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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 1, FEBRUARY 2009
Day-Ahead Price Forecasting of Electricity Markets
by Mutual Information Technique and Cascaded
Neuro-Evolutionary Algorithm
Nima Amjady and Farshid Keynia
Abstract—In a competitive electricity market, price forecasts are
important for market participants. However, electricity price is a
complex signal due to its nonlinearity, nonstationarity, and time
variant behavior. In spite of much research in this area, more accurate and robust price forecast methods are still required. In this
paper, a combination of a feature selection technique and cascaded
neuro-evolutionary algorithm (CNEA) is proposed for this purpose. The feature selection method is an improved version of the
mutual information (MI) technique. The CNEA is composed of
cascaded forecasters where each forecaster consists of a neural network (NN) and an evolutionary algorithm (EA). An iterative search
procedure is also incorporated in our solution strategy to fine-tune
the adjustable parameters of both the MI technique and CNEA.
The price forecast accuracy of the proposed method is evaluated
by means of real data from the Pennsylvania-New Jersey-Maryland (PJM) and Spanish electricity markets. The method is also
compared with some of the most recent price forecast techniques.
Index Terms—Cascaded neuro-evolutionary algorithm (CNEA),
iterative search procedure, mutual information (MI), price forecast.
I. INTRODUCTION
W
ITH the introduction of restructuring into the electric
power industry, the pricing of electricity in the electricity markets has become very important [1]. Accurate day
ahead price forecast in the spot market helps power suppliers
to adjust their bidding strategies to achieve the maximum
benefit. Similarly, consumers can derive a plan to maximize
their purchased electricity from the pool, or use self production
capability to protect themselves against high prices. On a
short time scale, transmission bottlenecks may prevent a free
exchange among different regions resulting in extreme price
volatility or even price spikes in the electricity market, e.g., the
price spikes of the Pennsylvania-New Jersey-Maryland (PJM)
and California markets in 1999 and 2000, respectively [2], [3].
Besides, volatility in fuel price, load uncertainty, fluctuations
in the hydroelectricity production, generation uncertainty (outages) and behavior of market participants also contribute to
electricity price uncertainty [1].
Manuscript received January 25, 2008; revised August 26, 2008. First published December 09, 2008; current version published January 21, 2009. Paper
no. TPWRS-00027-2008.
The authors are with the Department of Electrical Engineering, Semnan University, Semnan, Iran (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPWRS.2008.2006997
In a power market, the price of electricity is the most important signal to market participants and the most basic pricing
concept is the market-clearing price (MCP) [1], [3]. Generally,
when there is no transmission congestion, MCP is the only price
for the entire system. However, when there is congestion, the
zonal market clearing price (ZMCP) or the locational marginal
price (LMP) could be employed. ZMCP may be different for
various zones, but it is the same within a zone. LMP can be
different for different buses. LMP is the sum of generation marginal cost, transmission congestion cost, and cost of marginal
losses, although the cost of losses is usually ignored [1]. When
there is no congestion, LMP is the same as MCP. When there is
congestion, transmission line constraints are considered in order
to balance supply and demand at each bus. The marginal cost of
each bus is the LMP.
The importance of electricity price forecasting on the one
hand, and its complexity on the other hand, motivates many
research works in the recent years. Stationary time series
models such as auto-regressive (AR) [4], dynamic regression
and transfer function [5], [6], auto-regressive integrated moving
average (ARIMA) [7], and nonstationary time series models
like generalized auto-regressive conditional heteroskedastic
(GARCH) [8] have been proposed for this purpose. However
most of the time series models are linear predictors, while
electricity price is generally a nonlinear function of its input
features. So, the behavior of the price signal may not be completely captured by the time series techniques [3]. To solve this
problem, some other research works proposed neural networks
(NNs) and fuzzy neural networks (FNNs) for electricity price
forecast [2], [9]–[14]. NNs and FNNs have the capability of
modeling the nonlinear input/output mapping functions. The
appropriate selection of input features is a key factor for the
success of FNN or NN based price forecast methods. Besides,
efficiency of these methods is usually dependent on the correct
tuning of their adjustable parameters, e.g., the number of hidden
nodes of the NNs. Some other price forecast methods have been
also proposed in recent years. In [15], combination of fuzzy
inference system (FIS) and least-squares estimation (LSE) has
been proposed for prediction of LMP. In [16], electricity price
series is decomposed by wavelet transform with the aim of
finding less volatile components and then each subseries is
separately forecasted by the ARIMA technique. Combination
of similar days (SDs) and NN techniques are proposed for LMP
prediction in [17]. In [18], a mixed model has been proposed for
the price forecast, where weekends and weekdays are modeled
by separate ARIMA time series techniques. Weighted nearest
0885-8950/$25.00 © 2008 IEEE
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AMJADY AND KEYNIA: DAY-AHEAD PRICE FORECASTING OF ELECTRICITY MARKETS
neighbors (WNNs) method has been presented for electricity
price forecast in [19].
In this paper a new prediction strategy is proposed for dayahead price forecasting of electricity markets. The contribution
of the paper can be summarized as follows.
1) Presentation of a new feature selection technique based on
MI method. The proposed technique can rank candidate inputs according to their information value for the prediction
of electricity price signal.
2) Presentation of a new forecast method based on CNEA.
The CNEA is composed of 24 cascaded forecasters in
which each forecaster consists of an NN and EA. In the
CNEA, the 24 hourly time series of electricity price are
separately modeled.
3) The adjustable parameters of both the feature selection
technique and CNEA are adaptively fine-tuned for each
forecast horizon by an iterative search procedure.
The paper is organized as follows. In Section II, the proposed feature selection technique is described. The CNEA is explained in Section III. In Section IV, the proposed search procedure for fine-tuning of the adjustable parameters is described. In
Section V, the proposed approach has been applied to the PJM
and Spanish electricity markets and the results are discussed.
The proposed method has also been compared with some of
the most recent price forecast techniques. A brief review of the
paper and future research are described in Section VI.
