Electricity Price Forecasting

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 23, NO. 2, MAY 2008

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A Statistical Approach for Interval Forecasting of the Electricity Price Jun Hua Zhao, Student Member, IEEE, Zhao Yang Dong, Senior Member, IEEE, Zhao Xu, Member, IEEE, and Kit Po Wong, Fellow, IEEE

Abstract—Electricity price forecasting is a difficult yet essential task for market participants in a deregulated electricity market. Rather than forecasting the value, market participants are sometimes more interested in forecasting the prediction interval of the electricity price. Forecasting the prediction interval is essential for estimating the uncertainty involved in the price and thus is highly useful for making generation bidding strategies and investment decisions. In this paper, a novel data mining-based approach is proposed to achieve two major objectives: 1) to accurately forecast the value of the electricity price series, which is widely accepted as a nonlinear time series; 2) to accurately estimate the prediction interval of the electricity price series. In the proposed approach, support vector machine (SVM) is employed to forecast the value of the price. To forecast the prediction interval, we construct a statistical model by introducing a heteroscedastic variance equation for the SVM. Maximum likelihood estimation (MLE) is used to estimate model parameters. Results from the case studies on real-world price data prove that the proposed method is highly effective compared with existing methods such as GARCH models. Index Terms—Data mining, electricity market price forecasting, interval forecasting, support vector machine.

I. INTRODUCTION LECTRICITY price forecasting is essential for market participants in both daily operation and long-term planning analyses, such as designing bidding strategies and making investment decisions. Because the electricity price is a stochastic time series with very high uncertainty [1], it is very difficult to predict the exact value of the future price. Consequently, in addition to predicting the price value, predicting the distribution of the future price, i.e., the prediction interval, becomes highly meaningful. Price intervals can effectively reflect the uncertainties in the predication results. Generally speaking, a prediction interval is a stochastic interval, which contains the true value of the price with a preassigned probability. Because the prediction interval can quantify the uncertainty of the forecasted price, it can be employed to evaluate the risks of the decisions made by market participants.

E

Manuscript received January 16, 2007; revised October 30, 2007. Paper no. TPWRS-00012-2007. J. H. Zhao and Z. Y. Dong are with the School of Information Technology and Electrical Engineering, The University of Queensland, St. Lucia QLD 4072, Australia (e-mail: [email protected]; [email protected]). Z. Xu is with the Centre for Electric Technology, Ørsted*DTU, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark (e-mail: [email protected]). K. P. Wong is with the Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong (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.2008.919309

There are two major challenges for accurately electricity price interval forecasting of the electricity price: 1) to estimate the prediction interval, the value of the future price should be accurately forecasted. However, this is difficult because the electricity price is a nonlinear time series, which is highly volatile and cannot be properly modeled by traditional linear time series models; 2) in addition to the value, the variance of the price should also be accurately forecasted. This is because it is essential to estimate the price distribution so as to estimate the prediction interval. In practice, the price distribution is usually unknown; however, an estimated distribution can be commonly assumed for analysis. In this case, it will be essential to know the variance in order to predict intervals. Unfortunately, forecasting the variance is even more challenging because the variances of the price can be time varying. The electricity price is therefore a heteroscedastic time series [2]. Because of the heteroscedasticity, the variance of the price at each time point should be estimated individually. However, in forecasting electricity price at each time point, we can have only one observation of the price, which is obviously insufficient to estimate its variance. In the past, extensive research has been conducted to forecast the value of various electricity price series. Regression models [3] and neural networks (NNs) [4], [5] have been employed to solve this problem. Wavelet techniques are applied to improve the performance of price forecasting [6]. Time series modelbased approaches including autoregressive integrated moving average (ARIMA) [7] and transfer function models [8], [9] are also applied to forecast the market clearing price. Recently, support vector machine (SVM) has also been employed in electricity market price forecasting and achieved satisfactory results [10]. However, these methods do not have the ability of forecasting the price variance and therefore cannot be directly applied for interval forecasting. Based on the above methods, several techniques are proposed for interval forecasting. In [11], the resampling method is applied to estimate the load distribution and obtain the prediction interval. The major drawback of resampling is that it assumes that the price distribution is fixed, which is not always true in reality where the electricity price is heteroscedastic. In [1], an ARIMA model is employed to forecast the price value, while another one is trained to forecast the future forecast errors. The prediction interval is then estimated based on the price value and the forecasted error with very good results. However, an ARIMA model is basically a linear model, and its ability to accurately model the nonlinear electricity price is limited. The generalized autoregressive conditional heteroscedastic (GARCH) model [2] is widely accepted as an effective time

