CC BY
29Journal of Siberian Federal University. Engineering & Technologies 2 (2013 6) 150-165
УДК 621.311.24
Simulation of Wind Speed in the Problems of Wind Power
Sergey N. Udalov and Natalya V. Zubova*
Novosibirsk State Technical University 20 K. Marksa, Novosibirsk, 630092 Russia
Received 15.03.2013, received in revised form 22.03,2013,accepted 31.03.2013
The article is devoted to research of mathematical models of wind speed and development a way to improve the efficiency of wind power plants based on fuzzy logic using fuzzy model of the wind speed. Simulation of the wind speed is a fairly difficult task, sinoe this source of energy is constantly changing in time and space. Four basic models of wind speed were identified at the end of research work: determinate, probability, spectral and fuzzy. Every one finds their own field of application. Thus, from the energy point of view, the mo del probabilistic distribution ofWeibull is most applicable at h level of technical and economic development. Deterministic model allows to determine the power generated by wind turbines at a given average wind speed. The spectral model should apply in th ose studie s where necessa ry to account for gusts of wind and sudden changes. Fuzzy model of the wind is the most convenient and relevnnt for modeling off wind turbine control, it is allows to form a flexible control system.
Keywords: distribution function, fuzzy sets, membership function, wind speed, probability, spectral density.
Introduction
Tlie revival of i nterest in the us e of wind p ower is now connected with the opportunity to determine the feasibility of convhrting it into alectricity. Econrmicaily, this eccurs when the feasibility of wind speeds exceeding 5 m/s. And this imposes a significant limitation on the development of wind energy. An area which has a rich wind, as a rule, do not need in electricity because of remote location of industrial facihties and reside ntial areas.
Most wind turbines are used tor gene rate electricity in power grid, as well as offline. When the wind speed u0, air density p and swept area A, wind turbine has an output power [1]:
3 pu(
P = C 0, (1)
T 2 w
where Cp is the fraction of nhe upstweam wind powee, which is crptured by the rotor blades andcalled the power coefficient o0 the rotoi or rotor efficiency.
From (1) can be seen that the power P is proportional to the swept area A and the cube of the velocity u0. Power coefficient Cp depends on the design of rotor and wind speed. Since the wind
© Siberian Federal University. All rights reserved
* Corresponding author E-mail address: zubna85@gmail.com
velocity is not constant, and the power is highly dependent on it, the choice of the optimal rotor design is lawgely determined by the requirements of the; consumer of enorgy. Usually, average power per unit area, which turbine extracts from wind is proportional to Cp, air density and the cube of the mean velocity i.e. P ~ Cpp(m)3.
From the expres sion (1)) we can see that the wind speed is flue most critiral data ne eded to appraise the power potential of a candidate site;. The wind is never steady in any state. It is influenced by weather conditions, topography, and relative neight abone the surface;. The wind speed vrriea by the minute, hour, day, season or year. Therenore, the annual mean speed needs to be averaghd over 10 or more yeans. Such a long team average raises tta confidence in assersing the energy-capture potential of a site. However, long-term measurements are expensive, and most projects cannot wait that long. In this situation, the short term, say one year, data is compared with a nearby site having a long term data to predict the long term annual wind speed at the site under consideration.
Mathematical model of wind speed
Probabilistic model of the wind speed
Taa varihtkm in wind speed are best described by the Weibul[ probability distritrutinn function 'h' with two paraImntert, the shape raraiiaeter 'k' and the scale prramerer 'n'. The probability of wind speed being ' u' during time interval is given bg the followi na expression [2]:
h(u) = (kS(-s(k-n)e7' , o < u < №. c c
Fig. 1 is the plot of h vtrsus u toe three different values of k. The cutve on the lhft with k=1 has a heavy baas to ehe left, where most days are windless (u=0). Tfe curve on the; right with k=3 looks more tike a normal bell shape distribution, -where some days have high wind and enual number of days have low wind. The curve in the middle with k=2 is a typical wind distribution found at most sites. In this distribution, more days have lower than the mean speed, while few days have high wind. The value of k determi nes the shape of the curve, hence i s calle d the "shape paramete r".
