S. Nojavan, K. Zare / Electrical Power and Energy Systems 48 (2013) 83–92
2.5. Uncertainty model
Combination with (17) and assuming W0 = W yields
For a price-taker generation company, the market price is an
uncertain variable. Thus, the uncertainty model of market price is
necessary and needs to be considered since it has direct impact
on the expected profit and optimal bidding strategy. In this part,
a model is suggested for representing the uncertainty of market
prices. The related robustness and opportunity functions will be
derived based on this model.
Here, the market price of electricity is uncertain for which the
following Weighted Average Squared Error (WASE) information
gap model is considered:
a2 ¼
Uða; kÞ ¼ fk : eW 1 e0 6 a2 g; a P 0
ð12Þ
where a and W represent uncertainty variable and variance–covariance matrix between hourly market prices, respectively. Note that
the variance–covariance matrix W can be estimated using historical
data. This model states that the WASE of price forecasting is not
greater than a2. It should be emphasized that WASE allows for
the convenient expression of the deviations of actual price from
the expected values, which in turn allows for making robust and/
or beneficial decisions concerning these deviations.
1
PWP0
4l 2
ð18Þ
And
1
a
¼ pffiffiffiffiffiffiffiffiffiffiffi0ffi
2l
PWP
ð19Þ
Substituting (19) in (17) results in:
PW
e ¼ a pffiffiffiffiffiffiffiffiffiffiffi0ffi
PWP
ð20Þ
Thus, the lowest profit [for worst-case error the ‘‘’’ sign is selected
in (20)] is
T
pffiffiffiffiffiffiffiffiffiffiffiffi X
min FðP; aÞ ¼ P ðkÞ0 a PWP 0 fC t ðPt Þ þ C startup t¼1
U t ð1 U t1 Þ þ C shutdown ð1 U t Þ U t1 g
Using (21), a can be computed for a given profit target FK as
P ðkÞ0 T
X
fC t ðP t Þ þ C startup U t ð1 U t1 Þ þ C shutdown ð1 U t Þ U t1 g F K
t¼1
aðP; F K Þ ¼
pffiffiffiffiffiffiffiffiffiffiffi0ffi
PWP
2.6. Deriving robustness function
ð22Þ
^ ðP; F k Þ is related to the profit lower than the maxThe function a
imum profit and works as a risk-averse mechanism. In other
^ ðP; F k Þ measures the protection level of the decision
words, a
against experiencing low profits. Thus, a low value of this function,
which corresponds to a low profit target, Fk means that the associated decision is highly robust against low market prices. It is
^ ðP; F k Þ increases with the decrease of Fk. Note that
expected that a
this condition represents the worst profit for the producer.
^ ðP; F k Þ is related to low
As mentioned, the robustness function a
market prices and represents the largest value of the uncertainty
variable in a way that the minimum value of the electricity selling
^ ðP; F k Þ, could be
profit is greater than a desired profit target F k a
computed according to the procedure described as follows.
Inserting (2) in (5) to find the minimum value of the profit
yields
MinFðP; eÞ ¼ P ððkÞ0 þ e0 Þ e
T
X
fC t ðP t Þ þ C startup U t ð1 U t1 Þ þ C shutdown ð1 U t Þ
t¼1
U t1 g
ð13Þ
Subject to : eW 1 e= 6 a2
ð14Þ
where a2 is a variable bounding the value of the WASE. Eqs. (13)
and (14) can be rewritten as follows:
MinFðP; eÞ ¼ P ðkÞ0 e
T
X
fC t ðPt Þ þ C startup U t ð1 U t1 Þ
ð15Þ
Since the optimization problem is convex, the first-order optimality condition for the associated Lagrangian problem (15) can
be stated as
ð16Þ
where l is the Lagrangian multiplier of the constraint (14).