307
Based on (1) and (2), the entropy is often considered a measure
of uncertainty. As an example, let the random variable represent the presence of a disease D. If there is no uncertainty about
or disdisease presence, i.e.,
, then the
ease absence, i.e.,
equals zero. If however, there is high uncertainty
entropy
about the presence or absence of disease, i.e.,
, then the entropy
equals 1. Generally,
diseases are equally possible
,
if any one of
achieves its maximum value
. This value
then
represents the highest possible uncertainty about the random
variable .
of two discrete random variables
The joint entropy
and with a joint probability distribution
is defined
by
(3)
indicates the total entropy of random variables
where
and [21]. When certain variables are known and others are
not, the remaining uncertainty is measured by the conditional
entropy [22]
II. PROPOSED FEATURE SELECTION TECHNIQUE
Feature selection is a process commonly used in machine
learning, wherein a subset of features available from data is selected for application of a learning algorithm. The best subset
contains the least number of key features contributing to accuracy; while discarding the remaining unimportant features.
This is an important stage of pre-processing and is one way of
avoiding the curse of dimensionality. One of the most recently
developed feature selection techniques is mutual information,
which is based on the entropy concept. This technique has been
frequently used for feature selection in classification problems
such as pattern recognition, cancer classification, medical image
processing, etc. [20]–[23]. Here, a new formulation of the mutual information technique is proposed for the feature selection
of electricity price forecast.
of a continuous random variable with
The entropy
is defined as follows [21], [22]:
probability distribution
(1)
If is a discrete random variable with values
and probabilities
is defined as follows:
then
(4)
The joint entropy and the conditional entropy have the following
relation [22]:
(5)
This, known as the “chain-rule,” implies that the total entropy
of random variables
and is the entropy of
plus the remaining entropy of for a given . This matter is graphically
shown in Fig. 1 [21]. The information found commonly in two
random variables and is of importance in our work, which
between the two
is defined as the mutual information
variables
(6)
, respectively,
(2)
If the mutual information between two random variables is
large, the two variables are closely related and vice versa. If the
mutual information becomes zero, the two random variables
are totally unrelated or the two variables are independent.
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 1, FEBRUARY 2009
Fig. 1. Representation of mutual information and different entropies.
The mutual information and the entropy have the following
relations, as shown in Fig. 1:
(7)
(8)
(9)
(10)
(11)
measures the a priori uncertainty of the random variable
measures the conditional a posteriori uncertainty
of after is observed [22]. According to (7), the mutual information
measures how much the uncertainty of
is reduced if
has been observed. If
and
are indepenbased on (9). Consedent, then
quently, their mutual information is zero, i.e., observing does
not reduce the uncertainty of . If, however, the two random
and becomes completely related, which occurs
variables
, then according to (11), mutual information bewhen
.
tween these two variables reaches to its maximum value
.
becomes known
Now, suppose that the random variable
resulting in its uncertainty becoming negligible. If the two variand are tightly related then the mutual information
ables
is high and vice versa. So by observing , the unsignificantly (insignificantly)
certainty of random variable
decreases. In the price forecast process, we assume that the
(like the lagged
candidate set of input features
values of the price, load demand or available generation) are
owning more mutual informaknown. The candidate input
with the target variable
is a better candition
the uncertainty
date as input feature, since by employing
of
reduces more than using the other candidate inputs. It is
noted that for the feature selection of electricity price forecast,
target variable becomes the next hour price. Hence, the mutual
can assign a value to each candidate
information
to forecast the target variable . In other words, we
input
can rank the candidate inputs based on their mutual information
with the target variable or their information value for the forecast process.
However, the mutual information algorithm suffers from
the curse of dimensionality for the price forecast of electricity
markets. We explain this problem in the form of an example.
For instance, considering the daily and weekly periodicity
characteristics of the electricity price signal, its candidate set
of input variables at least includes lagged values of price up to
and lagged values
200 hours ago
of load (as the most important price driver [1]) up to 200
, totally 400 candidates.
hours ago
, etc.), lagged
This set contains short run trend (
values
indicating daily
representing weekly periodicity [24].
periodicity and
The auto-regression part (the lagged values of the price signal)
and only one exogenous variable, i.e., load are included in this
candidate set [2]. More exogenous variables such as available
generation can be also considered in the candidate set provided
that their data be available in the corresponding electricity
market [3]. In the next step, a training period (for preparation of
training samples) should be selected. A long training period
may cause inaccuracies because price characteristics vary with
time. Besides, a longer results in more computation burden.
On the other hand, a short training period may cause volatile
estimates [16]. In [18], the effect of different training periods
on the price forecast accuracy has been evaluated. In this paper,
previous 50 days is used as a training period as recommended
training samples.
in [2], [16], which results in
Based on the above explanations, 400 probability distribution functions for the candidate inputs and one probability distribution function for the target variable should be constructed
where each probability distribution is based on 1200 sample
data. This input data is required by the mutual information technique. Besides, joint probability distribution between each candidate input and the target variable should also be computed
as shown in (6). So, implementation of the mutual information
technique for the electricity price forecast requires high computation burden. Since the price signal has a time variant mapping
function with respect to its input features, the feature selection
process should be regularly repeated for each forecast interval
(e.g., once every day) to avoid using irrelevant input variables.
Hence, we changed the formulation of the mutual information
technique for the price forecast process so that it can be implemented by a reasonable computation burden.