Wind speed, mph
Fig. l.Weibull probability distribution function with scale parameter c=10 and shape parameter k=1,2 and 3 (plots 1-3 respectively)
wind speed, mph
Fig. 2. Weibull probability distribution with shape parameter k=2 find the scale parameters ranging; from 8 to 16 miles per hour (mph)
Fig. 3. Rayleigh distribution of hours/year compared with measured wind-speed distribution
The Weibull distribution with k=l is called the expone ntial distribution which is generally used in the reliability studies. For k=3, it approaches the normal distribution, often called the Gaussian oe the bell-shape distribution.
Ft*. 2 shows the distribution cuives corresponding to k=2 with different values of c ranging from 8a to 16 mph (1 mpd = 0,446 m/sf For greatea -values ot c the curves shift right to the higher wind speeds. That is, the higher the c, the more number of days have high winds. Since this shifts the distribution of hours at a higher speed scale, the c is called the scale parameter. At most sites the wind speed has the Weibull distribution with k=2, which is specifically known as the Rayleigh distribution.
The actual measurements data taken at most sites compare well with the Rayleigh distribution, as seen in figure 3. The Rayleigh distribution is then a simple and accurate enough representation of the wind speed with just one parameter, the scale parameter 'c'.
Summarizing the characteristics of the Weibull probability distribution function:
к = 1 - makes it the exponential distribution, h = Xe-Xu, где X = c_1; к = 2 - makes it the Rayleigh distribution h = 2X2ue-<X">); (2)
к = 3 - makes it approach a normal bell-shape distribution..
Since most wind speed sites would have the scale parameter ranging from 10 to 20 miles per hour (about 5 to 10 m/s), and the shape parameter ranging from 1.5 1o 2.5 (rarely 3), our discussion in the
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k=3
wind speed, mph
Fig. 4. Weibull distributions: a) 10 mph, 6) 20 mph.
wind speed, mph
following will center around those ranges of c and k. Fig. ¿1 display s the number of hours on the vertical axis versus the wind speed on the horizontal axis with distributions of different scale parameters c=10, and 220 mph and shape parameters k=1.5, 2 and 3.
The resultingdislribution lhw for any area ran be uied ao aetermina the potential of wind power and annual powte genarationin the execution phase of technical - economic calculations.
Deterministic model of the wind speed
Mode speed is defined as the speed corresponding to the hump in the distribution function. This is the speed the wind blows most of the time.
Mean speed over tine period is defined as the total area under the h-u curve integrated from u=0 to co, divided by tee total namber of hours in the period ( 8760 id the period is one year):
1 ™
U =-f hudu .
mean gygQ J
In general, for the Weibull function can be obtained:
Umean = cr (1 +1) = c[(i)l], (3)
U"mean = c" r (1 + (4)
k ■
If n = 3 expression (4) can be rewrite:
UL, = c3 r (1 + (5)
k '
from which we can to derive an expres sion for wind energy.
The parameters c and k are defined on the phase approximation of the Weibull distribution of meteorological observations. For example, if Umean and U 3mean are known, the parameters c and k are defined by the equations (3) and (5). Umean and UU can simply define by modern methods of primary processing ef meteorological informatron wihhout referring to ehe results of numerous individual msaeurement s. The Weibull distribution pasametrr k ss dimeneionlehs.
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Dimensionless parameters, ¡allowing to operate with distribution functions, regardless of the actual values of wind speed, convenient in maoy cases, for txample, wlien the average wind speed is known only. At fht approximated Weibull distribution psramattr k, as a rule;, is in the range 1.6 ...3.0. The value c«2Cems,aii ./to is not; more than 1% differeni foom the conresponding value in the Rayleigh distribution, w lieie k consi, and thenefore is ts postible te sliow that:
3 k
U
mean
A number of studict verified the hypothesis that the value of k depends only on topography of this region and the general characteristics of the wind. Then h you know only the mean wind speed and the results of long-term meteocologirrl observations, you can evaluate the Osaquency and duration oC periods of calm.
Foa the Raeleigh distrhbutioo with k=2, the Gamme function can fe Ournher approximated to the following:
Umean = 0,9C .
