Nonlinear, Optimal Control of a Wind Turbine Generator - IEEE Xplore

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Nonlinear, Optimal Control of a Wind Turbine Generator Ranjan Vepa, Member, IEEE

Abstract—In this paper, the design of a nonlinear rotor-side controller (RSC) is developed for a wind turbine generator based on nonlinear, H2 optimal control theory. The objective is to demonstrate the synthesis of a maximum power point tracking (MPPT) algorithm. In the case of a doubly fed induction generator, it is essential that the RSC and the MPPT algorithm are synthesized concurrently as the nonlinear perturbation dynamics about an operating point is either only just stable or unstable in most real generators. The algorithm is validated based on using nonlinear estimation techniques and maximizing an estimate of the actual power transferred from the turbine to the generator. The MPPT algorithm 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. Index Terms—Control system, induction generators, Kalman filtering, nonlinear estimation, nonlinear filters, simulation, state estimation, tracking, tracking filters, wind power generation.

I. INTRODUCTION FFICIENT extraction of the power from a wind turbine has been the subject of several recent research investigations. The use of a doubly fed induction generator (DFIG) is the preferred option for large-scale electromechanical conversion of wind power to mechanical power. The DFIG requires a two-sided controller, a rotor-side controller (RSC) to control the speed of operation and the reactive power, and a grid-side controller (GSC) using a grid-side voltage-source converter, which is responsible for regulating the dc-link voltage as well as the stator terminal voltage. While the RSC is expected to 1) minimize or regulate the reactive power and hold the stator output voltage frequency constant by a form of current control; 2) regulate the rotor speed to maintain stable operation; and 3) alter the speed set point to ensure maximum wind power capture, the objective of the GSC is to ensure regulation of the dcvoltage bus, and thereby indirectly control the stator terminal voltage. Equivalently, it can be designed to control the power factor. A typical block diagram of DFIG control systems illustrating the rotor-side and grid-side subsystems and their multiloop structure is shown in Fig. 1. In the system shown, the stator phase voltages and the stator, rotor, and grid phase voltages are

E

Manuscript received March 20, 2010; revised May 25, 2010, August 20, 2010, and October 1, 2010; accepted October 4, 2010. Paper no. TEC-001192010. R. Vepa is with the School of Engineering and Material Science, Queen Mary University of London, London, E1 4NS, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TEC.2010.2087380

Fig. 1. Block diagram of DFIG control systems illustrating the rotor-side and grid-side subsystems and their multiloop structures.

assumed to be measured. It is usual to connect the GSC to the grid via chokes to filter the current harmonics. An ac crowbar is generally included to avoid dc-link overvoltages during grid faults. Only a fraction of the power available in the wind is converted to useful power by a wind turbine. This fraction, which is theoretically limited by the so-called Betz limit (about 58%), is 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]. To achieve maximum conversion, the turbine must necessarily operate at an optimum tip speed ratio, which to a very large extent depends on the variation of the power coefficient with respect to the tip speed ratio, a relationship that can only be determined experimentally. In the case of most of the current wind turbine designs, this can be achieved by appropriately controlling the rotational speed of the electric generator [2]. When the primary variables can be measured, then several maximum power point tracking (MPPT) [3] algorithms have been developed. The concept of MPPT was first introduced in the design of solar panels for spacecraft in the 1970s with the objective of maximizing the power transfer from the power source [4]. MPPT algorithms generally belong to three classes of algorithms: 1) based on maximizing the power generated at the grid side; 2) based on maximizing the power transfer from the wind to the generator at the rotor side; and 3) based on maximizing the total power transfer to the grid. What is required in the latter case is to be able to set the speed of the generator at a point where the power transfer from the wind to the generator is maximized; i.e., the source impedance of the generator is matched to the output impedance of the turbine in accordance with the maximum power transfer theorem

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in electrical circuit theory. Examples of MPPT algorithms for wind turbines are presented in [5]–[9]. Classical versions of MPPT algorithms employ a maximum power search algorithm (i.e., hill-climbing method) [10]–[12] for dynamically driving the operating point to the maximum power point. The hill-climbing method seeks to optimize either the estimated power transferred from the turbine to the generator or the power transferred from the wind to the turbine. Most methods have relied on maximizing the latter, and this does not guarantee that the actual power transferred from the turbine to the generator is also optimized. Another drawback of these methods is the significant estimation errors in the realtime computation of the wind power captured by the turbine or the power generated by the turbine, which can often result in high-frequency power fluctuations that are highly undesirable to say the least. Several schemes have been proposed to alleviate the problems due to the errors in the estimation by using real-time estimates of the wind speed in addition to estimating the power coefficient, while other schemes use relatively robust control techniques based on fuzzy logic (see [13] for example). Another approach [14] has been proposed to deliberately introduce a small probing signals or noise to determine the optimum point accurately. In this paper, the design of a nonlinear RSC is developed for a wind turbine generator based on nonlinear, H2 optimal control theory. The objective is to demonstrate the synthesis of a MPPT algorithm for transferring the maximum power from the turbine to the generator. In the case of a DFIG, it is essential that the RSC and the MPPT algorithm are synthesized concurrently, as the nonlinear perturbation dynamics about an operating point is either only just stable or unstable in most real generators. The algorithm is validated based on using nonlinear estimation techniques and maximizing an estimate of the actual power transferred from the turbine to the generator. The estimation method is based on the unscented Kalman filter (UKF) and compared with the traditional extended Kalman filter (EKF). The implementation of the algorithm is carried out by modeling the real wind velocity profiles from a white-noise generator and using shaping filters that are derived from approximations of the Kaimal wind velocity spectrum. The simulation was completely executed in the MATLAB environment. The simulation was based on executing the UKF estimator as well numerically integrating, in parallel, the process model step by step using ode45.m, about the steady equilibrium solution without linearizing the dynamics. The feedback controller was embedded within the main program. The MPPT algorithm is successfully demonstrated in cases when significant levels of wind disturbances are present. In particular, the actual power transferred is compared with maximum available power in the wind, and it is shown that the maximum power is transferred from the wind to the generator by the turbine. II. ELECTROMECHANICAL MODEL OF A WIND TURBINE There have been a number of papers on the subject of modeling of a wind turbine driving an induction generator under turbulent or stochastic wind conditions [15]–[18]. In this