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series method for forecasting the time-changing variance. Unfortunately, it is also a linear model, hence encountering the similar difficulty as the ARIMA-based approach does in [1]. In this paper, a novel approach is proposed to forecast the prediction interval of the electricity price series. To effectively handle the electricity price, the proposed method is developed to be a nonlinear and heteroscedastic forecasting technique. As a well-known data mining method, SVM is employed to forecast the price value. SVM is considered as a candidate of the best regression technique because it can accurately approximate any nonlinear function. Particularly, SVM has excellent generalization ability to unseen data, and it outperforms many NN techniques by avoiding the over-fitting problem [12]. To deal with the uncertainty in price forecasting, we propose a statistical forecasting model for SVM to explicitly model the price variance and derive the maximum likelihood equation for the model. A gradient ascent-based method for identifying the parameters of the proposed model has also been developed. The established model can be used to forecast both the price value and variance. The prediction interval is then constructed based on the forecasted price value and variance. Comprehensive experiments on real-world datasets have been conducted, through which it is demonstrated that the proposed method can effectively deal with nonlinearity and heteroscedasticity; therefore, it is suitable for electricity price forecasting. The rest of this paper is organized as follows: In Section II, problem formulation and several evaluation criteria are firstly proposed. Afterwards, the proposed nonlinear conditional heteroscedastic forecasting (NCHF) technique is discussed in detail in Section III. Comprehensive case studies are conducted in Section IV to demonstrate that the NCHF model is highly effective. Section V finally concludes this paper. II. PROBLEM FORMULATION In this section, we firstly introduce the concept of heteroscedasticity, and the Lagrange multiplier (LM) test, which can be used to mathematically examine the heteroscedasticity of a time series. The formal definition of the prediction interval is then presented. Finally, three measures are introduced to evaluate the performance of our method. A. Heteroscedasticity and Prediction Interval From the statistical point of view, a time series consists of the observations of a stochastic process. Generally, a time series can be assumed to be generated with the following statistical model: (1) where is the random variable to forecast, and denotes the at time . is a -dimensional observed value of of represents an exexplanatory vector. Each element can also planatory variable which can influence . Note that contain the lagged values of and , because is usually corand the previous noises related with its predecessors . The mapping can be any linear or nonlinear function. According to (1), the time series contains

Fig. 1. Electricity prices of the Australian NEM in May 2005 [15].

two components: is the deterministic component determining the mean of ; and is the random component, which is also known as noise. is usually assumed to follow a normal distribution with a zero mean. We therefore have (2) Because has a zero mean, the mean of is completely determined by and is usually selected as the forecasted [13]. On the other hand, because is a detervalue of ministic function, the uncertainty of purely comes from noise . Therefore, estimating is essential for estimating the uncertainty of . is The statistical model (1)–(2) assumes that the variance constant. This model is therefore named as the homoscedastic of a time series is usually time-changing, model. In practice, of which the characteristic is termed as heteroscedasticity. The formal definition of the heteroscedastic time series is given as follows. Definition 1: Assuming a time series generating model (3) (4) (5) If a time series is generated with the model (3)–(5), it is a heteroscedastic time series [14]. can also be either Similar to linear or nonlinear. Note that the definition of heteroscedasticity in this paper is a generalization of that in [14], because and can be both nonlinear in our model. According to (5), the variance of the heteroscedastic time series is time-changing and determined by the previous noises and the explanatory vector. A good example of a heteroscedastic time series is the electricity price of the Australian National Electricity Market (NEM) as plotted in Fig. 1, where it can be observed that the uncertainty/variance changes significantly in different periods. This observation clearly indicates that, even using the same forecasting technique, market participants may still face different risks in different time periods. Measuring these different risks is essential for market participants to make proper decisions. To verify the speculation from our visual observation, the LM test [16] can be employed to mathematically test the heteroscedasticity of the NEM price series. In the experiments, we

