WIND TURBINE POWER CURVE EXPONENTIAL MODEL WITH DIFFERENTIABLE

Background. The main characteristic of a wind turbine is its power curve. Getting measurement data off powerful wind turbines is a way harder than measuring characteristics of wind turbines for individual/home use. A yet significant gap is that all wind turbines have a few similarities in their power curves but they do not have a formalized description, which could help in selecting better turbines fitting specific areas (without precise measurements in a vicinity of cut-in and cut-out speeds). Objective. As there is a straight lack of mathematical description of wind turbine power curves, the goal is to obtain a model of such curves. Methods. A power curve is of seven parts. Factual power curves remotely remind trapezia with curvilinear flanks. Because of inertia, the curvilinearity is severer for those wind turbines whose output power is greater. As blades of industrial wind turbines are too massive, their inertia makes those lag effects, that could be modeled by using natural smoothness of power curves. For describing that smoothness along with the curvilinearity, we use two increasing and two decreasing exponential functions for the flanks. Results. A wind turbine power output function consists of two zero parts, one rated-out part, and the suggested four exponential parts. The cut-in parts are described with two increasing exponential functions whose exponential growth factors are equal. The cut-out parts are described with two decreasing exponential functions whose exponential decrease factors are equal also. Such equal factors ensure strong differentiability of the power curve within those parts. Conclusions. The exponential model is for a general description of the wind turbine power curve. Having differentiable cut-in and cut-out parts, it suggests the “natural smoothing” that happens in reality due to highly-inertial wind turbine blades. The model is not necessarily to be used to fit some experimental data, but rather for patterning power curves.


Introduction
Wind power is the most promising source of renewable energy. It is produced by wind farms consisting of wind turbines [1,2]. The wind farm productivity is mainly determined by characteristics of those turbines imposed on wind statistics of an area where the farm is built [1,3,4]. While the wind statistics cannot be influenced and changed, the wind turbine with better characteristics can be selected for fitting the area wind statistics [5,6]. Thus, before deploying a wind farm, the wind turbine characteristics should be modeled [4,5,7,8].
The main characteristic of a wind turbine is its power curve. This is a wind turbine power output function ( ) w s that shows an amount of kilowatts/megawatts produced at a range of wind speeds s . This range might be called active. When a power curve is plotted, the wind speed changes slowly, not abruptly. Acceleration of the wind speed change (caused by wind gusts, blasts, hurricanes, etc.) badly influences on the power output, but it has been studied less [9,10]. As close to an ideal case, the power curve would be rectangular or trapezoid having very steep flanks. The left flank, where the output power starts off zero, appears at a cut-in wind speed. The right flank, where the output power drops down back to zero, appears at a cut-out wind speed [1,6]. Factual power curves remotely remind trapezia with curvilinear flanks, whereon the nominal (rated-out) power appears at a speed greater than a cut-in speed, and the real drop to zero (a wind turbine turn-off) appears at a speed greater than a cut-out speed.
A few examples of power curves in [6] are of industrial turbines. For those wind turbines whose output power is greater, the left flank becomes more gently sloping. Their power curves remind trapezia less. And the right flanks become not so steep at all. An explanation is the turbines of the greater power are bigger, and their huger blades are much massive and inertial, taking longer periods to slow down and stop. Obviously, getting measurement data off more powerful wind turbines is a way harder than measuring characteristics of wind turbines for individual/home use. All the more that wind farms are not built of small turbines, whereas individual/home (non-industrial) use does not require building wind farms.
A yet significant gap is that all wind turbines have a few similarities in their power curves but they do not have a formalized description, which could help in selecting better turbines fitting specific areas (without precise measurements in a vicinity of cut-in and cut-out speeds). Indeed, it would be far easier to get a power curve knowing just a rated-out power along with cut-in and cutout wind speeds. Then a whole wind farm is possible to be modeled for its power output depending on wind statistics of a given area [5,6].
However, one should be aware of the wind power curve depends on the wind speed acceleration and turbulence [4,10,11]. The power curve is locally distorted worse as the wind speed acceleration grows higher. Apparently, small accelerations and turbulence always exist, whichever area for the wind farm is chosen. But they are very specific for different areas. This is why the known approaches to modeling power curves being tied to data of a given area are not correct for a general use. For instance, the power curves in [6] represented officially in product descriptions will pretty change for every turbine type on different areas. The change will be more considerable for Vestas V112-3.0 MW [2,3,10].
There are parametric and non-parametric techniques in power curve modeling methodology [11,12]. According to [11,13], the models based on the concept of power available in the wind, like the probabilistic and polynomial models, do not give accurate results. This is because of the fact that the fraction of wind power that is converted to electrical power depends on several other parameters like rotational speed of the turbine, turbine blade parameters, and the efficiencies of the mechanical transmission system and generator efficiency. The models based exclusively on the shape of the power curve, like the linearized segmented model and the model with the Weibull's parameters, perform poorly because the performance of the wind turbines with different design parameters and ratings cannot be modeled using a single set of general equations [11].
The modeling methods in which characteristic equations are developed based on the actual power curve of the wind turbine appear the best for wind resource estimation and for identifying potential wind farm sites/areas [4,5,11,12]. This also aids to make the right choice of turbines while the wind farm is projected [6]. Nevertheless, the actual po-wer curve requires long antecedent data on the wind speed and turbine power. Thus, a power prediction without the historic data should be based on some generalizations, without tying to a definite site/area. If a sizeable number of training and testing data is available, then non-parametric techniques based on data mining techniques and neural networks perform well. The performance of the wind turbine power curve modeled using four and five parameter logistic expressions is reported to outperform the linearized segmented model and the models based on neural network, fuzzy logic and data mining algorithms [9,11,13,14]. Thus, it is expected that such a outperformance shall exist for poorer initial and antecedent data.

