State Estimation of DFIG using an Extended Kalman Filter with an Augmented State Model

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1 State Estimation of DFIG using an Extended Kalman Filter with an Augmented State Model Mridul Kanti Malaar Department of Electronics and Electrical Engineering Indian Institute of Technology Guwahati, Assam- Praveen Tripathy Department of Electronics and Electrical Engineering Indian Institute of Technology Guwahati, Assam- Srinivasan Krishnaswamy Department of Electronics and Electrical Engineering Indian Institute of Technology Guwahati, Assam- Abstract This paper presents a novel method for state estimation in doubly fed induction generators (DFIGs) using an Extended Kalman Filter. In this wor, the conventional nonlinear state space model of a DFIG has been augmented with additional states in order to mae rotor position and speed estimates more robust to disturbances. The effectiveness of this method has been tested for various transient cases using MATLAB/Simulin. Index Terms Doubly fed induction generator, rotor position and speed estimation, Extended Kalman Filter. V i ψ Ψ L s,l r,l m R s,r r P, ω s NOMENCLATURE Stator or rotor voltage. Stator or rotor current. Stator or rotor flux component. Stator or rotor flux vector. Stator, rotor and magnetizing inductance. Stator and rotor resistance. Number of pole of the machine. Rotor and Synchronous speed of the machine respectively(electrical rad/s). Rotor position angle from stator stationary reference frame. ˆ Estimated rotor position angle from stator stationary reference frame. ε r Angle estimation error. Number of samples in discrete time systems. Subscripts α, β Stationary stator reference frame coordinate. a, b Rotor reference frame coordinate. d, q Synchronous reference frame coordinate. s, r Stator and rotor quantities. Superscripts ˆ Estimated value ----//$. c IEEE I. INTRODUCTION Doubly fed induction generators (DFIG) are widely used for constant frequency variable speed wind energy conversion systems (WECS). The stochastic behavior of wind flow and the nonlinearity of wind turbine model gives rise to various challenges in design and control of WECS []. Field oriented control (FOC) is commonly used to control the dynamics of DFIG which enables decoupled control of active and reactive power. The performance of FOC strategy depends on the accurate information of rotor position. Usually, both rotor position and rotor speed can be acquired using encoders. However, the use of an encoder introduces limitations on machine size, drive cost, reliability, mounting, robustness and noise thus affecting the performance of vector control strategies. The development of sensorless algorithms for accurate estimation of the rotor position and speed therefore a widely research topic. Open loop speed and position estimators have been dealt with in [], []. The use of closed loop observers have been discussed in [] []. They are mainly based on the model reference adaptive systems (MRAS). The performance of an MRAS based observer is highly sensitive to the inductance value []. Another possible way of estimating the dynamic states of a DFIG is by the use of recursive methods lie the Extended Kalman Filter []. In this paper, we propose an Extended Kalman Filter based method that uses an augmented state model in order to dynamically obtain an accurate estimate of rotor position and speed. The remainder of this paper is organized as follows: Section II deals with the dynamic model of a DFIG system. Extended Kalman Filter and the proposed augmented state model for the Extended Kalman Filter is presented in Section III and Section IV respectively. The effectiveness of the proposed extended alman filter model is demonstrated through simulation results in Section V. Finally the conclusions are summarized in Section VI.