Performing the derivations of (16) results in
e0 ¼
1
WP0 and a2 ¼ eW 1 e0 :
2l
a^ ðP; F K Þ ¼ maxaðP; F K Þ
P;U
P ðkÞ0 T
X
fC t ðP t Þ þ C startup U t ð1 U t1 Þ þ C shutdown ð1 U t Þ U t1 g F K
t¼1
¼ max
P;U
pffiffiffiffiffiffiffiffiffiffiffi0ffi
PWP
ð23Þ
Expression (23) represents the robustness function which
should be maximized subject to the constraints (6)–(11).
2.7. Deriving opportunity function
^
The opportunity function bðP;
F w Þ is related to high market
prices or high electricity selling profit and evaluates the possibility
of high profit from high prices. In other words, this is the immunity
^
against windfall ‘‘benefit’’; thus, a small value of bðP;
F w Þ is desirable. The opportunity function is the smallest value of a (best case)
so that the electricity selling profit can be possibly as high as Fw.
Note that this condition represents the best profit for the producer.
The corresponding mathematical formulation can be represented
by
ð24Þ
P;U
e
re;l fP e0 þ lða2 eW 1 e0 Þg ¼ 0
Note that FK represents the profit target which the producer is willing to face. Then, the robustness function and, therefore, a robust
bidding strategy (risk-averse model) can be derived by providing
the largest possible a (worst case) for a given profit target FK, as
shown in (23).
^
bðP;
F w Þ ¼ minaðP; F w Þ
t¼1
þ C shutdown ð1 U t Þ U t1 g þ minfP e0
: eW 1 e0 6 a2 g:
ð21Þ
ð17Þ
where Fw is generally bigger than Fk. The highest profit can be defined using the ‘‘+’’ sign in (21) for the best-case error as
T
pffiffiffiffiffiffiffiffiffiffiffiffi X
max FðP; aÞ ¼ P ðkÞ0 þ a PWP0 fC t ðPt Þ þ C startup U t
t¼1
ð1 U t1 Þ þ C shutdown ð1 U t Þ U t1 g
ð25Þ
Similar to the robustness function using (25) for a given profit
target Fw, the a can be computed. Note that Fw represents the highest electricity selling profit with which the producer is willingly to
be faced. Then, using (25), the opportunity function can be derived
for a given profit target Fw, as shown in (26).
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S. Nojavan, K. Zare / Electrical Power and Energy Systems 48 (2013) 83–92
Table 1
Computational size of the problem.
Table 2
Data of the generation thermal unit and technical constraint.
Number of binary variables
Number of real variables
Number of constraints
T
T
6T
^ F W Þ ¼ minaðP; F W Þ
bðP;
Capacity
Maximum capacity
100 (MW)
P
PG,min
a
b
c
Maximum output
Minimum output
Fixed cost
Linear cost coefficient generation
Quadratic cost coefficient generation
Rup
Rdown
Tup
Tdown
Cstartup
Cshutdown
U0
X on
0
Ramp up limit
Ramp down limit
Minimum up time
Minimum down time
Start up cost
Shutdown cost
Initial value of state of thermal unit
Time duration for which unit has been ON at
time 0
Time duration for which unit has been OFF at
time 0
100 (MW)
20 (MW)
100 ($)
80 ($/MW h)
0.1 ($/
(MW)2 h)
50 (MW/h)
50 (MW/h)
3 (h)
2 (h)
100 ($)
100 ($)
0
0 (h)
G,max
P;U
F W P ðkÞ0 þ
¼ min
T
X
fC t ðPt Þ þ C startup U t ð1 U t1 Þ þ C shutdown ð1 U t Þ U t1 g
t¼1
P;U
pffiffiffiffiffiffiffiffiffiffiffiffi0
PWP
ð26Þ
The constraints are the same as those of the robustness function
^
which are represented in (6)–(11). It is expected that bðP;
F w Þ increases with the increase of Fw.
The robustness and opportunity functions are mixed-integer
nonlinear programming problems which can be solved using
SBB/CONOPT [36] under GAMS [37]. Table 1 provides the size of
each problem, which is expressed as the number of binary variables, real variables and constraints.