Problem statement
As there is a straight lack of mathematical description of wind turbine power curves in the case of missing long antecedent data, the goal is to obtain a model of such curves. Such a model shall allow plotting a power curve without direct measurements but only by giving a nominal power along with a few crucial points (e. g., amounts of kilowatts/megawatts produced at cut-in and cut-out wind speeds). For achieving the said goal, the three tasks are to be accomplished: 1. To list those crucial points and initial data that should be given before modeling.
2. Based on the list, to substantiate a consistent generalized model of the wind turbine power curve compatible with power curves of real turbines.
3. To show how well the substantiated model fits some real measurements.
The list of initial data is presumed to be a set of input arguments for making the model. The said compatibility with real turbines' power curves implies not only true fits, but smoothing the right flank also, because that steepness must disappear after zooming in. Within the item #3, a procedure of how to fit the measurements will be explained. After that, the suggested model shall be discussed and a conclusion will be given with an outlook for further research.  The cut-in speed for Vestas The cut-out speed for Vestas The rated-out speed for Vestas The This is very naive, but the most expedient.
While those crucial wind speeds may be unofficially changed for fitting the model best, the nominal (rated-out) power rated-out w is given fixed. It is of the initial data wherein steepnesses for both flanks may be given as well.

An exponential model of the wind turbine power curve
The wind turbine power at each wind speed is influenced by a great deal of factors, both stochastic and predetermined [2,3,7,11,13]. The curvilinearity of the power curve flanks is implicitly formed in a similar way. That can be thought of as it is an average of a stochastic process with a normal distribution. All the more that logistic functions suit very accurately for the curvilinearity [4,9,11,12,14]. That is why the curvilinearity is going to be generally described with exponents.
Based on the list of crucial points and the sketch in Figure 2, a wind turbine power output function is stated implicitly as cut-in rated-out rated-out 1 2 2 cut-in rated-out rated-out 2 2 2 2 rated-out rated-out As blades of industrial wind turbines are too massive, their inertia makes those lag effects, that could be modeled by using natural smoothness of power curves. For describing that smoothness, we can obviously take two increasing exponential functions: by some exponential growth factor cut-in r and rated-out by some exponential growth factor rated-out r . Clearly, function (6) satisfies condition (2), and function (7) satisfies condition (4). Constants 1 K and 2 K are found from conditions (3) and, subsequently, rated-out 1 rated-out cut-in cut-in Plugging (5) into (7) gives us: rated-out 2 rated-out cut-in rated-out rated-out rated-out 2 rated-out and, subsequently, On the fifth and sixth parts of function (1) we have: cut-out 0 rated-out 3 2 2 cut-out 0 rated-out 4 2 2 For these parts we can take two decreasing exponential functions similar to (6) and (7): (1 e x p (( ) )) s s r K    by some exponential decrease factor cut-out r and rated-out by some exponential decrease factor 0 r . Clearly, function (14) satisfies condition (10), and function (15) satisfies condition (12). Constants 3 K and 4 K are found from conditions (11) and (13). Plugging (11) into (14) gives us: and, subsequently, And, finally, plugging (13) into (15) gives us: and, subsequently, rated-out Factors cut-in r , rated-out r , cut-out r , 0 r may be theoretically given as the initial data. However, their magnitudes are going to be constrained to a property of strong differentiability.

