Abstract-- Probabilistic models for the power output of Wind. Electric Conversion Systems (WECS) are considered. Wind speeds are modeled using Weibull ...
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Probability Density Functions for Power Output of Wind Electric Conversion Systems Manish Mohanpurkar, Student Member, IEEE, and R. G. Ramakumar, Life Fellow, IEEE
Abstract-- Probabilistic models for the power output of Wind Electric Conversion Systems (WECS) are considered. Wind speeds are modeled using Weibull distribution and probability density function for the power output of WECS is derived using the transformation theorem. Variable portion of the power output characteristic is modeled using different functions and the results are compared. Probability density functions for the combined power output of two, four and eight systems are presented and discussed. Index Terms-- Wind Electric Conversion, Weibull distribution, Scale parameter, Shape parameter, Probability density functions.
I. NOMENCLATURE Symbols v Instantaneous wind speed, m/s Vc Cut-in wind speed of WECS, m/s Vf Furling wind speed of WECS, m/s Vr Rated wind speed, m/s f(v) Density function of wind speed F(v) Distribution function of wind speed f(p) Density function of output power of WECS α Scale parameter of Weibull distribution of wind speed β Shape parameter of Weibull distribution of wind speed Pr Power rating of the WECS, kW pow Availability of the WECS qow Unavailability of the WECS F1 Probability the wind speed is between Vc and Vf F2 Probability that wind speed is between Vr and Vf h Hub height, m Dp Derivative of p w.r.t v
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II. INTRODUCTION
NDUSTRIAL, commercial and domestic energy needs are mainly satisfied by electrical energy. Up-till now, the primary generation has been using non-renewable resources such as coal, oil, natural gas, and diesel fuels. Depletion, environmental concerns, rising costs of extraction, and geopolitical implications are discouraging their usage, and renewable energy resources are gaining momentum to meet growing demands. Solar-thermal, solar-radiation, wind, geothermal, tidal and wave energies are all being considered
The work reported in this paper was supported by the Engineering Energy Laboratory and the PSO/Albrecht Naeter Professorship in the School of Electrical and Computer Engineering at Oklahoma State University, Stillwater, Oklahoma.
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worldwide. Technological and operational limitations are present with all these resources in the areas of resource variability, resource assessment, economic viability, penetration levels and public acceptability issues. Wind energy has been one of the most promising and widely employed resource over the millennium. Currently, significant penetration levels have been achieved by generation from wind into national electricity grids of many countries. Earliest ever use of wind power is estimated to be about 5000 years ago, for sailing ships along the Nile river in Egypt. Europeans utilized wind power for agricultural purposes such as grinding grains and pumping water supposedly in the 1700s. First ever windmills to generate electricity in rural America were installed in the 1890s. Large scale wind power development in the U.S.A has been going on since the late 1970s, sparked by the first Arab oil embargo of 1973. By 1979, a 2 MW experimental machine was installed jointly by Department of Energy and NASA on the Howard Knob Mountain in North Carolina. Major factors that have accelerated the development of Wind Electric Conversion are: high-strength fiber composites for constructing large low-cost blades, falling prices and improvements in power electronics, variable-speed operation to capture maximum energy and lower the stresses, improved plant operation, economy of scale and accumulating field experience leading to improved capacity factors [1]. As wind farms become larger, their outputs need to be evacuated to the grid for utilization and this will require adequate transmission facilities. Interconnection standards have to be followed in order to integrate effectively. Technical standards that are to be adopted by the industries originate from standards the developed by the IEEE or the IEC [2]. Wind is naturally variable and stochastic. This has a direct impact on the power output of Wind Electric Conversion System (WECS). Several approaches have been employed to forecast wind [3-4]. The most widely used model for wind speeds is the Weibull model. Rayleigh distribution is a special case of Weibull distribution and it has also been used [5]. In this paper, Weibull distribution is employed to model wind speeds of medium time scale which extends from monthly to annual. Probabilistic and chronological techniques are well established procedures to assess performances of generating systems. The latter one is computationally tedious and requires large data sets, and so the first one is preferred [6]. Probability density function of power output of WECS is obtained by combining the models for wind speed and power output characteristic using the transformation theorem. Parameters of a GE 1.5s WECS rated at 1.5 MW as given in [14]