section, the electromechanical model used in this study is briefly summarized. A. Mechanical Model The mechanical model of the wind turbine is described by Jeq dωm /dt + Beq ωm = Twt − Tm el

(1)

where Jeq is the equivalent total inertia of the generator shaft, ωm is the mechanical speed of the generator shaft, Beq is the equivalent total friction coefficient, Twt is the torque extracted by the turbine from the wind, and Tm el is the mechanical equivalent of the electromagnetic load torque. The electromagnetic load torque Tm el = (P /2) Tel may be estimated from the electromagnetic reaction torque of the electric generator per pole pair Tel . The torque extracted by the wind turbine Twt is related to the total power absorbed by the turbine from the wind that may, respectively, be expressed as 1 Twt = Pw /ωm and Pw = ρ πR2 Cp (λ) U 3 . (2) 2 In (2), ρ is the density of the air at the hub of the turbine, R is the rotor radius, Cp (λ) is a power coefficient that is a function of λ = Rωm /U , the tip speed ratio and U is the wind velocity. Thus, the mean torque may be expressed as Cq (λ) = Cp (λ)/λ. (3) Twt = ρ πR3 Cq (λ) U 2 /2, There are several approximations [19], [20] of Cp (λ) in use and a typical approximation is given by 125 −16.5 1 1 Cp (λ) = 0.44 − 6.94 exp = + 0.002. , λi λi λi λ (4) Typically, depending on the approximation used, the maximum power coefficient varies over the range 0.44 ≤ Cp (λ)m ax ≤ 0.492, and the corresponding tip speed ratio varies over 6.9 ≤ λm ax ≤ 8.8. B. Nonlinear Dynamic Electromechanical Model The basic equations of the dynamics of the doubly fed induction machine can be established 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 ωe = P ωm , which is the product of the number of pole pairs P and the rotor’s mechanical speed ωm . However, when all of the stator and rotor quantities are transformed to a stationary frame (the d–q frame), using the Park–Blondel transformation, in terms of the stator’s and rotor’s voltage-pulsation frequencies ωs and ωr , respectively, the dynamic equations reduce to a set of four with constant coefficients. Moreover, ωr = ωs − ωe and can be found by measuring ωs and ωe , and is also the slip frequency (the ratio s = ωr /ωs

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Furthermore, the active and reactive components of the power at stator terminals are given by Qs = vq s ids − vds iq s .

Ps = vds ids + vq s iq s

(7a)

The active and reactive components of the power at rotor terminals are given by Qr = vq r idr − vdr iq r .

Pr = vdr idr + vq r iq r

(7b)

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 Tel = (φds iq s − φq s ids ) .

(8)

Assuming that the stator flux is stationary in the d–q frame and neglecting the stator’s resistive voltage drop vds = 0. Fig. 2.

Relationships between the d–q frame and the stator and rotor phases

(9a)

The q-component of the stator voltage may be expressed as vs = vq s .

is the slip). Thus, the phase angles relating the directions of the d–q frame and the phase angles of the first of the three stator phases, A, B, and C, θs and the first of the three rotor phases, a, b, and c, θr satisfy the relation θr = θs − θe , where θe is the rotor’s electrical angle, as illustrated in Fig. 2. In modeling the stator of DFIG, the generator convention that positive direction of electromagnetic torque is in the direction opposing to the direction of rotation is used, while in modeling the rotor of DFIG, the motor convention is used. The dynamic equivalent circuit of generator in synchronous rotating reference frame, the d–q frame, is used to set up the model equations. The four dynamical equations of the DFIG with constant coefficients in the d–q frame are vds = Rs ids +

vdr = Rr idr +

dφds dφq s − ωs φq s vq s = Rs iq s + + ωs φds dt dt (5a) dφdr dφq r − ωr φq r vq r = Rr iq r + + ωr φdr . dt dt (5b)

The stator fluxes are related to the stator and rotor currents in the d–q frame as φds = Ls ids + Lm idr

φq s = Ls iq s + Lm iq r .