Strong differentiability for the flanks' proper curvilinearity
For completing the "natural smoothing", second and third parts of function (1), and fifth and sixth parts of this function should be differentiable within intervals, on which they are defined (except, probably, for endpoints of those intervals). Therefore, equalities cut-in rated-out cut-in rated-out The derivative of function (9) is 2 rated-out rated-out cut-in rated-out rated-out cut-in rated-out rated-out ( ) It is easy to see that values (20) and (21) The derivative of function (17) is  and cut-out 0 r r  , fits the measured power curves of wind turbines Enercon E82 E2 and Nordex N90/2500, where parameters (25) are easily found by a looped search by a step 0.01 (such accuracy is sufficient). However, fitting at cut-in and rated-out parts does not seem so good everywhere (the right flank in Figure 3) because of sharp changing of the power around the "corners". Consequently, finding a curve by (26) might have been accompanied with an additional condition of that leaps of the first derivative at each of points (24) be minimized.
For fitting more around the "corners", exponential growth/decrease factors may be taken dif- will not be the joint points. Nevertheless, the generalized model by those equal factors is still consistent because it should rather be "slower" around the "corners" due to probable wind dynamics and turbulence (the plotted power curves are static because their real measurements were conducted while the wind speed was changed in non-dynamic mode). It is worth to notice that this is not about to fit real measurements at all costs. While parameters (25) are searched/evaluated with their corresponding loops, where subsets (29) and (30) are evaluated separately, a few versions for them may come out. Then a parameter's evaluation that enables rounding the "corners" is accepted. Thus, evaluations of speeds cut-in s and cut-out s will be less than their nominal magnitudes (if they are available), and evaluations of speeds rated-out s and 0 s will be greater than their nominal magnitudes (if they are available). According to the said, the approximation in Figure 4 is more universal, unlike the approximation in Figure 3, which will work accurately on areas with no wind gusts (and where the wind speed changes very slowly).

Discussion
The exponential model reflects the sketch in Figure 2, where the flanks are really overextended. Such an overextension will exist due to the inertia of the blades. Their inertia dramatically grows when wind changes dynamically. Therefore, a static representation of the power curve is substituted with the proposed model suggesting that the strong differentiability is preferred to independently measured states. Despite the non-differentiability at points (24), the power curve exponential model becomes very smooth if appropriate parameters (25) are adjusted.
While adjusting, speeds cut-in s and cut-out s may be decreased. Speeds rated-out s and 0 s may be increased. This leads to that these speeds will differ from those ones declared by a wind turbine manufacturer. However, it is not a demerit but just a modifier.
An essential merit of the proposed model is that its power curve's natural smoothness is totally consistent with the inertia. It concerns especially industrial wind turbines. Eventually, the model allows calculating more reliable amounts of the expected power output based on wind statistics.

Conclusions
Exponential model (1) with parts (8), (9), (16), (17), by cut-in rated-out r r  and cut-out 0 r r  , is for a general description of the wind turbine power curve. It has differentiable cut-in and cut-out parts. Unlike static power curves plotted mostly by changing the wind speed non-dynamically, the model suggests the "natural smoothing" that happens in reality due to highly-inertial blades (compare the "natural smoothing" in Figure 2 to realbut-static measurements in Figure 1).
If a researcher/projector of a wind farm wants to select the best wind turbines under the known wind statistics, then parameters (25) are varied along with the nominal power until the expected power of the wind farm reaches its maximum (e. g., see [2,4,5,7] by [6]). The corresponding configuration of parameters (25) and rated-out w is compared to the existing wind turbines, whereupon a closest match is selected.
The research may be furthered with appending the condition of that leaps of the first derivative at each of points (24) be minimized. For this, conventions of cut-in rated-out r r  and cut-out 0 r r  will be canceled. One should remember, however, that the model is not necessarily to be used to fit some experimental data, but rather for patterning power curves. Practically, this will do also when a nominal power along with amounts of kilowatts/megawatts produced at cut-in and cut-out wind speeds are given.