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and distributions from typical wind sites have been used to generate the results discussed in this paper. Combined power output density functions of several WECS are also studied. III. ANNUAL WIND SPEED MODELING Wind is air in motion relative to the surface of the earth. The wind vector is considered to be composed of a steady wind plus fluctuations about the steady wind. The primary cause of air motion is uneven heating of the earth by solar radiation. Uneven heating is mainly caused by the nonhomogeneous nature of earth’s surface which causes differential absorption of solar energy. It is estimated that about 2% of all solar energy reaching the earth is converted into wind energy [7]. This leads to unequal atmospheric temperature, density, and pressure, which in turn creates forces that move air from one location to another. Spatial variations are the variations in wind at different locations and in meteorological terms it is known as planetary scale variation. As per origins they are classified into subtypes such as trade winds, monsoons, synoptic scale motions, mesoscale wind systems and convective scale motion [8]. Atmospheric variations occur over a wide range of time scales (seconds to months) and space scales (meters to thousands of kilometers) which are interrelated. For wind modeling, time scale variations of interest have 3 sub-types i.e. long-term variability, seasonal and monthly variability and diurnal variations. Long-term variability involves differences in trends on an annual basis whereas monthly and seasonal are over several months and diurnal consider daily variation. Longterm variations of wind speeds are modeled using distributions such as Weibull and Rayleigh. Rayleigh distribution is a subset of Weibull distribution and mathematically a Weibull distribution with a shape parameter of 2 is a Rayleigh distribution. Weibull distribution, being a 2 parameter distribution, has greater flexibility and hence it is used to model wind speeds in this paper. The probability density function and the probability distribution for annual wind speeds are given by (1) and (2) respectively: β
f(v)=( β )×v(β–1)× exp(–(v/α)β)
(1)
F(v)=1– exp(–(v/α)β)
(2)
α
where α is the scale parameter and β is the shape parameter. The scale parameter is directly related to the mean of the samples considered whereas the shape parameter describes the variability of samples about the mean. In addition, a higher value of shape parameter (i.e. in the range of 2.5 to 3) implies a lower variation of speeds about mean and a lower value (i.e. in the range of 1.2 to 1.5) implies a larger variation about the mean [9]. IV. MODELING OF POWER OUTPUT CHARACTERISTICS OF WECS Power output characteristic is the plot of power output versus wind speed at the hub height of WECS. It accounts for the overall efficiency of the aerodynamic, mechanical and electrical systems involved in the energy conversion process. Power curve depends on external atmospheric conditions such
as air pressure depending on the height above sea level as well as on any possible changes in the aerodynamic shape of rotor blades, caused by dirt or ice. Control methodology also determines the nature of the characteristic during its operation. Different control strategies are incorporated in WECS such as pitch-control (passive type), stall control (active control), active stall control and yaw control. In general, the power output characteristic has two distinct regions: one in the interval between cut-in and rated wind speeds in which the power output increases with wind speed and the other in the interval between rated wind speed and furling wind speed when the power output is maintained constant at the rated value. A typical power output characteristic of a WECS is shown Fig. 1 as given in [10]:
Fig. 1. General power output characteristics of a Wind Electric Conversion System
The region between cut-in wind speed and rated wind speed is known as ‘maximum power output region’ and the one between rated wind speed and furling wind speed is called ‘power regulation region’. While operating in the maximum power output region the control system strives for maximum generation, while in regulation mode it ensures constant generation even though excessive wind power is available which is spilled. The maximum power output region is described by different relations between wind speed and power output such as linear, square, cubic, etc. This is solely a function of the type of WECS used e.g. Darrieus (vertical axis) turbine generally has a linear velocity relationship; only a very few two bladed turbines exhibit a cubic relationship [11]. As the wind speed goes above the furling speed, control system turns the rotor away from the wind flow to ensure mechanical stability and avoid excessive turbulence and stresses. A general form of equation describing the power output versus wind speed characteristic is given below using an exponent “n” [12]: p= Pr× (vn–Vcn)/( Vrn – Vcn) for Vc≤v≤Vr for Vr≤v≤Vf Pr 0 else (3)
where Vc,Vr and Vf are cut-in wind speed, rated wind speed and furling wind speed respectively and Pr is the rated power output of WECS. Different values of exponent “n” correspond to different curves in the maximum power output region, according to the turbine type. Usual values of Vc ,Vr ,and Vf are in the range of 3 to 3.5 m/s, 10 to 13 m/s and around 25 m/s. These values vary for different manufacturers.
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V. PROBABILISTIC MODEL FOR THE POWER OUTPUT OF WECS As mentioned earlier, the power output of WECS can be considered to be a random variable and a suitable probability density function is a standard method of modeling it. Depending on the nature of the random variable, the probability density function can be discrete, continuous or mixed. Mixed density functions have both discrete and continuous components. The probability density function for the power output of WECS is obtained by combining the power output equation and Weibull density function using the transformation theorem [13]. The resulting density function is mixed and is given below: [qow+pow(1-F1)]δ(p) f(p) =
pow(1/Dp) ×(β/αβ) v(β-1)×exp(–(v/α)β)
for
p=0
for 0