3. Solution methodology
The optimal bidding strategy problem is formulated as a mixedinteger nonlinear problem and can be solved using SBB/CONOPT
[36] under GAMS [37]. The procedure of simulation and result derivation is as follows.
(1) The profit maximization problem is simulated based on (5)
and constraints (6)–(11) by considering the forecasted market price data.
(2) For some values of Fk with a fixed step, which is less than the
maximum profit, the robustness function (23) with constraints (6)–(11) is simulated and the results represented
the robustness level.
(3) For some values of Fw with a fixed step, which is higher than
the maximum profit, the opportunity function (26) with
constraints (6)–(11) is simulated and the results are derived.
^ ðP; F K Þ
(4) While for some values of Fk and Fw, values of P, U and a
^
and bðP;
F w Þ are derived, the day-ahead prices are calculated
^
^ ðP; F K Þ and bðP;
using (20) and (2) by instituting the a
FwÞ
instead of a, respectively. Then, the output power and dayahead price pairs are used to build the step-wise bidding
strategy.
4. Case study
In this section, numerical simulations are conducted to illustrate the work of the proposed method. The thermal unit producer
participated in a day-ahead market; thus, the time horizon of this
study contains 24 periods (1 day).
4.1. Data
The data of the generation thermal unit and technical constraint
are provided in Table 2. The variance–covariance matrix for the
day-ahead market is provided in Table 3. The prices of working
days in the Iberian market from January 2008 to October 2008
are used to construct the variance–covariance matrices using
appropriate MATLAB functions. The price profile of one day of Iberian market is considered as the forecasted market price profile for
the study horizon, which is depicted in Fig. 1. In these simulations,
the profit step for Fk and Fw is considered equal to $600.
The problem associated with a single instance of profit target, as
characterized by 24 binary variables, 24 real variables and 144 con-
X off
0
4 (h)
straints, is solved using an Intel (R) Celeron (R) CPU, 2-GHz and
0.99 GB of RAM computer in the average solution time of about
20 ms.
4.2. Results
The deterministic expected value problem consisting of maximizing the profit (5) subject to the constraints (6)–(11) and considering the market price forecasts is provided in Fig. 1. The maximum
profit in this case is equal to $9582.50. Note that the value of the
robustness and opportunity functions related to the maximum
^
^ ð$9582:50Þ ¼ bð$9582:50Þ
profit is equal to zero, i.e. a
¼ 0. Also,
the hourly expected profit considering deterministic price profile
is shown in Fig. 2. In some hours, the generating unit is going to
shut-down mode and so, the related profit is negative.
The results of simulations related to the robustness function are
useful if the producer decided on a risk-averse strategy and the
opportunity function-related strategy can be used when the producer chose to be in a risk-taking decision making mode.
4.3. Results of robustness function
^ ðP; F k Þ versus electricity selling profit targets Fk
The robustness a
are depicted in Fig. 3, in which it is clear that robustness increases
with the decrease Fk, as expected. If the generation company
desires risk-averse decision and higher robustness bidding strategy, less profit should be obtained and vice versa; if the generation
company obtains less profit, its decision will be more robust.
For example, if the profit target Fk is equal to $8382.5, according
^ ðP; $8382:5Þ ¼ 0:273, which means
to Fig. 3, it is clear that the a
that if the weighted square error of the forecasted hourly market
prices equal to 0.273, therefore, the profit of the generation company with this bidding strategy will not be less than $8382.5.
In the risk-averse decision, the day-ahead price errors are calcu^ ðP; F K Þ values by instituting the a
^ ðP; F K Þ instead of a in
lated using a
Eq. (20). After computing of errors, the day-ahead prices in the riskaverse decision are calculated using calculated errors and Eq. (2).