As seen from (2), (3), (4), and (6), the basic definitions of
the mutual information technique use logarithms in base 2
[14]–[17]. So, this fact motivates using binomial distributions
for the inputs and output. For this purpose, at first, all the
candidate inputs and target variable are linearly normalized in
, denoted by “Normalization” in the data
the range of
preparation part of Fig. 2. Then, the median of each normalized
variable is computed. Half of the values of the normalized
variable are more than its median which are rounded to 1 and
the other half are less than it which are rounded to 0. After this
process, a binomial distribution is obtained for each candidate
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AMJADY AND KEYNIA: DAY-AHEAD PRICE FORECASTING OF ELECTRICITY MARKETS
309
Fig. 2. Structure of the proposed forecast strategy.
input and the target variable. The joint probability distribution
of these binomial distributions can be obtained as shown in
Table I. To find the joint distributions, an auxiliary variable
corresponding to joint probability
is defined as
follows:
TABLE I
JOINT PROBABILITY DISTRIBUTION BETWEEN CANDIDATE
INPUT Y AND TARGET VARIABLE X
(12)
The auxiliary variable
can have four values ranging from 0
counts number of data points (out of all
to 3.
samples) in which
.
and
are similarly
defined. So, the four states of the joint probability
can be easily computed as shown in Table I. Besides, the two
and
can be also determined as
state probabilities
follows:
(13)
(14)
(15)
(16)
Based on the above individual and joint probabilities, the previous (6) can be written as shown in (17) at the bottom of the
page. In the proposed feature selection technique, we compute
and target
mutual information between the candidate input
variable , i.e.,
, by means of (17) and then rank the
candidate inputs based on their mutual information. This formulation of the mutual information technique can be implemented
with reasonable computation burden and acceptable accuracy.
III. PROPOSED FORECAST METHOD
For day ahead prediction, two approaches including the iterative forecasting and direct forecasting can be used [25], [26].
In iterative forecasting, a single forecaster with one output node
is employed and the forecast values are iteratively used as inputs for the next forecasts. On the other hand, if the number of
output nodes is equal to the length of the forecast horizon (here,
24 hours ahead), then the direct forecasting approach is used in
which we forecast the future values directly from the forecaster
outputs. In [26], it has been discussed that the first approach may
generate more prediction errors and the second approach can introduce serious round off errors. To modify these imperfections,
we apply the idea of cascaded forecasters as shown in the forecast part (CNEA) of Fig. 2. As seen, the CNEA block consists
(17)
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 1, FEBRUARY 2009
of 24 consecutive forecasters, where each forecaster has a single
output allocated to predict the price of one hour of the next day.
In other words, the 24 hourly time series have been separately
modeled instead of the complete time series of the price.
The larger homogeneity of the 24 hourly time series in comparison with the complete one, as well as the fact that 24 onestep-ahead forecasts are calculated everyday instead of 24 individual forecasts with prediction horizons varying from 1 to 24
hours, will allow an improvement in the accuracy of the forecasts [18]. The structure of cascaded forecasters can be considered as a combination of direct and iterative forecasting, where
price of each hour is directly predicted and each forecaster has
only one output.
Each of the 24 forecasters of the CNEA is composed of an
NN part and EA part (Fig. 2). The NN part of all 24 forecasters has multilayer perceptron (MLP) structure and Levenberg–Marquardt (LM) learning algorithm, which is an efficient
training mechanism for the prediction tasks. The LM algorithm
trains a neural network 10 to 100 times faster than the usual
gradient descent back propagation method [24]. This algorithm
is an approximation of Newton’s method and computes the approximate Hessian matrix. Mathematical details of this learning
algorithm can be found in [27]. According to Kolmogorov’s
theorem, the MLP can solve a problem by using one hidden
layer provided it has the proper number of neurons [28]. So,
one hidden layer has been considered in the MLP structure of
all 24 NNs.
The MLP network is good at capturing global data trends [2],
[10]. However, its forecast capability can be enhanced if we add
the local search ability to it. An appropriate solution for this purpose is EA. The results of the training phase of each NN, i.e.,
its weights, are given to the corresponding EA part (Fig. 2). An
MLP network usually searches the solution space in a special
direction based on its learning algorithm (like the steepest descent). The EA searches around the final solution of the learning
algorithm in various directions as much as possible to find a
better solution. Since a local search around the solution of the
NN is required, we use EA with momentum as the EA part of
the forecaster, which has smoother search paths avoiding from
sudden changes. In the EA with momentum, evolution from a
generation to the next one is performed as follows:
(18)
(19)
where
is an adjustable parameter (weight) of the correindicates its change. The subscripts
sponding NN and
and
represent two successive generations (parent and
child, respectively) of the EA. is a small random number
separately generated for each adjustable parameter.
is the
and is
momentum constant. In our examinations
selected in the range of (0, 0.1) for all generations of the EA.
is the obtained value from
At the beginning of the EA,
the learning algorithm of the NN for the adjustable parameter
and
. In each cycle, the EA repeats (18) and
(19) until the next generation of all adjustable parameters is
obtained. Then the error function of the NN (its validation
error) is evaluated for the new generation. If the child has
less error than its parent, the parent is replaced by the child,
otherwise the parent is restored and the next cycle of the EA is
executed. So, at the end of the EA, the best examined solution
among all generations will be selected. In the rare cases where
the EA part cannot find a better solution, the NN’s one will
be restored. We considered 100 generations for the EA in our
solution strategy for day ahead price forecasting.
IV. ITERATIVE SEARCH PROCEDURE
The proposed forecast strategy has two main adjustable paand number of hidden
rameters: number of input features
. As with many other forenodes of the NNs denoted by
cast methods, the efficiency of the proposed prediction strategy
(including the feature selection technique and forecast method)
is dependent on the appropriate adjustment of its parameters.
It is noted that the MI based feature selection technique can
only rank the candidate inputs, but the number of selected candidates is a degree of freedom and should be separately deteris considered for all 24 NNs of the forecast
mined. The same
method to decrease the number of adjustable parameters. Usually the adjustable parameters of load or price forecast methods
are selected based on experience or heuristics. Here, an iterative
search procedure is proposed, which can automatically adjust
and
(Fig. 2) with minimum reliance on the heuristics.
This procedure can be summarized as the following step-by-step
algorithm.
. Select
candidate inputs
1) Set an initial value for
owning the highest MI values (absolute value) with the
is selected equal
target variable. The initial value of
to
[2].
2) By the selected inputs, training samples can be constructed
based on the data of the 50-day training period. Hence,
for each of 24 forecasters, 50 training samples related to
its respective hour are generated. Each forecaster can be
trained (including the learning algorithm of the NN part
and evolution of the EA part) by its own training samples.