(6a)

The rotor fluxes are related to the stator and rotor currents in the d–q frame as φdr = Lr idr + Lm ids

φq r = Lr iq r + Lm iq s .

(6b)

In the aforementioned equations, Rs , Rr , Ls , and Lr are, respectively, the resistances and self-inductances of the stator and rotor windings and Lm is the mutual inductance between a stator and a rotor phase when they are fully aligned with each other.

(9b)

To achieve the desired stator conditions, voltage-source converters with a constant dc-link voltage are connected on the grid side to the ac grid through a transformer. Then (5a) are modified and expressed as dφds − ω s φq s (10a) dt dφq s + ωs φds vs − eq s = Reqs iq s + (10b) dt where eds and eq s are the d–q components of the voltages generated by the voltage source converters and Reqs is the total equivalent stator resistance. The inputs eds and eq s are expressed in terms two auxiliary control inputs e¯ds and e¯q s , defined by the GSC, as −eds = Reqs ids +

eds = −Reqs ids + e¯ds +

Reqs φds + ωs φq s Leqds

eq s = vs − Reqs iq s + e¯q s +

(11a)

Reqs φq s − ωs φds . (11b) Leqq s

In (11a) and (11b), Leqds and Leqq s are equivalent d-axis and q-axis stator inductances. Equations (10a) and (10b) are decoupled and reduce to Reqs dφq s . φq s + Leqq s dt (12) The control inputs e¯ds and e¯q s are defined in terms to optimum linear proportional–integral (PI) controllers in the innermost loop of the GSC, thus facilitating the arbitrary choice of reference values for φds and φq s . Assuming a GSC is in place and choosing a stator-fluxoriented reference frame, the d-axis is aligned with the statorflux-linkage vector φs ; thus, φds = φs and φq s = 0. It follows from (6a) and (5a) that −¯ eds =

Reqs dφds φds + Leqds dt

iq s = −

Lm iq r Ls

− e¯q s =

vds = Rs ids +

dφs . dt

(13)



The q-component of the stator voltage may be expressed as vq s = Rs iq s + ωs φs

(14a)

φs = Ls ids + Lm idr =

vq s − Rs iq s = Lm im s ωs

TABLE I DEFINITIONS AND TYPICAL VALUES ASSUMED FOR THE GENERATOR’S PARAMETERS

(14b)

where im s is defined as im s ≡

vq s − Rs iq s . ωs Lm

(14c)

Hence, it follows that ids = −idr

Lm vq s − Rs iq s + Ls Ls ωs

(15a)

which may be expressed in terms of im s as ids = Lm

im s − idr . Ls

(15b)

The electromagnetic reaction torque given by (8) and the reactive power at the stator terminal given by the second of (7a) may also be expressed in terms of im s as Tel = φs iq s = −

L2m im s iq r Ls

(16a)

The definitions of the resistances and inductances and their typical values assumed in this paper are listed in Table I.

Qs = vq s ids − vds iq s = ωs φs ids = ωs L2m im s

im s − idr . Ls

(16b)

The rotor flux components given by the first of the (6a) and (6b) are Ls Lr − L2m φdr = Lr (17a) idr + L2m /Ls im s Ls Lr Ls Lr − L2m φq r = Lr (17b) iq r . Ls Lr Defining the mutual inductance coupling coefficient σ as σ = (Ls Lr − L2m )/Ls Lr

(18)

the rotor voltage equations given by the first of the (5a) and (5b) are expressed as

vq r

didr − σsωs Lr iq r (19a) vdr = Rr idr + σLr dt diq r Lm L2 −s vs = Rr + s m2 Rs iq r + σLr Ls Ls dt + σsωs Lr idr .

(19b)

The total reactive power given by (16b) is Q = vq r idr − vdr iq r + ωs L2m im s

im s − idr Ls

(20a)

where the current im s defined by (14e) is im s ≡

(vs − Rs iq s ) vs Rs = + iq r . ωs Lm ωs Lm ωs Ls

Hence, the total reactive power given by (20a) may be expressed as vs L2 Rs + iq r Q = vq r idr − vdr iq r + ωs m Ls ωs Lm ωs Lm vs Rs × + iq r − idr . (20c) ωs Lm ωs Lm

(20b)

C. Steady-State Electromechanical Model In steady state, assuming that 0 vdr = vdr vq r = vq0r idr = i0dr iq r = i0q r s = s0

(21)

where the superscript ‘‘0’’ refers to the steady-state condition, (19a) and (19b) may be expressed in matrix form as ⎡ ⎤ 0 vdr

0 Rr −σs0 ωs Lr idr ⎣ ⎦= . L 0 m −σs ωs Lr Rr s i0q r vq0r − s0 vs Ls (22) In the aforementioned equations, Rr s is defined as Rr s = Rr + s0

L2m Rs . L2s

(23)

The electromagnetic torque given by (16a) in steady state is vs L2 Rs 0 Tel0 = − m + iq r i0q r . (24) Ls ωs Lm ωs Ls It is customary to refer the mechanical motion dynamics also to the electrical reference speed and write the mechanical equation of motion given by (1) as dωe P P + Beq ωe = (25) Jeq Twt − Tel . dt 2 2 0 In steady state, Twt = Twt , ωe = ωe0 and

dωe = 0. (26) dt Since Rs 1, for given steady-state electrical rotor speed ωe0 , the solution for i0q r may be obtained iteratively. The control


inputs are selected so that the total reactive power given in (20c) is equal to zero in steady state. Expanding and rearranging the terms in (20c), one obtains i0dr . Thus, steady-state control inputs 0 and vq0r may be found, and the solutions for i0q r and i0dr may vdr be iteratively updated.

