day-ahead electricity price forecasting based on panel cointegration and particle filter

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Electric Power Systems Research 95 (2013) 66– 76

Contents lists available at SciVerse ScienceDirect

Electric Power Systems Research

jou rn al h om epa ge: www.elsev ier .com/ locate /epsr

ay-ahead electricity price forecasting based on panel cointegrationnd particle filter

.R. Lia, C.W. Yua,∗, S.Y. Renb, C.H. Chiuc, K. Mengd

Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong KongInstitute of Textiles & Clothing, The Hong Kong Polytechnic University, Hong KongSun Yat-Sen Business School, Sun Yat-Sen University, ChinaCentre for Intelligent Electricity Networks, The University of Newcastle, Australia

r t i c l e i n f o

rticle history:eceived 3 March 2012eceived in revised form 7 June 2012ccepted 18 July 2012vailable online 2 October 2012

eywords:rice forecastinganel data

a b s t r a c t

An accurate forecasting of energy price is important for generation companies (GENCOs) to developtheir bidding strategies or to make investment decisions. Nowadays, day-ahead electricity market isclosely associated with other commodity markets such as fuel market and emission market. Under suchan environment, day-ahead electricity price is volatile and its volatility changes overtime due to theuncertainties from the multi-market. This paper proposes a two-stage hybrid model based on panelcointegration and particle filter (PCPF). Panel cointegration (PC) model, which utilizes information ofboth the inter-temporal dynamics and the individuality of interconnected regions, provides powerfulforecasting tool for electricity price. Particle filter (PF) has achieved significant successes in tracking

ointegration modelarticle filter

applications involving non-Gaussian signals and nonlinear systems. This paper has two main focuses: (1)To expand the dimension of electricity price dataset from time series to panel data so that the dynamicsof interconnected regions can be analyzed simultaneously and considered as a whole. (2) Regarding themodel coefficients as a time-varying process, PF is used to forecast electricity price adaptively. In the casestudy, the proposed PCPF model is applied to the real electricity market data of PJM in the year 2008.Promising results show clearly the superior predicting behavior of the proposed modeling.

. Introduction

Nowadays, day-ahead electricity market is closely associatedith other commodity markets such as fuel market and emissionarket. This evolution has changed the roles of market participants

nd complicated the analysis of electricity price [1]. Day-aheadlectricity price is volatile and its volatility changes overtime dueo the uncertainties from the multi-market. GENCOs always adjustheir bidding strategies to achieve their maximum benefits accord-ng to their analysis. Similarly, consumers would derive a plan toptimize their purchased electricity from the pool, or use othernancial derivatives to protect themselves against high prices2,3].

The available forecasting methods can be broadly classifiednto three categories: system simulation models, market equilib-ium analysis, and time series models [4,5]. System simulation
odels usually concentrate on detailed insight of price formation
6]. Factors such as actual dispatch according to system oper-ting requirements and transmission constraints are considered.

∗ Corresponding author.E-mail address: [email protected] (C.W. Yu).

378-7796/$ – see front matter © 2012 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.epsr.2012.07.021

© 2012 Elsevier B.V. All rights reserved.

Market equilibrium analysis, on the other hand, involves economicsand game theory [7]. In addition to the forecasted prices, thesetwo categories always come up with general equilibrium or marketstrategic behaviors. Time series models are widely used to forecastelectricity price. Electricity price is forecasted through statisticalmethods with little attentions paying to the reasons of the pricechanging. This type of methods can be divided into three subtypes,namely regression based models, stochastic time series models andintelligent learning models.

Regression based models analyze the assumed relationshipbetween electricity price and a number of independent variablesthat are known or estimated [8]. These methods overcome serialcorrelation problems. However, they do not always work well inpractice since they assume the variables are stationary or stationaryafter the application of statistical techniques such as differencing.

Stochastic time series models are proposed to deal with nonsta-tionary time series. Both autoregressive moving average (ARMA)and autoregressive integrated moving average (ARIMA) modelswork by iteratively identifying a parametric model from hypoth-
esized models and estimating the corresponding parameters basedon observations. When the series have high volatility and pricespikes, GARCH model [5,9] is a good alternative because it con-siders the conditional variance as time dependent. However, the
dx.doi.org/10.1016/j.epsr.2012.07.021

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dx.doi.org/10.1016/j.epsr.2012.07.021

Systems Research 95 (2013) 66– 76 67

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X.R. Li et al. / Electric Power

dentification and estimation of these models can be badly distortedy outlier effects.