Using the calculated prices and the output power in the risk-averse
^ ðF K ÞÞ; kða
^ ðF K ÞÞÞ and FðPða
^ ðF K Þ ¼ 0Þ;
decision, Fig. 4 shows FðPða
^ ðF K ÞÞÞ curves versus profit target FK. This figure illustrates the
kða
robustness expected profit and the expected profit versus electricity
selling profit targets Fk. It is obvious that the robustness expected
profit is higher than the expected profit. The difference between
^ ðF K ÞÞ; kða
^ ðF K ÞÞÞ and FðPða
^ ðF K Þ ¼ 0Þ; kða
^ ðF K ÞÞÞ shows the extra
FðPða
profit of using robust bidding curve instead of the bidding without
considering the price uncertainty. For example, according to Fig. 3,
88
S. Nojavan, K. Zare / Electrical Power and Energy Systems 48 (2013) 83–92
Table 3
Variance–covariance matrix for the day-ahead market.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
22
17
17
15
16
14
11
13
11
14
13
12
9
6
4
1
2
5
4
8
11
7
2
1
17
17
16
15
15
14
11
13
10
12
10
9
7
4
3
1
2
4
3
8
13
11
1
1
17
16
18
17
17
16
11
13
11
13
12
11
9
5
4
2
0
3
2
8
11
11
1
2
15
15
17
21
22
19
12
13
11
12
11
11
10
6
5
4
1
1
2
1
0
4
2
0
16
15
17
22
26
21
13
14
12
12
12
12
11
7
7
5
2
0
1
0
4
2
2
0
14
14
16
19
21
20
12
13
11
11
11
11
9
6
6
4
2
1
1
2
0
3
2
0
11
11
11
12
13
12
12
13
10
11
9
9
7
5
4
3
2
0
0
4
1
1
3
1
13
13
13
13
14
13
13
23
21
23
20
19
15
11
7
6
4
3
4
15
14
9
1
1
11
10
11
11
12
11
10
21
26
27
25
23
18
14
8
7
5
4
6
20
18
8
1
2
14
12
13
12
12
11
11
23
27
38
34
30
25
20
13
11
7
6
11
36
36
14
1
6
13
10
12
11
12
11
9
20
25
34
35
32
27
22
14
12
9
7
11
33
32
12
1
6
12
9
11
11
12
11
9
19
23
30
32
32
28
23
16
14
10
8
11
29
28
8
2
6
9
7
9
10
11
9
7
15
18
25
27
28
27
23
16
15
11
9
10
23
22
5
2
5
6
4
5
6
7
6
5
11
14
20
22
23
23
22
16
14
11
10
11
23
25
5
3
7
4
3
4
5
7
6
4
7
8
13
14
16
16
16
16
14
12
11
11
17
16
3
4
6
1
1
2
4
5
4
3
6
7
11
12
14
15
14
14
16
14
13
12
14
13
3
4
5
2
2
0
1
2
2
2
4
5
7
9
10
11
11
12
14
13
13
13
11
9
2
5
5
5
4
3
1
0
1
0
3
4
6
7
8
9
10
11
13
13
15
15
14
11
3
6
5
4
3
2
2
1
1
0
4
6
11
11
11
10
11
11
12
13
15
19
27
26
10
9
9
8
8
8
1
0
2
4
15
20
36
33
29
23
23
17
14
11
14
27
80
93
42
19
26
11
13
11
0
4
0
1
14
18
36
32
28
22
25
16
13
9
11
26
93
127
65
26
34
7
11
11
4
2
3
1
9
8
14
12
8
5
5
3
3
2
3
10
42
65
95
18
17
2
1
1
2
2
2
3
1
1
1
1
2
2
3
4
4
5
6
9
19
26
18
85
13
1
1
2
0
0
0
1
1
2
6
6
6
5
7
6
5
5
5
9
26
34
17
13
18
Fig. 3. Robustness curve.
Fig. 1. Forecasted price data for the study horizon.
Fig. 2. Hourly expected profit considering deterministic price profile.
Fig. 4. The robustness expected profit and the expected profit versus profit targets
(FK).
S. Nojavan, K. Zare / Electrical Power and Energy Systems 48 (2013) 83–92
Fig. 7. Day-ahead bidding strategy for hour 7.
Fig. 5. Opportunity curve.