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TABLE II DEFINITIONS AND TYPICAL VALUES OF THE WIND TURBINE PARAMETERS

D. Nonlinear Perturbation Dynamics Subtracting (22) from (19a) and (19b), respectively, Δvdr = Rr Δidr + σLr

Δvq r

dΔidr dt

− σs0 ωs Lr Δiq r − σΔsωs Lr iq r (27a) L2 dΔiq r + σs0 ωs Lr Δidr = Rr + s0 m2 Rs Δiq r + σLr Ls dt +

L2 Δs m2 Ls

Rs iq r + σΔsωs Lr idr .

0 , Δvq r = vq r − vq0r , Δidr = idr − i0dr , Δvdr = vdr − vdr

Δωe , Δωe = ωe − ωe0 . ωs (28)

0 in steady state and that ΔTwt = Twt − Given that Twt = Twt Δωe satisfies the following: dΔωe P P P + Beq Δωe = Jeq ΔTwt − Tel + Tel0 . (29) dt 2 2 2

0 , Twt

The electromagnetic torque is vs L2m Rs 0 0 Tel = Tel − 2iq r + Δiq r Δiq r . + Ls ωs Lm ωs Ls (30) The wind turbine perturbation torque ΔTwt = ΔTwt (U, ωe ) is the function of two variables, the wind speed U and the rotor speed ωe . Given that the wind speed, U = U0 + Δuf is the sum of a mean component U0 and a fluctuating component Δuf , the wind turbine perturbation torque can be considered to be made of two components ΔTwt (U, ωe ) = (Twt (U, ωe ) − Twt (U0 , ωe )) 0 + Twt (U0 , ωe ) − Twt U0 , ωe0 = ΔTwt |ω e + ΔTwt |U =U 0

dΔidr = −σLr iq r Δωe dt

σLr

dΔiq r dt

− Rr Δidr + σs0 ωs Lr Δiq r + Δvdr (33a) L2 = + σLr idr + m 2 Rs iq r Δωe − σs0 ωs Lr Δidr ωs Ls L2 − Rr + s0 m2 Rs Δiq r + Δvq r . (33b) Ls

The complete nonlinear equations for the perturbation states used for the design of the nonlinear RSC may be expressed in state-space form as ⎡ ⎤ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ 0 0 Δωe 1 Δωe Δv dr d⎢ 1 ⎢ ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ 1 0 + ⎣ 0 ⎦ αu ⎣Δidr ⎦ = A⎣Δidr ⎦ + ⎣ ⎦ Δvq r dt σLr 0 1 Δiq r Δiq r 0 (34a) where ΔTwt /Δuf |ω e αu = (34b) Δuf Jeq is the disturbing angular acceleration on the rotor due to windspeed-fluctuating component and A is a matrix of functional coefficients. The complete characteristics of the wind turbine are summarized in Table II. III. TURBULENT WIND COMPONENT FILTER

(31)

where the first component is evaluated at the current rotor speed ωe . Hence, without making any assumptions that the perturbations are small, the equation for mechanical motion may be expressed as

dΔωe P ΔTwt

= Jeq − Beq Δωe dt 2 Δωe U =U 0 vs P 2 L2m Rs 0 2iq r + Δiq r + + 4 Ls ωs Lm ωs Ls × Δiq r + ΔTwt |ω e

σLr

(27b)

The perturbation states, inputs, and variables are defined as

Δiq r = iq r − i0q r , Δs = s − s0 = −

while the electrical machine perturbation equations are

(32)

Turbulence refers to wind-speed fluctuations over short time scales. Wind speed can be described as the sum of a mean wind speed U0 and a fluctuating component Δuf . The standard deviation of the turbulent component σu depends on the turbulence intensity IU and the average wind speed, and is given by the product σu = IU × U0 . The turbulence intensity IU may be modeled as [12] a + 15/U0 ∗ IU = I (35) a+1 where I ∗ = 0.18 for higher turbulence sites and I ∗ = 0.16 for lower turbulence sites with corresponding values for a of 2 and 3, respectively.



TABLE III ROTOR AVERAGING FILTER COEFFICIENTS [19]

could be made to fit the Kaimal spectrum over a specified band of frequencies by choosing the filter constants appropriately. The Butterworth filter developed in this paper represents an approximation of the Kaimal spectrum and was chosen for the simulation. The reason for the use of this filter is that it facilitates easier simulation. In this paper, we assume that the filter constants could be uncertain parameters and validate the control law for a range of filter parameters (1 < k < 20). The corresponding state equation of the shaping filter is xL ωc w(t), y = z(t) (40) z(t) ˙ = − √ z(t) + ωc σu kU0 k Fig. 3. Comparison of the Kaimal spectrum with the first-order Butterworth spectrum, k = 1, 10, and 20.

