Space-Time Wind Speed Forecasting for Improved Power System Dispatch Xinxin Zhu 1, Marc G. Genton 1, Yingzhong Gu 2 and Le Xie 2 January 16, 2012 Abstract In order to support large scale integration of wind power, state-of-the-art wind speed forecasting methods should provide accurate and adequate information to enable efficient scheduling of wind power in electric energy systems. In this article, space-time wind forecasts are incorporated into power system economic dispatch models. First, we proposed a new space-time wind forecasting model, which generalizes and improves upon a so-called regime-switching space-time model by allowing the forecast regimes to vary with the dominant wind direction and with the seasons. Then, results from the new wind forecasting model are implemented into a power system economic dispatch model, which takes into account both spatial and temporal wind speed correlations. This, in turn, leads to an overall more cost-effective scheduling of system-wide wind generation portfolio. The potential economic benefits arise in the system-wide generation cost savings and in the ancillary service cost savings. This is illustrated in a test system in the northwest region of the U.S. Compared with persistent and autoregressive models, our proposed method could lead to annual integration cost savings on the scale of tens of millions of dollars in regions with high wind penetration, such as Texas and the Northwest. Key words: Power system economic dispatch; Power system operation; Space-time statistical model; Wind data; Wind speed forecasting. Short title: Improved Power System Dispatch 1 Department of Statistics, Texas A&M University, College Station, TX 77843-3143, USA. E-mail: {xzhu, genton}@stat.tamu.edu 2 Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843-3128, USA. E-mail: Lxie@ece.tamu.edu, gyzdmgqy@gmail.com The work of the first two authors was supported in part by NSF grant DMS-1007504. The work of the last two authors was supported in part by NSF ECCS grant 1029873. This publication is based in part on work supported by Award No. KUS-C1-016-04 made by King Abdullah University of Science and Technology (KAUST)
1 Introduction 1.1 Wind Energy Renewable energy, in particular wind energy, is rapidly penetrating into electric power systems throughout the world. In Demark, wind has become one of the largest electricity sources, supplying 21% of the electricity demand in 2010. In Portugal, 18% of the electricity consumption was generated by wind in 2010, along with 16% and 9% in Spain and Germany, according to the World Wind Energy Association (WWEA, 2010). The US Department of Energy (DOE) published a report in 2008 to describe a model-based scenario in which wind energy provides 20% of the US electricity demand by 2030 (DOE, 2008). China also is pursuing a total capacity of 150 Gigawatts (GW) by 2020, 250 GW by 2030 and 450 GW by 2050 (CREIA, 2010); see the review by Zhu and Genton (2012) for more information about world wide wind energy. Large scale penetration of wind power poses profound challenges to today s power system scheduling due to the high variations and limited predictability in wind. Power system operations require the balance of total supply and demand of electricity in near real-time. Unlike conventional fossil fuel generation, wind power is not fully dispatchable. A coal power plant, for example, can be turned on or off, or can adjust its output based on dispatch commands. However, wind power cannot be fully controlled by power system operators because wind farms cannot increase their power generation when there is not sufficient wind, only reduce their outputs. Also wind cannot be stored like coal, natural gas or atoms for future power generation. Therefore, highly accurate wind speed forecasting and advanced power system scheduling are needed. Otherwise the full potential of wind power would be largely offset by the balancing and ancillary cost provided by fast responsive fossil fuel units. 1.2 Power System Scheduling The basic objective of power system scheduling is to maintain the supply and demand balance 1
at a minimum cost, subject to transmission constraints and plausible contingencies. Before high penetration of renewable energy, such as wind and solar, the uncertainty in power system scheduling primarily came from the demand side (Xie et al., 2011). With the high presence of intermittent wind power, the uncertainty comes from both the demand and supply sides. Load forecasting has been an active area of research for more than four decades. Stateof-the-art load forecasts can achieve high accuracy in the day-ahead stage. In contrast to load forecasting, the variable wind generation is very difficult to be accurately predicted at 24 36 hours ahead, only short-term (hours ahead) prediction is of acceptable confidence; see Monteiro et al. (2009) and Marquis et al. (2011). An overview of the major technical challenges for power system operations in support of large-scale wind energy integration are given by Xie et al. (2011). It is pointed out that the impact of wind power integration appears in scheduling, frequency regulations, and system stabilization requirements, and new computationally efficient methods for improving system performances by using prediction and operational interdependencies over different time horizons are needed. Xie and Ilić (2009) proposed a look-ahead optimal control algorithm for dispatching the available generation resources with the objective of minimizing an objective function comprising of both generation and environmental costs, based on the prediction of the available output from the intermittent resources. Lower economic and environmental costs of generation outputs can be achieved by a look-ahead dispatch algorithm. 1.3 Wind Speed Forecasting Highly accurate wind speed prediction is a crucial solution to reducing the uncertainty from the supply side in the power system scheduling. Compared to long-term prediction, shortterm forecasting is more accurate and reliable, and also it is essential to effective power system operation planning. Hours ahead wind forecasting allows conventional power sources to have enough time to start and provide power as demanded on time. Typically, it is between 2
1 to 10 hours, but for quick resources, it can be under 1 hour; see Genton and Hering (2007). In their discussion, it was pointed out that wind power forecasts by converting wind speed forecasts based on a power curve is more general than predicting wind power generation directly. In this article, the focus is on short-term wind speed forecasting. Extensive research has been devoted to short-term wind speed forecasting, among which statistical models are the most competitive. Unlike black-box approaches such as neural networks and fuzzy logic, statistical models give more interpretable results and achieve high accuracy. Statistical models for short-term wind speed forecasting that incorporate spatial information are found to be more accurate than conventional time series models. Zhu and Genton (2012) reviewed the statistical models for short-term wind speed forecasting from conventional time series models to space-time models. Gneiting et al. (2006) proposed a regime-switching space-time diurnal (RSTD) model to forecast 2-hour-ahead wind speed at Vansycle, Oregon, in the northwest of the US and it outperformed persistence forecasts and autoregressive forecasts, by 29% and 13% in terms of root mean squared error (RMSE) in July 2003, for instance. However, the RSTD model relies on local geographic features. To eliminate these constraints, Hering and Genton (2010) generalized the RSTD model by treating wind direction as a circular variable and including it in their model, coined a trigonometric direction diurnal (TDD) model. The TDD model obtained similar or better forecasting results than the RSTD model without requiring prior geographic information. In order to support large scale integration of wind power, state-of-the-art power forecasting methods should be able to provide accurate and adequate information to help advanced technologies in power system operation utilize wind power in an efficient way. In this article we propose a first attempt to incorporate wind forecasts from a new space-time model into a power system dispatch. In summary, the main contributions of this article are the following: 1) A new space-time statistical model for short-term wind speed forecasting is proposed. This model generalizes the RSTD model by allowing the forecast regimes to vary with 3