This is a very simple relation between nhe scele parameter c and Shmean, which can be used wfth reasonafle accueancy. For enompte, most sites are repo efed m terms of their mean wind speeds. The c parametee in the corresponding Rayleigh disfribution is then c = t/mean/0,9, and k = 2. Thu si. We have thce Rayleigh disiribution of the site; using the generally repo rted mean speed as Rollows:
u .2
■ rV -(--)2
2u - (_) 2u U h(u)=—e c =-—-e mean .
c2 (U )2
mean
Thus, the deterministic model to calculate the approximate value of power generated by wind turbines, ait a certain overage wind speed.
Spectral model of the wind spee d
From a sy ttem point of view, tine wind speed represents the; main exogenous rignal applied to the wind turbine aed deaermines its behavior. Ies erratic variation, highly dependent on she given site and on nhe aamospheric conditions, makrs nhe wind speed quite difficult to model. "Usually the thermic equilibrium (if the atmosphere nearby Earth is arsumed [3]. Thenefore, Surbubence cesulrs mainly from the friction between air and ground, due to the ground roughnesi. When designing wind turbine, the history of the wind speed extreme values (gusts) is considered for the mechanical structure design and also for control purposes.
Wind near the Earth's surface is generally modeled by a spatial (3D) speed distribution. Assuming tdet thee turbine is equipped with a vane (or yawing equipment) and that changes in wind direction are sufficiently slow, then the turbine rotor is maintained eormal to the wind and wind turbine analysis requires only the loagitudinal wind speed being aynthesized/modeled. Thus, in ehe presendbook only scalar (1D) wind speed models will be used. Ass ehe interest here is focueed on wind turbine behavior
^ectaim/- Svv(f)
f \ Low frcqucncy
' | component
Tucbulence ,•«,
aomponent ; >e 1 I 1 % t 1 h 1
1 III I t + / * Crtmucnfd e ^f wh a' ll l l II ll
0
1<Tdcys 4dayh T4h 10h ft fiaOmir. lOmin ninie lmifll ()s 11 5a Fig. 5. Van der Hoven's opectral modelof the wind speed
in normal operating regimes, the developed models will not include extreme operating conditions like wind gusts.
Wind dynamics result from combining meteorological conditions with particular features of a given site. Thus, wind speed is modeled in tire literature as a non-stationary random process, yielded by superposing two components [3 - 5]:
u(t) = us (t) + u, (t), (6)
whete us(i) - is she low-frequency component (describing long term, low-frequency variations); u,(t) - is the tuebulenoe component (corresponding to fast, high-frequancy variations).
These components can be identified in Van der Hoven's large band (six decades) model (Fig. 5). The spectral gap of around 0.5 MHz suggests that the turbulence component can be modeled as a zero average tandom process (thers is lirtle energy in the rpectsal range beiween 2 ( and 10 min). us(t) is considertd constant (equal to the average wind speed) when viewed ht the turbulence time scale. Averaging is usually performed on a 10-min time window [3].
The low-(rtquency component corresponds to the vtry slow wind speed variations and charaaterizes the site nrom the energy viewpoint. It can be modeled as a Weibull's distribution or a Rayleigh's distribution see expression (2).
Fast wind speed variations (typically occurring within 10 min) are modeled by the turbulence component. This is mathematically described as a zero average normal d^tribution, whose standard deviation, c, depends on the current value of tire hourly average, us. The turbulence intenrity is a mearure of the global level of hurbulence, depends ou the graund surface roughness and is defined as:
It = ^ , (7)
t u
s
The mathematical descriptiou of the turbulence's dynamical properties, u,(t), can be obtained by using two kinds of spectra: von Karman's and Kaimal's respectively. According to [3], Kaimal's
spectrum reflects better the correspondence to experimental data, when turbulence is present. But von Karman's spectrum is mooe consistently theotetically founded (an analyticat connection with the correlation function is provited) and allows a realistic representation of turbuSence data in wind
tunnels. The von Karman's model for the longitudinal component of the turbulence is:
4 f•L
f • Suu ( f ) _ _US m
-ô-_-L-5 ' (8)
(1 + 70.8(f • L)2)6 US
wliere Shef) is the power sprctral de nsity, L, is the length ot turbulence, specific to the site (ground roughness), and f is the frequency in Hz. Kaimal's spectral model has the form:
4/A
f-S^f ) = us
2 j 5 ' (9)
(1 + 6/-^) US
One can note that in both models the power spectral density is influenced by the turbulence intensity, It, which determinee the tuebulence"level" (i.e., ies variance, c2) and the turbulence length, Lr, which impresses the turbulence dynamic properties (the apecteat function bandwidth). Both these parameters are adopted according to various standands. For example, in the Danish standard (DS 742 200a), )he followibg relationr are used to compute these parameters:
It =-V- , (10)
la=(—)
zn
i 0
iii^d respectively
L, =
150m, z > 30»î 5 • zm, z < 30m
(11)
vvhe;re ^ is the height from ground where the wind speed is computed and z0 is the roughness length.