A commonly used model for the power spectral density of turbulence is the Kaimal model [19] −5/3 3 f xL 2 xL (36) 1+ S (f ) = σu U0 2 U0 where σu is the variance, xL is the turbulence length scale (with a proposed maximum of 600 m), and U0 is the mean wind velocity at the hub height. There is difficulty in specifying a uniformly valid approximation to the aforementioned spectrum, as it is not a function of f 2 . Nevertheless, it is possible to approximate it and construct a shaping filter that may be used to simulate the wind. A curve fit of Kaimal’s spectral approximation is σu2 (xL /U0 )

5/3

(1 + (3/2) (f xL /U0 ))

≈

σu2 (xL /U0 )

1 + k (ωxL /2πU0 )2 (37) where k ≈ 10 to accurately model the mid-frequency range (0.01–1.0 rad/s), k ≈ 40 to model the low-frequency range (1.0 rad/s). Given a Butterworth-type-shaping filter transfer function of the form xL 1 2πU0 √ ωc = × (38) Gv v (s) = σu U0 xL 1 + k (s/ωc ) S (f ) =

xL 1 . U0 [1 + k(ω/ωc )2 ]

Δuf = Krotor

s2 T4 + sT3 + 1 y s2 T2 + sT1 + 1

(41)

is used to represent this effect. The state-space representation for the rotor-averaging filter, given by (41), is

1 1 0 x˙ 1 0 T2 x1 = + z. (42) x˙ 2 x2 T2 −1 −T1 T2 1 Thus, the composite state-space model of the turbulent wind component filter is ⎡ 2πU T ⎤ ⎡ ⎤ 0 2 ⎡ ⎤ √ − 0 0 z˙ ⎥ z 1 ⎢ ⎢ xL k ⎥⎣x ⎦ ⎢ ⎥ ⎢ ⎥ ⎣ x˙ 1 ⎦ = 1 0 0 T2 ⎦ T2 ⎣ x 2 x˙ 2 1 −1 −T1 ⎡ ⎤ 1 U0 ⎢ ⎥ + 2πIU U0 ⎣ 0 ⎦ w (t) kxL 0 Δuf =

1 [ T4 Krotor (T2 − T4 ) Krotor (T3 T2 − T4 T1 ) ] T2 ⎡ ⎤ z ⎣ (43) × x1 ⎦ . x2

the power spectral density may be approximated by Φv v (ω) = σu2

where w (t) is the unit intensity white noise. The wind acting on the rotor area of a wind turbine can be modeled by a single wind speed. The rotor has the effect of filtering and smoothing the wind turbulent variations. A typical second-order filter of the form

(39)

The Kaimal spectrum given by (36) and the first-order approximation given by (39) are compared in Fig. 3. The parameters k = 1, 10, and 20 were approximately obtained by, respectively, matching the normalized magnitudes of 0.7, 0.5, and 0.3 of the Butterworth filter to that of the Kaimal spectrum. Although not perfectly fit, it is seen that that the Butterworth spectrum

Typical values of the parameters of the wind disturbance model are given in Table III. IV. NONLINEAR OPTIMAL CONTROL At this stage, it is desirable to design an optimal controller so as to maximize the robustness of the controller. The approach adopted is to generalize the linear quadratic Gaussian controller. This is done by generalizing the design of the linear quadratic


regulator (LQR) to the nonlinear plant, while the linear Kalman estimator is replaced by 1) a classical extended Kalman estimator and 2) the UKF. The reason for implementing both the classical extended Kalman estimator and the UKF is to be able to compare the estimates obtained by both methods. For the purposes of controller design, the controllable part of the plant, given by (34a) is assumed to be in the form x˙ 1 = A11 (x1 ) x1 + B11 w + B12 u.

(44)

When the plant is assumed to be linear, i.e., when A11 (x1 ) is a constant matrix, the LQR is given by ˆ1 u = −K∞ x

K∞ = R−1 BT2 X∞

(45)

where X1∞ satisfies an algebraic Riccati equation (ARE) given by X1∞ (A11 − B12 K∞ ) + (A11 − B12 K∞ )T X1∞ +X1∞ B12 RBT12 X1∞ + Q = 0.

(46)