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will verify that the electricity price is heteroscedastic by performing the LM test on real-world price datasets. To quantify the uncertainty of predicting the heteroscedastic price at each time point, we expect to construct a prediction interval, which contains the future value of the price with any preassigned probability. We give the following definition. Definition 2: Given a time series which is generated with model (3)–(5), an level prediction interval (PI) of is a stochastic interval calculated from , such that . is usually assumed to be normally disBecause noise tributed, also follows a normal distribution. The -level prediction interval can therefore be calculated as (6) (7) where is the conditional mean of and usually estimated . In (6) and (7), is the confidence level and with is the critical value of the standard normal distribution. Now to calculate the prediction interval, the only remaining problem is from historical data, which is one of our major to estimate contributions in this paper. B. Performance Evaluation Before proposing our forecasting approach, several criteria are introduced for performance evaluation. Given historical of a time series and the corobservations , mean absolute responding forecasted prices percentage error (MAPE) is defined as

variance of the price series, so as to forecast the PI. To accomplish these objectives, we propose the nonlinear conditional heteroscedastic forecasting (NCHF) model as follows: (11) (12) (13) (14) (15) In the above model, the time series is a nonlinear function of its predecessors , the previous noises , . In (11) and (15), are and the explanatory variables in (11) is slightly difuser-defined parameters. Note that in (1). In (11), does not contain and ferent from that anymore. The variance in the proposed model is assumed to be a linear function of and . Note that (15) is called “linear” not because it is linear in and but rather because it is linear in the parameters and . This is similar to the linear regres, and the sion [18]. Given an observed time series , corresponding observed explanatory variables the objective of the NCHF model is to estimate and the , and . Subsequently, the forecasted mean and parameters variance of the price can be given as

(16)

(8)

(17)

MAPE is a widely used criterion for time series forecasting. It will also be employed to evaluate the proposed method in the case studies. Two criteria are also introduced to evaluate the interval foreof a time casting. Given historical observations and the corresponding forecasted -level prediction series , the empirical confidence [17] intervals and the absolute coverage error (ACE) are defined as

, and are the estimations of , and . Fiwhere nally, the prediction interval can be calculated based on the forecasted mean and variance. By applying this method iteratively, we can easily obtain multiple step forecast. In the proposed NCHF model, the explanatory variables include the market variables relevant to the price, such as demand and generation capacity. The essential difference between NCHF and GARCH is that the proposed model assumes that the price is a nonlinear function of the explanatory variables, while GARCH only models the linear relationship between the price and explanatory variables. Therefore, the proposed NCHF model is able to accurately capture the nonlinear patterns of the electricity price, which cannot be modeled by ARIMA and GARCH. Normal distribution is the basic assumption of many widely accepted/established econometrics time series models, such as ARIMA and GARCH. In this paper, we assume the normal distribution because of the same reason. There is plenty of literature in support of this assumption. A comprehensive discussion on why normal distribution is assumed in statistical and econometrics time series analysis lies beyond the scope of this paper and can be found in [13], [18], and [22]. Moreover, it is demonstrated in [22] that, even if the actual distribution is non-normal,

MAPE

frequency ACE

(9) (10)

where is the number of observations, which fall into the forecasted PI, divided by the sample size. It should be as close to as possible. III. PROPOSED INTERVAL FORECASTING APPROACH A. Intuition Behind Our Approach As stated in the preceding section, the proposed approach should be able to handle nonlinear and heteroscedastic time series. It must be able to accurately forecast both the value and