Fig. 6 comparatively presents the spectral functions at Equations 8 and 9 for the same values of parameters z, z0 and For ¡an easier analysis of von Karman's and Kaimal's spectra, in Fig. 7a one can see the corresponding power spectral densities, Suu(f), whereas Fig. 7b shows the Bode diagrams oh the non-integer-order shaping filter outputting the turbulence component when fed with a white noise [6].
Fuzzy model of the wind speed
Handling fuzzy information in the problems of wind energy is achieved by using linguistic variables. As part of the linguistic approach the values of variables are allowed not only the number but also the words and sentences of natural language, as well as a mathematical tool used to formalize the theory of fuzzy sets.
Normalized spectrum
0.3,-1—!-
-N, \
/ j / / /' /
/[Hz]
Fig. 6. Comparison between the von Karman's (Equation 8 - solid line) and Kaimal's (Equation 9 - dashed line) normalized spectra (z=30 m, z0=0.01 m, us=10 m/s , Danish standard DS472)
Fig;. 7. Von Karman's (solid line) vs. Kaimal's (dashed line) spectral models (z=30 m, z0=0.01 m, ur=10 m/s , Danish standard DS472): a) power ipectral densities; b) shaping filter gains
One of the main steps is to build membership functions that describe the semantics of the basic values of the variables used in the model when you create a fuzzy model of decision-making. These functions are characterize the uncertainty such as "approximately equal", "average", "is in the range", "like an object", etc and used to specify sets of properties.
The process of fuzzy modeling is leased on a quantitative representation of the system variables in the form of fuzzy membership functions.
It is known that wind speed in the interval setting can be represented by the Beaufort scale (Table 1) [7]. At the srme charasteristics of the wind speer is given as both a linguistic evalurtirns: weak, steong, variable, erc. [8].
When information about wind speed given lhke an interval, for example, the wind is strong, its speed is u = [11, 14] m / s and the resulting solution of the expected power generation interval estimates are obtained, which enlails a rignificant dsawback - it is impossibls ro determine which value of the variable is more or lesr seliably.
Table 1. Strength of the wind on the Beaufort scale and its impact on wind turbines
points Beaufort Wind speed, m/s Characteristics of wind power Observed effects The impact of wind on wind turbines The conditions for wind turbines production, with an average wind speed
2 1,8 - 3,6 Light breeze Wind felt on face, leaves rustle, vanes begin to move not Bad for all wind turbine
3 3,6 - 5,8 Gentle breeze Leaves, small twigs in constant motion; light flags extended Begin to turn low-speed turbine Satisfactory to the pumps and some aerogenerators
4 5,8 - 8,5 Moderate breeze Dust, leaves and loose paper raised up; small branches move Begin to rotate the wheel aerogenerators Good for aerogenerators
5 8,5 - 11 Fresh breeze Small trees begin to sway Capacity of wind turbines up to 30% of the project Very good
6 11 - 14 Strong breeze Large branches of trees in motion; whistling heard in wires Maximal power Valid
7 14 - 17 Moderate gale Whole trees in motion; Resistance felt in walking against the wind Maximal power Valid
8 17 - 21 Fresh gale Twigs and small branches broken off trees Some wind turbine off The maximum permissible
9 21 - 25 Strong gale Slight structural damage occurs; slate blown from roofs All wind turbine off Invalid
We use the possibility of representation of wind speed by fuzzy variables. The Beaufort scale imagines corresponding characteristic membership functions of linguistic variables of wind speed. Moreover, membership functions are chosen from the following considerations: for the border interval of values of wind speed, the famous Beaufort scale, each linguistic variable is assigned a value of belonging | = 0.5, at these points the values of wind speed will have equal weight in relation to the neighboring variable. With a value of | =1 the velocity in each band is equal to (umax - umin)/2 t9].
