## 1. Introduction

Conventionally, energy from multiple sources, such as coal, natural gas, nuclear power, wind, and tides, goes from a power plant to an electrical grid or network for some form of storage, stabilization, and regulation before eventual transmission and distribution through substations to end users. For fossil fuels and non-renewable energy sources, the actual energy input into the grid per unit time is known hence “static”, while that of renewable sources is “dynamic”. Typically, the energy profile includes three major parts: base load, middle load, and peak load [

1]. One major challenge of limiting wind power diffusion is that wind sources cannot serve as a base load due to intermittency [

1,

2,

3,

4,

5]. Since the base load is that fraction of total energy demand that must be available constantly to avoid power outage, it must be static. In energy surplus conditions where supply exceeds demand, the challenge for wind energy is one of optimization. For energy deficit conditions when demand exceeds supply, maximizing wind power input into the grid and meeting energy demand are key.

The relationship between wind speed and wind power based on the wind power equation is described by a turbine specific non-linear transformation curve termed turbine power curve (TPC) and is turbine-specific. One method of wind resource assessment is to fit the wind speed characteristics of the select location into the Weibull distribution probability distribution function (PDF). The frequency of occurrence of different wind speed bins obtained from the PDF is then used to estimate the total power generated for the corresponding wind speed and time duration [

5,

6]. The sum of all the bins is the total estimate of wind energy output for such a study site. Another method is to use the wind power density (WPD), which does not consider the turbine characteristics, but rather gives an estimate of the available wind power based on wind speed only [

7]. Both methods are capable of expressing long-term estimates, which has been shown to sometimes vary from actual field observations. For example, wind speed and power pairs collected from an observational 800 kW wind turbine show deviations of the measured power output (referred to as the actual power curve henceforth APC) from the TPC creating a dispersion belly [

8]. Similar dispersion in other studies [

3,

5,

6,

9] suggest that if the time series were extracted, APC fluctuations would make estimations of wind energy source input into the electrical grid a technical problem for energy grid operators.

Table 1 shows some reviewed studies that have identified fluctuations of the TPC from the APC. Specifically, the turbine model, location, time steps considered, and scope of studies are included in the table. Although there are marked differences in these deviations, the surveyed studies all point to occurrence of deviations occuring irrespective of turbine location, turbine type or specific climatic conditions.

Wind turbine control system operators require wind speed prediction times in the range of seconds ahead. Wind speed prediction can be via numerical weather prediction (NWP) methods as summarized in previous works [

17,

18,

19] and [

1] who developed an adaptive neural fuzzy inference system (ANFIS) for wind speed forecasting. Another approach is through statistical methods as described in Reference [

3] and used in Reference [

8] to develop a linear prediction model effective for wind speed forecasting or the fractionally-integrated Auto-Regressive Integral and Moving Average (f-ARIMA) method adopted in Reference [

20]. Statistical methods are ideally suited for short-term predictions, and common forms include regression based methods such as auto-regressive (AR), auto-regressive and moving-average, (ARMA) [

21], auto-regressive integral and moving-average (ARIMA) [

22]; exponential smoothing (single or double); the persistence model; and neural network fitting (Artificial Neural Network). However, studies on wind speed prediction still continue because there does not seem to be any industry-wide accepted, short-term wind forecasting system for wind power. While studies on wind power and wind turbine performance exists (see

Table 1), more recently the use of stochastic models for wind power related studies have increased. Wind turbine fatigue load estimation was conducted by Reference [

23] using a stochastic model, especially in wind farms, a condition that is certain to affect turbine performance and power output. In Reference [

24], actual measured data was used to construct a stochastic model to monitor wind turbine vibrations and could prove significant for understanding and monitoring turbine operations/maintenance. As regards to the operation and control of turbines, better modeling that incorporates wind uncertainty and accommodates deviations from “normal” turbine working conditions [

25] are essential for performance characterization and fault diagnosis. Such better modeling is seen in Reference [

26] where a stochastic approach was applied to model energy conversion and power production from a wind farm under various wind conditions including turbulent wind fluctuations. In another dimension, Reference [

27] combined a computational fluid dynamic (CFD) approach with blade element momentum (BEM) theory to propose operating conditions for wind turbines during icing conditions.

When errors from wind power estimation occur, the base load may have to be increased [

28], and large amounts of usable wind energy may have to be disposed of [

29]. In addition, power outages may occur while financial losses in the energy market would be incurred [

30]. Ultimately, grid systems may be jeopardized, thereby affecting energy quality and stability [

3]. Thus, a combination of approaches from previous studies is adopted to analyze actual turbine performance and its impact on grid input. Measurements from a single turbine such as conducted by References [

3,

13] is used to investigate the occurrence of very short-term wind variation and turbine power output similar to the studies by References [

15,

16]. Three different time-steps and averaging times greater than 10 s intervals used by Reference [

15], but less than 1 h used by Reference [

5] was adopted. This was then extended to investigating estimation errors and possible methods for reducing these errors based on a form of polynomial curve belonging to the parametric modeling technique of wind power curves seen in Reference [

31]. The results from this study will be of significant interest for electrical grid management, wind power optimization, and wind turbine controllers and operators, as well as the wind energy industry. First, insight into the time series of wind power estimation is provided for energy grid operators who play an essential role in ensuring an uninterrupted power supply and maintaining grid quality. Second, the analysis provides real-time information for wind turbine controllers to guide decision making in short-term time scales during turbine operation, as opposed to previous studies that provided such information in the range of 1 h, 6 h, or even daily. Third, by investigating estimation errors, the wind energy industry is able to overcome some of the marketing and policy challenges affecting wind power optimization and diffusion.