Intelligent learning models derived from Neural Networks (NN)nd data mining have also been studied. Artificial neural networksANN) are defined as information processing systems which haveommon specific characteristics associated to biological networks.NN is capable to model nonlinear input/output mapping func-

ions. Its families have strong fault tolerance ability though theysually require long training time. A standard ANN is a group of

nterconnected neural processing units imitating the brain acti-ation. Recent studies on ANN focus on the determination of theest forecasting model by comparing various neural architectures,pplying several decomposition techniques or selecting properransfer functions [10]. Wu and Shahidehpour [11] proposed aybrid time-series and adaptive wavelet neural network (AWNN)odel, composed of linear and nonlinear relationships of prices and

xplanatory variables, for day-ahead price forecasting. AWNN wassed to present the nonlinear, nonstationary impact of load seriesn electricity prices. Amjady [12] developed a fuzzy neural networkFNN) which combined fuzzy logic and standard ANN to provide

ore accurate results than ARIMA, wavelet-ARIMA, multilayererceptron, and radial basis function neural networks. Kernel-ased machine learning method such as support vector machineSVM) [9] has shown good accuracy and efficiency in some real-orld problems. Furthermore, relevance vector machine (RVM) [9]as been proved outperforms SVM in both the forecast accuracynd computational efficiency. However, the performance of theseachine learning models relies on heuristics, e.g. the choices of

ernel and penalty functions.The integration of different commodity markets leads to uncer-

ainties in both time and space dimensions. Furthermore, electricityrice features high volatility, non-stationary and non-linear pat-erns [12]. Having a robust and accurate prediction model foray-ahead electricity price is important under these circumstances.owever, rare literature has studied both inter-temporal dynamicsnd inter-regional interactions of uniform day-ahead price amongifferent interconnected regions. Furthermore, little attention haseen paid to develop methods that can handle both linear andonlinear problems simultaneously. Therefore, a novel panel coin-egration and particle filter (PCPF) model is proposed in this papero predict day-ahead electricity price.

The rest of this paper is organized as follows: In Section 2, theroblem formulation and the evaluation criteria are described. Sec-ion 3 presents the forecasting framework of the proposed PCPF

odel. A case study is conducted in Section 4 to demonstrate theffectiveness of the PCPF model. Section 5 concludes the paper.

. Problem formulation

A pool-based electricity market (EM) with N interconnectedegions is studied in this paper. In this kind of market (e.g. PJM),he operator coordinates the movement of electricity through thenterconnected power grid. The uniform price in the day-ahead

arket is affected significantly by the regional loads because elec-ricity price is calculated based on the consideration of the entireird [4]. On the other hand, the variability of the uniform pricean influence the energy-usage patterns and introduce new trends.ig. 1 shows a snapshot of seven regional loads of PJM from Jan 1o Jan 7, 2008.

It can be seen that significant loading differences, which changerom time to time, exist among regions. Besides, significant individ-
al intra-day variations can be observed. In other words, both the
nter-temporal dynamics and the inter-regional interactions existn the panel. In addition, nonlinear patterns exist in the relation-hip among the uniform day-ahead price and the loads of different

Fig. 1. Loads of 7 regions from Jan 1, 2008 to Jan 7, 2008.

regions. In this case, using time series models such as regressionbased models and stochastic time series models cannot capture thecomplex nonlinear behavior. On the other hand, using intelligentmethod such as NN will yield mixed results and the efficiency willvary from case to case.

The proposed PCPF model is introduced to tackle these difficul-ties by using a two-stage forecasting framework. It has two featuresthat differentiate it from other existing techniques. First, it makesprediction by using historical loading data of different regions inthe pool and constructs the regional loading data together with theuniform day-ahead price as a panel [13]. Using panel data, boththe impacts of inter-temporal dynamics and inter-regional load-ing differences on the uniform day-ahead price can be taken intoaccount. Secondly, PF is applied as a post-processor to effectivelyhandle the nonlinearity and the volatility of electricity price. A casestudy on PJM day-ahead electricity market is conducted. The mainreason of selecting PJM is due to the fact that its market partici-pants’ decisions are affected not only by the uncertainties in theday-ahead electricity market, but also by the fuel market and theemission market in the Regional Greenhouse Gas Initiative (RGGI)scheme. GENCOs’ bidding strategies and investment plans havebeen changed after the implementation of RGGI in year 2009 [14].Through the case study, it can be demonstrated that the proposedmethod can effectively deal with volatility and nonlinearity in apool environment with power resources allocated dispersedly indifferent regions.

The data in the period Jan 1, 2007–Dec 31, 2007 is used forthe estimation of the models, while the data in the period Jan 1,2008–Dec 1, 2008 is used for out-of-sample testing. To assess andcompare the performance of the models, weekly mean absolutepercentage error (WMAPE), WMAPE for period j(WMAPEj), dailymean absolute percentage error (DMAPE), weekly mean absoluteerror (WMAE) and weekly root mean square error (WRMSE) indicesare adopted in this paper.

WMAPE = 1168

7∑t=1

24∑j=1

|XAj,t

− XFj,t

|

XAj,t

(1)

WMAPEj = 17

7∑t=1

|XAj,t

− XFj,t

|

XAj,t

(2)

68 X.R. Li et al. / Electric Power Systems Research 95 (2013) 66– 76

Stationarity

Test

Fail

Pass

Pass

PC

EsitmationPCPF

Forecasting

Input variables

(price & loads)

Difference

Operator

First Stage Second Stage

Cointegration

test

Panel

Construction

odel f

D

W

W

wp

3

3

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Fail

Fig. 2. Two-stage PCPF m

MAPE = 124

24∑j=1

|XAj,t

− XFj,t

|

XAj,t

(3)

MAE = 1168

7∑t=1

24∑j=1

|XAj,t − XF

j,t | (4)

RMSE =

√√√√ 1168

7∑t=1

24∑j=1

(XAj,t

− XFj,t

)2

(5)

here XAj,t

is the actual value and XFj,t

is the forecasted value of theredicted variable.

