Definitions and typical values assumed for the generator’s parameters.
Abstract
In this chapter, the design of a nonlinear rotorside controller is described for a variable pitch wind turbine based on nonlinear, H2 optimal control theory. The objective is to demonstrate the synthesis and application of a maximum power point tracking (MPPT) algorithm. In the case of a variable pitch wind turbine, the blade collective pitch angle is controlled to ensure that the turbine is not overloaded. In the case of such turbines the blade pitch may be treated as unknown input and the actual pitch angle is estimated in real time from torque measurements. The algorithm uses a nonlinear estimation technique and maximizes an estimate of the actual power transferred from the turbine to the generator. It is validated by simulating the windturbine’s dynamics. It is shown that the MPPT algorithm performs within prescribed error bounds both in the case when no disturbances are present, as it is an indicator of the validity of the algorithm and in cases when significant levels of wind disturbances are present.
Keywords
 control system
 Kalman filtering
 nonlinear estimation
 nonlinear filters
 simulation
 state estimation
 induction generators
 tracking
 tracking filters
 wind power generation
1. Introduction
The extraction and regulation of the power from the wind by a wind turbine followed by the capture of this power by a generator has been the subject of several recent research investigations. The use of a doubly fed induction generator (DFIG) is one of most popular options for largescale electromechanical conversion of wind power to electrical power. The DFIG employs a twosided controller, a rotorside controller (RSC) to control the speed of operation and the reactive power, and a gridside controller (GSC) using a gridside voltage source converter which is responsible for regulating the DC link voltage as well as the stator terminal voltage. The rotorside controller is expected to (i) minimize or regulate the reactive power and hold the stator output voltage frequency constant by a form of current control, (ii) regulate the rotor speed to maintain stable operation, and (iii) alter the speed set point to ensure maximum wind power capture. The role of the gridside controller is to ensure regulation of the DC voltage bus, and thereby indirectly control the stator terminal voltage. In the case when the generator is feeding an ACgrid, it can be designed to control the power factor. In a typical system, the stator phase voltages and the stator, rotor, and grid phase voltages are assumed to be measured. It is usual to connect the gridside converter to the grid via chokes to filter the current harmonics. An ACcrowbar is generally included to avoid DClink overvoltages during grid faults.
It is well known [1] that only a fraction of the power available in the wind is captured by a wind turbine. There is further reduction in the actual power converted to useful power by the generator. The fraction of the power captured by the wind turbine, which is theoretically limited by the so called Betz limit (about 58%), known as the power coefficient is primarily a function of the tip speed ratio, and is usually less than a certain peak value which is about 45% [1]. Maximum energy conversion is possible when the turbine operates at an optimum tip speed ratio which depends on the variation of the power coefficient with respect to the tip speed ratio. The relationship between the power coefficient and the tip speed ratio can be best determined experimentally. In the case of most of the current horizontal axis wind turbines operating at optimum speed, this can be accomplished by indirect control of the rotational speed. The indirect control of the speed is realized by directly controlling the reaction torque of the electric generator [2]. When the principal variables can all be measured, then one could employ one of a large number of maximum power point tracking (MPPT), algorithms have developed. The concept of maximum power point tracking was first introduced in the design of solar panels for spacecraft in the 1970s with the objective of maximizing the power transfer from the photovoltaic power sources.
In a recent paper [3], the design of a nonlinear rotorside controller for a wind turbine generator was developed based on nonlinear,
When applying this algorithm to a real wind turbine, it was found that for purposes of ensuring that the turbine was not overloaded, the collective pitch angle of the turbine’s blades could be controlled so as to be able to limit the maximum power captured by the wind turbine. When the pitch of the blades is controllable there are two control inputs to consider. While the blade pitch angle can be used to regulate the capture of the power from the wind by the wind turbine rotor, controlling the generators reaction torque allows for the power to be smoothly converted into electrical energy. For such variable pitch wind turbine, it was found that in order to implement the algorithm developed in ref. [3], it was essential to either measure the blade pitch angle or the torque on the turbine shaft, which is then used to estimate the true wind turbine aerodynamic torque and the blade pitch angle. In the latter case only a model of the closed loop collective pitch angle dynamics is essential. As there were no other benefits of measuring the blade pitch angle, the second option was preferred. The blade pitch angle was then considered as an unknown input to the torque, and it was estimated from the measurements.
In this chapter the modified MPPT algorithm, in the presence of unknown inputs to the aerodynamic torque, is successfully demonstrated both in the case when no disturbances were present, as it is a prerequisite for successful implementation, and in cases when significant levels of wind disturbances are present.
2. The electromechanical model of a wind turbine
There have been a number of papers on the subject of modelling of a wind turbine driving an induction generator under turbulent or stochastic wind conditions [4–7]. In this section the electromechanical model used in this study which is identical to the model used in ref. [3], is briefly summarized.
2.1. The mechanical model
The mechanical model of the wind turbine is described by,
where, as defined in ref. [3],
In the above expression
There are several approximations [8, 9] of
Typically depending on the approximation used the maximum power coefficient varies over the range, 0.44 ≤
2.2. The dynamics and control of the pitch angle