Then, validation errors of the forecasters are evaluated.
is kept constant and
is varied at a neighborhood
3)
around its previously selected number. For each value of
in the neighborhood, training phase of the forecasters
is repeated and their validation error is evaluated. The value
resulting in the least validation error is selected and
of
fixed.
,
is varied at a neighborhood
4) With the fixed value of
around its previously selected number. For each value of
in the neighborhood, training samples are constructed.
Then training phase of the forecasters is repeated and their
resulting in
validation error is evaluated. The value of
the least validation error is selected and fixed.
and
in the current cycle
5) If the obtained values for
coincide with their previous values, the iterative search
procedure is terminated. Otherwise go back to step 5.
The validation set is a part of training samples which is removed from training phase and so becomes unseen for NN.
Thus, the error of NN for validation set or validation error can
be a measure of the NN’s error for forecast horizon. Validation
set should be as similar as possible to forecast horizon so that
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AMJADY AND KEYNIA: DAY-AHEAD PRICE FORECASTING OF ELECTRICITY MARKETS
311
the validation error can be a true representative of the prediction
error. We considered the day before the forecast day as the validation set, since it includes both the short run trend and daily
periodicity characteristics of the price signal [2]. The other alternatives, such as the same day in the previous week, can be
also considered as the validation set. So, each of 24 forecasters
training samples and validated
is really trained by
by one unseen validation sample (the last sample, related to the
respective hour in the day before the forecast day). The mean
absolute percentage error (MAPE) for 24 validation samples is
calculated and considered as the validation error in this paper
(20)
and
are actual and forecasted prices of
where
is number of samples. Here,
.
hour , respectively;
The error of validation samples is also used for early stopping
of the training phase of the NNs and avoiding from overfitting.
For a better illustration, the sample results of the iterative
search procedure for the PJM electricity market is shown in
Fig. 3. The validation set is March 12, 2006. In the right and left
vertical axes of this figure, the results of the MAPE with fixed
(step 3 of the algorithm) and MAPE with fixed
(step 4 of
the algorithm) are shown. Two consecutive transitions (from left
to right and right to left) represent one cycle of the procedure. As
seen, from top to bottom, slope of lines decreases, until we reach
to a horizontal line indicating the convergence point of the procedure where more cycles of the algorithm again reach the same
point and so the procedure is terminated. It cannot be claimed
that this procedure finds the global optimum of the adjustable
parameters since it is based on local searches. It can only find
appropriate values for these parameters. However, to enhance
the efficiency of the procedure, we add the adaptive search capability to it. In our initial implementation of the iterative search
of
or
procedure, a fixed radius of neighborhood (e.g.,
was considered in the local searches (steps 3 and 4 of the
algorithm). This results in slow convergence since a different
radius of neighborhood may be suitable for each cycle of the
procedure. Besides, appropriate adjustment of these parameters
depends on experience. So, we implemented the iterative search
procedure with an adaptive neighborhood. It is explained in the
is 20 while
form of an example. Suppose the initial value of
is fixed (step 3 of the algorithm). The procedure increases
by one and evaluates the validation error. If
results in a lower error than
, the procedure proceeds
; otherwise the proceand evaluates the next value
and stops searching values of
.
dure returns
is returned
The search is continued until the value of
where after that the validation error begins to increase. Then, the
step by step and similarly evaluate the
procedure decreases
until the validation
validation error for
error begins to increase and searching the values of
is stopped (the value of
with minimum validation
found in the
error is returned). Between the two values of
and
, reincrease and decrease directions (
spectively), i.e., two local minimums, the better one with the
Fig. 3. Sample results of the iterative search procedure.
least validation error is selected. If no better value can be found
in the both directions, the procedure returns the previous value
. The adaptive neighborhood is similarly executed
of
(step 4 of the algorithm). The adaptive
in the local search of
neighborhood increases the convergence rate of the procedure.
Besides, it provides an unsymmetrical search capability in the
two directions.
We executed the iterative search procedure with an adaptive
neighborhood for many validation sets in different electricity
markets, but it always converges in a few iterations. For instance,
in Fig. 3, it converges in five iterations. Table II from top to
bottom represents obtained results for this figure. This table includes ten half cycles or five cycles. In each half cycle, the local
or
as described in the
search varies only one parameter (
algorithm. The procedure terminates when
and
, i.e., their previous values are
obtained which are indicated in the last two rows of Table II.
The effect of adaptive neighborhoods can be seen in Table II.
changes from 15 to 27 (80% inFor instance, in cycle 1,
crease) indicated in the third and fourth rows, while in cycle
changes from 24 to 25 (about 4% increase) indicated in
4,
the ninth and tenth rows. Table II also represents the efficiency
of the proposed procedure where validation error shrinks from
13.54% at the beginning of the procedure to 4.63% at the end of
it (a three times decrease in the validation error).
It is noted that in NN based price forecast methods, the prediction error usually has a nonlinear behavior with respect to
the number of input nodes and number of hidden nodes. So, the
correct tuning of these parameters is a difficult task, which is
usually performed based on the experience and heuristics. For
instance, in Fig. 4, the nonlinear behavior of the validation error
is shown (
is kept constant) for the same
with respect to
validation set of Fig. 3. By considering both degrees of freedom
and
, we reach to Fig. 5, representing the complex behavior of the validation error with respect to the adjustable parameters. The proposed procedure can automatically and adaptively fine-tune the parameters with minimum reliance on the
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 1, FEBRUARY 2009
TABLE II
EXECUTION OF THE ITERATIVE SEARCH PROCEDURE WITH
THE ADAPTIVE NEIGHBORHOODS(FROM TOP TO BOTTOM)
the optimum solution in the next one. So, when the procedure
converges at the end of the iterations, the probability of not
finding the optimum solution is low.