Thus, for each linguistic variable, we define the value of belonging to the whole interval (Table 2). Fuzzy numbers and intervals, which are most often used to represent fuzzy sets can be described in the form of analytical approximation using the so-called (L - R) -functions [10].
Graphically, these characteristics may be represented by a family of fuzzy triangular function (Fig. 8). From here you can see that in the section | = 0.5, this characteristics described by the interval values, as indicated on the Beaufort scale.
Considering the membership function ofwind speed (Fig. 8), it is necessary to consider membership in the interval from 0 to 1. So, finish the construction of each value of fuzzy variable, which will have a base value of | = 0.
Table 2. Fuzzy variables as a characteristic of wind power
Wind speed u, m/s Affiliations Characteristics of wind power
1,8 0,5
2,7 1 Light breeze
3,6 0,5
3,6 0,5
4,7 1 Gentle breeze
5,8 0,5
5,8 0,5
7,15 1 Moderate breeze
8,5 0,5
8,5 0,5
9,75 1 Fresh breeze
11 0,5
11 0,5
12,5 1 Strong breeze
14 0,5
14 0,5
15,5 1 Moderate gale
17 0,5
17 0,5
19 1 Fresh gale
21 0,5
light gentle moderate fresh strong moderate fresh breeze breeze breeze breeze breeze gale gale
wind speed, mph
Fig. 8. Fuzzy values of wind
In general, each of these functions can be described analytically by the following expression.
M x) =
0, c¡ < x < a¡ x - a
bi - ai c¡ - x c. - b
at < x < bt, ( 12)
b, < x < c,
where m, bj h cp - numerical parameters (L - R - function that take real values.
The parameters at and c, characterize the basse on th- triangle, and the parameter bi - its top (Fig. 9). Atyou can see, -his membtrtltip function generates animodat normd conivax fuzzy set with the catrinr - the in-esnal (a,-, c(, boundaries (a,-, i,)\{khS, tbh ausdi mode bf
s r tind s i - left-handed dnd right spread of ttlties maximum value crf belonging.
In describing the fuzzy scale Beaufort, the membership runction takes the form of symmet-lctl trianntes, i.e. so = sa .
Table 3 shows the values of numerical parameters (L - R) - functions that characterize the linguistic variables of wind speed.
Corresponding analytical expressions can be fined for linguistic variables of wind speed knowing the general expression (L - R) - function (12) and numerical parameters.
Oftan, in real peoblems it is necessar- to split the memaeoship function of fuzzy set on the soc alled a - levels, so thaf if a f-zzy set A is defined on the underlying seet X and a e [ 0, 1 ], then ttte classical eet Aa, defined iy thee expression (13)), called the setof a - the level ot A.
4, ={ies (*)>«., ha
Figure 10 shows an illustration a - levels, Or OC2, . . . , tt„ fuszy set with a symmetric triangular membeaship function.
It is should be noted that, eachfuzzy set can be represented by asetofa-levels. This repteaentation of fuzzt sets ustd in solving practical proflems, {particularly ie the construction of fuzzy models with different levels of fti-zinesc [11].
Fig. 9. Graph a triangular membership function
Table 3. Numerical parameters of (L - R) - functions fuzzy Beaufort scale
Characteristics of wind power a, b, ci
Light breeze 0,9 2,7 41,5
Gentle breeze 2,5 4,7 6,9
Moderate breeze 4,45 7,15 9,85
Fresh breeze 7,25 9,75 12,25
Strong breeze 9,5 12,5 15,5
Moderate gale 12,5 15,5 18,5
Fresh gale 15 19 23
In further studies, the entire area of belonging from 0 to 1 should be split into multiple levels, so for clarity, will cross a few (four) points of the values of n = (0...1);; each section (Fig;. 10) will characterize the level of membership aj.
Knowing a given level of membership) and the corresponding wind speed, we can calculate the typical powe r pso duced by wind turbines with respect to a given spee d, which will take place at the same a - level.