The dynamics of variable pitch wind turbine blades plays a key role in the capture and regulation of the power from the wind by the wind turbine rotor. In the case of a horizontal axis wind turbine, there are up to five blades which are assumed to be equispaced and to lie with the plane of the rotor disc. The most popular choice for most variable pitch wind turbines is a three bladed rotor. The dynamics of a variable pitch wind turbine blade can be expressed either in a frame that is fixed in the blade or in a frame that is fixed to the rotor disc. It is convenient to represent the aerodynamic forces in a frame fixed to the blade, while the wind inputs and gusts are more easily represented in a frame fixed to the rotor disc plane. In most practical horizontal wind turbine designs, the rotor disc plane usually aligns itself normal to the wind direction. Thus both frames of reference are used in the dynamical analysis of wind turbines and are transformed from one to the other as and when this is required.
The rotor dynamic model is typically described in terms of nondimensional quantities so that the general rotor configurations can be analyzed without the need to specify the size. Because of the similarity between the mechanical designs of the rotor for a helicopter, the development of the model closely follows the methodology outlined by Padfield [10] and Fox [11]. The important rotor blade properties of interest are the aerodynamic forces and moments acting on the individual blades, as well as the rotor thrust and torque which are related to the blade forces. Each blade is assumed to be fully controllable in pitch with the root of the blade offset from the rotor axis. The Lock number is an important nondimensional aerodynamic parameter, and is used to characterize the rotor dynamics parameters. The aerodynamically coupled flappitch equations of motion of a single blade are derived in a rotating frame as function of the azimuth angle. To derive the equations of motion of all the blades as a single unit, the coefficients in the equations may be expressed in terms of the so called multiblade coordinates. This is done by expanding all trigonometric functions such as products of sine and cosine functions as the sums of relevant sine and cosine terms. Thus the fixed frame equations of motion obtained by applying multiblade coordinate transformations will represent the dynamics of the rotor disc containing
Broadly, the approaches to pitch control may be classified into two groups. In a direct pitch controlled system, the controller monitors the windturbine’s power output at every sampling instant. When the power output exceeds an upper bound, the blade pitch is altered to lower the power generated by the turbine. Increasing the pitch attitude generally reduces the power output. When the maximum power output of the turbine is within the safe operating limits, the pitch angle is reduced to zero.
The second approach to pitch control involves operating the wind turbine with the blades pitched at angle just below the stall angle. The geometry of the blade profile and twist, however are aerodynamically tailored to ensure that when the induced wind speed is high, the angle of attack also increases and the blade begins to stall. The stalling of the lift generated restricts the magnitude of the lift generated and consequently the power generated is also limited. In an actively stall controlled turbine, the pitch of the blade is maintained just below the critical stall angle as long as the power generated by the wind turbine is within the safe operating limits, and increased beyond the critical value when it is desired to stall the generation of lift on the blade. Thus, when the generator is overloaded, the controller will pitch the blades in the opposite direction from what a pitch controlled machine does, in order to make the blades go into an increased state of stall.
The approximation to
Thus the discrete dynamics of the pitch angle may be expressed as,
The model may be used to design control laws for both active pitch controlled and active stall controlled windturbines. The demanded blade pitch angle
To design an active stall controller, the first step is to model the section lift and drag coefficients of the blade when the section angle of attack exceeds the stall angle. Modes of both the section lift and drag coefficients of the blade when the section angle of attack exceeds the stall angle have been presented by Tangler and Kocurek [13] and by Tangler and Ostowari [14] based on a model developed by Viterna and Corrigan [15]. These are then substituted into the expression for the power coefficient developed on the basis of the blade element momentum theory (see for example Vepa [16], Section 4.4.1). Once the expression for the power coefficient is found, the commanded blade angle is found by requiring the error between the actual power generated, estimated from the power coefficient, and the maximum power is a minimum.
2.3. The nonlinear dynamic electromechanical model
The basic equations of the dynamics of the doubly fed induction machine can be established as done in ref. [3], by considering the equivalent circuit of a single stator phase and a single rotor phase and the mutual coupling between the stator and rotor phases. The voltage vector consisting of the voltages applied to each stator and rotor phases is related to the voltage drops across the resistances of these phases and the rate of change of the fluxes linking the stator and rotor phases. The fluxes in turn are related to the current vector via a matrix of inductances which are not constant but period functions of time with the period equal to the rotor’s electrical speed,
The phase angles relating the directions of the
The dynamical equations of the DFIG relating the voltages in the stator and rotor and in the
The stator fluxes are related to the stator and rotor currents in the
The rotor fluxes are related to the stator and rotor currents in the
In the above equations, as defined in ref. [3],
At the stator terminals, the active and reactive components of the power are given by,
At the rotor terminals, the active and reactive components of the power are given by,
The active and reactive powers exchanged by the generator and the grid are respectively the sum of the active and reactive components of the power at the stator and rotor. The electromagnetic reaction torque may be expressed as,
Assuming that the stator flux is stationary in the
Thus, from equations (8a),
Eliminating
The electromagnetic reaction torque given by equation (10) and the reactive power at the stator terminal given by the second of equations (9a) may also be expressed in terms of
Defining the mutual inductance coupling coefficient
and using equations (11c) with
The total reactive power is
The definitions of the resistances and inductances and their typical values assumed in this chapter are listed in Table 1.
Number of poles  6  


