Non-stationary Wind Speed Research Articles

Non-Gaussian non-stationary wind speed simulation based on time-varying autoregressive model and maximum entropy method

Accurate simulation of non-Gaussian nonstationary wind speeds is a prerequisite for the wind resistant design of some nonlinear structures. Due to its efficiency, the time-varying autoregressive (TVAR) model has been extensively employed for simulating non-Gaussian nonstationary processes. Nevertheless, these simulation techniques based on TVAR exhibit suboptimal performance when confronted with nonstationary and highly non-Gaussian processes. Furthermore, they are unable to replicate the bimodal characteristics of specific wind speeds. This paper presents a new method for simulating univariate non-Gaussian nonstationary wind speeds using the TVAR model and the maximum entropy method. Herein, the connection between the statistical moments of input and output processes in TVAR is firstly derived. Secondly, the maximum entropy method is utilized to reconstruct the probability density function of input process and the time-varying translation function is determined. Finally, the translation process theory is applied to generate the input process, which is then input into the TVAR model to output the non-Gaussian nonstationary wind speed. The numerical results demonstrate that the proposed method exhibits superior simulation accuracy for nonstationary and strongly non-Gaussian wind speed processes. Furthermore, it is capable of capturing the bimodal characteristics of certain hardening non-Gaussian nonstationary wind speeds and possesses a broader range of applications.

A straightforward method for estimating evolutionary power spectral density of non-stationary typhoon wind speed

PurposeA precise estimation of the evolutionary power spectral density (EPSD) of typhoon wind speed is a difficult and significant undertaking in the analysis of turbulence effects on large-expansive structures. A majority of the prevailing EPSD estimation techniques rely on complex signal processing methodologies, such as wavelet decomposition, Hilbert–Huang transformation and time-varying autoregressive moving average (ARMA) model. However, these approaches often pose challenges in terms of comprehensibility and practical implementation for engineers. In light of this issue, the present study introduces a straightforward and effective EPSD estimation method tailored specifically for typhoon wind speed, aiming to facilitate its understanding and application in engineering contexts.Design/methodology/approachFirstly, the mathematical model of a uniformly modulated non-stationary process is employed to represent the typhoon wind speed. Secondly, the reverse arrangement test serves as an auxiliary tool in conjunction with wavelet transform or empirical mode decomposition, aiding in the determination of the optimal slowly varying mean wind speed. Thirdly, Kernel regression technique is utilized to discern the time-dependent standard deviation of wind speed fluctuations. Finally, the power spectral density (PSD) of wind speed residuals is computed to facilitate the estimation of the EPSD.FindingsFirstly, the reverse arrangement test-assisted approach enables the determination of an optimal time-dependent mean from the candidate results obtained through discrete wavelet transform (DWT) and empirical mode decomposition (EMD). Secondly, the application of the Kernel regression technique facilitates accurate identification of the time-dependent variance from the fluctuating wind speed data. Thirdly, due to the influence of the extreme weather, the Gaussianity of the reduced turbulent fluctuations in typhoon wind is easily disturbed, resulting in the obvious non-Gaussian features.Originality/valueThis paper employs the mathematical model of uniformly modulated non-stationary process to characterize typhoon wind speeds and then proposes a straightforward and efficient method for estimating the EPSD of typhoon wind. The accuracy and efficacy of the presented estimation method are verified using the field-measured wind speed data from Typhoon Rammasun. The proposed EPSD estimation method for typhoon wind exhibits suitability for engineering applications owing to its simplicity and computational efficiency.

Mathematical Analysis of the Wind Field Characteristics at a Towering Peak Protruding out of a Steep Mountainside

Wind field characteristics in a complex topography are significantly influenced by the nature of the surrounding terrains. This study employs onsite measurements to investigate the wind field characteristics at a towering peak protruding out of a steep mountainside, where butterfly−lookalike landscape platform will be constructed; the impact of the surrounding topography on the wind flow is highlighted. The results showed that the blocking effect of the mountains in the mountainous side of the valley caused a significant drop in the mean wind speed from that direction. The stationary test (reverse arrangement test) indicated that the wind speed had a strong nonstationary characteristic, necessitating the employment of a steady and nonstationary wind speed model to assess the wind turbulence characteristics. The three directions’ wind turbulence integral scales were critically influenced by the occurrence of the wind speedup effect, unexpectedly resulting in the vertical turbulence integral scale being the greatest of the three. Furthermore, the measured wind turbulence properties under both wind speed models showed certain variations from the recommended specifications. Consequently, the impact of the local terrain and the speedup effect on the wind characteristics must be thoroughly evaluated to ensure the structural stability of structures installed at a similar topography.