the dominant wind direction in each season instead of fixing the forecast regimes based on prior geographic information. In the original application of the RSTD model, it was straightforward to define west and east forecast regimes due to prevailing westerly winds. However, for other situations where the winds have more complicated patterns, the number and position of the forecast regimes are difficult to decide. In the new model, the best position of the forecast regimes are detected by rotating the regime dividing angles until the minimum prediction mean absolute error (MAE) for each season is reached. We name this new model RRSTD for rotating RSTD. 2) We formulate an power system economic dispatch model which incorporates the spacetime wind forecast information. Then numerical simulations are conducted in a representative test system derived from the northwest of the US and the results demonstrate economic benefits from improved wind forecasts. This article is organized as follows. In Section 2, we first introduce our new space-time statistical model for short-term wind speed forecasting, the RRSTD model, and then describe persistence and autoregressive models as reference models along with the TDD model for later comparisons. In Section 3, the newly proposed RRSTD model is then applied to a spatio-temporal wind data set from the northwest of the US. Its prediction MAE values for each month are compared with other models. In Section 4, we propose an economic dispatch model which incorporates available short-term space-time wind power forecasts. An illustrative power system economic dispatch example for the northwest of the US is presented, which quantifies the potential savings in both generation costs and ancillary services in the proposed dispatch model. Concluding remarks are provided in Section 5. 2 The Rotating RSTD Model In this section, we propose a new space-time statistical model for short-term wind speed forecasting problems, the RRSTD model. This model is motivated by the need to generalize 4
the aforementioned RSTD model, which builds separate forecasting models for each fixed forecast regime predefined by the user and based on local geographic information. In the RRSTD model, we define variable forecast regimes to allow the forecast regimes to vary with the dominant wind direction and with the seasons. Detailed descriptions of the RRSTD model are given in this section, along with a brief introduction to two reference models and the aforementioned TDD model for later comparisons. 2.1 RRSTD Model Description Let y s,t and θ s,t be wind speed and direction at site s and time t, where s = 1,..., S, t = 1,..., T, and θ s,t [0, 360 ) with 0, 90, 180, 270 indicating southerly, easterly, northerly and westerly winds, respectively. The objective is to predict y s,t+k, the k-step-ahead wind speed at site s, k = 1, 2,.... When k = 1 for example, depending on the resolution of the wind data, it is 1-hour-ahead forecasting for hourly wind data, and 10-minute-ahead forecasting for 10-minute wind data. To simplify, we illustrate the RRSTD model in the setting of forecasting wind speed k-step-ahead at a site s 1. Since wind speed is non-negative and has large values with low probabilities (right skew distributed), we assume that Y s1,t+k follows a truncated normal distribution (Gneiting et al., 2006) with a center and scale parameters µ s1,t+k and σ s1,t+k: Y s1,t+k N + (µ s1,t+k, σ s1,t+k). To predict y s1,t+k precisely, the key lies in modeling µ s1,t+k and σ s1,t+k appropriately. Generally, seasonal and diurnal patterns are seen in winds. In the RRSTD model, we fit the diurnal pattern with two pairs of harmonics as D s1,h = d 0 + d 1 sin ( 2πh ) + d 2 cos 24 ( 2πh ) + d 3 sin 24 ( 4πh ) + d 4 cos 24 ( 4πh ), (1) 24 where h indicates the hour of a day, h = 1, 2,..., 24, and the coefficients are estimated by the least squares method. Then, the center parameter is modeled as µ s1,t+k = D s1,t+k + µ r s 1,t+k, (2) 5
where µ r s 1,t+k is the residual wind speed after removing the diurnal pattern. The residual, µ r s 1,t+k, is modeled by a linear combination of historical wind speed residuals, up to p-step lags, of itself as well as its neighbors (to take the spatio-temporal correlations in wind into account), allowing the linear coefficients to vary with the dominant wind direction and season by defining variable forecast regimes: S p ( µ r s 1,t+k = α 0 + α s,j θs,t, θm(t+k)) µ r s,t j, (3) s=1 j=0 where α 0 and α s,j, s = 1,..., S and j = 0,..., p, are linear coefficients. Here θ m(t+k) [0, 360 ) defines the forecast regimes based on the prevailing wind direction in the season, m(t + k), that time t + k belongs to. Here θ s,t is the current wind direction at site s used to indicate the direction of nearby future wind. The site s {1,..., S} is located at the upstream of wind and indicates the wind source; see Section 3.3 for details on its selection. The meaning of the above model is that, for a certain season, if the future wind direction at the target site, which is estimated by θ s,t, falls into a predefined forecast regime, a particular space-time linear model will be applied to estimate µ r s 1,t+k, and the forecast regimes are based on the dominant wind in that season. For example, if θ m(t+k) = {26, 206 } Aug and s = s 2, then, in August the RRSTD model fits two separate models for the center parameters µ r s 1,t+k : model 1, when the current wind direction at site s 2 is between 26 and 206, or θ s2,t [26, 206 ); model 2, when θ s2,t [206, 360 ) [0, 26 ); see Figure 1 (left panel). The dimension of θ m(t+k) indicates the number of regimes that are defined. For θ m(t+k) = {60, 196, 295 } Aug and s = s 2, three separate models are built for the three forecast regimes divided by these angles; see Figure 1 (right panel). The scale parameter σ s1,t+k is modeled as where b 0, b 1 > 0 and v s1,t is the volatility value: v s1,t = { 1 2S σ s1,t+k = b 0 + b 1 v s1,t, (4) S s=1 1 } 1/2. (µ r s,t i µ r s,t i 1) 2 i=0 6
The key point of the RRSTD model is how to decide the number and the position of the regimes. For locations that have significant prevailing wind, this can be determined practically. The RSTD model is a special case of the RRSTD model with θ = {0, 180 }, motivated by the westerly prevailing wind in the northwest area of the US. For other situations, we propose that θ m(t+k) be chosen by minimizing the prediction MAE for each season/month after deciding the number of regimes. The predictors in equation (3) are selected by the Bayesian Information Criterion as in Hering and Genton (2010). The coefficients in (3) along with b 0, b 1 in (4) are estimated by means of the continuous ranked probability score method; see Gneiting and Raftery (2007) for more details. With the estimated predictive distribution N + (µ s1,t+k, σ s1,t+k), we take the median of the truncated normal distribution as the wind speed forecast k-step-ahead at s 1, defined as ( z 0.5 + = µ s1,t+k + σ s1,t+k Φ {1 1 µs1 )}/,t+k + Φ 2, σ s1,t+k where Φ( ) is the cumulative distribution function of a standard normal distribution. 2.2 Persistence, Autoregressive, and TDD Models To evaluate the performance of the RRSTD model, we compare its forecasts to other models, including the persistence (PSS), autoregressive (AR), and the TDD models, by the prediction MAE values in Section 3 and the economic benefits to power system dispatch in Section 4. The main ideas of the three reference models are introduced briefly: PSS assumes the future wind speed is the same as the current one, or ŷ s1,t+k = y s1,t. An AR(p) model estimates µ r s 1,t+k in (3) as a linear combination of the previous p wind speed residuals from the same location only, or µ r s 1,t+k = α 0 + p i=0 α iµ r s 1,t i. For the scale parameter, a GARCH(1,1) model is used instead of (4). The TDD model is similar to the RRSTD model except that (3) has the form: S p S p µ r s 1,t+k = α 0 + α s,j µ r s,t j + {β s,j cos(θs,t j) r + γ s,j sin(θs,t j)}, r (5) s=1 j=0 7 s=1 j=0