Stator resistance  0.95 

Stator inductance  94 

Magnetizing inductance  82 

Rotor resistance  1.8 

Rotor inductance  88 

Stator phase voltage  380 V 
Grid frequency  50 Hz  
Nominal mechanical rotor speed  100 rads/sec  
Rated maximum power  100 kW 
In steady state, assuming that,
where the superscript ‘0’ refers to the steadystate condition, and subtracting the steadystate components from (14a) and (14b), the electromechanical perturbation equations are obtained. The perturbation states, inputs and variables are defined as:
Given that
Using equation (11a) and introducing the steady state and perturbation variables, the expression for the electromagnetic torque is,
The wind turbine perturbation torque
where the first component is evaluated at the current rotor speed,
while the electrical machine perturbation equations are,
Assuming that
equations (21) and (22) may be partially decoupled, and (22a) may be treated independently. Thus the complete nonlinear equations for the perturbation states used for the design of the nonlinear rotorside controller may be expressed in state space form as,
where
is the disturbing angular acceleration on the rotor due to wind speed fluctuating component and
Wind turbine blade disc radius  6 m  

Number of blades  3  

Nominal wind speed  10 m/s 

Wind power at nominal wind speed  ~10 kW 
Gearbox ratio  10  

Rotor inertia  40 kgm^{2} 

Viscous friction coefficient  0.07 Nms/rad 
Cutin wind speed (m/s)  3.5 m/s 
The modeling of the turbulent wind component and the control of wind turbine is based entirely on [3] and will not be repeated here. The complete characteristics of the wind turbine are summarized in Table 2.
3. The measurements and nonlinear state estimation
The dynamic model of the wind turbine that must be employed for purposes of state estimation is not only not linear but also involves the estimation of large dynamic signals. The Kalman filter which was formulated in the 1960s is primarily applicable to linear systems. To overcome the limitations imposed by the requirement of linearity, it was subsequently, empirically, extended and applied to nonlinear systems. A number of approaches such as the extended Kalman filter (EKF) have been proposed in the literature to extend the application of the traditional Kalman filter for nonlinear state estimation. However, the stability of these extended formulations is not guaranteed unlike the linear Kalman filter. Thus the EKF may diverge if the consecutive linearizations are not a good approximation of the linear model over the entire uncertainty domain. Nonetheless the EKF provides a simple and practical approach to dealing with essential nonlinear dynamics.
The UKF has been proposed by Julier, Uhlmann, and DurrantWhyte [17], and used in ref. [3]. It can overcome the limitations of applying the Kalman filter to nonlinear systems. The UKF based on the unscented transformation of the statistics of a random variable. It provides a method of calculating the mean and covariance of a random variable undergoing a nonlinear transformation
As in ref. [3], given a general discrete nonlinear dynamic system in the form,
where
It is possible to estimate the unknown inputs only when the matrix product
4. The MPPT outer loop controller
Several algorithms for achieving maximum power tracking and control have been proposed for a number of power systems [21, 22]. There have been a number of MPPT controllers proposed recently for wind turbines based on maximizing the net power captured by the generator [23–26]. A recent book on the topic has covered the optimal control based strategies quite extensively [27]. There have also been a few methods based on some form of optimal estimation of the wind speed [28]. A nonlinear controller based MPPT method has also been applied to wind turbines [29]. Several of the optimal control strategies may be efficiently implemented for a wind turbine provided that highly reliable nonlinear estimation algorithms are used to estimate the states of the wind turbine in operation. In this section, one such approach is briefly outlined and implemented as in ref. [3]. The system now includes the independently controlled variable pitch blades while in ref. [3], only a turbine with fixed blades was considered.
It is assumed that the induction machine is controlled in a manner so as to ensure variablespeed operation over a wide range input conditions, so it is possible to exercise direct control of the system’s tip speed ratio. The wind power captured by the windturbine is estimated from the state estimates by the equation,
The wind turbine torque
The condition for maximum power capture is,
Thus the instantaneous torque speed ratio or the
In evaluating
To determine the rotor frequency at which maximum power is extracted from the wind by the turbine, the rotor frequency is assigned an initial value