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the time series models based on normal distribution can still achieve satisfactory performance. In the following sections, the normal assumption will also be demonstrated to be reasonable by obtaining promising results on real price data. The NCHF model (11)–(15) is proposed because of two reasons. First, the main idea of the NCHF model is similar to the model (3)–(5), and it is therefore appropriate in modeling nonlinear and heteroscedastic time series and suits the objective of this paper. On the other hand, model (3)–(5) is modified into model (11)–(15) because of mathematical convenience. In particular, we change (5) into (15) because it is then easier to calculate the likelihood of the model. We will discuss this problem in more detail in following sections. Training of the NCHF model can be divided into two major steps. 1) Any available nonlinear regression technique can be emfrom historical data. ployed in the NCHF to estimate We select SVM because of its excellent ability in handling nonlinearity and over-fitting. 2) and cannot be estimated using a regression technique is unknown. Instead, we debecause the true value of rive the likelihood function for the NCHF model and use the maximum likelihood estimation (MLE) criterion to es, and . The gradient ascent method is used to timate , and , which maximize the likefind out the optimal , and are used as the lihood function. The resultant estimates of their actual values. With the NCHF model, the nonlinear patterns of the price can be well captured by SVM. The heteroscedastic (15) is introduced to model the time-changing variance. Therefore, the NCHF model can effectively handle both nonlinearity and heteroscedasticity, hence satisfying the requirements of interval forecasting of the electricity price. This will be justified in the experiments. B. Support Vector Machine SVM is a data mining method developed by Vapnik et al. at Bell Laboratories [19]. It provides reliable regression functionality for the proposed NCHF model and is briefly reviewed for completeness. This method received increasing attention in recent years due to its excellent performance in both classification and regression. Generally speaking, SVM method is based on Vapnik’s work on statistical learning theory [12]. Unlike previous regression methods such as neural networks which minimize the empirical risk for the learning machine so as to achieve high forecasting accuracy, SVM tries to achieve a balance between the empirical risk and the learning capacity of the learning machine. This idea leads to the principle of structural risk minimization, which is the basis of SVM. Theoretically, SVM is a global optimal learning method and can effectively handle the over-fitting problem suffered by NN and other regression methods [12]. It has been proven that SVM is suitable for estimating the nonlinear functional relationship, and it has excellent performance in electricity market price forecasting [10]. In . Because SVM our research, SVM is applied to estimate is a deterministic method, which cannot handle the price uncertainty, (15) is introduced and we will discuss how to estimate

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 23, NO. 2, MAY 2008

the parameters of the NCHF model, so as to perform interval forecasting. C. Estimating the NCHF Model As introduced in Section III-A, constructing the NCHF model and estimating , and . involves two steps: estimating If we consider as the response variable (the output of SVM) and as the predictor variables (the inputs to can be well approximated by SVM), a nonlinear function SVM as the estimate of . The remaining problem is how we , and for the NCHF model. can estimate In practice, we never know the true values of . Therefore, data mining methods, such as SVM and regression tree, cannot and . be applied to estimate the relationship between , and , MLE, which is a statistical estimaTo estimate tion method, is employed in our approach. The main idea of MLE is to firstly derive the likelihood function, which represents the probability that the historical data can be observed given the NCHF model and a set of its parameters. The parameter values, which maximize the likelihood function, are then selected through an optimization process as the maximum like, and . lihood estimates of be the parameters to be estiFormally, let , we demated. Given the historical time series note the likelihood function of the NCHF model as (18) Likelihood (18) is known as the unconditional likelihood function, which represents the probability that is observed given the NCHF model in (11)–(15) and parameters . However, it is difficult to directly obtain (18) for the NCHF model. We therefore introduce the following lemma to decompose (18). generated by the Lemma 1: Given a time series model (11)–(15), we assume that , and are smaller than and . The following equation holds:

(19) Lemma 1 is based on the Bayesian theory. According to Lemma 1, unconditional likelihood (18) can be obtained by multiplying the unconditional joint distribution of the first observations with the conditional distributions of the last observations. For computational convenience, an alternative likelihood function is employed instead of unconditional as deterministic values likelihood. By considering , the conditional likelihood function is

(20) In (20), because

follows a normal distribution, are treated as constants and the uncer-

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ZHAO et al.: STATISTICAL APPROACH FOR INTERVAL FORECASTING

tainty is introduced only by According to (11), we have

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, which is normally distributed.

(21) where is the estimate of . The conditional density function of can therefore be given as (22) at the bottom of the page. Substituting (22) into (20), we reach the following theorem. as the obserTheorem 1: Denote and the relevant explanatory varivations of a time series ables obtained until time , the conditional log likelihood for the NCHF model is given as (23) at the bottom of the page. The MLE of can now be considered as the value that maximizes (23). To calculate the log likelihood (23), a problem unsolved is how to obtain and . According to (11), we have

method is the gradient ascent optimization. To utilize this optimization method in our approach, we introduce the following lemma. Lemma 2: Given a sample of a time series and explanatory is assumed to be generated variables with the model (11)–(15). Denote