V. NUMERICAL RESULTS
Fig. 4. Curve of the validation error with respect to
N (N
N
Fig. 5. Curve of the validation error with respect to
and
= 25 and
= 30).
of the validation error occurs for
N
N
N
= 25).
(the minimum
heuristics (only initial value of
is required in the step 1 of
the algorithm). This iterative search procedure can be easily extended for any number of adjustable parameters. For this purpose, in each step of each iteration, one parameter is varied while
the others are kept constant.
It is noted that if the proposed search procedure has only one
cycle, then it possibly traps in a local minimum and misses the
optimum solution. However, the search process of the proposed
procedure is repeated in each iteration with the refined values of
the adjustable parameters. Thus, if the procedure misses the optimum solution in one iteration, then it has the chance of finding
The proposed method is examined for the day ahead electricity markets of mainland Spain and PJM. Besides, we compared the method with at least seven recently published price
forecast techniques. Our forecast strategy used in all examinations of this paper can be briefly summarized as follows (Fig. 2).
At first, the candidate inputs and target variable are linearly nor. The MI based feature selection
malized in the range of
technique selects the input variables. By the selected inputs,
training and validation samples are constructed based on the
data of 50 days ago. Each of 24 forecasters of the CNEA is
trained by its respective 49 training samples and validated by
one validation sample (the last sample). The adjustable parameand
, are fine-tuned by the iterative search proters, i.e.,
cedure. The candidate inputs for the PJM electricity market include lagged values of price, load, and available generation and
for the Spanish electricity market include lagged values of price
and load (each variable up to 200 hours ago), which are publicly available data on their websites [29] and [30], respectively
(no specific data has been used). The whole proposed price forecast strategy (including the feature selection technique, 24 forecasters of the CNEA and iterative search procedure) has price
data up to the midnight of last night. In general, if the th forecaster requires the hourly prices of the forecast day, the predicted price values for these hours by its previous forecasters
for the second forecaster is the preare used. For instance,
dicted price by the first forecaster.
It is noted that considering the congestion information can affect LMP forecast. Congestion occurs when a transmission line
flow exceeds its limit. So line flow and line limit information
together could reveal line flow congestion and its severity [1]. If
we have a large number of lines, considering line flows and line
limits highly increases the number of inputs of the price forecast
method. Processing of this large number of inputs (even after
feature selection) is a complex task and may even deteriorate
the accuracy of the price forecast method since the forecasters
(Fig. 2) should also learn the input/output mapping function between these added inputs and output. The other alternatives such
as congestion index [1] and congestion price [29] have been also
proposed to include the congestion information in the price forecast method. However, most of research works in the area do not
consider the congestion information in the LMP forecasting like
[15] and [17].
The first verification for the method proposed in this paper
has been performed by means of real data of the PJM electricity
market in year 2006. The obtained results for some test days and
weeks are shown in Table III and compared with the results of
[17], which proposes combination of SD and NN techniques for
day ahead price forecasting. For the sake of a fair comparison,
the same test days and weeks of [17] have been considered in
have been directly quoted
Table III and the results of
from [17]. Besides, the same MAPE definition of this reference
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AMJADY AND KEYNIA: DAY-AHEAD PRICE FORECASTING OF ELECTRICITY MARKETS
TABLE III
MAPE OF THE PROPOSED METHOD AND SD + NN PLUS THE MAPE
IMPROVEMENT FOR THE PJM ELECTRICITY MARKET IN YEAR 2006
313
TABLE V
SELECTED FEATURES PLUS THEIR RANKS AND NORMALIZED MI VALUES FOR
A TEST DAY OF THE FIRST EXAMINATION (FEBRUARY 10, 2006)
TABLE IV
ERROR VARIANCE OF THE PROPOSED METHOD AND SD + NN PLUS THE
VARIANCE IMPROVEMENT FOR THE PJM ELECTRICITY MARKET IN YEAR 2006
has been used for the results of Table III, denoted by
in this paper
(21)
(22)
and
are as defined for (20) and
indicates the average of actual prices as defined in (22).
indicates forecast horizon. For daily and
equals to 24 and 168, respectively. For the
weekly MAPE,
five test days and two test weeks of Table III, daily MAPE and
weekly MAPE are used, respectively [17].
The forecasting error is the main concern for power engineers; a lower error indicates a better result. For all test days
and weeks of Table III, the proposed method outperforms
. Besides, improvement in the MAPE of the proposed
method with respect to
is shown in the last column of
Table III. In addition to the mean error, stability of results is another important factor for the comparison of forecast methods.
In Table IV, variance of the prediction errors, as a measure
of uncertainty, for the forecast days and weeks is presented.
have been quoted from
Again, the variance values of
[17]. The proposed method also has lower error variances than
in all test days and weeks indicating less uncertainty
in the predictions of the proposed method. Improvement in the
error variance of the proposed method with respect to
is shown in the last column of Table IV.
where
Now, sample results for the feature selection part are presented. For the sake of conciseness, obtained results for only one
of the test days (February 10, 2006) are represented in Table V.
, , and
in Table V indicate price, load, and available
generation of hour , respectively. In this Table, selected features (columns 1 and 4) plus their ranks (columns 2 and 5) and
normalized MI values (columns 3 and 6) are shown. The reported results have been obtained after the convergence of the
value. As seen, out of
iterative search procedure with final
600 candidate inputs considered for the PJM electricity market
(200 lagged values of price, load, and available generation),
only 28 inputs are selected, which indicates high filtering ratio
of the proposed feature selection technique.
For the other test days of this examination, the final
value
varies from 24 to 32. Among the 28 selected features of Table V,
11 inputs are from the auto-regression part and 17 inputs are
from the exogenous variables. The 17 inputs include 12 and 5
features from the load and available generation, respectively.