Fig. 6. The opportunity expected profit and the expected profit versus profit targets
(FW).
Fig. 8. Day-ahead bidding strategy for hour11.
^ ðP; F k Þ ¼ 0:896, the profit target Fk equal to $5982.50 and
for a
^ ðF K ÞÞ;
according to Fig. 4, it is clear that the robustness FðPða
^ ðF K ÞÞÞ is equal to $4693.842 and the FðPða
^ ðF K Þ ¼ 0Þ; kða
^ ðF K ÞÞÞ is
kða
equal to $4114.085. Thus, the extra profit of the robustness strategy
is equal to $4693.842 $4114.085 = $579.757.
4.4. Results of opportunity function
^
The opportunity value bðP;
F w Þ versus the electricity selling
profit targets Fw is shown in Fig. 5. This figure is relevant if the generation company decides to take a risk. According to Fig. 5, it is
clear that the opportunity value increases as Fw increases and vice
versa; if the generation company obtains high profit, its decision
will be more risk-taker. For example, if the profit target Fw is equal
^
to $13182.5, according to Fig. 5, it is clear that the bðP;
F w Þ ¼ 0:739,
which means that if the weighted square error of the forecasted
hourly market prices equal to 0.739, therefore, the profit of selling
energy will be at least equal to $13182.5.
In the risk-taking decision, the day-ahead price errors are calcu^
^
lated using bðP;
F w Þ values by instituting the bðP;
F w Þ instead of a in
Eq. (20). After computing of errors, the day-ahead prices in the risk-
Fig. 9. Day-ahead bidding strategy for hour 16.
89
90
S. Nojavan, K. Zare / Electrical Power and Energy Systems 48 (2013) 83–92
taking decision are calculated using calculated errors and Eq. (2).
Using the calculated prices and output power in the risk-taking
^ W ÞÞ; kðbðF
^ W ÞÞÞ and FðPðbðF
^ W Þ ¼ 0Þ;
decision, Fig. 6 shows FðPðbðF
^ W ÞÞÞ curves versus the profit target Fw, which illustrates the
kðbðF
opportunity expected profit and the expected profit versus electricity selling profit targets FW, respectively. It is obvious that the opportunity expected profit is higher than the expected profit. In other
words, this figure provides the extra profit of the risk-taking, which
^ W ÞÞ; kðbðF
^ W ÞÞÞ and
is equal to the difference between FðPðbðF
^ W Þ ¼ 0Þ; kðbðF
^ W ÞÞÞ. For example, according to Fig. 5, for
FðPðbðF
^
bðP;
F W Þ ¼ 1:077, the profit target FW is equal to $14982.50 and
according to Fig. 6, it is clear that the opportunity expected profit
is equal to $16543.708 and the expected profit is equal to
$15814.069. Thus, the profit of the risk is equal to
$16543.708 $15814.069 = $729.639, which can be delivered, if
the generation company use this bidding strategy.
Fig. 10. Day-ahead bidding strategy for hour 21.
Fig. 11. Calculated day ahead price profile for different profit targets.
4.5. Risk-based optimal bidding strategy curves
The resulting risk-based optimal bidding strategy curves for
hours 7, 11, 16 and 21 are reported in Figs. 7–10, respectively. The
x-axis represents the bidding price whereas the y-axis represents
the amount of the power that the generation company bids to the
day-ahead market. It can be observed that these curves are made
using the results of robustness and opportunity functions, as indicated in these figures. The results of simulations related to the
robustness function are useful if the producer decided on a riskaverse strategy and the opportunity function-related strategy can
be used when the producer choose to be in a risk-taking decision
making mode. The breaking line in Figs. 7–10 show that the right
hand of this line is achieved from the opportunity and the left hand
is achieved from the robustness function. The final number of blocks
in the bidding strategy curves is an output of the proposed
approach. Note that this number is always less than or equal to
the number of different profit targets used for solving the problem.
In this case study, 20 different profit targets are imposed. Finally,
Figs. 11 and 12 show the calculated day-ahead prices and the output
power of generation unit for three profit targets using expressions
(20) and (2) and the value a obtained from solving (23) and (26)
subject to (6)–(11), respectively.