. Proposed PCPF model

.1. PCPF introduction

Both PC and PF models have achieved successes in their owninear or nonlinear domains. However, none of them is a univer-al model that is suitable under all circumstances. For example,n one hand, the approximation of PC models for complex nonlin-ar problems may not be adequate. On the other hand, using PFo model linear problems has yielded mixed results [15]. Since its difficult to fully realize the characteristics of the data in a realroblem, it is reasonable to consider day-ahead electricity serieso be composed of both linear autocorrelation structure and non-inear patterns. Combining different models, different aspects ofhe underlying patterns may be captured. PCPF which has both lin-ar and nonlinear modeling capabilities can be a good strategy forractical use.

As it is unlikely to include all market fundamentals into a priceorecasting model, the proposed model takes the most importantactors into account in the first stage and then treats the others asncertainties which will be handled by PF. As the change in regional

oads is the key factor that affect the uniform electricity price, thisodel uses historical regional loads and price as input variables to

redict the price although other factors can be incorporated easily.his model is based on a two-stage architecture shown in Fig. 2.

.2. First stage: Panel data construction and analysis

The dynamics of electricity price and loads are always non-tationary due to the discrete changes in participants’ strategiesrom different regions so that individual time series analysis can-ot simultaneously reflect electricity consumption conditions in
ifferent regions. In view of the disadvantages of time series data,he panel data is used to identify both the impacts of inter-temporalynamics and inter-regional load differences on the uniform day-head price.
or electricity forecasting.

The panel data is a set of sample values which combines cross-sectional and time-series data sets. Mathematically, the panel datahas N cross-sections (i.e. number of regions in EM) and T numberof days in sample. The panel data is constructed as:

Pj,t = ˇ0,j +m∑

l=1

˛0,j,lPj,t−l +N∑

i=1

ˇi,jLi,j,t + �j,t;

Li,j,t = ei,j +n∑

l=1

˛i,j,lLi,j,t−l + ui,j + ıi,j,t;

(i = 1, 2, . . . , N; j = 1, 2, . . . , 24; t = 1, 2, . . . , T;

�j,t∼(0, �2� ); ui,j∼(0, �2

u ); ıi,j,t∼(0, �2ı

))

(6)

where ˇ0,j, ˇi,j, ei,j are coefficients for cross-section and ˛0,j,l, ˛i,j,l

are coefficients for time-series. l is the time lagged value in orderto capture the distinct profile of each inter-day period. m andn are the number of time lagged items Pj,t−l and Li,j,t−l , respec-tively. The advantage of having panel data as compared to a singlecross-section or series of cross-sections with non-overlappingcross-section units is that it allows us to test and relax the assump-tions that are implicit in cross-sectional analysis [16]. To estimatethe long-run equilibrium relationship and the short-run adjust-ment relationship among variables, three procedures need to becarried out: panel stationarity test, panel cointegration (PC) test,and PC estimation.

The establishment of PCPF model among variables requires totest whether the panel data is (i) stationary (integrated of orderzero) or non-stationary (integrated of order one); and (ii) cointe-grated. In general, electricity prices and loads are non-stationaryvariables that cannot be directly estimated [12]. Hence, cointegra-tion is an alternative to describe their relationships in econometricanalysis which indicates a long-term equilibrium among vari-ables. Therefore, the cointegration test and PC estimation willbe carried out if the panel data is non-stationary; otherwise thecoefficients of the panel model can be directly estimated accordingto (6).

3.2.1. Panel stationarity testUnit root test is a conventional econometric method to test

the stationarity of time series by examining the existence of unitroots. Recent literature [17] finds that panel-based unit root testsare much more powerful than the basic tests such as AugmentedDickey–Fuller test [18] which are based on individual time series.
The panel unit root test methods including Levin, Lin, and Chu(LLC) [19] and PP–Fisher Chi-square [20] are suitable candidatesfor examining the common unit root process and individual unitroot process, respectively.

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.2.2. Panel cointegration testPC technique involves not only long-run relationship among

on-stationary variables, but also short-term fluctuation of station-ry variables, which can help to achieve high forecasting precision.

In this paper, Johansen Fisher Panel Cointegration (JFPC) test21] is used to examine the existence of determined cointegrationelationship among the variables. The JFPC test allows the exis-ences of both stationary and non-stationary variables in the panel.his method permits more than one cointegration relationship andence it is generally applicable in testing the panel data. Onceointegration relationships are ascertained within the constructedanel data, the coefficients of PC model can be estimated accordingo (7) described in the next sub-section. Otherwise, the variables inhe panel will be processed by another operators [13,20] in whichj,t , Li,j,t in (6) will be replaced by �Pj,t, �Li,j,t and then the recon-tructed panel data will be re-tested.

.2.3. Panel cointegration estimationBased on the validated cointegration within a panel of N cross-

ections, PC model can be established. PC can lead to a betternderstanding of the nature among different component series andlso improve long term forecasting with an unconstrained model.

