短时负荷预测

134 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 27, NO. 1, FEBRUARY 2012 Short-Term Load Forecasting Based on a Semi-Parametric Additive Model Shu Fan, Senior Member, IEEE, and Rob J. Hyndman Abstract—Short-term load forecasting is an essential instrument in power system planning, operation, and control. Many operating decisions are based on load forecasts, such as dispatch scheduling of generating capacity, reliability analysis, and maintenance plan- ning for the generators. Overestimation of electricity demand will cause a conservative operation, which leads to the start-up of too many units or excessive energy purchase, thereby supplying an un- necessary level of reserve. On the other hand, underestimation may result in a risky operation, with insufficient preparation of spinning reserve, causing the system to operate in a vulnerable region to the disturbance. In this paper, semi-parametric additive models are proposed to estimate the relationships between demand and the driver variables. Specifically, the inputs for these models are calendar variables, lagged actual demand observations, and historical and forecast temperature traces for one or more sites in the target power system. In addition to point forecasts, prediction intervals are also estimated using a modified bootstrap method suitable for the complex seasonality seen in electricity demand data. The proposed methodology has been used to forecast the half-hourly electricity demand for up to seven days ahead for power systems in the Australian National Electricity Market. The performance of the methodology is validated via out-of-sample experiments with real data from the power system, as well as through on-site implementation by the system operator. Index Terms—Additive model, forecast distribution, short-term load forecasting, time series. I. INTRODUCTION L OAD forecasting is a key task for the effective operationand planning of power systems. The forecasting accuracy has significant impact on electric utilities and regulators. Over- estimation of electricity demand will cause a conservative op- eration, which leads to the startup of too many units supplying an unnecessary level of reserve or excessive energy purchase, as well as substantial wasted investment in the construction of excess power facilities. On the other hand, underestimation may result in a risky operation and unmet demand, persuading insuf- ficient preparation of spinning reserve, and causes the system to operate in a vulnerable region to the disturbance. Load forecasting is usually concerned with the prediction of hourly, daily, weekly, and annual values of the system demand and peak demand. Such forecasts are sometimes categorized as Manuscript received August 12, 2010; revised January 10, 2011 and March 26, 2011; accepted July 06, 2011. Date of publication August 15, 2011; date of current version January 20, 2012. Paper no. TPWRS-00655-2010. The authors are with the Business and Economic Forecasting Unit, Monash University, Clayton, VIC 3800, Australia (e-mail: Shu.Fan@monash.edu; Rob. Hyndman@monash.edu). 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.2011.2162082 short-term, medium-term, and long-term forecasts, depending on the time horizon. In terms of forecasting outputs, load fore- casts can also be categorized as point forecasts (i.e., forecasts of the mean or median of the future demand distribution), and density forecasts (providing estimates of the full probability dis- tributions of the possible future values of the demand). Various techniques have been developed for electricity demand forecasting during the past few years [1]. Statistical models are widely adopted for the load forecasting problem, which include linear regression models, stochastic process models, exponential smoothing, and ARIMA models [2]–[7]. To incorporate the nonlinearity of electricity demand series, artificial neural networks (ANNs) have also received substantial attention in load forecasting with good performance reported [1], [8]–[14]. Neural networks have been shown to have the ability not only to learn the load series but also to model an unspecified nonlinear relationship between load and weather variables. Recently, machine learning techniques and fuzzy logic approaches have also been used for load forecasting or classification and achieved relatively good performances [15]–[18]. Although substantial attention has been paid to short-term load forecasting, a few researchers have proposed interval and probabilistic approaches for long-term forecast horizons; notably [29], [17], and [6]. Electricity demand nowadays is nonlinear and volatile, and is subject to a wide variety of exogenous variables, including prevailing weather conditions, calendar effect, demographic and economic variables, as well as the general randomness inherent in individual usage. How to effectively integrate the various fac- tors into the forecasting model and provide accurate load fore- casts is always a challenge for modern power industries. The purpose of this study is to develop short-term load fore- casting (STLF) models for regions in the National Electricity Market (NEM) of Australia. This paper follows a regression methodology, but focuses on the nonlinear relationships be- tween load and various driving variables. This study aims to allow nonlinear and nonparametric terms within the regression framework. In particular, semi-parametric additive models are proposed to estimate the relationships between load and the exogenous variables, including calendar variables, lagged load observations, and historical and forecast temperature traces for one or more sites in the target power systems. In addition to point forecasts, forecasting distributions are also estimated based on a bootstrap method. The proposed methodology has been used to forecast the half- hourly electricity demand for up to seven days ahead for power systems in the NEM. The performance of the methodology is validated via out-of-sample comparisons using real data from the power systems. The proposed forecasting software has been 0885-8950/$26.00 © 2011 IEEE FAN AND HYNDMAN: SHORT-TERM LOAD FORECASTING BASED ON A SEMI-PARAMETRIC ADDITIVE MODEL 135 used by Australian Energy Market Operator (AEMO) to forecast half-hourly electricity demand in Victoria and South Australia for system scheduling and planning, with satisfactory accuracy. II. METHODOLOGY A. Model Establishment The proposed semi-parametric additive model is in the regression framework but with some nonlinear relationships and with serially