where θ r s,t is the residual of wind direction after removing a diurnal pattern fitted similarly to the formulation in (1). The PSS is reasonable in short-term wind speed forecasting problems because wind lasts in time. However, due to the high variation in wind, it works better for very short-term forecasting, such as 10-minute-ahead prediction. The AR(p) model predicts future wind speed by a linear combination of historical wind speeds based on a stationarity assumption. It can capture the temporal correlation in wind patterns and usually outperforms the persistence in short-term wind speed forecasting problems. The TDD model includes wind direction as circular variables to take into account the spatial correlations in wind and it was found to be more accurate than the PSS and AR(p) models in 2-hour-ahead wind speed forecasting, and similar or better forecasts than the RSTD model; see Hering and Genton (2010). 3 Numerical Experiments 3.1 Wind Data The data sets considered here are 10-minute wind speed (m/s) and direction (degrees) records from three meteorological towers located at Vansycle (Oregon), Kennewick (Washington), and Goodnoe Hills (Washington) in the northwest of the US. The period is from 1st August to 30th November 2002, and from 25th February to 30th November 2003. Missing data are imputed by linear interpolation described in Gneiting et al. (2006). This area has westerly prevailing winds due to its special geographical features. Vansycle is close to the Stateline Wind Energy Center, 39 km away from Kennewick to the southeast and 146 km away from Goodnoe Hills to the east. Detailed information about the data and the three sites can be found in Gneiting et al. (2006). These three locations are along the Columbia River, which runs from east to west along the boundary of the states of Washington and Oregon, with high terrain in both north and south. 8
Due to the relative position of the 3 sites and the westerly prevailing winds, the winds at the three locations are spatially correlated. When wind is blowing from west, Goodnoe Hills is at the upstream while Vansycle and Kennewick are at the downstream. Depending on the wind speed, the future wind speed at the latter two sites will be highly correlated with the wind blowing at the former site currently. This indicates that including wind information from neighbors is of great benefits in improving wind speed prediction accuracy regarding to the setting considered here. The data are divided into two parts as follows: Training set: data from 1st August to 30th November 2002. The regime dividing angles θ in (3) are trained by minimizing the prediction MAE values, as well as s, the wind source indicator. The linear models for the center parameter are obtained for each forecast regime and monthly, in order to take into account the seasonality in wind. Testing set: data from 25th February to 31st November 2003. The trained models are evaluated during this period. The parameters in the models are estimated by the data up to 45-day earlier as suggested by Gneiting et al. (2006). 3.2 Exploratory Data Analysis In this subsection, an exploratory analysis is carried out on the relationship between wind speed and wind direction intending to determine the number of forecast regimes and how to divide the regimes in the RRSTD model. The wind roses in Figure 2 give a view of how wind speed and wind direction are distributed each month from August to November 2002 at Vansycle (top row), Kennewick (middle row), and Goodnoe Hills (bottom row). In a wind rose, each petal indicates the frequency of winds blowing from a particular direction and the color bands in each petal show the ranges of wind speed. As we can see from the wind roses, at Goodnoe Hills (bottom row), which is located along the north bank of the Columbia River, there are high frequencies and large speed ranges in the west direction in August, September, and October. This indicates that in 9
these months winds are mostly from the west and the wind speeds are higher than those from other directions. In November, the east wind has higher frequency than other months and it balances the westerly wind. These wind patterns are consistent with the comment in Subsection 3.1 that the dominant wind is westerly. Based on this information, two forecast regimes would be appropriate for each month to forecast wind speed at Goodnoe Hills. At Vansycle (top row), northwesterly wind has high frequency in all of the four months. Also wind speeds are larger in directions with higher frequency. In August, September and October, most of the winds blow from west, while there are low frequencies of easterly wind. In November, a spike appears in the southwest direction with wind speed in the main range from 0 to 5 (m/s) and the east and northeast winds have slightly higher frequencies than other months. This also indicates that two forecast regimes would be appropriate for each month to forecasting wind speed at Vansycle. At Kennewick (middle row), high frequency of wind happens in the north direction slightly to west in all four months and the other frequencies concentrate in the opposite direction. Since Kennewick is along the southwest bank of the Columbia River, these wind patterns are consistent with the geographic features near this location. Here again, two forecast regimes would be appropriate for each month to forecasting wind speed at Kennewick. Based on the discussion above and similarly to Gneiting et al. (2006), we consider two equally divided regimes resulting in a two-regime RRSTD model. The two regimes are divided by a diameter in Figure 1, θ m(t+k) degrees away counterclockwise from the south. 3.3 Training Data Results In this subsection, the two-regime RRSTD model for each month is trained by minimizing the prediction MAE of the forecasts in the training set at all three locations for 1-hour-ahead and 2-hour-ahead forecasting problems. The gains of using the RRSTD model rather than the RSTD model are presented by the MAE values in each month. 10
Since the best two-regime forecasting design in the RRSTD model is defined as the one that results in the minimum MAE value, we calculate the MAE value resulting from a dense number of possible two-regime forecasting designs, and get the best two-regime RRSTD model as the one that reaches the smallest value of MAE. This is equivalent to rotating the dashed line in Figure 1 until we get the smallest value of MAE, and the resulting two regimes are the best forecast regimes. To illustrate the training procedure, we describe in detail how the two-regime RRSTD model is trained to predict 1-hour-ahead wind speed at Vansycle in August, while implementations are similar for other months, locations and forecasting horizons. To simplify the exposition, we define some notations based on this particular wind data set. The label for site s has values V, K and G. As a result, the wind speeds and directions at time t at Vansycle, Kennewick and Goodnoe Hills are denoted by y V,t, y K,t, y G,t, θ V,t, θ K,t, and θ G,t, respectively, and the goal here is to predict y V,t+6 (1-hour-ahead is equal to 6-step-ahead for 10-minute data). The objective of the training procedure is to find the best two-regime dividing angle θ m(t+6) among the set of possible two-regime dividing angles δ = 1, 2,..., 180, (where m(t + 6) is August in this setting), the wind source indicator, and the models for each forecast regime in (3). Specifically, the training procedure is as follows: 1) Let δ = 1, which results in two equally divided forecast regimes. 2) Find the corresponding s. Divide the series of 10-minute wind speed data at Vansycle in August {y V,t } into two subsets based on the wind directions 1-hour-behind at Vansycle: {y V,{t:θV,t 6 regime1}} and {y V,{t:θV,t 6 regime2}}, and calculate the difference between the two medians of wind speeds in the subsets. Repeat this based on the 1-hour-behind wind directions at Kennewick and Goodnoe Hills. Then s is the site that has the largest value of difference. For example, for δ = 8, s = G, or Goodnoe Hills; see Figure 3. 11