2 II. STRUCTURE AND MODELING OF DFIG BASED WIND TURBINE SYSTEM The DFIG system comprises of a wound rotor induction machine and a bac-to-bac power converter. The stator terminal of the machine is directly connected to the grid and the rotor terminal also connected to the grid via power converter. The rotor operating speed range of the generator is normally limited to ±% of the synchronous speed. The Fig. shows that the basic structure of the DFIG coupled with wind turbine and grid. The mechanical power P mech produced by the wind turbine is given by the following equations as, P mech T t ω t () T t C p(λ, β) ρar λ ωt () where T t,a,c p (λ, β) and ρ are the turbine aerodynamic torque, rotor swept area, power coefficient for a variable pitch (β) wind turbine and air density respectively. The tip speed ratio is given by λ Rωt V w,whererω t and V w stands for turbine rotor radius, turbine speed and wind speed respectively. Main Grid by di αs di βs di αr di βr ( R s L r i αs + L m σl s L i βs + R r L m i αr r + L m L r i βr + L r V αs L m V αr ) () ( L m σl s L i αs R s L r i βs L m L r i αr r +R r L m i βr + L r V βs L m V βr ) () (R s L m i αs L s L m i βs R r L s i αr σl s L r L s L r i βr L m V αs + L s V αr ) () ( L s L m i αs + R s L m i βs + L s L r i αr σl s L r R r L s i βr L m V βs + L s V βr ), () where leaage factor, σ L m L sl r. The electromagnetic torque and the mechanical motion are given by T e P L m(i αr i βs i αs i βr ) () d P J (T e T t ) neglecting frictional loss () Eq. ()-() have been used to obtain the dynamic model of the DFIG. III. EXTENDED KALMAN FILTER Gear box DFIG DC lin RSC GSC Grid filter Fig. : Wind turbine with doubly fed induction generator The machine equations are represented in (α, β) frame []. The flux-linage equations are as follows: ψ αs L s i αs + L m i αr () ψ βs L s i βs + L m i βr () ψ αr L r i αr + L m i αs () ψ αr L r i βr + L m i βs () The differential equations of the stator and rotor of the DFIG with stator and rotor current as state variables can be expressed Extended Kalman Filter is widely used for estimating unmeasured states in a nonlinear process. Consider a nonlinear system with the following state space equations, x() f{x( ), u( )} + w( ) () z() h{x()} + v() () where, x() and z() be the process state vector and measurement output vector respectively. w and v are assumed zero mean normal distributed noise perturbations. These perturbations could be model uncertainties or external disturbances. The process and measurement noise covariance matrices are given by: cov(w) E { ww T } Q cov(v) E { vv T } R The Extended Kalman Filter is a two step algorithm. The first step involves the prediction, and the second one is the correction step. For the Extended-Kalman Filter, the predictor step is given by, ˆx() f{x( ), u( )} P () F( )P ( )F T ( )+Q()

3 where, the Jacobians are defined as, f(x, u) F() x () x( ),u( ) H() h(x) x () x( ) ˆx() and P () are apriori states, and the corrector step is given by, K g () P () H T () { H()P () H T () +R()} ˆx() ˆx() + K g (){z() h{x() }} P () {I K g ()H()}P () In the above equations P () is an estimate of the covariance of the measurement error and K g () is called the Kalman gain. After both the prediction and correction steps have been performed then ˆx() is the current estimate of the states and ẑ() can be calculated directly from it. Both ˆx() and P () are stored and used in the predictor step of the next time period. IV. AUGMENTED STATE MODEL OF DFIG The dynamic model of DFIG is a set of continuous-time nonlinear differential equations as expressed by Eq. ()-(). Based on the nonlinear dynamic model of DFIG, the following process state vector and measurement output vector can be defined []: x [i αs ; i βs ; i αr ; i βr ; ; ] [x ; x ; x ; x ; x ; x ] () z [i αs i βs i αr i βr ] T () By using the state vector in Eq. (), the steady state error is persistent while estimating the rotor position in presence of transient dynamics. To mitigate the steady state error we partially augmented the state vector with integral of currents as given by: [ x i αs ; i βs ; i αr ; i βr ; ; ; i αs ; i βs ; ] i αr ; i βr [x ; x ; x ; x ; x ; x ; x ; x ; x ; x ] () To further mitigate the steady state error, we augment the state vector with integral of square of currents which is as given below: [ x i αs ; i βs ; i αr ; i βr ; ; ; i αs ; i βs ; i αr ; i βr ; i αs ; i βs ; i αr ; ] i βr [x ; x ; x ; x ; x ; x ; x ; x ; x ; x ; x ; x ; x ; x ] () Eq. (), () and () are denoted as EKF, EKF normal and EKF improvement. The EKF normal is a modification of EKF with addition of integral of currents. The EKF improvement is an modification of EKF normal with addition of square of integral of currents. The addition of integral states by augmenting the state equations will reduce the steady-state error. For implementation of Extended Kalman Filter, the DFIG model should be discretized. Using bacward Euler s method, dx(t) x() x( ) ; ( )Δt t Δt, Δt we can get discrete form of machine state equations are as follows: x () x ( ) + Δt dx () x () x ( ) + Δt dx () x () x ( ) + Δt dx () x () x ( ) + Δt dx () { Kt P x () x ( ) + Δt J {x ()x () x ()x ()} K } tp J T t () x () x ( ) + Δtx () () where,k t P L m. The discrete time integral states are defined using bilinear transform as: x i+ () x i+ ( ) { } xi ()+x i ( ) +Δt x i+ () x i+ ( ) { x +Δt i ()+x } i ( ) () ()