Motivated by the LQR problem, which is characterized by an optimal control law that is obtained by solving an ARE, Cloutier et al. [21] proposed an approach to the nonlinear regulation problem for the input-affine system (49) with quadratic cost function of the type used for LQR synthesis. The advantage of this method is that the characterization of the resulting feedback controller (45) has the similar structure to the LQR problem, and it is obtained by solving the corresponding ARE by replacing the state matrix by its “frozen” equivalent. Naturally, one can expect that a proper choice of A11 (x1 ), such that A11 (x1 ) x1 in (44) is exact, plays a significant role in getting good solutions to the regulation problem. In this paper, such an approach is adopted to synthesize the regulator and the matrix A11 (x1 ) is chosen such that when the state vector is the true equilibrium solution, the use of the matrix A11 (x1 ) results in the exact linear approximation of the nonlinear system of state equations. For the nonlinear plant, the matrix A11 (x1 ) is evaluated at a particular “frozen state” x1 , and is used to define the matrix X1∞ at that state. To start the synthesis, initially the simplest approximation is used by evaluating A11 (x1 ) when x1 has reached steady state. Thereafter, A11 (x1 ) is evaluated at the current time step. V. MEASUREMENTS AND NONLINEAR ESTIMATION The complete model of the wind turbine and generator may now be assembled, as it consists of the dynamics of the wind turbine speed perturbation Δωe , the two generator rotor current perturbation states Δidr and Δiq r , and the three winddisturbance states z, x1 , and x2 . For the application considered in this paper, a minimum set of measurements must be made. Typically, these would be the mechanical generator shaft speed ωm , the angular acceleration dωm /dt, the mean wind speed U0 , and the rotor phase currents, which may be used to estimate rotor currents in the d–q frame. Employing only these measurements, a UKF is used to estimate all the states of the model. Only some of the aforementioned measurements are used for the purposes of nonlinear state estimation, while the measurement of dωm /dt and an estimate of ωm is used to filter the mechanical speed rate independently by using a first-order mixing filter.

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The UKF is a nonlinear EKF, which has been proposed by Julier et al. [22] as an effective way of applying the Kalman filter to nonlinear systems. The UKF gets its name from the unscented transformation, which is a method of calculating the mean and covariance of a random variable undergoing nonlinear transformation y = f(w). In essence, the method constructs a set of sigma vectors and propagates them through the same nonlinear function. The mean and covariance of the transformed vector are approximated as a weighted sum of the transformed sigma vectors and their covariance matrices. The details may be found in the paper by Julier et al. [22]. Given a general discrete nonlinear dynamic system in the form xk +1 = fk (xk , uk ) + wk , zk = hk (xk ) + vk

(47)

where xk ∈ Rn is the state vector, uk ∈ Rr is the known input vector, zk ∈ Rm is the output vector at time k. wk and vk are, respectively, the disturbance- or process-noise and sensor-noise vectors, which are assumed to Gaussian white noise with zero mean. Furthermore Qk and Rk are assumed to be the covariance matrices of the process noise sequence wk and the measurement noise sequence vk , respectively. The unscented transforma= tions of the states are denoted as, fkUT = fkUT (xk , uk ), hUT k (x ), while the transformed covariance matrices and crosshUT k k covariance are, respectively, denoted as, Pfk f = Pfk f (ˆ xk , uk ), xh− x ˆ− x ˆ− = Phh = Pxh− Phh− k k k , and Pk k k , uk . We have chosen to use the scaled unscented transformation proposed by Julier [23], as this transformation gives one the added flexibility of scaling the sigma points to ensure that the covariance matrices are always positive definite. The UKF estimator can then be expressed in a compact form. The state-time-update equation, the propagated covariance, the Kalman gain, the state estimate, and the updated covariance are, respectively, given by UT xk −1 ) x ˆ− k = fk −1 (ˆ

(48a)

ˆ− = P k

(48b)

f Pfk −1

+ Qk −1

ˆ xh− (P ˆ hh− + Rk )−1 Kk = P k k x ˆk =

x ˆ− k

+ Kk [zk −

hUT x− k (ˆ k )]

ˆk = P ˆ − − Kk (P ˆ hh− + Rk )−1 KTk . P k k

(48c) (48d) (48e)

Equations (48a)–(48e) are in the same form as the traditional Kalman filter and the EKF. The state-time-update equation, the propagated covariance, the Kalman gain, the state estimate, and the updated covariance in the case of the EKF are, respectively, given for purposes of comparison in the form x ˆ− xk −1 ) k = fk −1 (ˆ

(49a)

ˆ − = Φk −1 Pk −1 ΦTk −1 + Qk −1 P k

(49b)

ˆ − HTk + Rk )−1 ˆ − HTk (Hk P Kk = P k k

(49c)

x ˆk =

x ˆ− k

+ Kk [zk −

hk (ˆ x− k )]

ˆ k = (I − Kk Hk )P ˆ− P k Φk −1 = ∇fk −1 (ˆ xk −1 )|k −1 , Hk = ∇hk (ˆ x− k )|k .

(49d) (49e) (49f)



Thus, higher order nonlinear models capturing significant aspects of the dynamics may be employed to ensure that the Kalman filter algorithm can be implemented to effectively estimate the states in practice.