3) Given s, divide the wind speeds in August {y V,t } into two subsets: {y V,{t:θs,t 6 regime1}} and {y V,{t:θs,t 6 regime2}}. For each subset, fit the predictive distribution N + (µ V,t+6, σ V,t+6 ) through equations (2) to (4) and use its median as the point forecast of y V,t+6. 4) Calculate the MAE value of the forecasts in August. The above procedure is carried out for δ = 0, 2,..., 180, resulting 180 corresponding prediction MAE values, as shown in Figure 4. The blue dashed line indicates the position of the best two-regime dividing angle, or θ, that has smallest MAE value. From the plots, we can see that there is no significant pattern in the MAE values when the dividing angle changes from 0 to 180. In August, the MAE values have several dramatic falls and downs, and small values happen when δ is around 8 to 20, as well as 120 to 130 and the smallest value is at 8. In September, October and November, the MAE curves are smooth in the middle part and have small values at the two ends, and the smallest values are at 9, 175 and 29. The resulting two regimes are close to the west-east regime, which are reasonable according to the prevailing westerly wind. The training results of the two-regime RRSTD model for 1-hour-ahead and 2-hour-ahead wind speed forecasting in the 4 months at Vansycle, Kennewick and Goodnoe Hills are shown in Table 1. For 1-hour-ahead forecasting at Vansycle, all the s have the same value G, or Goodnoe Hills, which is consistent with the trained two forecast regimes. The trained two forecast regimes in each month are all close to west-east regime, indicating that the dominant wind is from west. In this case, Goodnoe Hills is at the upstream and is a good indicator of the wind source for Vansycle. Smaller or equal prediction MAE values are achieved by the RRSTD model for all of the four months compared to the RSTD model. This means that although the westerly wind dominates this area, a simple west-east forecast regime may not be the best forecasting model. Since wind power generation is the cubic of wind speed, small improvements in wind speed are valuable. The wind roses in Figure 2 show how the two-regimes are adjusted with wind direction and season, from the RSTD model 12
(the vertical dotted line) to the two-regime RRSTD model in 1-hour-ahead (dashed lines) and 2-hour-ahead forecasting problem at all of the three locations. The resulting linear models for the residual of the center parameter in (3) in predicting 1-hour-ahead wind speed at Vansycle from August to November 2002, or µ r V,t+6, are: August: September: 8 >< µ r V,t+6 = α 0 + P 1 j=0 α V,jµ r V,t j + α Kµ r K,t if θ G,t [8, 188 ), >: β 0 + P 4 j=0 β V,jµ r V,t j + β Kµ r K,t + β Gµ r G if θ G,t [188, 360 ) [0, 8 ), 8 >< µ r V,t+6 = α 0 + P 3 j=0 α V,jµ r V,t j + P 1 j=0 α K,jµ r K,t j if θ G,t [9, 189 ), >: β 0 + P 1 j=0 β V,jµ r V,t j + P 2 j=0 β K,jµ r K,t j + P 6 j=0 β G,jµ r K,t j if θ G,t [189, 360 ) [0, 9 ), October: 8 >< µ r V,t+6 = α 0 + α V µ r V,t + P 2 j=0 α K,t jµ r K,t j + P 3 j=0 α G,jµ r G,t j if θ G,t [0, 175 ) [355, 360 ), >: β 0 + β V µ r V,t + β Kµ r K,t + β Gµ r G,t if θ G,t [175, 355 ), November: 8 >< µ r V,t+6 = α 0 + P 7 j=0 α V,jµ r V,t j + α Gµ r G,t if θ G,t [29, 209 ), >: β 0 + P 5 j=0 β V,jµ r V,t j + β Kµ r K,t + P 1 j=0 β G,jµ r G,t j if θ G,t [209, 360 ) [0, 29 ). (6) (7) (8) (9) 3.4 Testing Data Results In this subsection, the trained two-regime RRSTD models for each month are applied to forecast wind speed in the same months in the testing set at all 3 locations, as well as the PSS, AR and TDD models. The prediction MAE values from different models are compared. In the testing procedure, the two-regime RRSTD models for the months in the training set are applied to the same months in the testing set based on the assumption that the monthly pattern in wind remains similar from one year to another. Specifically, the trained models in (6) to (9) are implemented to forecast 1-hour-ahead wind speed at Vansycle from August to November. Due to the lack of data from May to July in the training set, the two-regime RRSTD model of August is used to forecast wind speed from May to July along with August in the testing set, since these months are all in the same summer season. 13
The assumption that monthly patterns remain similar for training and testing sets is challenged by the high variations in wind and the limitation of available data. This means that the monthly wind pattern in the training set may be different from that in the testing set and at least several years of wind data would be needed to model the monthly patterns accurately. Given that the two-regime RRSTD model is still reasonable for the training data, the best two-regime dividing angles trained from August to November 2002 may not be the best dividing angles for the months in 2003. As a result, the forecasts may not be as good as one would expect. In order to evaluate the potential performance of the RRSTD model with our data set in the northwest of the US, an oracle RRSTD (ORRSTD) model is constructed. The main idea of the ORRSTD model is based on the following questions: if the true wind speeds in the testing set were known, what would be the best two-regime dividing angles and what would be the resulting wind speed forecasts for each month? To do this, two cases are studied based on including information from the training set or only using the testing set. For the first case, only the s and model in (3) are trained from the training set for each dividing angle each month, while the best dividing angle is chosen to be the one that has the smallest prediction MAE value in that month of the testing set instead of the training set. For the second case, the s and model in (3) are trained from the 45-day training period and the dividing angle is defined as the same as the first case. These two cases of the ORRSTD model are denoted by ORRSTD02 and ORRSTD03, based on whether the training results are from the data in 2002 or 2003. The MAE values of 1-hour-ahead forecasts at Vansycle in each month of the testing data from ORRSTD02 and ORRSTD03 with the two-regime dividing angle ranging from 0 to 180 are plotted in Figure 5. The dashed vertical lines indicate the positions of the best two-regime dividing angles. Comparing plots of August to November in Figures 4 and 5, we can see the differences in how the MAE values change with the two-regime dividing angle, which indicate the different monthly patterns in wind between 2002 and 2003. 14