Study on Non-Stationary Wind Speed Models and Wind Load Design Parameters Based on Data from the Beijing 325 m Meteorological Tower, 1991–2020

As the capital of China and a densely populated major city, the characteristics of Beijing’s near-surface wind field change significantly with the increase in the density of underlying urban structures. The high randomness of natural wind makes it extremely difficult to develop a universally applicable wind-resistant load design code based on topographic factors and architectural features. This article takes the wind speeds recorded at 15 different height levels within the urban area by the 325 m meteorological tower in Beijing from 1991 to 2020 as the research subject. It quantifies the wind speed trends at different heights and introduces time-varying functions to establish a non-stationary wind speed model based on the optimal model. Additionally, it compares the basic wind speeds and wind pressure height variation coefficients obtained from measurements with the standards. The results show that, during the past 30 years of urbanization, the near-surface wind speed in the Beijing area has shown a decreasing trend. The model incorporating time-varying functions exhibited the best fit and demonstrated good predictive capabilities, with its calculated basic wind speeds being relatively high. The wind pressure height variation coefficient values in Beijing are between Class C and Class D terrains, being closer to Class C at lower altitudes. The conclusion reveals that urbanization has a significant impact on wind speeds, primarily concentrated at lower height levels, and that the basic wind speeds calculated based on standards underestimate the actual conditions when this impact is not considered. Although the average wind speed’s wind profile index across the entire time series is mostly greater than the fixed value of 0.3 given by Class D, this represents an overestimated wind profile index for maximum wind speeds.

A novel framework for wind energy assessment at multi-time scale based on non-stationary wind speed models: A case study in China

Wind energy assessment at multi-time scale is essential for the effective utilization of offshore wind resources. Conventionally, energy assessment factors are estimated based on parametric models under the assumption of wind speed stationarity. However, given the subjection of seasonal variability and long-term trends, the hypothesis is violated, especially in the context of climate change. In this paper, to advance the conventional approach, wind speeds were modelled using the decomposition-based method to consider the non-stationarity, and a novel framework for the estimation of energy factors at multi-time scale was designed. Using long-term ERA5 data, wind energy potential across the sea area along Chinese continental coastline was investigated. It is demonstrated that the established non-stationary models are superior to commonly used stationary models, allowing the estimation of energy assessment factors at arbitrary time scales after one-time parametrization of parametric distributions. Results confirmed that wind energy exhibits apparently temporal and spatial variability. The difference between annual averaged wind power density at different nodes exceeds 1000 W/m2. Moreover, significant long-term trends of wind energy characteristics were detected. The increase can be up to 16 W/m2/year in Taiwan Strait, emphasizing that an adequate consideration of the non-stationarity of wind speed in future research is required.

Non-stationary buffeting responses of a twin-box girder suspension bridge with various evolutionary spectra

The effects of evolutionary wind spectra with different modulation functions on the nonstationary buffeting responses of suspension bridges are uncertain. After considering the nonlinear buffeting force of a twin-box girder, the buffeting responses of a cross-sea suspension bridge under four nonstationary wind speed models with two uniform modulation and two nonuniform modulation functions were investigated in this paper. Through the evolutionary spectral theory, the nonstationary wind speed models at the bridge site with four typical modulation functions were generated and then validated from the autocorrelation and power spectrum density. The results show that the mean and root-mean-square error (RMSE) values of vertical and horizontal wind speeds by using nonuniform modulation functions (NMF1 and NMF2) were much larger than those by using uniform modulation functions (uMF1 and uMF2). Moreover, most of the peak and RMSE values for the torsional and lateral displacement under the NMF1 are the largest, while the RMSE values of the vertical displacement without the modulation function are the largest. With the increase of the circular frequency γ or decrease of the initial phase θ in the cosine function of time-varying mean wind speeds, the RMS values in three displacement responses of the bridge deck become larger.