0

[~ zt (~)] =

1; (yt01

0 0

q i=1 q i=1

0 f (y 02 ; . . . y 0 01 ; ~x 01) t

t

i et010i )2 ; . . . (yt0r

p

t

0 f (y 0 01; . . . y 0 0 t

r

t

r

p

;~ xt0r )

i et0r0i )2

the derivative of conditional log likelihood with respect to is thus given by

(24) Therefore, as the estimate of

can be calculated as (27)

(25) Replacing the in (15) with and substituting (25) into (15), can be given as the estimate of

Consequently, based on (27) in Lemma 2, the gradient of the log likelihood function can be calculated analytically

(28)

(26) Given the sample , the log likelihood (26) of the NCHF model can now be calculated via several steps. First, selecting an initial numerical value for and setting as 0, the sequence of condican be iteratively calculated tional variances with (26) and employed to calculate conditional log likelihood (23). Second, an optimization algorithm should be performed to get the MLE of that maximizes (23). A simple optimization

Summarizing the discussions above, the main procedure of training the NCHF model is presented in Table I. D. Time Series Forecasting Using the NCHF Model , and are obtained, foreAfter the estimates of using the NCHF model is casting the PI of a time series straightforward. The estimated mean of , which is also the forecasted value of , can be calculated with (16). The forecasted variance is obtained with (17). Finally, to forecast the PI

(22)

(23)

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Fig. 2. Real electricity prices and the corresponding forecasted prices given by NCHF and ARIMA.

of , we can employ as the estimates of , and use (6) and (7) to obtain the forecasted lower and upper bounds of the PI.

TABLE I MAIN PROCEDURE OF TRAINING THE NCHF MODEL

IV. CASE STUDIES A. Introduction to the Experiments In the experiments, the effectiveness of the proposed NCHF forecasting approach is tested using real-world datasets. The Australian NEM is chosen for the case studies in this paper. Historical and real-time data of the NEM regional reference price (RRP) are published at National Electricity Market Management Company (NEMMCO) website. These data are used as the experimental data in this paper. Detailed information about NEM can be found in [15]. The major objectives of the experiments are: 1) To perform the Lagrange Multiplier (LM) test and verify that the NEM electricity price is heteroscedastic, and the NCHF model is therefore a suitable tool for forecasting such a time series. 2) To compare the NCHF with the ARIMA, which is a wellestablished linear model, we show that NCHF is powerful in handling the nonlinearity of the price series. 3) The GARCH model is a heteroscedastic model and widely used in interval forecasting in financial analysis. We compare the GARCH model with the NCHF model to demonstrate that the NCHF model has a better performance in forecasting PI. 4) User-defined parameters and can influence the performance of the NCHF model. We analyze the impact of and give a discussion on selecting proper and . Some previous work on NEM can be found in [23]. B. Experiment Results The LM test [16] is the standard hypothesis testing for heteroscedastic effects in a time series. The LM test gives two measures, -value and LM statistic, which are the indicators of heteroscedasticity. In particular, the smaller -value is, the stronger heteroscedastic effects are present in the time series. Moreover, we can also conclude that the time series is heteroscedastic when the LM statistic is greater than the critical

value. More detailed discussion of the LM test can be found in [16]. The LM test is performed on five price datasets from the Australian NEM, and the results obtained are shown as follows. As illustrated in Table II, setting the significance level as 0.05 and as 20, the -value of the LM test is zero in all five months. Moreover, the LM statistics are significantly greater than the critical value of the LM test in all occasions. These two facts strongly indicate that significant heteroscedasticity exists in the means that the variance electricity price. In the test, is correlated with its lagged values up to at least . In other words, the electricity price at 20 time units before time can still influence the uncertainty of the price at time . To validate that NCHF is able to handle the nonlinear pattern of the electricity price, we apply both NCHF and ARIMA on the price datasets of May 2005, August 2004, and December 2004

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TABLE II RESULTS OF LM TEST

TABLE III PERFORMANCES OF NCHF AND GARCH ON FORECASTING PI

Fig. 3.

p; q; r

versus ACE in May 2005.