So, load is a more important price driver than available generation for this test case. Between load and price variables, the
selected load features are slightly more than the selected price
features, but the price features overall have higher ranks. Similar results have been obtained for the other test days. It is noted
that all 24 forecasters of the CNEA use the same set of 28 input
features (for February 10, 2006) based on the sliding window
technique. The first forecaster of the CNEA uses the input feaor the price of the first
tures shown in Table V and predicts
hour of the forecast day. Then, the input features of Table V are
shifted by one hour ahead and used by the second forecaster of
(the price of the second hour of the
the CNEA to predict
forecaster of the chain of foreforecast day). In general, the
casters of the CNEA uses the selected inputs of Table V that are
hours ahead.
shifted by
We considered a large candidate set of input variables including the effects of short-run trend, daily and weekly periodicities so that the maximum information content of the input
data is utilized (in other words, no effective input variable is
missed). Then, this candidate set is refined by the proposed feature selection technique to remove candidate inputs with low information value. Although the effect of weekly periodicity may
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 1, FEBRUARY 2009
be less than the effect of daily periodicity in the price time series, however it is still useful for the price forecast. For instance,
is among the selected inputs, but
as seen from Table V,
with a lower rank than
. This is due to the fact that load
demand is an important price driver and it has weekly periodicity behavior [24]. Thus, we can expect that the effect of weekly
periodicity is also observed in the price time series. We also included price data of coal and natural gas in the feature selection
process of the PJM electricity market. Coal and natural gas have
the highest portions among different fuels in the PJM electricity
market. We examined different test periods of year 2006; however, prices of coal and natural gas were never selected in competition with the initial candidate inputs including the history of
electricity price, load demand and available generation. When
the fuel prices have normal changes (like those of year 2006),
they may be less effective than the initial candidate inputs. However, sudden changes in the fuel prices and especially fuel price
spikes can be effective in the unexpected changes of electricity
price and even electricity price spikes may be generated. This
is a matter that demands further research especially on the electricity price spike analysis and forecasting. In the next examinations, only the final results of the whole proposed method
are evaluated and compared with the other price forecast techniques.
The second examination of this paper has been performed on
the Spanish electricity market. Obtained results for four weeks
corresponding to four seasons of year 2002 are presented in
Tables VI and VII. The fourth week of February, May, August, and November are selected for winter, spring, summer,
and fall seasons, respectively, which is the test period considered in [2], [16], [18], and [31]. In the second and third columns
of Table VI, results of ARIMA and ARIMA plus wavelet transform have been quoted from [16]. The wavelet transform was
used to decompose the price signal into less volatile components. The fourth column represents the obtained results from
the FNN (quoted from [6]), in which instead of decomposing
the price values, its solution space is softly divided into subspaces and each functional relationship of the price is implemented in one subspace. In the fifth column of Table VI, obtained results from an MLP neural network with LM learning
algorithm, quoted from [31], are shown. In the sixth column,
results of mixed model are represented (quoted from [18]). In
this reference, a separate ARIMA model has been presented
for each hour of the forecast day while weekdays and weekends have been separately modeled. It is noted that although
the idea of individual modeling of the 24 hourly time series
of electricity price has been presented in [18], their prediction
method is based on the ARIMA time series. However, in the
CNEA, the combination of NN and EA has been proposed as the
forecast block. Besides, the other parts of our forecast strategy
including the MI based feature selection and iterative search
procedure for fine-tuning of adjustable parameters are new and
not seen in the previous works. Finally, in the last column of
Table VI, the obtained results of the proposed method are represented. The weekly MAPE values in Table VI are in terms
in (21) with
. The proposed method
of
has slightly more weekly MAPE than the wavelet-ARIMA and
FNN in the winter test week and than the mixed model in the
TABLE VI
WEEKLY MAPE VALUES IN TERMS OF PERCENTAGE (%) FOR FOUR
WEEKS OF THE SPANISH ELECTRICITY MARKET IN YEAR 2002
TABLE VII
ERROR VARIANCE FOR FOUR WEEKS OF THE SPANISH
ELECTRICITY MARKET IN YEAR 2002
spring test week. However, in all other test cases of Table VI,
the proposed method has lower weekly MAPE than the other
techniques. Moreover, the average of the weekly MAPE values
of the proposed method is considerably less than all other techniques (indicated in the last row of Table VI). Improvement
in the average MAPE of the proposed price forecast strategy
with respect to ARIMA, wavelet-ARIMA, FNN, NN, and mixed
model is 46.59%, 34.40%, 29.25%, 40.29%, and 42.79%, respectively.
In Table VII, the variance of the prediction errors for the
ARIMA [16], wavelet-ARIMA [16], FNN [2], and proposed
method for the four test weeks are presented. Although the error
variance of the proposed method is slightly more than the other
techniques in the winter and spring test weeks, but the average
variance value of the method is less than the others. Improvement in the average error variance of the proposed price forecast
strategy with respect to ARIMA, wavelet-ARIMA, and FNN
is 60.87%, 43.75%, and 33.33%, respectively. For the NN and
mixed model, the error variance has not been presented in the
respective references.
Another important characteristic of the proposed method,
which can be seen from Table III, Table IV, Table VI, and
Table VII, is its less sensitivity to the electricity market
volatility than the other examined price forecast techniques. For
instance, volatility of the Spanish electricity market increases
in summer and fall seasons as discussed in [2] and [16]. So,
the price forecast methods generally encounter more prediction
errors in these seasons, which can be seen from Table VI.
However, the weekly MAPE of the proposed method has less
seasonal variation than the other techniques. Table VII shows
that the error variance of the proposed method also has less
variations.
We also executed the proposed method for all 52 weeks
of year 2002 for the Spanish electricity market. The obtained
weekly MAPEs and error variances for 52 weeks are close to
the results of Tables VI and VII, respectively. For instance,
average of weekly MAPEs for 52 weeks is 5.38% versus 5.32%
in Table VI (average of the four test weeks). Average of error
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AMJADY AND KEYNIA: DAY-AHEAD PRICE FORECASTING OF ELECTRICITY MARKETS
variances for both 52 weeks and four weeks (Table VII) is
0.0036. These results indicate the robustness of the proposed
method and its performance in a long run for a complete year.