Wind speed prediction utilizing dynamic spectral regression broad learning system coupled with multimodal information

As the integration of wind energy into the power system increases, accurate wind speed prediction becomes crucial to ensure the reliable and economically efficient operation of the grid. The non-stationary, chaotic, and nonlinear characteristics of wind speed pose significant challenges for prediction models in uncovering the dynamic evolution process. To address these challenges, we proposed a wind speed prediction method based on the dynamic spectral regression broad learning system coupled with multimodal information (DSR-BLS). Firstly, we proposed a frequency density clustering-based mode decomposition (FDCMD) algorithm, which automatically transforms the non-stationary wind speed into multiple relatively stationary modal components. Next, we proposed the dynamic spectral regression (DSR) algorithm, which is based on dynamic-inner latent variable modeling and spectral regression. DSR can extract features and reconstruct the dynamic characteristics of wind speed through non-uniform embedding in the phase space. Finally, DSR-BLS is proposed to enhance the deterministic point prediction accuracy of the original BLS by utilizing multimodal features and dynamic features. The experiments show that DSR-BLS outperforms the comparative prediction methods in multi-step ahead prediction results.

A novel hybrid algorithm based on Empirical Fourier decomposition and deep learning for wind speed forecasting

As the demand for renewable energy is increasing, wind speed forecasting (WSF) becomes increasingly popular and essential for wind power generation. Several methods have been developed in the literature for WSF. However, WSF accuracy is challenging to obtain due to its non-stationary and non-linear character. To overcome this issue, this study suggests a novel integrated forecasting model for WSF. In the majority of cases, decomposition techniques are essential for removing noise and extracting patterns from non-stationary and non-linear wind speed time series. Among these decomposition techniques, the most widely used decomposition methods based on frequency domain theory are the Fourier decomposition method (FDM), the empirical wavelet transform (EWT), and the variational mode decomposition (VMD). However, FDM decomposition gives inconsistent results, EWT has mixed problems, and VMD is unsuitable for non-stationary time series. Thus, a novel combined model based on empirical Fourier decomposition (EFD) techniques, long short-term memory (LSTM) neural networks, and the Grey Wolf optimizer (GWO) is proposed in this paper. Initially, the wind speed time series is decomposed using EFD into a set of sub-series and a residual, which is a stationary sub-series that can be easily modelled by a recurrent neural network (RNN). Further, the matching LSTM is used to forecast each sub-series and residual, while the GWO optimizes the sub-series estimated output, resulting in improved prediction precision and stability. In order to assess the model performance, five wind speed datasets from various places in India are employed. The findings show that the proposed model is the top performing model among the other ten models; The average statistical score obtained from all datasets reduces 11.01% in the mean absolute error (MAE), 10.68% in the root mean square error (RMSE) and 10.73% in the mean absolute percentage error (MAPE). Further, regularity conditions for universal consistency of proposed model is also proved mathematically.

A wavelet transform based stationary transformation method for estimating the extreme value of the non-stationary wind speeds

The rational estimation of extreme wind speeds is crucial. As reported, in the Chinese area, the measured wind speed variation from different meteorological observatories shows a general non-stationary trend in the past 40 years due to urbanization factors or temperature variation. Thus, the non-stationary extreme value analysis (TNGEV) of the measured wind speed is required. However, TNGEV is relatively complicated. Under such a background, in this paper, a novel simplified wavelet transform-based stationary transformation method (PNGEV) is proposed for the extreme value analysis of the non-stationary wind speeds. This paper firstly derives the proposed time-varying extreme value distribution parameter form, combining the wavelet transform and moving average. Secondly, a case study is provided to illustrate and verify this proposed method (PNGEV), using the 40-year daily maximum wind speeds recorded from 1981 to 2020 in Baoshan District Shanghai, China. The effectiveness of the proposed method (PNGEV) is verified by comparing it with TNGEV in the return period and the risk of failure. By comparison, it can be figured out that the proposed PNGEV is practicable and outperforms TNGEV in the stability of the return period calculation, as well as the time-varying extreme value distribution establishment.