PC model is expressed as follows:

�Pj,t = Pj,t − Pj,t−1 = Cj + ˘ECMj,t−1 +m∑

l=1

˚j,l�Pj,t−l

+N∑

i=1

n∑l=1

�i,j,l�Li,j,t−l + �j,t

where ECMj,t−1 = Pj,t−1 −N∑

i=1

�i,j,t−1Li,j,t−1 + �j,t−1,

∏=

⎛⎜⎜⎜⎜⎜⎝

∏1 ∏

2

. . . ∏J

⎞⎟⎟⎟⎟⎟⎠

Jr×Jr

(7)

here Cj is the common coefficient for each trading period j. Js the number of periods in a day. The dimension of each sub-

atrix ˘j within the diagonal matrix∏

is r × r. The sub-matriceselate �Pj,t to the error correction item ECMj,t−1. �i,j,t−1 is theoefficient for error correction. r denotes the number of regres-ors (variables in the regression) in the explanation item ECMj,t−1hich reflects long-term cointegration relationship for the panel,

.e. it describes a kind of long-term adjustment for deviating thequilibrium relationship. Besides long term adjustment, the PCodel also involves short-term fluctuation items �Pj,t−l, �Li,j,t−l .

he coefficient matrix ˚j,l captures the dynamics within timeomain while �i,j,l correlates the variability of the price withoth the inter-temporal dynamics and the inter-regional interac-ions among regional loads. Adding the error correction features tohe panel data, the disadvantage of traditional forecasting modelshich lose the long-term information collected from variables can

e overcome.Maximum likelihood estimator is employed to predict the

oefficients. A variable elimination procedure is then processedo reduce the redundancy and complexity of the model. Theoefficients with relatively higher statistical significance (based onhe corresponding standard error and T-statistics) are inputted to
he second stage. The filtered parameters of the PC model are usedo formulate the PCPF model. As the cointegration relationship isssumed dynamic, the responses of price to various market funda-entals may change continuously. The coefficients estimated in the
s Research 95 (2013) 66– 76 69

PC model can be regarded as a time-varying process so that the PFcan adaptively give forecasting similar to agents’ learning accordingto subtle rule modifications described in the second stage. It shouldbe noted that the PC model itself can provide electricity price fore-casting directly. The performance comparison between the PC andPCPF models will be investigated in the case study.

3.3. Architecture of the PCPF model

In this paper, a hybrid model is applied to forecast the day-aheadelectricity price using the following state space representation:

Pj,t = X ′j,t

j,t + εj,t Measurement Function

j,t = fj,t(j,t−1, �j,t) Transfer Function(8)

where

εj,t∼i.i.d. N(0, �2εj,t

), �j,t = (�j1t , �j2t , �jkt, . . . , �jKt)′,

�j,t∼Nk(0, ˙j), E(εj,t�j,t) = 0, and ˙j = diag{�2�j,k

}

In the measurement equation, matrix Pj,t represents the uniformday-ahead prices in period j on day t. There are K input variables∈ {1, �Pj,t−l, �Li,j,t−l, Pj,t−l, Li,j,t−l} in each element of the matrixXj,t , K is the number of regressors after the application of a variableelimination procedure carried out in (7). These regressors (vari-ables) will be used for forecasting in the second stage. Notice thatboth εj,t and �j,t follow Gaussian distribution i.i.d. N(0, �2

εj,t) and

Nk(0, ˙j), respectively and independently. In the transfer equation,fj,t(·) is a nonlinear function where j,t will be obtained recursivelyvia PF processing. The coefficients j,t are not unknown constantsbut latent stochastic that follow random walks. PF achieves this byobtaining an optimal approximation of posterior distribution forj,t .

In the first stage, we let PC to model both the inter-temporaldynamics and the inter-regional interactions of uniform day-aheadprice among different regions through the cointegration analysisbased on panel construction; then the residuals from the PC modelwill contain only the nonlinear pattern. To handle the nonlinearpattern and also uncertainties, the coefficients matrix j,t in thetransfer equation is considered as a set of time-varying particlesfollowing random walks so that it enables the coefficients to reactto the arrival of new observations. The disturbance terms εj,t followthe fitting distribution of estimation residues and the variance ofthe process noise can be estimated from the variances of the par-ticles, regarding to the previous PC coefficients at time (t − 1) asfollows:

�2�j,k

= �2�j,k(t−1)

(1g

− 1)

(9)

where g is called as the “forgetting factor” and takes values betweenzero and one. Different from Kalman filter [22], the stochastic vari-ables in j,t are considered nonlinear and estimated by PF for eachperiod j simultaneously on day t.

3.4. Second stage: Particle filter forecasting

Particle filter (PF) is particularly successful in dealing with non-linear and non-Gaussian problems [22,23]. Unlike the extendedKalman filters, which only use the mean and variance to describethe distribution of a state, PF utilizes sequential Monte Carlomethod to approximate the optimal filtering, using particles to rep-
resent the probability density function (PDF) of a state. The numberof particles is equal to the number of regressors described in the lastsub-section. The main task in this stage is to estimate j,t basedon the arrival of the new observations Pj,t so that the day-ahead


Forecasted

StopYes

Estimate

Update

Weights

Propose

Particles…

… t=T?