## 2. Data and Methodology

Wind speed and wind power pairs for an operational wind turbine were sourced in 15 s intervals for a period of 14 days each for 2 seasons. The first set of measurements were taken in August 2016, representative of the summer season, while the second set was collected in February 2017, representative of the winter season. The turbine is located on a hill approximately 440 m above mean sea level, a Nordex N-50 800 kW turbine with a hub height of 46 m. The coefficient of performance (Cp) ranged from 35% at low wind speed to about 45% at between 5 m/s and 10 m/s, before gradually reducing to about 21% at wind speed above 10 m/s (see

Figure S1). The turbine serves mostly for educational and demonstration purposes with real time performance data available.

For the two periods considered, the dataset was initially divided into two parts (dataset A and dataset B) for both summer and winter measurements. The first dataset, comprising about 80% of the datasets, was used for aggregation into 1-min, 5-min, and 15-min time steps using the mathematical notation below.

where

X is the value considered (i.e., wind speed and wind power),

i is the time, and

n is the number of observations. The second set (20%) was used for model development.

Three case scenarios were established to typify possible variations of daily wind speed irrespective of time of day using 60 time steps per scenario.

Scenario A–u > 3.0 from t_{1}–t_{60}:

Wind speed range above cut-in speed for the 60 time steps

Scenario B–u < 3.0 at t_{1}:

Wind speed startup lower than cut-in speed

Scenario C–u > 3.0 between t_{1}–t_{n}, u < 3.0 at t_{n+}_{1}:

A range of lower wind speed between higher values

where t_{1} = time step 1, t_{n} = time step at arbitrary time n, and t_{n+}_{1} = wind speed at the next time step after time t_{n}; all scenarios were considered using 60 time steps.

The actual power curve (APC) is derived from the wind power values recorded from the corresponding speed for the time series by taking the instantaneous power output from the turbine. The turbine power curve (TPC) is obtained by using a nonlinear transformation function for wind power as stated in two below. It is important to state here that the TPC can actually be divided into three regimes, the non-linear (also cubic) transformation of wind speed to wind power when incident wind speed ranges between 3.0–15 m/s, the linear regime during which the power output is constant for a rated wind speed of 15–25 m/s, and the saturated regime where power output is negligible due to turbine shut-down for too high wind speeds greater than 25 m/s.

To minimize estimation errors, an effective power curve (EPC) was proposed for the performance evaluated using the following criteria: mean absolute error (MAE), mean square error (MSE), and root mean square error (RMSE) [

32].

where

n is the number of samples,

p is the actual power output (APC) and wind speed, and

$\widehat{p}$ is the modelled output (EPC).

## 4. Conclusions

Time series of wind turbine performance and energy yield were examined under three different scenarios using two representative seasons (i.e., winter and summer). First, a 1-min interval time series comparison of actual turbine output (APC) and theoretical turbine output (TPC) was considered. The observed data show that wind direction may significantly impact turbine power output; likewise turbine momentum may sustain wind power production despite low wind speed. Knowledge of this effect and its dynamics is very useful for guiding the actions of wind turbine controllers and operators. The 5-min and 15-min interval time series appear more suited for addressing delays in turbine response, developing energy storage systems, energy scheduling, and load management, and bear greater significance in electrical grid quality and stability. All the cases and scenarios considered exhibit relatively large estimation errors both for each time step and for the total energy yield in kWh. These errors are major underestimations of actual power and could directly hinder wind power diffusion and optimization, especially in terms of acceptance, adoption, and commercialization.

An effective power curve (EPC) was proposed based on turbine performance over a given period. The EPC produces fewer estimation errors relative to the TPC when used for estimating power production in the 15-min interval time series. The EPC afforded an easier and more direct estimation of turbine power production, with an estimation error of less than 5% (average value) during “good” wind conditions. Better estimates during good wind conditions offer optimal value in terms of wind power exploitation and harnessing. Generated energy does not have to be disposed-off while energy from alternate sources (non-stochastic/intermittent) can be stored or reserved Energy Storage Systems (ESS). However, under poor wind conditions where the performance of the EPC is less remarkable other strategies may be required in addition. Typically, a change in turbine blade operating conditions such as turbine shut-down or sustaining a certain rotational speed may be more beneficial for accurate estimations. Another strategy can require operator changes to the turbine tip speed ratio (TSR), thus creating a better define estimate. These strategies are however best suited for defined environmental conditions and relatively higher wind speeds (higher mean value but less time-step fluctuations). Importantly, studies on turbine performance during poor wind and wind ramp events would be scaled up in the near future, requiring a combination of approaches (computational fluid dynamic modeling, machine learning, and or support vector mechanism) validated with field observations to increase estimation accuracy. This line of knowledge provides a great motivation for our future studies based on the foundation provided in this study.

Since estimation errors and fluctuations occur irrespective of turbine type, location, and size, an essential step by energy producers would be to create their turbine or wind farm EPC before grid connection. By combining the turbine EPC approach with accurate short-term wind prediction tools such as the ARMA, ARIMA, and exponential smoothing methods, wind power prediction and estimation for very very short term (>5 min) and very short term (10 min–6 h) can be done more accurately. This would foster wind power optimization, positively influence the wind energy market, and improve wind power diffusion. Also, by minimizing estimation errors, policy support for increasing wind power share in energy portfolios may be more easily achieved.