A novel decomposition-based approach for non-stationary hub-height wind speed modelling

An accurate description of hub-height wind speed characteristics is indispensable to offshore wind resource assessment and structure reliability analysis. However, given the assumption of stationarity in wind speeds that is violated, the commonly used stationary statistical models will lead to bias, and a non-stationary frequency analysis is required. In this paper, a novel decomposition-based non-stationary modelling approach was proposed. To decompose time series into deterministic and stochastic components, a procedure was designed by combining signal decomposition methods and recurrence quantification analysis and the performances of seven signal decomposition methods were evaluated under various represent non-stationary scenarios via numerical experiments. Then the non-stationary model was established by aggregating the modelled two components. Compared with other methods, the proposed approach is superior, which is of good self-adaption to data and relies on no hypotheses and explanatory covariates, guaranteeing the simplicity and reliability of the constructed models. Additionally, the SSA-based procedure is capable of capturing complicated non-stationary patterns while preserving the higher-order moments of the underlying stochastic process. The capacity of the proposed approach was verified using wind speed data at six positions distributed along China's coastline. Results emphasize the importance of the consideration of non-stationarity and the necessity of this study.

First Passage Dynamic Reliability Analysis of Non-Stationary and Non-Gaussian Buffeting Random Dynamic Responses of Long-Span Suspension Bridge

In the complex mountain wind environment, especially in the strong wind, fluctuating wind speed is a non-stationary and non-Gaussian random process. With the increase of suspension bridges’ span, they are more sensitive to wind-induced vibration response. In this paper, a non-stationary and non-Gaussian buffeting reliability analysis method for long-span bridges is presented. Firstly, the non-stationary wind speed is simulated by the modulation function based on the stationary wind speed model. Secondly, the load is obtained by simulated wind speed, the obtained loads are loaded on the finite element model by ANSYS and the whole bridge time-domain analysis is performed, the time history of displacement response is obtained. Thirdly, based on the first excursion failure criterion of random vibration, the samples of the non-stationary and non-Gaussian displacement time history are transformed into a standard Gaussian process through a modified Fleishman approximation method, and then the non-stationary Poisson distribution method is used for structural reliability analysis. Finally, a mountainous long-span suspension bridge is used as the engineering background, and the proposed reliability analysis method is applied to analyze the reliability of the bridge. The influence of different wind attack angles and modulation functions on the dynamic reliability of the bridge is further studied. The results indicate that the displacement response obtained by the transformation of non-stationary and non-Gaussian random process is less reliable than that obtained by traditional analysis method. Its dynamic reliability is maximum at 1[Formula: see text] wind attack angle, the reliability of negative wind attack angle is lower than that of positive wind attack angle, and decreases with the increase of wind attack angle. The reliability obtained by using different modulation functions is lower than that of traditional method. If the traditional analysis method is still used for reliability analysis, it will produce unsafe consequences and reduce the engineering safety reserve.

An improved multi-taper S-transform method to estimate evolutionary spectrum and time-varying coherence of nonstationary processes

Obtaining the accurate evolutionary spectrum and time-varying coherence of multivariate nonstationary processes is essential in engineering applications. However, estimating the time-varying coherence is challenging and has been rarely studied for a single realization of a nonstationary process. To address this issue, an improved multi-taper S-transform (MTST) method is developed to reduce the variance of the evolutionary spectrum and time-varying coherence estimates with a sufficient time–frequency resolution. This new method uses the different number of tapers (NoT) at each frequency to achieve the trade-off between the resolution and variance of the estimated spectrum and coherence. The nonstationary process defined by Wold-Cramer decomposition with a time-varying coherence is considered in this study. First, the analytical formulae of the bias and variance of the evolutionary spectrum of the improved MTST are deduced. The NoT is then analytically derived when time–frequency sine and Hermite tapers are adopted. Subsequently, an iterative procedure is proposed to determine the optimal NoT at each frequency without manual tuning. The proposed technique is finally applied to a nonstationary wind speed process and an earthquake ground motion with different spectral characteristics. Results demonstrate that the improved MTST technique outperforms Priestley’s method, the wavelet-based method, and the MTST method with fixed NoT by estimating a more accurate evolutionary spectrum and time-varying coherence.