Wind roses in Figure 6 show the difference of two-regime divisions in different models for each month in the testing data. The vertical dotted lines give the west-east RRSTD model, or the RSTD model, the blue dashed line gives the two-regime RRSTD models with minimum prediction MAE value for each month, the pink dashed line is from the ORRSTD02 model and the cyan dashed line is from the ORRSTD03 model in the 1-hour-ahead forecasting problem. From May to August, there are not many differences in the two-regime division between the ORRSTD02 and ORRSTD03 models, which have a northeast-southwest tworegime division, and are away from the dividing angle of the RRSTD model. In September, the resulting two-regime from the three RRSTD models are different, with 9, 136 and 64 from the south, respectively. In October and November, the dividing angle in the ORRSTD02 model is close to the one in the RRSTD model and away from that in the ORRSTD03 model. The training results of ORRSTD02 and ORRSTD03 are shown in Table 1. For 1-hourahead forecasting at Vansycle, the values of s are different from those of the RRSTD model, which have Goodnoe Hills as the wind source indicator for all months in the training set. In the ORRSTD02 model, Vansycle is chosen to be the wind source indicator from May to September. This is reasonable according to the two-regime divisions for these months, which are mainly northeast and southwest forecast regimes, and since Vansycle is located in the southeast of the area of interest. In October and November, the wind source indicators are Kennewick and Goodnoe Hills in the ORRSTD02 model. Differences in training results are also found for other locations, which indicate that due to wind variation, the RRSTD model trained in 2002 may not be as good as one would expect to predict wind speed in 2003. The ORRSTD03 model is trained from data up to 45-day earlier to forecast wind speed at time t + k, which allows the model structure to vary with t instead of a fixed model for each month. This means that there is one s value for each t, and due to space limitation, the s values in the ORRSTD03 model are not listed in Table 1. Then 1-hour-ahead and 2-hour-ahead forecasts at all three sites in the testing set are performed by the PSS, AR(p), RSTD, TDD and RRSTD models, as well as the oracles, 15
ORRSTD02 and ORRSTD03. The AR model is fitted with order at most 9 based on the Akaike Information Criterion. Tables 2 display the prediction MAE values of the forecasting results from the above models for each month as well as the overall MAE values. From the results, we can see that space-time models always achieve better forecast results than the PSS and AR models with smaller MAE value. The differences between the spacetime models are very small. On average, the results from the TDD model are slightly better than those from the RSTD and RRSTD models. The results from the ORRSTD02 and ORRSTD03 models have smaller MAE values than those from the TDD and RSTD model. This indicates that, if the assumption that monthly patterns between each year were similar or if there were enough data to model the monthly patterns appropriately, then the RRSTD model would be able to improve the forecasting accuracy from the RSTD and TDD models. 4 Integrating Wind Power into a Power System In this section, we propose a first attempt to incorporate space-time wind forecasts into a power system dispatch. In terms of potential economic benefits in the system-wide generation cost saving, as well as the ancillary service cost saving, the performances of the aforementioned forecasting models are compared. First, a test system based on the Bonneville Power Administration (BPA) system, which covers the area where the wind data were collected, is introduced and studied for power system operation with space-time wind forecasts. Second, we formulate the power system dispatch problem which incorporates advanced spatio-temporal correlated wind forecast. Finally, an illustrative example is simulated and analyzed, and the results from different forecasting models are compared. 4.1 Power System Specification in the BPA Region Power system economic dispatch is used by system operators in generation scheduling, which determines the generators output so as to maintain the supply and demand balance, as well 16
as to minimize the total system operating cost while satisfying the security constraints. In this subsection, a detailed procedure of power system dispatch is introduced based on the BPA system which covers the area of Vansycle, Kennewick and Goodnoe Hills. Established in 1937, BPA is a nonprofit agency located in the Pacific Northwest area. About one-third of the electric power used in the northwest comes from BPA. BPA also operates and maintains about 75% of the high-voltage transmission network (15,212 circuit miles) in its service territory (BPA, 2010). The service territory of BPA includes Idaho, Oregon, Washington, western Montana and small parts of eastern Montana, California, Nevada, Utah and Wyoming. The major missions of BPA in operating electric energy are two folds: 1) act as an adequate, efficient, economical and reliable power supply; and 2) maintain a transmission system that is capable of integrating different kinds of power resources, providing electricity service to BPA s customers, providing interregional interconnections, and maintaining electrical reliability and stability. In order to balance the power demand, the output of every generator in the power system has to be dispatched over different time frameworks (i.e. day-ahead, hour-ahead, and 5-10 minutes-ahead). This process is called generation scheduling. The BPA scheduling procedure is shown in Figure 8. In the generation scheduling process, the system operator at BPA will schedule generators to meet the expected demand at several time scales. All the scheduled generation must be within their output capacity, as well as within their ramping capacity which refers to the maximum change of generation output between two consecutive time intervals. For example, a natural gas generator s ramping capacity can be 15% of its maximum output in 10 minutes. Given the fact that it takes several hours to start up or shut down many large generators (e.g. nuclear, coal), a day-ahead schedule (or pre-schedule) process is required to plan the generators operations over the next 24 hours. Based on day-ahead forecast, the pre-schedule is completed before 2 pm the day before the day-of-delivery (or the day on which the real- 17
time operation takes place). However, the day-ahead load forecast and day-ahead wind forecast have relatively low accuracies so a real-time schedule which is one hour-ahead in BPA is required to pick up the mismatch between the near-term forecast and the day-ahead forecast. Real-time schedule is established on hour-ahead forecast, which has to be completed 20 minutes before the hour-of-delivery (the hour when real-time operation takes place). Within each hour, the available wind generation, as well as the electricity demand, still varies from second to second. Such imbalance between total supply and total demand will cause the degradation of the electricity frequency, which has very stringent requirement for the safety of many appliances. In order to maintain the system electrical frequency at 60Hz, automatic feedback control loops are installed at many generators speed governors, which is referred to as the automatic generation control (AGC) mechanism. Such a mechanism is very similar in principle to the cruise speed control in automobiles. 4.2 A Power System Dispatch Model with Space-Time Wind Forecasts The power system economic dispatch problem is formulated in this subsection. Table 4 lists the notations and the mathematical formulation of the model is described as follows: min : C Gi (P Gi ) + C Wi (P P Gi,PW k W k i ) + C Ri (P Ri ) (10),P Ri i i G i W i G subject to P Gi + PW k i = P Di, (11) i G i W i D P Ri R D + R W, (12) i G F F max, (13) P Gi P 0 G i P R i T, i G W, (14) P min G i P Gi P max G i, (15) 18