Fig. 12. Output power profile for different profit targets.
S. Nojavan, K. Zare / Electrical Power and Energy Systems 48 (2013) 83–92
5. Conclusion
This paper a price-taker generation company considered to participate in day-ahead electricity energy market. While making
optimal bidding strategy for producer, factors such as characteristics of generator and market price uncertainty need to be considered because of having direct impact on the expected profit and
bidding curve. The market price considered an uncertain variable
and it is assumed that the generation company could forecast the
market prices. In this study, the uncertainty model of market price
is considered based on the concept of weighted average squared
error using a variance–covariance matrix. Information gap decision
theory is used to develop the optimal bidding strategy of generation companies. This paper demonstrated that risk-averse or risktaking decisions might affect expected profit and bidding curve
to day-ahead electricity energy market. A case study is used to
illustrate the proposed approach.
91
The opportunity function addresses the propitious aspect of the
uncertainty and evaluates the possibility of achieving profits. Here,
^ is the minimum value of a which can be tolerated in order to enb
able the possibility of high electricity selling profit as a result of
decisions, P. In other words, the opportunity function is the least
value of a for which the electricity selling profit can possibly be
as high as a given value Fw. This function is the immunity against
^ is desirable. A high value
windfall profit. Thus, a high value of b
^
of bðP;
F w Þ indicates a situation in which the profit is achievable
in the presence of high market prices. This function is related to
determining risk-taking optimal bidding strategy. The corresponding mathematical formulation can be represented through the following minimization problem:
^
bðP;
F w Þ ¼ minfa : maxðFðP; kÞÞ P F w g:
a
P
ðA:4Þ
where Fw is generally bigger than Fk.
Appendix A. IGDT background
A.3. Uncertainty model
Uncertainties may be harmful and lead to lower profits or may
be beneficial and lead to higher profits. IGDT addresses these two
conflicting issues using two immunity functions: robustness and
opportunity.
An IGDT decision problem is specified by three components:
objective function, performance requirements and uncertainty
model. These issues are explained as follows.
The uncertainty is represented by an information gap model.
The information gap model of uncertainty U(a, k) assumes the prior
information about the uncertain vector k. In this paper, it is assumed that the price of energy in the day-ahead market is uncertain. Also, the following weighted average squared error model is
considered:
Uða; kÞ ¼ fk ¼ k þ e : eW 1 e0 6 a2 g; a P 0:
ðA:5Þ
A.1. Objective function
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For a set of decision variables P, uncertainty variable a and
uncertain variable k, the objective function F(P, k) expresses the input/output structure of the system for which the decision is applied. In this paper, the objective function is the electricity
selling profit function with which a generation company is faced.
A.2. Performance requirements
The performance requirements which describe the requirements from the objective function can be expressed in terms of
profit function. These requirements are evaluated based on robustness and opportunity functions. The functions for the electricity
selling problem can be stated as follows:
a^ ¼ max
fa : minimum profitðleast profitÞwhich is not less
a
than a given profit targetg fRobustnessg
ðA:1Þ
^ ¼ minfa : maximum profitðbest profitÞwhich is bigger
b
a
than a given profit targetg fOpportunityg
ðA:2Þ
The robustness function addresses the harmful face of the
uncertainty and expresses the greatest level of uncertainty at
which the electricity selling profit cannot be less than a given value
Fk. In other words, this function describes the risk-aversion possibility of an optimal bidding strategy. This means that a large value
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of a
through an optimization problem as follows:
a^ ðP; F k Þ ¼ maxfa : minFðP; kÞ P F k g
a
P
ðA:3Þ
^ ðP; F k Þ, the decision will be robust, riskFor a large value of a
averse and insensitive to the uncertainties. On the other hand, if
a^ ðP; F k Þ is small, the decision will be fragile and sensitive to the
variation of uncertain variable; therefore, its related decision will
not be more consistent.
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