The averages of MAPE values for working days and weekends
of year 2002 in the Spanish electricity market are 5.22% and
5.79%, respectively. As seen, the average of MAPE values
for working days (5.22%) is slightly less than the average of
MAPE values for all days (5.38%) while the average of MAPE
values for weekends (5.79%) is slightly more than the average
of MAPE values for all days (5.38%). As a comparison, in
[19], weighted nearest neighbors technique has been proposed
for day-ahead electricity price forecasting and examined on the
Spanish electricity market. The best average MAPE value of
[19] for the working days of year 2002 is 8.45% compared with
our 5.22% average MAPE value for the working days of this
year.
We also modeled the calendar effect in our price forecast
strategy in three different ways. In the first alternative, the
partitioning method of [18] is used in which the proposed
method separately trains and forecasts the data of working
days and weekends. In the second alternative, one calendar
indicator ranging from 1 to 7, is used to encode days of week
[1]. The calendar indicator is added to the selected inputs of the
feature selection technique. In the third alternative, seven extra
zero-one input factors indicating day of the week are added
to the selected inputs. However, in all of these alternatives,
the price forecast accuracy of the proposed method degrades.
By partitioning working days and weekends, the available
data of the proposed method for working days and weekends
decreases. The information content of the calendar indicators,
introduced by the second and third alternatives, are included in
the load and price information.
In addition to day-ahead price forecast, we executed the proposed method for week ahead (168 hours ahead) price forecast.
The MAPE values for week ahead price forecast of the proposed method for the two test weeks of Table III are 7.14% and
8.98%, respectively, and for the four test weeks of Table VI are
9.15%, 8.38%, 9.12%, and 10.32, respectively. Although the
week-ahead MAPE values of the proposed method are larger
than its day-ahead MAPEs, they are comparable with day-ahead
MAPEs of the other techniques, which indicate performance of
the proposed method for one week forecast horizon.
In the last examination, the proposed method is compared
with the mixed model and another ARIMA model proposed by
Contreras et al. [7] for two other test weeks of the Spanish electricity market. The obtained results are shown in Tables VIII and
IX, where the daily mean errors of the mixed model and Contreras ARIMA model (quoted from [18] and [7], respectively)
are calculated like the MAPE in (20), i.e.,
appears in the
. For the sake of a fair comdenominator instead of
parison, we used the same definition of the daily mean error
for the results of the proposed method in Tables VIII and IX.
These tables show that in 10 out of 14 test days, the proposed
method has less daily mean error than both the mixed model and
Contreras ARIMA. Besides, the proposed method has considerably lower average value of daily mean errors than the two other
methods. Since the variance data of the other methods are not
315
TABLE VIII
DAILY MEAN ERRORS FOR MAY 25–31, 2000
OF THE SPANISH ELECTRICITY MARKET
TABLE IX
DAILY MEAN ERRORS FOR AUGUST 25–31, 2000
OF THE SPANISH ELECTRICITY MARKET
Fig. 6. Hourly prices (dark), price forecasts (green), and absolute value of forecast errors (red) for March 5, 2006 of the PJM electricity market.
available, comparison of the variances cannot be accomplished
in this case.
To also give a graphical view about the forecast accuracy of
the proposed method, its results for March 5, 2006 of the PJM
electricity market (owning the highest MAPE value in Table III)
and the fall’s test week, 2002 of the Spanish electricity market
(with the highest weekly MAPE in Table VI) are shown in
Figs. 6 and 7, respectively. Although the worst test periods
have been purposely selected, however, the forecast curve
acceptably follows the real price curve in the both figures. Only
some minor deviations in the morning and evening peaks of
the test day of Fig. 6 and peaks and valleys of the test week of
Fig. 7 are observed.
Total set-up time of the proposed method including the MI
based feature selection, training of the 24 forecasters
of the CNEA and fine-tuning of the adjustable parameters (with
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 1, FEBRUARY 2009
TABLE XI
OBTAINED MAPE VALUES FROM THE TWO ADDITIONAL EXPERIMENTS
INCLUDING HOURLY LOADS OF THE FORECAST DAY
TABLE XII
MAPE OF THE PROPOSED PRICE FORECAST STRATEGY WITH AND WITHOUT
EA ADJUSTMENT FOR THE PJM ELECTRICITY MARKET IN YEAR 2006
Fig. 7. Hourly prices (dark), price forecasts (green), and absolute value of forecast errors (red) for the fall’s test week of the Spanish electricity market.
TABLE X
MAPE OF THE PROPOSED METHOD WITH
CORRELATION ANALYSIS FOR THE PJM AND
SPANISH ELECTRICITY MARKETS
the adaptive neighborhood in the iterative search procedure) was
about 40 min on a Pentium P4 3.2-GHz personal computer with
1 GB of RAM memory. This set-up time is acceptable within a
day-ahead decision making framework.
Correlation analysis has been recently used by some researchers for feature selection of price forecasting [6], [15],
[26]. However, correlation analysis is a linear feature selection
technique while MI is a nonlinear criterion. The electricity
price has a nonlinear and multivariate input/output mapping
function and so a nonlinear forecast method (CNEA) is proposed to predict it. Using a linear analysis technique for feature
selection of a nonlinear signal and combination of it with a
nonlinear forecast method is incoherent, which degrades the
forecast accuracy of the whole strategy. For a better illustration
of this matter, we replaced the feature selection technique of the
proposed strategy with correlation analysis (the other parts of
the strategy are kept unchanged) and examined it on the same
test days and weeks of the PJM and Spanish electricity markets.
The obtained results are shown in Table X. From this table, it
can be seen that the forecast errors of the proposed strategy
with the correlation analysis are higher than the forecast errors
of the proposed strategy with the MI technique.
In the previous experiments, only lagged values of price, load
and available generation are considered as the input features of
the CNEA. We perform two additional experiments to evaluate
the effect of next day load values on the price forecast accuracy of the proposed method. In the first experiment, load value
(actual load) of each hour of the forecast day is added to the selected inputs of the corresponding forecaster of the CNEA. In
the second experiment, load of the next hour is added to the candidate inputs. Then, the processes of feature selection, training
of the forecasters of the CNEA and fine-tuning of the adjustable
and
) are repeated with the extended set of
parameters (
the candidate inputs including 401 candidates for the Spanish
electricity market. The obtained results from these two experiments for the four test weeks of the Spanish electricity market
are shown in Table XI.