P min G i 0 P Ri P max G i, (16) P Gi + P Ri P max G i, (17) P min W i P k W i P max W i, (18) P k W i ˆP k W i. (19) In the proposed formulation, the objective function (10) is to minimize the power system operating costs which include costs of generation and costs of providing reserve and regulation services. The decision variables include the generation dispatch point for each generator P Gi, the dispatch point for wind generators PW k i, the dispatch point for regulation and reserve capacity P Ri. Constraints of this problem are system and individual units operating constraints for security and reliability purposes. The energy balance equation (11) requires the total generation to be always equal to the total load in steady state. The system reserve and regulation requirements (12) are determined by reliability requirement component of load R D and reliability requirement component of wind generation R W. The load component is a linear function of actual system-wide load level for each interval as a practice of major independent system operators (ERCOT, 2010). The wind component is given by the deviation between the actual wind generation production potential and the wind generation forecast. Although this is not exactly the way how a system operator estimates the reliability requirement caused by wind forecast uncertainty, this is what the system operator wants to estimate, which serves as a lower bound for the reliability requirement due to wind forecast uncertainty. The transmission line capacity limitations (13) contribute to network transmission congestion. The ramping constraints of generators are described by (14). The upper bounds and lower bounds of conventional generators outputs are provided by (15). The available reserve and regulation capacity constraints are given by (16). The system reserve is determined by a reliability requirement component associated with load R D and a reliability requirement component associated with wind generation R W. The load component is a linear function of actual system-wide load level for each interval as a practice of 19
major system operators. The wind component is given by the deviation between the actual wind generation production potential and the wind generation forecast. This approach of quantifying system reserve requirement approximates the actual empirical-based approach to quantifying reserve requirement, and serves as a lower bound for the reliability requirement due to wind forecast uncertainty. The capacity constraints of each generator for providing both energy and reserve services are in (17). The upper and lower bounds of wind farms power output are described by (18). The wind forecast for each wind farm is provided by (19). This is where the improved space-time forecasts enter in the optimization problem. 4.3 A Realistic Illustrative Example In this subsection, numerical simulations are performed in a test system representative of a realistic BPA system. The wind speed forecasts in Section 3 are converted into wind power forecasts with a 2.5 MW Nordex power curve, and put, along with other variables, into the power system dispatch model which is setup based on the BPA system. According to the economic dispatch results, different wind forecasting models are compared in terms of potential savings in both generation cost and ancillary services. Vansycle, Kennewick and Goodnoe Hills are located in the Columbia River Basin. The electric power grid of this area is operated by BPA as introduced in Section 4.1. Our simulation system is revised from the IEEE Reliability Test System (RTS-24) (Grigg et al., 1999). The generators are assigned as different technology based power resources such as hydro power, coal power, wind power, natural gas power and nuclear power. The generator capacity portfolio (installed capacity percentage of different technologies) is configured according to the generation portfolio of the practical BPA system (BPA, 2010); see Table 3. The network typology of the simulation system is presented in Figure 9. The load profile used in the simulation is scaled from the historical load profile of the BPA system (BPA, 2007). Fourteen typical days in seven months of different seasons are selected 20
for simulation. The duration of simulation for each day is a typical power system operation period of 24 hours (T = 144). Different wind forecast methods described in Sections 2.2 and 2.3 are implemented in the simulation. The operating interval T of generation scheduling is 10 minutes. Wind profiles during the selected fourteen days are collected from the BPA system, which are scaled into the simulation system. For example, the wind generation potentials at the three locations (n = 3) on Aug. 15th are presented in Figure 10. Wind generation over the maximum generation capability of wind turbines has to be curtailed for security purpose. One-hour-ahead forecast data are used as inputs of the established simulation system. Generator parameters are configured according to Gu and Xie (2010). In the simulation, the minimum output levels of conventional generators P min G i and wind generators P min W i assumed to be zero. The capacities of transmission lines F max are assumed to be 999 MW. The total installed generation capacity is 4,000 MW of which 290 MW are the capacity of wind generation, which is about 7.3% (representative of a realistic scenario). Ramping are rates and marginal costs of different generators are shown in Table 5. Bus number (the number of the electrical node where the generator is located), type (what technology is used), capacity (Cap.: the total power capacity of the generator, in MW), marginal cost (M.C.: the marginal generation cost which indicates the cost increment due to generation increase, in $/MWh), and ramp rate (RP.: the capability of a generator to change its output per minute in normalized per unit value) of each generator are listed in Table 5. 4.4 Analysis of Economic Dispatch Results We present in Table 6 the economic dispatch performance under different kinds of wind forecast methods. The wind observation (OB), i.e. the true value, has the lowest system operating cost for all fourteen days. Among different methods, the total operating costs by using PSS are relatively higher over the fourteen days. The AR model which only considers 21
temporal wind correlations results in an operating cost second to the highest. Other spacetime methods such as RSTD, RRSTD, and TDD have a relatively better performance in the power system economic dispatch. To be noted, the wind speed spatial correlations may vary from day to day due to the change of wind patterns and directions. It is difficult to say that one space-time approach is always outperforming all the other approaches, even the PSS model and AR model. For most of the days, space-time wind forecast approaches (i.e. RSTD, RRSTD and TDD) have relatively higher cost savings than other approaches such as the PSS and AR models. Among the fourteen days, Aug. 15th is selected for a detailed study in the rest of this section. The system operating results of this day are presented in Table 7. The row Energy Market Cost infers the generation cost of all the generator units in the perspective of the system operator. The rows Regulation Cost and Reserve Cost indicate the total cost of providing regulation services and providing reserve services of all the units. The row Cost Reduced refers to the cost reduction (in %) by applying different wind forecast models compared to the cost when using the PSS model. According to the results of the simulation system, the space-time wind forecast models (such as RSTD, RRSTD, and TDD) can increase the actual wind resources utilization, reduce the system-wide generation cost, the system ancillary services (including regulation and reserve services) cost, the wind generation deviation penalty and thus reduce the total system operating costs. It can be observed in Table 7 that the system-wide operating cost using the RRSTD model is 3.0% lower than using the PSS or AR models. One of the advantages of space-time wind forecasts such as the RSTD, RRSTD and TDD is the reduction in wind generation deviation. As is observed in Table 7, using the RRSTD model, the wind generation deviation penalty has been reduced by almost 60% compared to the PSS model and by 24% compared to the AR model. The reduction in wind generation deviation is because the space-time wind forecasts can increase the wind forecast accuracy and reduce the overestimation of available 22