Effect of nonstationary extreme wind speeds and ground snow loads on the structural reliability in a future Canadian changing climate

Extreme wind speeds and ground snow loads for many regions across North America exhibit non-stationarity in projected future climates. The adequacy of conventional return period design values estimated in the past for the future design life needs to be investigated. Nonstationary extreme value analysis methods were used in several previous studies to evaluate the nonstationarity of extreme wind and snow loads based on the latest future projections for Canada in the RCP8.5 (where RCP denotes the representative concentration pathway) emission scenario. These methods have not been considered in nonstationary reliability analyses. This study will examine the effect of including these nonstationary extreme value analysis methods in the reliability analysis. Investigations were conducted by applying the design value estimated in past stationary climate to the reliability analysis considering nonstationary climatic load effects in the future design life. Two non-stationary extreme value analysis methods were then used to determine the design level value in the nonstationary future climates for mitigating the climate change impact. These two approaches target different risk levels. Both nonstationary methods provide design level values greater than the conventional return period values and can maintain the probability of failure close to the target value at the end of the design life. Analyses were also conducted to examine the climate change impact on the structural reliability for considering principal wind load effects or snow load effects with the companion live load effects.

State-monitoring for abnormal vibration of bridge cables focusing on non-stationary responses: From knowledge in phenomena to digital indicators

Input-output characteristics in an event of the abnormal vibration of a kilometer-level railway-highway-combined cable-stayed bridge's cables under in-service winds are deeply investigated. The knowledge of commonality that cables' abnormal vibration exists multiple independent non-stationary sections in time series with high-amplitude and quasi-single-frequency is summarized. An automatic extraction of non-stationary section of abnormal vibration is conducted via Gaussian mixture modeling of the envelope of time series signal. A new process of modal parameter identification is presented using the data of non-stationary section after bandpass filtering and its last descent. It is demonstrated that the identified damping ratio of cables with abnormal vibration will be below 0.05%. Each abnormal non-stationary vibration's frequency, damping ratio, and wind speed are clustered into corresponding natural vibration modes. Correlation model of wind speed-vibration frequency is fitted by data centroid of clusters to predict the possible-occurrence frequency of an abnormal non-stationary vibration before it is fully completed.

Influence of coherent structure on turbulence characteristics of 0814 strong typhoon Hagupit

In order to explore the formation mechanism of the nonstationary wind speed in the eye wall regions and eye region of a typhoon, taking the measured data of No.0814 strong typhoon Hagupit as the research object, we identify and extract the coherent structure of the measured wind field by discrete wavelet transform combined with hypothesis testing, and analyze the influence of the coherent structure on the nonstationary and non-Gaussian wind speed and study the influence of the coherent structure on the wind turbulence characteristics. The results show that for the 0814 typhoon Hagupit process, the coherent structure is the main reason that leads to the nonstationary and non-Gaussian fluctuating winds in the typhoon field, and with the coherent structures being exacted from the original winds, the nonstationary and non-Gaussian characteristics of the nonstationary wind samples are significantly weakened. The average contribution of the coherent structure to the turbulence parameters ranges from 6.42% to 32.40%. The duration of the coherent structure is about 10.60% of the sample duration, however, its contribution to turbulent kinetic energy is up to 40.50%. The coherent structure has the characteristics of short duration and high energy content, which is the cause of the abnormally large turbulence parameter value. This study can provide a reference for the refined wind resistance design of structures under the action of nonstationary wind.

Field measurement analysis of wind parameters and nonstationary characteristics in mountainous terrain: Focusing on cooling windstorms

Cold fronts passing through the mountain areas may produce continuous windstorm and cooling processes, potentially impacting the mountain areas' infrastructure. Cooling windstorms belong to the dynamically-driven wind in the mountain wind system. To accurately estimate the wind field parameters of these storms and provide a basis for structural wind resistance design, cooling windstorms should be extracted and analyzed separately from the wind field observation data. Based on the long-term field measurement in complex mountain terrain, a framework for identifying cooling windstorms is established according to temperature and wind speed features. The mean and fluctuating wind parameters of these intense winds are then obtained. Strong nonstationary is detected for cooling windstorms. This paper focuses on the time-varying regularity of fluctuating parameters during intense wind with cooling based on the evolutionary power spectral density. From the perspective of turbulent energy, the basis for selecting the optimal wavelet decomposition levels is put forward when extracting the time-varying mean wind through discrete wavelet transform in the nonstationary wind speed model. This investigation can provide a reference for studying the nonstationary wind field.