Initialization(λ j,0 Pj,0:1 t=1 )

YesNeed

Resample?Resampling

Normalize

Observation

price (Pj,t )t=t+1 No

Evaluate the

Effective

Particle SizeKj,λ1

j,λ 2

j,λ

Kj,w1

j,w2

j,w

r forec

ed

wa

p

tavtpt

w

w

dttivifFi

ir

weights

Fig. 3. Particle filte

lectricity price Pj,t+1 can be obtained using (8) for each period j onay t.

A feature of PF is to approximate the posterior distribution ofj,t by a collection of K weighted particles:

= {kj,t, wk

j,t}K

k=1(10)

here wkj,t

is the weight of the particle kj,t

, and the posterior prob-bility of the random event j,t is approximated as follows:

(j,t |Pj,0:t) ≈K∑

k=1

wkj,t−1ı(j,t − k

j,t−1) (11)

We therefore have a discrete weighted approximation to therue posterior p(j,t |Pj,0:t). It is the conditional probability that isssigned to j,t after all available measurements up to day t (abbre-iated as Pj,0:t) have been observed. Dirac-delta function ı(·) meanso perform the integral function. Since it is difficult to draw sam-les from p(j,t |Pj,0:t), importance function q(·) is usually adoptedo generate K particles as follows:

kj,t∼q(j,t |

kj,0:t−1, Pj,0:t) =

K∑k=1

wkj,t−1p(j,t |

kj,t−1) (12)

here the weights in (11) are updated accordingly as:

kj,t ∝ wk

j,t−1

p(Pj,t |kj,0:t , )p(k

j,t|k

j,t−1)

q(kj,t

|kj,0:t−1, Pj,0:t)

(13)

The importance function q(·), known as a proposal conditionalistribution, is important in the performance of the PF. In general,he closer the importance function q(·) to the distribution of p(·),he better the approximation is. The aim of choosing the optimalmportance function, q(j,t |k

j,0:t−1, Pj,0:t), as q(·) is to minimize theariance of the true weights so that degeneracy problem is dimin-shed in one way. The details of choosing the optimal importanceunction can be found in [24] and is outside the scope of this paper.or each period j, the forecasting architecture of the PF is illustrated
n Fig. 3.
The PF algorithm is illustrated for each period j in the follow-ng pseudo-codes and its main task is to estimate the state j,t

ecursively from the observations Pj,t .

price (Pj,t+1 )λ j,t

asting for period j.

(1) Initialization:• Set t = 1, wk

j,t= 1/K

(2) Prediction:For k = 1: K• Generate particles k

j,t∼q(j,t |k

j,0:t−1, Pj,0:t) according to (12)

• Assign each particle an updated weight wkj,t

according to (13)

• Normalize the weights according to {wkj,t

}K

k=1= wk

j,t/∑K

kwk

j,t

• The PDF of kj,t

is approximated so that the estimated electricity pricePj,t+1 can be computed using (8)• End For

(3) Particle size evaluation:• Evaluate the effective size of particles by counting the percentage ofparticles with weights smaller than a certain value. If the percentage isbelow a predefined threshold, Resampling takes place; otherwiseproceed to Iteration

(4) Resampling:• Sample {uk}K

k=1∼U[0, 1] and K particles ̃kj,t

∼p(j,t |Pj,0:t )

• Replace {kj,t

}K

k=1with {̃k

j,t}K

k=1if uk < min{1, p(Pj,t |k

j,0:t)/p(Pj,t |k

j,t)}

• Assign particles an equal weight wkj,t

= 1/K(5) Iteration:

• For t = 1: T − 1• Set t = t + 1, obtain the day-ahead electricity price Pj,t observation andupdate Pj,0:t

• Go to Prediction• End For

A major problem with particles filtering is that the discrete ran-dom measure degenerates quickly. Besides adopting the optimalimportance function, Resampling is also used [15] since it can avoidthe increasing of the variance of the particle weights. In this step,K particles are drawn from the current particle set with probabili-ties proportional to their weights. Particles with higher importanceweights are replicated, while the others are discarded.

4. Case study

4.1. Basic data

Data of PJM’s day-ahead electricity market in the year 2008 isused to test the effectiveness of the proposed PCPF hybrid model.It is a market with 7 regions in which hourly prices are cleared.The inputs include time lagged values of price and loads which areavailable on the PJM’s website [25].

4.2. First stage: Analysis based on panel cointegration model

Data in the period Jan 1, 2007–Dec 31, 2007 are inputted intothe PC model for the coefficients estimation in the first stage. Model

X.R. Li et al. / Electric Power Systems Research 95 (2013) 66– 76 71

Table 1Panel unit root test results.