wind generation. The oracle space-time wind forecast models such as ORRSTD02 have of course better performance than the RRSTD in reducing the wind generation deviation. By training the RRSTD model as the ORRSTD02 model, the performance of the RRSTD model can be further improved. Besides, the results of space-time wind forecast models such as the RSTD, RRSTD, TDD, ORRSTD02, and ORRSTD03 reveal the advantage in the operating cost of ancillary services. For instance, the total cost of regulation services and reserve services are reduced by 3.0%, when using the RRSTD model. Given the same wind pressure patterns and system load patterns, the space-time wind forecast models accomplish higher wind resources utilization and a higher wind generation ratio than other models such as the PSS and AR models. This is because the increased accuracy of the space-time wind forecast models (e.g. the RRSTD), by considering spatial wind patterns correlations, decreases the wind generation that would be wasted by underestimation of available wind generation potentials. In Figure 11 (top panel), the actual wind generation output at Kennewick is presented. The curve OB depicts the case where there is no wind forecast error, which gives the highest utilization of wind generation as well as the best performance in the economic dispatch. The wind generation profile of the PSS model is with the lowest utilization of wind resources, while the wind generation profile of the AR model is with the second to the lowest utilization. As we can see, by using the RRSTD model, more wind generation can be integrated into the power system. This is because a high accuracy of wind forecasts avoids the underestimation of available wind generation during the dispatch process and reduces the potential waste of wind resources not being dispatched. The system overall reserve requirement takes into account the uncertainty in both wind generation (or wind forecast errors) and load (or load forecast errors). The selected reserve capacity is used to compensate the energy imbalance within time frameworks of half an hour to 2 hours. In Figure 11 (middle panel), the total system reserve requirement is compared between using the space-time wind forecast model (e.g. the RRSTD model) and the temporal wind forecast models (e.g. the PSS and the AR 23
models). It can be observed that by using the RRSTD model, due to the improved forecast accuracy, the overall reserve requirement can be reduced. Regulation services also help to compensate the energy imbalance of the system in order to keep the system frequency within a secure range. Unlike reserve services, regulation capacity is used to smooth out the short-time (1 minute to 10 minutes) frequency fluctuation and energy imbalance. It can be seen from Figure 11 (bottom panel) that the RRSTD model can decrease the system requirement for regulation services and therefore reduce the corresponding regulation cost. Because of the reduction in both energy balancing market (total generation cost) and ancillary services market by using the RRSTD model, the total system operating cost has been reduced by 3.0% compared with using the PSS model. Given the fact that the electric energy market is a significant market (multi billion annual sale in regions like Texas and BPA), 3.0% savings in operating costs means tens of millions of dollars of cost savings due to improved wind forecasts. 5 Discussion To support high penetration of wind power into the electric supply market, today s power system operation is greatly challenged by the high variations and limited predictability in wind. Advanced technologies in both highly accurate wind forecasting and algorithms that can help utilize wind energy efficiently are in need. In this article, we proposed a first attempt to incorporate space-time wind forecasting results into a power system dispatch. First, a new space-time model, the RRSTD model, was introduced in short-term wind speed forecasting problems. This model generalizes the RSTD model by allowing the forecast regimes to vary with the dominant wind direction and with the season without requiring much prior geographic information. Its forecasting results are as good as the RSTD and TDD models according to our numerical experiments, while better than the PSS and AR 24
models. Also this model has great potential to achieve better forecasts, if more information of monthly patterns in wind were available. Second, we formulated an economic dispatch model which takes into account the spacetime wind forecast information. With more accurate wind forecasts, the potential economic benefits were illustrated in a modified IEEE RTS-24 bus system. It was observed that using the RRSTD model reduced the wind generation deviation penalty by almost 60% compared to the PSS model and by 24% compared to the AR model. The advantages of space-time wind forecasting models appear in reducing the operating cost of ancillary services including regulation and reserve. The total cost of regulation services and reserve services were reduced by 3.0%, when using the RRSTD model. Given the same wind pressure patterns and system load patterns, the space-time wind forecast models accomplish higher wind resources utilization and generation ratio than other models such as the PSS and AR. References BPA (2007), BPA Total Transmission System Load, Bonneville Power Administration. BPA (2010), 2009 BPA Facts, Bonneville Power Administration. CREIA (2010), 2010 China Wind Power Outlook, China Renewable Energy Industries Association. DOE (2008), 20% Wind Energy by 2030: Increasing Wind Energy s Contribution to U.S. Electricity Supply, The US Department of Energy. ERCOT (2010), ERCOT Quick Facts, Electric Reliability Council of Texas. Genton, M. G. and Hering, A. S. (2007), Blowing in the Wind, Significance, 4, 11 14. Gneiting, T., Larson, K., Westrick, K., Genton, M. G., and Aldrich, E. (2006), Calibrated Probabilistic Forecasting at the Stateline Wind Energy Center: The Regime-Switching Space-Time Method, Journal of the American Statistical Association, 101, 968 979. Gneiting, T. and Raftery, A. E. (2007), Strictly Proper Scoring Rules, Prediction, and Estimation, Journal of the American Statistical Association, 201, 359 378.
Grigg, C., Wong, P., Albrecht, P., Allan, R., Bhavaraju, M., Billinton, R., Chen, Q., Fong, C., Haddad, S., Kuruganty, S., Li, W., Mukerji, R., Patton, D., Rau, N., Reppen, D., Schneider, A., Shahidehpour, M., and Singh, C. (1999), A Report Prepared by the Reliability Test System Task Force of the Application of Probability Methods Subcommittee, IEEE Transactions on Power Systems, 14, 1010 1020. Gu, Y. and Xie, L. (2010), Look-ahead Coordination of Wind Energy and Electric Vehicles: A Market-based Approach, in North American Power Symposium 2010, The University of Texas, Arlington. Hering, A. and Genton, M. (2010), Powering Up with Space-time Wind Forecasting, Journal of the American Statistical Association, 105, 92 104. Makarov, Y., Lu, S., McManus, B., and Pease, J. (2008), The Future Impact of Wind on BPA Power System Ancillary Services, in Transmission and Distribution Conference and Exposition, IEEE/PES. Marquis, M., Wilczak, J., Ahlstrom, M., Sharp, J., Stern, A., Smith, J., and Calvert, S. (2011), Forecasting the Wind to Reach Significant Penetration Levels of Wind Energy, Bulletin of the American Meteorological Society, 92, 1159 1171. Monteiro, C., Bessa, R., Miranda, V., Botterud, A., Wang, J., and Conzelmann, G. (2009), Wind Power Forecasting: State-of-the-Art 2009,. WWEA (2010), World Wind Energy Report 2010, World Wind Energy Association. Xie, L., Carvalho, P., Ferreira, L., Liu, J., Krogh, B., Popli, N., and Ilić, M. (2011), Wind Energy Integration in Power Systems: Operational Challenges and Possible Solutions, Special Issue of the Proceedings of IEEE on Network Systems Engineering for Meeting the Energy and Environment Dream, 99, 214 232. Xie, L. and Ilić, M. (2009), Model Predictive Economic/Environmental Dispatch of Power Systems with Intermittent Resources, in IEEE Power and Energy Society General Meeting, Galgary, Canada. Zhu, X. and Genton, M. G. (2012), Short-term Wind Speed Forecasting for Power System Operation, International Statistical Review, to appear.