Comparing the results of these two additional experiments
with the results of the original method shown in the last column
of Table VI (without including the next day load values) indicates that the MAPE values of the experiment 1 are lower than
those of the original method due to considering the information
content of the next day hourly loads. The MAPE values of experiment 2 are even lower than those of experiment 1. In the
second experiment, the added feature (next hour load) participates in the optimization process of the feature selection. The
added feature can be either selected or not for a forecast day
based on its information value. Evaluation of the effect of the
other noncausal features (such as the values of available generation or congestion probabilities for the forecast day) on the
price forecast accuracy needs further study, which we will consider it in our future research.
To evaluate the effect of the EA adjustment on the performance of the proposed price forecast strategy, its MAPE values
with and without the EA adjustment for the same test days
and weeks of Table III are shown in Table XII. As seen from
Table XII, the EA adjustment improves the forecast accuracy
of the proposed strategy in all test days and weeks with MAPE
improvements ranging from 5.66% to 12.80%.
Our proposed price forecast method in its original form has
not been designed for price spike forecasting. However, we add
a preprocessor and postprocessor before and after the proposed
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AMJADY AND KEYNIA: DAY-AHEAD PRICE FORECASTING OF ELECTRICITY MARKETS
TABLE XIII
MAPE OF THE PROPOSED PRICE FORECAST METHOD IN THREE
CASES FOR THE PJM ELECTRICITY MARKET IN YEAR 2006
price forecast method, respectively, to process price spikes [1].
The preprocessor converts prices as follows:
if
if
price spikes is decreased and accordingly a slight decrease in the
MAPE of all prices is also observed. If prices greater than 200
, then the
$/MWh are considered as the price spikes
MAPE values of the price spikes and all prices are decreased by
and
,
, the MAPE values of the price
respectively. With
spikes and all prices are decreased by
and
, respectively. So, by
adding the preprocessor and postprocessor, accuracy of the proposed price forecast method for prediction of price spikes increases, which can be considered as an additional benefit of the
proposed method.
(23)
VI. CONCLUSION
are original price and its processed value, reIn (23), and
is the upper limit of the price. If a price is higher
spectively.
, its processed value is
plus
times the logathan
. The postprocessor
rithm (base 10) of the ratio of price to
performs the inverse transform as follows:
if
if
317
(24)
and
are forecasted price and modified forecasted
where
price (after post-processor). We performed an extra experiment
values of 200
on the PJM electricity market with two
$/MWh and 150 $/MWh. The results of the proposed price
forecast method in three cases are presented in Table XIII.
In the first case, the proposed price forecast method in its
original form (without using preprocessor and postprocessor)
is examined by the data of the whole year 2006 of the PJM
electricity market. The MAPE values of this case for prices
greater than 200 $/MWh, for prices greater than 150 $/MWh
and for all prices of year 2006 are shown in the first column of
Table XIII, respectively. The MAPE values of the Table XIII
are calculated based on (20). There are 25 and 73 hourly prices
greater than 200 $/MWh and 150 $/MWh in year 2006 of the
PJM electricity market, respectively. So, in the Table XIII,
, 73 and 8760 (number of hours in year 2006) for
”, “
”, and “All Prices”, respectively.
“
In the second and third cases (shown in the second and third
columns of Table XIII), the preprocessor and postprocessor are
added to the original price forecast method with
and
, respectively. In other words, in the second and
third cases, prices greater than 200 $/MWh and 150 $/MWh
are considered as the price spikes, respectively. The MAPE
values of the second case for prices greater than 200 $/MWh
(considered as price spikes in this case) and “All Prices” are
separately reported in the second column of Table XIII, respectively. Similarly, the MAPE values of the third case for prices
greater than 150 $/MWh and all prices are mentioned in the
third column of Table XIII, respectively.
From Table XIII it can be seen that the MAPE values for price
” and “
”) are considerably higher
spikes (for both “
than the MAPE values for all prices, especially in the first case.
This can be expected since a price spike is an abnormal price
that is significantly higher than its expected value. However,
when we add the preprocessor and postprocessor, the MAPE of
Price forecast plays a major role in today’s electricity markets. Companies that trade in electricity markets make extensive
use of price forecast techniques either to bid or hedge against
volatility. However, price prediction has its own complexities.
At the same time, utility companies usually have limited and uncertain information for price prediction. Thus, companies that
trade in the electricity markets require efficient and robust price
forecast methods. In this paper, a new price forecast strategy
is proposed, which is composed of MI based feature selection,
CNEA as the prediction part and an iterative search procedure
to fine-tune the adjustable parameters. The proposed strategy
has been compared with seven of the most recently published
price forecast methods. These comparisons reveal the superior
forecast capability of the proposed method. Further research is
needed to expand the proposed method to the special case of
price spike forecasting.
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Nima Amjady was born in Tehran, Iran, on February 24, 1971. He received the
B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from Sharif University
of Technology, Tehran, in 1992, 1994, and 1997, respectively.
At present, he is a Professor with the Electrical Engineering Department,
Semnan University, Semnan, Iran. He is also a Consultant with the National Dispatching Department of Iran. His research interests include security assessment
of power systems, reliability of power networks, load and price forecasting, artificial intelligence, and its applications to the problems of power systems.
Farshid Keynia was born in Kerman, Iran. He received the B.Sc. degree in electrical engineering from S. B. University, Kerman, Iran, in 1996 and the M.Sc.
degree in electrical engineering from Semnan University, Semnan, Iran, in 2001.
Currently he is pursuing the Ph.D. degree in the Electrical Engineering Department of Semnan University.
His research focuses on short-term and midterm price and load forecasting
in deregulated electricity markets as well as feature selection and classification
algorithms.
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