correlated errors. In particular, the proposed models allow nonlinear and nonparametric terms using the framework of additive models [19]. Specific features of the models are summarized below: • previous demand observations are used as predictors; • temperatures from two sites are considered; • temperature effects are modeled using regression splines; • temperatures from the last three hours and the same period from the last six days are considered; • loads from the last three hours and the same period from the last six days are considered; • errors are serially correlated. A separate model for each half-hourly period is fitted. Since the demand patterns change throughout the day, better estimates can be obtained if each half-hourly period is treated separately. This procedure of using individual models for each time of the day has also been applied by other researchers [16], [20], [21]. The model for each half-hour period can be written as (1) where • denotes the demand at time (measured in half-hourly intervals) during period ; • models all calendar effects; • models all temperature effects where is a vector of recent temperatures at one location and is a vector of recent temperatures at a second location; • models the effects of recent demands; • denotes the model error at time . The logarithmic demand, rather than the raw demand, is mod- eled. A variety of transformations of demand from the Box-Cox (1964) class were tried and it was found that the logarithm re- sulted in the best fit to the available data. Natural logarithms have been used in all calculations. 1) Calendar Effects: includes annual, weekly, and daily seasonal patterns as well as public holidays: (2) where • takes a different value for each day of the week (the “day of the week” effect); • takes value zero on a non-work day, some non-zero value on the day before a non-work day, and a different value on the day after a non-work day (the “holiday” ef- fect); • is a smooth function that repeats each year (the “time of year” effect). The smooth function is estimated using a cubic regres- sion spline. A regression spline consists of several polynomial curves that are joined at points known as “knots”. Six knots are chosen at equally spaced times throughout the year for smooth function. The choice of knots is made automatically based on the forecasting performance. 2) Temperature Effects: The function models the effects of recent temperatures on the aggregate demand. Be- cause the temperatures at the two locations are often highly cor- related, these were not used directly. Instead, the average tem- perature across the two sites and the difference in temperatures between the two sites are both used. These will be almost uncorrelated with each other making it easier to use in statistical modeling. Then the temper- ature effects are included using the following terms: (3) where • is the maximum of the values in the past 24 h; • is the minimum of the values in the past 24 h; • is the average temperature in the past seven days; • is the maximum number of lagged days considered in the model; • is the maximum number of lagged half-hourly periods considered in the model. Each of the functions ( , , , , , , and ) is assumed to be smooth and is estimated using a cubic regression spline. In selecting the knots for the cubic regression splines, different knot positions and different numbers of knots are tried. Since it is not feasible to try all the possible knot positions, we consider knots equally spaced in the possible sample ranges, based on the finding that a minor change of knot position has little effect on the model performance. In this way, the computa- tional burden can be effectively reduced without sacrificing the model’s forecasting capacity. The knot positions that provided the best forecasting performance are selected in the model. In this paper, splines with knots at 9, 22, and 29 were used for , , and . For the functions and , knots at 2.2 and were used. 3) Lagged Demand Effects: We incorporate recent demand values into the model. By doing this, some of the serial correla- tions within the demand time series can be captured within the model, and the variations of demand level throughout the time can be embedded into the model as well. Demand effects, , model the effects of recent de- mands using the following terms: (4) 136 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 27, NO. 1, FEBRUARY 2012 where • is the maximum of the values in the past 24 h; • is the minimum of the values in the past 24 h; • is the average demand in the past seven days; • is the maximum number of lagged days considered in the model; • is the maximum number of lagged half-hourly periods considered in the model. Each of the functions ( , , , , and ) is assumed to be smooth and is estimated using a cubic regression spline. 4) Error Term: The error term will be serially uncorre- lated within each model, as the lagged demand terms remove se- rial correlations that might otherwise exist. Note that there will still be a small amount of correlation between residuals from the different half-hourly models. B. Forecasting Distribution Estimation Forecasting distributions convey more meaningful informa- tion than predicted point values, and are valuable in evaluating and hedging the financial risk accrued by demand variability and forecasting uncertainty. Forecasting distributions (and the pre- diction intervals that are obtained from them) provide an indi- cation of the forecast accuracy, and give useful information for system scheduling. They also enable planning and operating de- cisions to be made in a stochastic as opposed to a deterministic context. For instance, the upper bounds of prediction intervals can be used for developing conservative electricity generation plan and schedules, while the use of the lower bounds of predic- tion intervals reflects an optimistic attitude in scheduling with more attention to over-supply avoidance. Generally, there are two kinds of methods used in con- structing forecasting distributions and prediction intervals: parametric methods and nonparametric methods. Selection of the appropriate method usually depends on the model, compu- tational burden, number of available data, etc. Typically, the parametric method assumes the disturbance is an i.i.d normal variate with zero mean and finite variance. Some- times, another distribution can be assumed. In this paper, the normality of the model residuals are tested using the Anderson- Darling test and Pearson chi-square test [23]. The p-values ob- tained from the two tests