Figure 1: Regime dividing plots for θ m(t+k) = {26, 206 } Aug (left) and θ m(t+k) = {60, 196, 295 } Aug (right). The dashed line connects the south (0 ) and the north (180 ), with westerly wind to the left and easterly wind to the right. Separate models of µ r s,t+k are built in each regime. Figure 2: Wind roses of wind data from August to November 2002 at Vansycle (top row), Kennewick (middle row) and Goodnoe Hills (bottom row). Each petal indicates the frequency of wind blowing from a particular direction and the color bands in each petal show the ranges of wind speed. The vertical dotted lines give the west-east RRSTD model, or the RSTD model, while the blue dashed and solid lines give the two forecast regimes of the two-regime RRSTD models with minimum prediction MAE value for each month, for 1-hour-ahead and 2-hour-ahead forecasting problems.
Aug 2002 at Vansycle, 1 hour ahead Wind Speed at VS 0 5 10 15 20 Wind Speed at VS 0 5 10 15 20 Wind Speed at VS 0 5 10 15 20 Regime1 Regime2 Regime1 Regime2 Regime1 Regime2 Wind direction at VS Wind direction at KW Wind direction at GH Figure 3: Boxplots of 10-minute wind speed (m/s) at Vansycle in August 2002 based on the wind directions 1-hour-behind of itself, Kennewick and Goodnoe Hills for δ = 8. It suggests s = G (Goodnoe Hills) as the indicator of wind source. Aug Sep MAE 1.050 1.055 1.060 1.065 1.070 1.075 1.080 1.085 8 0 50 100 150 MAE 0.97 0.98 0.99 1.00 1.01 1.02 9 0 50 100 150 Two Regime Dividing Angle (degree) Two Regime Dividing Angle (degree) Oct Nov MAE 0.86 0.87 0.88 0.89 0.90 MAE 1.16 1.17 1.18 1.19 0 50 100 150 Two Regime Dividing Angle (degree) 175 29 0 50 100 150 Two Regime Dividing Angle (degree) Figure 4: Plots of 1-hour-ahead prediction MAE results based on the two-regime RRSTD model with dividing angle δ from 0 to 180 for each month at Vansycle. The blue dashed line indicates the position of the best two-regime dividing angle, or θ, that has smallest MAE value.
0 50 100 150 May at Vansycle 2003, 1 hour ahead June at Vansycle 2003, 1 hour ahead July at Vansycle 2003, 1 hour ahead Aug at Vansycle 2003, 1 hour ahead MAE 1.14 1.15 1.16 1.17 1.18 147 MAE 1.045 1.050 1.055 1.060 1.065 1.070 1.075 1.080 104 0 50 100 150 MAE 1.08 1.09 1.10 1.11 1.12 143 0 50 100 150 MAE 1.095 1.100 1.105 1.110 1.115 1.120 1.125 150 0 50 100 150 Two Regime Dividing Angle (degree) Two Regime Dividing Angle (degree) Two Regime Dividing Angle (degree) Two Regime Dividing Angle (degree) Sep at Vansycle 2003, 1 hour ahead Oct at Vansycle 2003, 1 hour ahead Nov at Vansycle 2003, 1 hour ahead MAE 1.095 1.100 1.105 1.110 1.115 1.120 1.125 136 0 50 100 150 MAE 1.22 1.23 1.24 1.25 1.26 1.27 138 0 50 100 150 MAE 1.155 1.160 1.165 1.170 1.175 1.180 1.185 1.190 147 0 50 100 150 Two Regime Dividing Angle (degree) Two Regime Dividing Angle (degree) Two Regime Dividing Angle (degree) May at Vansycle 2003, 1 hour ahead June at Vansycle 2003, 1 hour ahead July at Vansycle 2003, 1 hour ahead Aug at Vansycle 2003, 1 hour ahead MAE 1.14 1.15 1.16 1.17 1.18 1.19 145 0 50 100 150 MAE 1.040 1.045 1.050 1.055 1.060 1.065 1.070 1.075 136 0 50 100 150 MAE 1.085 1.090 1.095 1.100 1.105 1.110 1.115 109 0 50 100 150 MAE 1.10 1.11 1.12 1.13 1.14 150 0 50 100 150 Two Regime Dividing Angle (degree) Two Regime Dividing Angle (degree) Two Regime Dividing Angle (degree) Two Regime Dividing Angle (degree) Sep at Vansycle 2003, 1 hour ahead Oct at Vansycle 2003, 1 hour ahead Nov at Vansycle 2003, 1 hour ahead MAE 1.10 1.11 1.12 1.13 MAE 1.23 1.24 1.25 1.26 MAE 1.155 1.160 1.165 1.170 1.175 64 0 50 100 150 0 50 100 150 170 39 0 50 100 150 Two Regime Dividing Angle (degree) Two Regime Dividing Angle (degree) Two Regime Dividing Angle (degree) Figure 5: Plots of 1-hour-ahead prediction MAE results based on the ORRSTD02 (top) and ORRSTD03 (bottom) models with dividing angle δ from 0, to 180 for each month at Vansycle. The blue dashed line indicates the best two-regime dividing angle, or θ, that has smallest MAE value.
Figure 6: Wind roses from May to November 2003 at Vansycle (by row). The vertical dotted lines give the west-east RRSTD model, or the RSTD model, the blue dashed line gives the two-regime RRSTD models with minimum prediction MAE value for each month, the pink dashed line is from the ORRSTD02 model and the cyan dashed line is from the ORRSTD03 models in 1-hour-ahead forecasting problem. Figure 7: BPA transmission system including Idaho, Oregon, Washington, Montana and Wyoming (BPA, 2010). Figure 8: BPA scheduling procedure (Makarov et al., 2008).