and compare their performances. We present the results of these three months to show that our method is able to deal with the different patterns of different seasons in Australia. These three seasons represent the variation of demand and price over a year in Australia [20]. The data of 01–10 May 2005, 01–10 Aug. 2004, and 01–10 Dec. 2004 are used as the training data for both NCHF and ARIMA. The rest of the data are used as the test data. The experiment results are shown in Fig. 2. As observed, NCHF significantly outperforms ARIMA in all three months. The MAPE of NCHF in May 2005, August 2004, and December 2004 are 5.3%, 7.49%, and 6.16%, respectively. The averaged MAPE of NCHF in these three months is 6.32%, while the averaged MAPE of ARIMA is 14.37%. Moreover, we can clearly observe that the performance of ARIMA collapses when two spikes occur in December 2004. This is because ARIMA is a linear model and therefore cannot capture the nonlinear pattern of the electricity price in a volatile period. On the other hand, NCHF has excellent performance given these spikes. This is a strong proof of our claim that NCHF is able to accurately model the nonlinear pattern of the electricity price series. The major objective of NCHF is to forecast the prediction interval. To prove that NCHF is effective in interval forecasting, we compare NCHF with GARCH on realistic NEM datasets. The GARCH model [2] is a well-established heteroscedastic time series model. It is proven to be effective in modeling the

time-changing variance and forecasting PI in financial time series. The major drawback of GARCH is that it is also a linear model. We compare NCHF with GARCH to verify that NCHF is superior in forecasting PI on nonlinear time series. Similarly, we apply both the NCHF and GARCH models on the price datasets of May 2005, August 2004, and December 2004. The data of 01–10 May 2005, 01–10 Aug. 2004, and 01–10 Dec. 2004 are still used as the training data for both models. The rest of the data are the test data. The expected confidence , empirical confidence , and ACE are shown in Table III. As seen in Table III, the NCHF consistently outperforms the GARCH in all occasions, disregarding how much the expected confidence level is set. The ACE of the NCHF is consistently within 4% and usually around 1% for all datasets, which indicates that the PI calculated by the NCHF is highly accurate. On the contrary, the performance of GARCH is far from satisfactory. The ACE is usually above 20%. These results clearly demonstrate that the NCHF is superior in handling heteroscedasticity and forecasting the PI of the electricity price. This superiority certainly comes from the NCHF’s capability of modeling the heteroscedasticity and nonlinearity of a time series. The 95% level PIs given by both the GARCH and NCHF models are illustrated in Fig. 4. As clearly shown, in all three months, the PIs given by the NCHF perfectly contains the true values of the electricity price, while GARCH has a much worse performance. It should be noted that the GARCH fails in predicting the two spikes in December 2004. On the contrary, these two spikes fall well within the PIs forecasted by the NCHF. This indicates that the NCHF is reliable even with the presence of large price volatility. This characteristic is very important for market participants. In the period with large price volatility, the uncertainty involved in the price is greater, and it will increase the risks of market participants. Market participants are

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Fig. 4. The 95% level prediction intervals forecasted by NCHF and GARCH.

therefore more interested in estimating the uncertainty for decision making. NCHF provides an excellent tool for market participants to analyze the uncertainty of the price given large volatility. , and The NCHF model has three user-defined parameters . To further investigate the performance of the NCHF, another experiment is performed. In the experiment, the expected con%. The price data of 01–10 May 2005 fidence is set as and 11–31 May 2005 are employed as the training and testing data, respectively. The ACE of forecasting results against dif, and are plotted and shown in Fig. 3. ferent values of According to Fig. 3, performance of NCHF cannot be significantly influenced by . By changing from 1 to 8, ACE is always within 1%, which means that ACE is not sensitive to . Different from , ACE rapidly the lagged values of in jumps to 80% by setting a large . This discovery indicates that only the noises of the time points, which are close to time are correlated with . Incorporating more lagged values of can cause over-fitting, thus significantly degrading the performance of the NCHF. Similar to , ACE is also insensitive to according to Fig. 3. However, NCHF achieves a better performance when a small is set. Based on the above observations, we suggest

that small and , which are no greater than 4, should usually be selected. Careful selection of is especially important for obtaining a good performance. A thorough parameter selection . [21] may be performed to search the best values of To further investigate the effectiveness of the proposed NCHF model, the NCHF model is applied to perform 24-h ahead forecasting with the data of June 2005. The results are plotted in Fig. 5. As clearly illustrated in Fig. 5, the performance of NCHF is still promising in 24-h ahead forecasting. The MAPE and ACE are 8.4% and 4.38%, respectively, which are sufficiently good considering that the electricity price is highly volatile. ARIMA and GARCH models are also employed to conduct 24-h ahead forecasting with the same dataset. The MAPE of ARIMA is 16.32%, and the ACE of GARCH is 25.13%, which are clearly worse than NCHF. Moreover, we have also calculated the maximum percentage errors of NCHF and ARIMA, which are 18.6% and 37.84%, in this experiment of 24-h ahead forecasting. The above experiment is based on the data of Queensland market [15] of Australia. The NCHF model is also trained with the NSW market [15] data from November 2004 to January 2005 and tested on the data of November 2005 to January 2006.