Method Levin, Lin and Chu* PP–Fisher Chi-square**

Null hypothesis Common panel unitroot probability

Individual panel unitroot probability

Period (j) Li,j,t Pj,t Li,j,t Pj,t

j = 1 0.201 0.000 0.897 0.312j = 2 0.219 0.000 0.906 0.280j = 3 0.216 0.000 0.904 0.065j = 4 0.209 0.000 0.897 0.091j = 5 0.195 0.000 0.888 0.033j = 6 0.182 0.003 0.881 0.000j = 7 0.154 0.011 0.861 0.000j = 8 0.142 0.016 0.854 0.000j = 9 0.154 0.011 0.866 0.001j = 10 0.185 0.026 0.889 0.009j = 11 0.191 0.028 0.892 0.006j = 12 0.151 0.040 0.852 0.034j = 13 0.148 0.033 0.849 0.082j = 14 0.139 0.025 0.841 0.006j = 15 0.156 0.008 0.864 0.014j = 16 0.154 0.006 0.861 0.013j = 17 0.144 0.007 0.851 0.043j = 18 0.163 0.009 0.869 0.058j = 19 0.169 0.007 0.874 0.093j = 20 0.177 0.000 0.880 0.103j = 21 0.188 0.000 0.888 0.190j = 22 0.198 0.001 0.895 0.336j = 23 0.212 0.000 0.902 0.396j = 24 0.257 0.000 0.923 0.468

N

(paoCp

tuinhtl(rcf

otct

Table 3Estimated coefficients of PC model.

Variable 1,t(k) Coefficient Standard error T-statistics

k = 183 0.039 0.020 2.010k = 42 0.002 0.001 1.629k = 177 0.009 0.006 1.588k = 176 0.008 0.006 1.485k = 162 0.014 0.010 1.395k = 182 0.024 0.018 1.373k = 161 0.014 0.010 1.361k = 170 0.010 0.008 1.316k = 36 0.002 0.002 1.295k = 158 0.015 0.011 1.281k = 186 0.031 0.025 1.256k = 157 0.015 0.012 1.214

TP

N

ote: All tests assume asymptotic normality.* Significance at 10% level.

** Significance at 5% level.

6) is used to describe the linear pattern of day-ahead electricityrices based on panel construction. To examine the stationarity ofll the series of the panel variables and clarify the stochastic naturef the price dynamics, unit-root tests employing LLC and PP–Fisherhi-square [26] methods, as shown in Table 1, are used to test theresence of unit roots in the panel data in different time periods.

It can be seen that the results in different periods listed in Table 1end to be consistent. The LLC test assumes that there is a commonnit root process in the panel. This assumption can be rejected

f the probability is less than 0.1. Standing on 10% level of sig-ificance, null hypothesis of a common unit root process (i.e. theomogeneous panel series are non-stationary) can be rejected inhe price series but not in the load series. Similarly, standing on 5%evel of significance, null hypothesis of individual unit root processi.e. the heterogeneous panel series are non-stationary) cannot beejected in all the load series and most of the price series. Hence, itan be concluded that the panel model cannot be directly used fororecasting since not all the included series are stationary.

Based on the empirical results that each panel contains at least
ne panel unit root, JFPC is conducted to examine the cointegra-ion relationships among variables. JFPC is a robust trace basedointegration test with the null hypothesis for the number of coin-egration vectors. Standing on 10% level of significance, Table 2
able 2anel cointegration test results.

Null hypothesis: none cointegration

Period (j) 1 2 3

Probability 0.003 0.000 0.00Period (j) 7 8 9



Probability 0.000 0.000 0.00

ote: The significance of the test is at 10% level.

k = 155 0.016 0.013 1.209k = 185 0.030 0.025 1.194k = 184 0.027 0.022 1.190

shows that all the probabilities for the null hypothesis of nonecointegration are nearly zero. This suggests that cointegration rela-tionship exists in the constructed panel in all the periods. Hence,the coefficients of the PC model (7) can now be estimated.

The derived coefficients ˘, ˚j,l, �i,j,l in the PC model corre-spond to j,t in the PCPF model (8). The coefficients are predicted bymaximum likelihood estimator. Owing to space constraint, only theestimated coefficients with relatively higher statistical significancederived from the PC model (7) in period 1 are listed in Table 3. Thestandard error and T-statistics indicate the contributions of eachvariable to the PC model, which are also the evaluation criteria forselecting coefficients inputted to the second stage.

4.3. Second stage: Particle filter forecasting

As described in Section 3.2.3, the coefficient matrices obtainedin the first stage capture the dynamics of the price with both inter-temporal dynamics and inter-regional interactions among regionalloads. However, due to the nonlinear composition of the electricityprice and its variability, PF is adopted to capture the nonlinear pat-terns according to model (8). The efficiency of the proposed PCPFmodel is mainly dependent on the appropriate adjustments of itsparameters. There are two main adjustable parameters: number ofparticles K and the time-varying coefficient matrix j,t. To balancethe efficiency and accuracy, the size of particles (K) which affects thecomputation time dominantly is better to be controlled between100 and 200. In this study, K is equal to the number of coefficientsj,t after the variable elimination process. Different from the othertechniques, the proposed PCPF model is an adaptive forecastingtool which can automatically adjust the parameters with minimumreliance on the heuristics.