ωpk +1 = ωpk + Δk u k = Kω

VI. MPPT OUTER LOOP CONTROLLER

Δk +1 = Δk +

Pˆk −1 (ωpk −1 − ωpk −2 ) (Pˆ − Pˆ ) k −1

Several strategies for achieving maximum power tracking and control have been proposed for a variety of power systems [24], [25]. In addition to the studies mentioned in Section I [5], [14], there have been a number of MPPT controllers proposed recently for wind turbines based on some form of optimal control [26]–[29]. In fact, a recent book on the topic has covered the optimal-control-based strategies quite extensively [30]. There have also been a few methods based on some forms of optimal estimation of the wind speed [5], [31]. A nonlinear-controllerbased MPPT method has also been proposed for wind turbines [32]. Several of the optimal control strategies may be efficiently implemented for a wind turbine, provided 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. It is assumed that the induction machine is controlled in a manner so as to ensure variable-speed operation over a wide range input conditions; therefore, it is possible to exercise direct control of the system’s tip speed ratio. The wind power captured by the wind turbine is estimated from the state estimates by the ˆ m , where Tˆwt is an estimate of the turbine equation Pˆw = Tˆwt ω torque and ω ˆ m is an estimate of the mechanical speed. The wind turbine torque may be estimated from the mechanical equation of the shaft rotation by rearranging (25) to give Twt . It is a weighted linear combination of Tel , dωe /dt, and ωe . When Pˆw is maximum ˆwt ˆwt T d ˆ d T Pw ≡ Pˆw = ω ˆm + = 0. (50) dˆ ωm dˆ ωm ω ˆm The condition for maximum power capture is Tˆwt dTˆwt =− . ω ˆm dˆ ωm

by

(51)

Thus, the torque speed ratio is the negative of the incremental torque to incremental speed ratio. This is frequency-dependent and is determined when the wind power input is a maximum. It follows that in the nonlinear perturbation equations

ΔTwt

Twt =− . (52) Δωe U =U 0 ωe In evaluating ΔTwt /Δuf |ω e , the wind turbine torque perturbation due to a change in the wind speed, the power coefficient Cp (λ) is assumed to be a maximum. To determine the rotor frequency that would yield maximum wind power captured by the turbine, the actual rotor frequency is set to an initial value ωp0 = ωr 0 and updated at regular intervals. The update is filtered, driven by an estimate obtained by using the Newton–Raphson formula at each time step k, and is given

k −2

(uk − Δk )dt τf u

(53a)

ˆk k dTwt k Pˆk = ω ˆm + Tˆwt . k dˆ ωm (53b)

The frequency ωpk is the estimated rotor frequency at the maximum power point. The optimum time constant τf u ≈ 0.04 and the update gain Kω ≈ 0.27 were determined after several simulations. The turbine torque and its derivative with respect to the estimated mechanical rotor speed ω ˆ m must be evaluated accurately in order to ensure that the turbine operates at the peak power. The turbine torque peaks at a speed slightly less than the speed at which the turbine power is a peak. The full-control law for MPPT takes the form ωr = ωpk

(54)

where ωpk is given by (53a). The controller design in this paper is essentially to control the DFIG under normal operating conditions. The gridcode requirements may be succinctly summarized as: the gridconnected wind turbine should stay connected during grid faults and should assist in the recovery of the grid, as prescribed by the grid operators, as well as provide the voltage-dip handling and ride-through capability by the control of active and reactive power. Under fault conditions, the need to satisfy grid-code requirements, such as reactive power exchange and voltage control, fault-ride-through support in the presence of balanced faults and a well-defined dynamic behavior in the presence of unbalanced faults make the reconfiguration of the controller [33], [34] mandatory. However, certain requirements of the grid codes, such as the regulation of rotor d–q currents when faults are present, require the controller to be adequately robust, and the LQR methodology adopted here is capable of meeting the robustness requirements. Although this paper does not address the reconfiguration of control strategies required to meet fault ridethrough requirements, nonlinear optimal control and filtering are expected to play a key role in design of automatic controller reconfiguration systems that meet with the North American and European grid-code requirements. VII. TYPICAL SIMULATION-BASED RESULTS The initial equilibrium conditions were deliberately chosen so that the nonlinear perturbation dynamics of the turbo generator about the initial operating point were not stable. Therefore, the initial feedback controller was obtained by adopting the LQR methodology using a “frozen-state” model evaluated at the initial perturbation. Measurements of the rotor speed and the rotor d–q currents were simulated and all the perturbation states were estimated using both the UKF and EKF methodologies. In Fig. 4, the simulated electrical speed of the generator, which is thrice the mechanical speed, for a time step dt = 0.001s is illustrated and compared with the UKF and EKF estimates over a time frame of 20 000 time steps or 20 s in real time.


Fig. 4. Simulated electrical speed (in rad/s) response of the DFIG generator compared with the measurement, the UKF, and the EKF estimates.

9

Fig. 6. Comparison of the measured and estimated electrical speed rate (in rads/s2 ) corresponding to Fig. 4.

Fig. 7. Comparison of the measured and estimated electrical speed rate (in rads/s2 ) corresponding to Fig. 4, over the first 50 time steps.

Fig. 5. Comparison of the measured and UKF and EKF estimated electrical speed (in rads/s2 ) errors corresponding to Fig. 4.