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REFERENCES

Fig. 5. The 24-h ahead forecasted price and PI given by NCHF.

No significant difference is observed between the performances on two markets. To investigate the computational efficiency of the proposed method, the computational time of the proposed method on May 2005, August 2004, and December 2004 is compared with GARCH. The proposed NCHF is slightly slower than GARCH in training. Its computational time is only 20% longer than GARCH at most, which is acceptable in practice. On the other hand, NCHF and GARCH have similar computational time in performing forecasting. In summary, the computational efficiency of the proposed method is sufficient for real-world applications. V. CONCLUSION Forecasting the prediction interval of the electricity price is important for market participants to estimate their risks in decision making. Traditional time series methods cannot accurately model the nonlinear pattern of the electricity price. In this paper, we propose a novel forecasting approach, which can handle both nonlinear and heteroscedastic time series and thus is suitable for interval forecasting of the electricity price. Two major contributions of this paper are: 1) we introduced a new statistical model for SVM, which is originally developed as a deterministic data mining method. With the new statistical model, the proposed NCHF model is able to accurately forecast the time-changing variance of the price. 2) We derived the log likelihood function for the proposed NCHF model and demonstrated how to use the gradient ascent method to perform maximum likelihood estimation to obtain the model parameters. Accurate PI can be forecasted based on the forecasted mean and variance of the price given by the NCHF model. Comprehensive experiments on real-world price data are conducted to validate that NCHF is highly effective compared with well-established time series models, such as ARIMA and GARCH. This proposed method provides a useful risk management tool for market participants in a deregulated electricity market.

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JunHua Zhao (S’05) received the bachelor degree from Xi’an Jiaotong University, Xi’an, China, in 2003 and Ph.D. degree from the University of Queensland, St. Lucia, Australia, in 2007. He is now a postdoctoral research fellow at the School of Information Technology and Electrical Engineering, University of Queensland. His research interests include electricity market management and analysis, data mining, statistical methods, and their applications in electricity market research.

Zhao Yang Dong (M’99–SM’06) received the Ph.D. degree from The University of Sydney, Sydney, Australia, in 1999. He is now an Associate Professor at the School of Information Technology and Electrical Engineering, the University of Queensland, St. Lucia, Australia. His research interest includes power system security assessment and enhancement, electricity market, artificial intelligence and its application in electric power engineering, power system planning and management, and power system stability and control.

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 23, NO. 2, MAY 2008

Zhao Xu (S’00–M’06) received the Ph.D. degree in electrical engineering from the University of Queensland, St. Lucia, Australia, in 2006. He is now an Assistant Professor at the Centre for Electric Technology, Technical University of Denmark, Lyngby, Denmark. His research interest include demand side, grid integration of wind power, electricity market planning and management, and AI applications in power engineering.

Kit Po Wong (M’87–SM’90–F’02) received the M.Sc., Ph.D., and D.Eng. degrees from the University of Manchester Institute of Science and Technology, Manchester, U.K., in 1972, 1974, and 2001, respectively. He was with The University of Western Australia since 1974. Currently, he is a Chair Professor of the Department of Electrical Engineering, The Hong Kong Polytechnic University. His research interests include artificial intelligence and evolutionary computation applications to power system planning and operations. Prof. Wong received three Sir John Madsen Medals (1981, 1982, and 1988) from the Institution of Engineers Australia, the 1999 Outstanding Engineer Award from IEEE Power Chapter Western Australia, and the 2000 IEEE Third Millennium Award. He was General Chairman of IEEE/CSEE PowerCon2000. He has been an Editor-in-Chief of IEE Proceedings in Generation, Transmission, and Distribution and Editor (Electrical) of the Transactions of Hong Kong Institution of Engineers. He is a Fellow of IET, HKIE, and IEAust.

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