A continuous simulation of point forecasting in all periods from
Jan 1, 2008 to Dec 31, 2008 consumes 4 h and 48 min when the sizeof particles is fixed at 200, which indicates that the average time forone day-ahead forecasting is less than 1 min. All the computationtimes are measured on a Dell 2.66 GHz personal computer with 2 GB
4 5 60 0.001 0.001 0.014

10 11 120 0.000 0.000 0.000

16 17 180 0.001 0.000 0.000

22 23 242 0.000 0.000 0.000


20

60

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220

1 25 49 73 97 12 1 14 5

Pric

e [$

/MW

h]

Time [h]

Spring (17-24 Ja n, 2008)

C

week

Rd

cDsst9os

iocsScb

Actual PCPF P

Fig. 4. Spring

AM. Therefore, the proposed PCPF is practical within a day-aheadecision-making framework.

To illustrate the behaviors of the proposed modeling, resultsomprising of four weeks in January, May, September andecember (months 1, 5, 9, and 12) corresponding to the four

easons in 2008 are presented in Figs. 4–8. In this manner repre-entative results for the whole year are provided. It should be notedhat the accuracy for the weeks of spring and summer are around3% while of autumn and winter are around 95%. Comparing withther relevant studies [1,9,27], these results are accurate for a studypanning one whole year.

To demonstrate the superiority of the proposed model, forecast-ng results of the four selected weeks using the proposed PCPF andther techniques including PC, NN [10], FNN [12] and RVM [9] areompared in Figs. 4–8. The spring week is from 17 Jan to 24 Jan; the
ummer week is from 3 May to 10 May; the autumn week is from 22ep to 29 Sep; the winter week is from 21 Dec to 28 Dec. Note thatriteria for selecting the four representative weeks include unstableehaviors or drastic variations. Although these unsteady behaviors
0

40

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120

160

200

240

280

320

1 25 49 73

Pric

e [$

/MW

h]

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Summer (3-10

Actual PCPF PC

Fig. 5. Summer wee

NN FNN RVM

forecasting.

make forecasting difficult, the proposed PCPF model has good per-formance in daily forecasting with an average DMAPE below 5% ineach studied week.

For the spring week, Fig. 4 shows the comparison of the actualand forecasted prices. The prediction behavior of the proposed PCPFfor the spring week is relatively better with a WMAPE of 7.3% whileWMAPE of the PC technique is 15.6%. Hence, the postprocessor PFreduces the forecasting error by around 50%. Although the PC tech-nique has some over-predictions of the mild spikes (20% over theusual peak), it has a good performance in estimating the transi-tion tendency. The prediction behavior of NN in this representativespring week is less accurate than its counterparts since the pricein this selected week varies considerably. With the adjustmentsprovided by fuzzy logic, FNN (with a WMAPE of 14.0%) improvesthe forecasting performance of NN by around 32%. RVM shows a
moderate forecasting performance with a WMAPE of 14.3%.
Refer to Fig. 5, the reason of selecting this week is because it hasthe highest price in the whole year of 2008 (the peak was betweenhour 109 and hour 121). Owing to the seasonality, the number of

97 121 145e [h]

May, 2008)

NN FNN RVM

k forecasting.


0

20

40

60

80

100

1 25 49 73 97 121 145

Pric

e [$

/MW

h]

Time [ h]

Autumn (22-29 Sep , 2008)

C

n wee

cwW1ptFP6

gctaf1spm6o

Actual PCPF P

Fig. 6. Autum

rest is cut down by one half (i.e. two price peaks reduce to oneithin a day). The predictions of PC and NN are inaccurate withMAPE of 18.0% and 20.3%, respectively. RVM, with a WMAPE of

7.2%, forecasts slightly better than PC and NN. Observe that therice pattern in this week is particularly unstable, probably owingo the strategic behavior of the dominant players in the market.NN, with a WMAPE of 20.3%, reduces the error of NN by 26% whileCPF, with a WMAPE of 7.3%, reduces the error of PC by more than0%.

As shown in Fig. 6, the selected autumn week has relativelyood forecasting behaviors. The peaks have no drastic changes asompared with that appeared in the previous seasons. Because ofhe less anomalous fluctuations and higher autocorrelation in theutumn series, all the three techniques show good forecasting per-ormances. The values of WMAPE for PC, NN and RVM are 6.4%,1.68% and 9.8%, respectively. PC performs better than that in thepring and summer weeks due to the better price pattern. Com-aring with PC, NN and RVM, the forecasting by FNN and PCPF are
ore accurate most of the time. The WMAPE of FNN and PCPF are
.3% and 9.0%, respectively. However, the PCPF would sometimesver-estimate the wild peaks.

10

30

50

70

90

1 25 49 73

Pric

e [$

/MW

h]

Tim

Winter (21-2

Actual PCPF PC

Fig. 7. Winter week

NN FNN RVM

k forecasting.

As for the winter week, the prediction is difficult resultingfrom the significant changes in prices between hours within a day.Because of the seasonality, there are only one low crest and onehigh crest instead of two similar crests within a day. This drasticchange also leads to the difficulties in the forecasting. The valuesof WMAPE for the NN, FNN and RVM techniques are 13.3%, 10.2%and 10.9%, respectively. After a long evolution of particles, PCPFperforms stably with WMAPE of 5.6% while WMAPE of PC is 8.3%.