Stochastic configuration network based on improved whale optimization algorithm for nonstationary time series prediction

AbstractThe stochastic configuration network (SCN), a type of randomized learning algorithm, can solve the infeasible problem in random vector functional link (RVFL) by establishing a supervisory mechanism. The advantages of fast learning, convergence and not easily falling into local optima make SCN popular. However, the prediction effect of SCN is affected by the parameter settings and the nonstationarity of input data. In this paper, a hybrid model based on variational mode decomposition (VMD), improved whale optimization algorithm (IWOA), and SCN is proposed. The SCN will predict relatively stable data after decomposition by VMD, and parameters of SCN are optimized by IWOA. The IWOA diversifies the initial population by employing logistic chaotic map based on bit reversal and improves the search ability by using Lévy flight. The exploration and exploitation of IWOA are superior to those of other optimization algorithms in multiple benchmark functions and CEC2020. Moreover, the proposed model is applied to the prediction of the nonstationary wind speeds in four seasons. We evaluate the performance of the proposed model using four evaluation indicators. The results show that the R2 of the proposed model under four seasons are more than 0.999, and the root mean square error, mean absolute error, and symmetric mean absolute percentage error are less than 0.3, 0.17, and 13%, respectively, which are almost 1/10, 1/10, and 1/4 those of SCN, respectively.

Change point detection-based simulation of nonstationary sub-hourly wind time series

In this paper, we present a wind speed simulation method by detecting change points in multivariate nonstationary wind speed time series data. The change point detection method identifies changes in the covariance structure and decomposes the nonstationary multivariate time series into stationary segments. Parametric and nonparametric techniques are also provided to model and simulate new time series within each stationary segment. The proposed simulation approach retains the statistical properties of the original time series (as obtained from a Numerical Weather Prediction system) and therefore, can be employed for simulation-based analysis of power systems planning and operations problems. We demonstrate the capabilities of the change point detection method through computational experiments conducted on wind speed time series at five-minute resolution. We also conduct experiments on the economic dispatch problem to illustrate the impact of nonstationarity in wind generation on conventional generation and location marginal prices.

A novel hybrid approach based on variational heteroscedastic Gaussian process regression for multi-step ahead wind speed forecasting

Accurate wind speed forecasting is the key to safe and economic operation of electric power and energy systems. As a Bayesian nonparametric method, Gaussian process regression (GPR) has provided competitive forecasting results in recent years. However, conventional GPR model assumes that the noise obeys Gaussian distribution and the variance of the noise in the whole data set is a constant, which is not appropriate for some problems. Motivated by this, this study makes the first attempt to study the ability of the variational heteroscedastic GPR (VHGPR) model in wind speed forecasting. The Marginalized Variational (MV) approximation is employed to approximate the heteroscedastic Gaussian process in the VHGPR model. What’s more, the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) is employed to transfer the nonstationary wind speed series into a certain number of subseries with more regularity such that the forecasting performance of VHGPR can be enhanced. The proposed method is compared with the other twelve benchmark models for 1- to 3- step ahead wind speed forecasting. Experiments results on four real-world datasets demonstrate that VHGPR with the CEEMDAN decomposition strategy is able to obtain better forecasting results for wind speed time series.

A Combined Forecasting System Based on Modified Multi-Objective Optimization for Short-Term Wind Speed and Wind Power Forecasting

Wind speed and wind power are two important indexes for wind farms. Accurate wind speed and power forecasting can help to improve wind farm management and increase the contribution of wind power to the grid. However, nonlinear and non-stationary wind speed and wind power can influence the forecasting performance of different models. To improve forecasting accuracy and overcome the influence of the original time series on the model, a forecasting system that can effectively forecast wind speed and wind power based on a data pre-processing strategy, a modified multi-objective optimization algorithm, a multiple single forecasting model, and a combined model is developed in this study. A data pre-processing strategy was implemented to determine the wind speed and wind power time series trends and to reduce interference from noise. Multiple artificial neural network forecasting models were used to forecast wind speed and wind power and construct a combined model. To obtain accurate and stable forecasting results, the multi-objective optimization algorithm was employed to optimize the weight of the combined model. As a case study, the developed forecasting system was used to forecast the wind speed and wind power over 10 min from four different sites. The point forecasting and interval forecasting results revealed that the developed forecasting system exceeds all other models with respect to forecasting precision and stability. Thus, the developed system is extremely useful for enhancing forecasting precision and is a reasonable and valid tool for use in intelligent grid programming.