Fig. 5 compares the electrical speed error in the measurement and the UKF and the EKF estimates over the first 50 time steps. The comparisons in Fig. 5 clearly illustrate the superiority of the UKF over the EKF, and all the other results corresponding to the EKF estimates are not shown in the figures for the purposes of maintaining clarity. Experience in using this algorithm has underscored the importance of making accurate electrical speed-rate measurements and estimates. The electrical speed-rate estimation was done using measurements of the electrical speed rate and the earlier estimated electrical speed and independently processing these measurements in another first-order mixing filter. This approach provided precise estimates of the speed rate and facilitated the accurate estimation of the torque absorbed by the turbine from the wind. Fig. 6 illustrates the measured and estimated electrical speed rate, and it is clear from monitoring the speed rate that the machine is in steady state at the end of the time frame. Fig. 7 shows the electrical speed-rate measurement and the corresponding estimate obtained independently from the first-order mixing filter over the first 50 time steps. In Fig. 8, the simulated variation of the wind speed over the first 20 000 time steps is shown. In Fig. 9, the corresponding power transferred from the wind to the generator over the first 20 000 time steps is shown and compared with maximum available wind power at that particular maximum magnitude of the wind speed.

Fig. 8.

True wind speed over the first 20 000 time steps.

Fig. 10 illustrates the power-speed characteristic corresponding to Figs. 8 and 9. From Figs. 8–10, it may be observed that the power transferred by the turbine from the wind to the generator tracks the maximum available wind power. Figs. 11 and 12 illustrate the corresponding torque on the generator and the torque electrical speed characteristic. Also, the torque corresponding to the maximum available power is shown in these figures. VIII. DISCUSSION AND CONCLUSION In this paper, a nonlinear estimator is used to provide the rotor-side control inputs and also in a tracking controller that ensures that the desired maximum power operating point is exactly tracked. The uncontrolled DFIG is unstable, and thus, this necessitates the use of a controller before implementing a MPPT filter to track the maximum power transferred from the wind to the turbine. The RSC is synthesized by employing a H2 optimal control strategy while employing a “frozen” state



Fig. 9. Wind power transferred to the generator over the first 20 000 time steps and compared with maximum available wind power.

Fig. 10.

Turbine power characteristic over the operating speeds in rad/s.

Fig. 11. Turbine torque over the first 20 000 time steps and compared with the torque at the maximum available wind power.

dynamic model with different constant coefficients at each time step. The new MPPT filter is proposed and validated based on using nonlinear estimation techniques and maximizing an estimate of the actual power transferred from the turbine to the generator. The estimation method is based on the UKF, which is compared with the traditional EKF and shown to be superior. The validation of the algorithm is carried out by simulating real wind-velocity profiles from a white-noise generator and using shaping filters that are derived from approximations of the Kaimal wind-velocity spectrum. The MPPT algorithm is successfully demonstrated in the case where significant levels of wind disturbances are present. The advantage of using stochastic optimal control theory to derive the optimal control is twofold: first the stochastic system controlled by the control input generated by using deterministic control theory, i.e., not considering uncertainty inputs or noise to derive control input, results in a controller with the maximum

Fig. 12.

Turbine torque characteristic over the operating speeds in rad/s.

gain and phase margins for a given level of control effort. This is a well-known result of deterministic optimal control theory [35]. Second, stochastic system controlled by the control input generated using stochastic optimal control theory significantly reduces the dependence of the controller and the sensitivity of the closed-loop system to noise, although there may be a small loss in the gain and phase margins. This is also a well-known result [35] in stochastic optimal control theory (i.e., the case with uncertainty inputs or noise). The performance of the closedloop system using the control input generated by the deterministic optimal control theory is inferior to that of the stochastic control input. One can, therefore, conclude that the stochastic control gives a better result as compared to the deterministic control. The distinct advantages of nonlinear modeling over the use of a linear model are also twofold: first, one avoids the significantly tedium of linearization, which was done in this paper, to demonstrate the efficacy of the controller. Linearization is also not a unique method and resorting to it leads to additional questions concerning the method of linearization. Second, linearization is generally performed about an equilibrium solution. Thus, implicit in the linearization step is knowledge of the equilibrium solution, which may be either unknown at the design stage or variable and dependent on a host of environmental conditions. Thus, avoiding the linearization step also implies that one does not need to know the equilibrium solution to design the controller. This is a significant advantage in practice unless there are more than one equilibrium solutions. In particular, it has been demonstrated in this paper that the MPPT control filter, which acts as an outer loop controller, continuously seeks to maximize the power absorbed by the wind turbine. The MPPT control filter included a feedback signal estimated using the Newton–Raphson formula at each time step. 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 without a significant error, and to eliminate the influence of power transients. The maximum available power in the wind is computed 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 Cp (λ)m ax is used to calculate it. In steady state, the estimated


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Ranjan Vepa (M’06) received the Ph.D. degree in applied mechanics from Stanford University, Stanford, CA, in 1975. He is currently a Lecturer in the School of Engineering and Material Science, Queen Mary University of London, London, U.K., where since 2001, he has been the Program Director of the Avionics Program. He is the author of two books: the Biomimetic Robotics: Mechanisms and Control (Cambridge University Press, 2009) and the Dynamics of Smart Structures (Wiley, 2010). His research interests include design of control systems, and associated signal processing with applications in smart structures, robotics, biomedical engineering, and energy systems. In particular, the research interests include dynamics and robust adaptive estimation and control of linear and nonlinear aerospace and biological systems with parametric and dynamic uncertainties. Dr. Vepa is a member of the Royal Aeronautical Society, London and a Fellow of the Higher Education Academy, U.K.