It is should be noted that whrn the wind speed is less than the minimum operating wind (<4m / s), blades stationary and power generated by wind turbines is zero. Thus, the values oy power oytput oy wind turbines can be determined depending on wind speed and the speed of the membership of each of the respective linguistic variable.
In geneval, the domain oy the membership | = (0 ... 1), we have a set of fuzzy sets Xy wlaere l - tlae number od liaguistic variablas ohwind speed (very light, light,... vety stsong).
X, ={<vj,^Xl(vi)>, < v2,^Xl(v2)> , ... , < Vn,Vx(vn) >}, (14)
where n - number of levels of membership aj.
In accordvnce with (14) can ne noted that tlie power of wind turbines is a function ov wind speed:
Yk=f(X) = f {<vi, ^yl(v1)>, < MTl(v2)> , ... , < v„, fly (v„) >}. (15)
where k - a number of linguistic variable of the wind turbine power (very small, small and etc.).
Controller for improvement efficiency of wind turbine on the basis of the wind speed fuzzy model
Fuzzy modeling is a new modern technology that is used in various fields of science and technology. First of all, this technology is relevant in cases when you want to improve the adequacy of the model systems to take into account many different factors that influence the decision-making processes. In addition, mathematical models and formal management systems and processes increasingly complex and sometime system of fuzzy relations can to implement them without too much trouble [12].
In the wind power system fuzzy logic is used quite actively - fuzzy controllers used for wind turbine yaw control [13] changing the angle of attack and angle of the blade jammed, the rotor speed [14]. In this paper we propose a controller for wind turbines, which changes the blade length [15. Fuzzy model is used for realization of the wind speed, because it is most relevant in terms of impermanence energy source.
The proposed fuzzy controller is based on the idea of variable length blades, which arose as a result of the desire to increase power output of wind turbines in area 2 (area from start to rated speed of the wind turbine) [16]. The calculations showed that using of these blades can increase the production capacity up to 30%. The implementation of a fuzzy controller gives below in more detail.
The results ofthe development
Fuzzy controller designed for wind turbine Nordex N80/2500 kW with a radius of wheel 40 m. The study assumed that blades can be increased up to 48 m.
The length of the blade, the wind speed given by Beaufort scale and power of wind turbines are input variables for fuzzy controller. Mamdani algorithm is used for fuzzy output. The average of the maximum is used for defuzzification, which is defined as the arithmetic mean of left and right modal values. We get a specific value on which the length of blades changes to maximize power generation.
Rules were established for the proposed model - the control action (Table 4). For the linguistic variable power, P, used the following term-sets: BN - more than nominal value, SN - less than the nominal value, N-nominal value; membership functions shown in Fig. 11. For the linguistic variable wind speed, Vw, used the term-sets of the Beaufort scale; the membership functions shown in Fig. 11. For the linguistic correction of variable length, AL, use the following term-sets: Z - do not change, D - to decrease, I - to increase; the membership functions shown in Fig. 12.
Fuzzy controller is implemented in the program Matlab, using a special expansion pack Fuzzy Logic Toolbox. As part of this package, you can perform all actions necessary for the development and use of fuzzy models. With the help of fuzzy inference system editor FIS can set and edit the properties of high-level fuzzy inference system, such as the number of input and output variables, the type of fuzzy inference, defuzzification method, etc. We obtained a kind of summary table that can be used to assess the adequacy of the controller (see Fig. 13).
0 500 1000 1500 20(1(1 25(1(1 3(1(1(1 0 2 4 c 8 10 12 14 16
input variable "power input variable "wind"
Fig. 11. Membership function forthepower and the wind speed
output variable "dL
Fig. 12. Membership function tochange the length of the blade
Table 4. Rules for the fuzzy controller
№ rule Power, P Wind speed, Vb Correction oftCe blade length, AL
1 BN Gentle breeze 3,(5 -15,8 m/c D
2 N Z
3 IS Isi I
4 BN Moderate breeze 5,8 - 8,5 m/c D
5 N Z
6 SN I
7 BN Fresh breeze 8,5 - 11 m/c D
8 N Z
9 SN I
10 BN Strong breeze 11 - 15 m/c D
11 N Z
12 SN I
Fig. 13. Rules table in Matlab
Features of this software can also evaluate the effects of input variables on the value of the output variable.