are extremely small, indicating the model residuals are not normal. Another practical problem when using the parametric method is that the forecasts of the dependent variable usually rely on forecasts of the explanatory variables. For instance, the tem- peratures used for load forecasting in this study are forecasts provided by meteorological service. In such cases, the exact distribution of the load forecasts depend on the distribution of the explanatory variables and so is extremely difficult to deter- mine [24]. A widely used nonparametric approach is the bootstrap. As a distribution-free method, bootstrapping is robust against viola- tions of the normality assumption and offers a promising method of computing a forecast distribution. The bootstrap approach to the construction of forecasting distributions can be summarized as follows: • Estimate the forecasting model using historical data. • Create artificial sample sets by sequentially substi- tuting the randomly resampled residuals into the estimated model. • Re-estimate models based on the artificial sample sets, and obtain simulated forecasts for each forecast horizon by substituting the original data into the re-esti- mated models. • Resample another set of residuals, and substitute these into the original estimated model to obtain the simulated ac- tuals. • The differences between the simulated actuals and the sim- ulated forecasts are the simulated forecast errors, which can be used to construct empirical forecasting distribu- tions, by centering the simulated forecast errors around the point forecasts of the original model. For the distribution forecasts in this work, the above boot- strap implementations will not be directly applicable due to some practical problems. First, there are 48 different models es- timated for each half-hourly period within the day, and up to seven days ahead load forecasts are calculated with multi-pe- riod ahead forecasts in an iterative manner. By using the above bootstrap approach, the computational burden will be extremely heavy, making it impractical for an industrial application like STLF, which requires timely updates in a real-time operation. Second, the load forecasting models use temperature forecasts of up to seven days ahead from a meteorological service, while the information regarding the variances of the temperature fore- casts are not provided, making it difficult to incorporate this ad- ditional source of randomness. To address the above issues, a modified bootstrap method is proposed in this paper. Instead of bootstrapping residuals in the model learning procedure, we focus on the forecasting residuals in the real application with forecasted temperatures, which con- tains the uncertainties from both the models and forecasts of ex- ogenous variables. Specifically, the modified bootstrap method can be summarized as follows: • Estimate the forecasting model using historical data. • Calculate load forecasts using half-hourly temperature forecasts of up to seven days ahead, and calculate the forecasting residuals between the forecasts and the actual demands. • Accumulate the forecasting residuals for a contiguous pe- riod. • Bootstrap the forecasting residuals to obtain simulated forecast errors. Block bootstrapping is used since there are correlations between the forecasting residuals from different half-hourly models, and growing variances also result from multi-step ahead forecasts iteratively derived. • Construct empirical forecasting distributions by centering the simulated forecast errors around the original point fore- casts. • Continue accumulating the historical temperature forecasts and repeat the above procedures every time the model has been updated. The revised bootstrap implementation is fast and straight- forward, and incorporates the randomness from the model and the exogenous variables. A potential limitation of this boot- strap method is that it requires the accumulation of a substantial FAN AND HYNDMAN: SHORT-TERM LOAD FORECASTING BASED ON A SEMI-PARAMETRIC ADDITIVE MODEL 137 Fig. 1. Half-hourly demand data for Victoria from 1997 to 2009. Fig. 2. Half-hourly demand data for Victoria, January 2009. number of historical forecasting residuals before reaching a rea- sonable estimation of the forecasting distribution. In addition, the randomness in the estimated coefficients is not incorporated. III. MODEL IMPLEMENTATION A. System and Data Description The forecasts presented in this paper are for half-hourly “native demand” in Victoria, Australia, being the demand met by both scheduled and non-scheduled generators supplying the Victorian region of the NEM. Each day is divided into 48 half-hourly periods which correspond with NEM settlement periods. Victoria is the second most populous state in Australia. Ge- ographically the smallest mainland state, Victoria is the most densely populated state, and has a highly concentrated popula- tion of about 5.4 million in Melbourne, the state capital, and in nearby areas. Half-hourly demand and temperature data were obtained from the AEMO from 1997 to 2009. Time plots of the half-hourly demand data are illustrated in Figs. 1 and 2, which clearly show the intra-day pattern, the weekly seasonality, and the annual seasonality. Half-hourly temperature data for two different locations in high demand areas (Richmond and Melbourne Airport) have been considered in this study. The relationship between demand and average temperature is shown in Fig. 3, where a nonlinear relationship between load and temperature can be observed. Fig. 3. Half-hourly Victoria electricity demand (excluding major industrial de- mand) plotted against temperature (degrees Celsius). B. Variable Selection A highly significant model term does not necessarily trans- late into good forecasts. Instead, we need to find the best com- bination of the input variables for producing accurate demand forecasts. A separate model of the (1) for each half-hourly period has been estimated, i.e., 48 half-hourly demand models. For each model, the lagged demands, lagged and future temperatures, and calendar variables were selected through a cross-validation pro- cedure [25]. That is, the data have been separated into training and validation sets, and then the input variables are selected by minimizing the accumulated prediction errors for the validation data set. Here the mean absolute percentage error (MAPE) is used as the selection criterion. To select the input variables for the half-hourly demand model, we began with the full model including all demand, temperature, and calendar variabl