The frequency
where
5. Typical simulationbased results
To make the comparisons easy and to draw meaningful conclusions, the same example as the one considered in ref. [3] is also considered here with the exception that, in the case considered here, the blade pitch angle was assumed to be independently controlled. The initial equilibrium conditions were deliberately chosen so the nonlinear perturbation dynamics of the turbogenerator about the initial operating point were not stable. So the initial feedback controller was obtained by adopting the LQRbased methodology of Vepa [3] and using a model evaluated at the initial perturbation. Measurements of the rotor speed and the rotor
Figure 3 compares the electrical speed error in the measurement, with estimates of it obtained by using the UKF and the EKF. To make the comparison we have zoomedin over a time frame of the first 50 time steps. Quite clearly the UKF estimate converges rapidly to the measurement while the EKF estimate fluctuates in the vicinity of the measurement. From the comparisons shown in Figure 3, the superiority of the UKF over the EKF can be deduced. For purposes of maintaining clarity, all the other results corresponding to the EKF estimates are not shown in the figures.
Using this algorithm repeatedly has accentuated the need for making accurate electrical speed rate measurements and estimates. The electrical speed rate estimation was done using measurements of the electrical speed rate. From the previously estimated electrical speed and independently processing the measured electrical speed rate in another first order mixing filter, the estimates of the electrical speed are continuously updated. This approach provides precise estimates of the speed rate and facilitates the accurate estimation of the torque absorbed by the turbine from the wind. Figure 4 shows the corresponding power transferred from the wind to the generator over the first 20,000 time steps and compared with maximum available wind power at that particular maximum magnitude of the wind speed and zero blade pitch angle.
Figure 5 illustrates the power speed characteristic corresponding to Figure 4. From Figures 4 and 5, it may be observed that the power transferred by the turbine from the wind to the generator tracks the maximum available wind power. Figures 6 and 7 illustrate the corresponding torque on the generator and the torqueelectrical speed characteristic. Also shown on these figures is the torque corresponding to the maximum available power.
Figure 8 shows the growth of the estimated blade pitch angle and compared with the simulated blade pitch angle.
6. Discussion and conclusions
In this chapter, a nonlinear UKF is used to provide the rotorside control inputs and also in a tracking controller that ensures that the desired maximum power operating point is exactly tracked. As in ref. [3], the uncontrolled DFIG is unstable. Thus this necessitates the use of a stabilizing controller prior to implementing a MPPT filter. The MPPT filter tracks the maximum power point as the power is transferred from the wind to the turbine. The rotorside control laws are synthesized by employing a
The advantages of using stochastic optimal control theory and nonlinear optimal control are discussed in ref. [3]. In this chapter it has been demonstrated that the MPPT control filter which acts as an outerloop controller, continuously seeks to maximize the power absorbed by the wind turbine while the inner loop estimator continuously estimates and updates the states including the blade pitch angle (which was held fixed in ref. [3]). The MPPT control filter included a feedback signal estimated using the NewtonRaphson formula at each time step, just as in ref. [3]. One can estimate the wind power captured by the turbine and it indicates that the filter is seeking to operate within 5–9% of the simulated operating maximum power point by controlling the speed of the rotor after a 0.5 s delay, which allows the UKF to estimate the states and the unknown blade pitch angle without a significant error and to eliminate the influence of power transients. The maximum available power in the wind is computed, as in ref. [3], by assuming that the wind is frozen at its maximum magnitude before it encounters the blades and the deterministic formula for the maximum power coefficient
Finally the MPPT algorithm, based on the nonlinear state estimation using the UKF, is shown to perform, even when the blade pitch angles are dynamically varied and the introduction of the blade dynamics does not cause any additional instabilities when compared with the case of fixed blades considered in ref. [3]. The maximum power transfer achieved is less than in ref. [3], unless the blade is assumed to be fixed with the pitch angle set at zero.
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