Conclusion
In the paper reviewed the basic mathematical models of wind speed. Their area of use is defined. Controller for improvement efficiency of wind turbine on the basis of the fuzzy logic and using wind speed fuzzy model are designed. This model tested on adequacy in Simulink/Matlab. The idea of this controller can find a place in industrial area along with existing ones. Coordination of multiple controllers, if necessary, you can also implement a fuzzy controller.
References
[1] Твайделл Д., Уэйр А. Возобновляемые источники энергии. М.: Энергоатомиздат, 1990.
[2] Patel Mukund R. Wind and Solar Power Systems. L.; N.Y.; Washington, D.C.: CRC Press,
1999.
[3] Burton T., Sharpe D., Jenkins N., Bossanyi E. Wind energy handbook. John Wiley & Sons, New-York, 2001.
[4] Bianchi F., De Battista H., Mantz R. J. // Principles, Modelling and Gain Scheduling Design. Springer, London, 2006.
[5] Nichita C., Luca D., Dakyo B., Ceang E. // IEEE Transactions on Energy Conversion, 2002. 17(4). P. 523-529.
[6] Munteanu I., Bratcu A.J., Cutululis N-A., Ceanga E. Optimal Control of Wind Energy Systems. Springer-Verlag, London Limited, 2008. P. 300.
- 164 -
[7] Удалов С.Н. Возобновляемые источники энергии: Учебник. Новосибирск: Изд-во НГТУ, 2007. 432 с.
[8] Заде Л. Понятие лингвистической переменной и его применение к принятию приближённых решений. М.: Мир, 1976. 165 с.
[9] Udalov S.N. // Korus - 2005, 9th Korean-Russian International Symposium On Science & Technology, 2005. P. 234238.
[10] Кофман А. Введение в теорию нечетких множеств М.: Радио и связь, 1982. 432 с.
[11] Нечеткие множества и теория возможностей. Последние достижения / ред. Р. Р. Ягер. М.: Радио и связь, 1986. 408 с.
[12] Леоненков А.В. Нечеткое моделирование в среде MATLAB и FuzzyTECH. СПб.: БХВ-Петербург, 2005. 736 с.
[13] Kung Chris Wu, Rony K. Joseph, Nagendra K. // Mechanical and Industrial Engineering Department The University of Texas at El Paso, El Paso, TX 79968-0521, IEEE.
[14] Evgenije Adzic et al. // Acta Polytechnica Hungarica, 2009. Vol. 6. No. 1.
[15] Зубова Н. В. // Научный вестник НГТУ Новосибирск: Издательство НГТУ, 2010. № 1 (38). С. 159-163.
[16] Sharma R.N., Madawala U. // 16th Australasian Fluid Mechanics Conference, Crown Plaza, Gold Coast, Australia, December 2-7, 2007. P. 481-486.
Моделирование скорости ветра в задачах ветроэнергетики
С.Н. Удалов, Н.В. Зубова
Новосибирский государственный технический университет Россия 630092, Новосибирск, пр. К. Маркса, 20
Статья посвящена исследованию математических моделей скорости ветра и разработке способа повышения эффективности выработки ветроэнергетической установки на основе нечеткой логики с использованием нечеткой модели скорости ветра. Моделирование скорости ветра представляет собой достаточно сложную задачу, так как данный источник энергии постоянно изменяется во времени и пространстве. В результате исследований было выделено четыре основных модели скорости ветра: детерминированная, вероятностная, спектральная и нечеткая. Каждая из них находит свою область применения. Так, с энергетической точки зрения, на уровне технико-экономических разработок наиболее применима вероятностная модель или распределение Вэйбулла. Детерминированная модель позволяет определить мощность, вырабатываемую ветроустановкой при заданной средней скорости ветра. В тех исследованиях, где необходим учет порывов и резких изменений ветра, следует обратиться к спектральной модели. Нечеткая же модель ветра удобна и наиболее актуальна при моделировании процессов управления ВЭУ, так как позволяет сформировать достаточно гибкую систему управления.
Ключевые слова: функция распределения, нечеткие множества, функция принадлежности, скорость ветра, вероятность, спектральная плотность.