Fig. 8 shows WMAPEj of PCPF, PC, NN, FNN and RVM methodsfor the four test weeks. The circle axis denotes the plotting hourswhile the vertical axis refers to the error percentage. Among allthe compared techniques, the performance of NN is sometimes farfrom satisfactory. WMAPEj are usually above 15% and deviate fromthe actual price at price peaks or minima. One of the major rea-sons for these problems is that the available data are insufficientat some peaks or minima which represent weak statistical samplesfor an algorithm based on historical-data learning. FNN improvesNN’s performance significantly though it has some large deviations
in some hours. Similar to FNN, RVM forecasts stably except hour17 in spring and winter. WMAPEj of PC is within 30% and usu-ally around 20%. Comparing with PC, PCPF (with WMAPEj within
97 121 145e [h]

8 Dec, 2008)

NN FNN RV M

forecasting.


0

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PCPF PC NN FNN RV M

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2324 Summer

PCPF PC NN FNN RV M

0

0.05

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0.41

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PCPF PC NN FNN RV M

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PCPF PC NN FNN RV M

the fo

1dnhto

aimaihia

Fig. 8. WMAPEj for

0%) has better accuracy and its performance is stable for mostatasets. At some hours, PCPF even reduces the forecasted errorsearly to 1%. PCPF shows strong forecasting capability since itsybrid structure captures different aspects of the underlying pat-erns. This superiority is especially clear when comparing with thether techniques.

The maximum and minimum error percentages, WMAE, WMAPEnd WRMSE of PCPF are compared to those of PC, NN, FNN and RVMn Fig. 9. It can be observed that the highest values of the mini-

um error percentage and maximum error percentage are 0.45%nd 150%, respectively. In terms of the WMAE, WMAPE and WRMSE
ndices, NN are less accurate than the others. PC, RVM and FNNave similar levels of accuracy, while PCPF has the lowest predict-
ng errors. Referring to the computational time, PCPF, PC and RVMre much faster than NN and FNN. In summary, the forecasting

ur selected weeks.

performance of the proposed PCPF model outperforms the othertechniques in all the selected weeks, which shows the usefulnessand practical significance of the postprocessor PF proposed in thispaper.

The superiority of the PCPF technique is more evident whenreferring to Table 4. Notice that the performance of the PCPF tech-nique is generally better than that of the other techniques. Thevalues of WMAE and average WMAPE are smaller in all the sce-narios. Furthermore, FNN and PCPF improve the forecast capabilityof NN and PC through their hybrid structure, respectively. Whencomparing the improvements made by PCPF and FNN, PCPF shows
a stronger error correcting capability than FNN. These results con-firm the intuition that the hybrid model allows particles to evolveadaptively and to sequentially update a priori knowledge aboutsome predetermined state variables given by PC.


Fig. 9. Comparison of indices of the five techniques.

Table 4Summary of forecasting results.

Season Criteria PCPF PC NN FNN RVM

Spring

WRMSE 12.225 17.115 25.982 16.130 16.978WMAE 6.523 13.105 18.145 12.308 12.675WMAPE 0.073 0.156 0.206 0.140 0.143Max Error (%) 61.866 71.361 152.220 52.371 59.197Min Error (%) 0.004 0.018 0.077 0.092 0.457

Summer


Autumn


Winter


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– a case study of greenhouse gas reduction verification and marketing, FuelProcessing Technology 87 (2006) 179–183.

6 X.R. Li et al. / Electric Power

. Conclusion

Nowadays day-ahead electricity market is closely associatedith different commodity markets. Under this complex circum-

tance both inter-temporal dynamics and inter-regional features,s well as uncertainties from different markets due to GENCOs’ bid-ing strategies and investment planning would exist. This paperroposes a hybrid model based on PC and PF techniques to tacklehese difficulties in day-ahead electricity-price forecasting.

Two major contributions of this paper are: (1) a new statisticalethod is introduced to expand the dimension of electricity price

ataset from time series to panel data so that the dynamics of inter-onnected regions can be analyzed simultaneously and considereds a whole; (2) PCPF model is derived and illustrations are madeo show how particle filter can be used to enhance the forecastingccuracy in day-ahead energy market price. In the case study, theroposed method is applied to forecast the PJM day-ahead electric-

ty market. The study, based on the load data of the interconnectedegions, has demonstrated the effectiveness and efficiency of theroposed method.

ppendix A. List of symbols

j,t day-ahead price in period j on day t

i,j,t historical load of region i in period j on day tPj,t, �Li,j,t price and load data processed by difference operator

j,t error term for time-seriesi,j error term for cross-sectioni,j,t error term for cross-section and time-seriesj,t coefficients matrix in PCPFCMj,t−1 error correction term of PC modelj,t , �j,t error term of PC modelj,t disturbance term for time-series in PCPFj,t error term for transfer function in PCPF(·) probability density function(·) importance functionkj,t

particle associated with importance weightkj,t

importance weight for the kth particle

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