4 where, Δt is the sampling time and i varies from to. The derivatives used in Eq. () - () correspond to the Eq. () - (). The input and measurement state vectors are as follows: u [V αs V βs V αr V βr ] T () z [i αs i βs i αr i βr ] T () In this wor, the values of Q and R are determined with extensive simulation to achieve satisfactory results. Note that the initial value of P ( ) is chosen to converge the Extended Kalman Filter. The Q, R and P are given as follows: Q diag([.... e e e e e e e e e e ]); R diag([e e e e ]); P (initial) diag([]); V. SIMULATION RESULTS The effectiveness of proposed estimator using Extended Kalaman Filter is validated by simulation using MAT LAB/Simulin. For simulating the proposed algorithm, DFIG parameters have been taen as follows: HP, Hz, pole Phase; Stator : V, Δ connected,. A; Rotor : V, Y connected,. A and J. g m. Stator resistance (R s ). Ω, rotor resistance (R r ). Ω, stator inductance (L s ). H, rotor inductance (L r ). H and mutual inductance (L m ). H. The switching frequency of the bac to bac converter is Hz. Rotor Speed (rad/s) Wr Wr r eference Fig. : Speed profile of DFIG. In simulation, a step command is applied to achieve the speed control response, at time t s. Further, the stator terminal was short circuited at time t.s. It is assumed that after ms, the machine retrieved it s rated voltage. Again, at time t s, the step command is applied at almost equal to synchronous speed (.rad/s). Under above simulation scenario, the speed response of the DFIG is shown in Fig.. Torque (N m) T t T e Fig. : Turbine torque (T t) and DFIG Torque (T e) responses. Further, a step command is set to reduce the turbine torque by % at time t s. Fig. shows the torque response of the turbine. In this case small amount of white gaussian noise N (,σ ) is added to get turbine torque dynamics. The simulation stop time is fixed at s with sampling time, Δt.s. Continuously prediction and correction steps are iterated which allows the on-line estimation of the DFIG state vector. The estimated and actual rotor speeds are shown in the Fig.. It is observed that a small error is present in the estimated speed. The estimated and actual rotor positions are shown in Fig. to Fig.. It was observed that the proposed state model archives less error with introduction of both both integral of stator and rotor current vectors and integral of square of the stator and rotor current vectors as defined in the Eq. (). VI. CONCLUSION In this paper, we have proposed an Extended Kalman Filter based rotor speed and position estimator using an augmented state model. A comparative analysis of the performance of an Extended Kalman Filter was done for the following three types of state model: (a) without integral states (non-augmented state model) (b) with integral of stator and rotor current (partially augmented state model), and (c) with integral of stator and rotor current and integral of square of the stator and rotor currents (augmented state model). The performance

5 Rotor Speed (rad/s) Fig. : Estimated and actual rotor speeds of DFIG. Fig. : Estimated and actual rotor positions during step change of DFIG rotor speed Fig. : Estimated and actual rotor positions during starting of DFIG Fig. : Estimated and actual rotor positions during short circuit of stator terminals. of the estimators have been analyzed under various dynamic conditions. For the cases simulated, the proposed augmented state estimator showed the best performance. REFERENCES [] M. Tazil and V. Kumar et.al, Three-phase doubly fed induction generators: an overview, IET Electric Power Applications, vol., pp.,. [] L. Morel, H. Godfroid, A. Mirzaian, and J.-M. Kauffmann, Doublefed induction machine: converter optimisation and field oriented control without position sensor, Electric Power Applications, IEE Proceedings -, vol., no., pp.,. [] R. Datta and V. T. Ranganathan, A simple position-sensorless algorithm for rotor-side field-oriented control of wound-rotor induction machine, Industrial Electronics, IEEE Transactions on, vol., no., pp.,. [] M. Pattnai and D. Kastha, Comparison of mras based speed estimation methods for a stand alone doubly fed induction generator, in Energy, Automation, and Signal (ICEAS), International Conference on,, pp.. [] D. Forchetti, G. Garcia, and M. Valla, Adaptive observer for sensorless control of stand-alone doubly fed induction generator, Industrial Electronics, IEEE Transactions on, vol., no., pp.,. [] F. Dezza, G. Foglia, M. Iacchetti, and R. Perini, An mras observer for sensorless dfim drives with direct estimation of the torque and flux rotor current components, Power Electronics, IEEE Transactions on, vol., no., pp.,. [] G. Marques and D. Sousa, Sensorless direct slip position estimator of a dfim based on the air gap pq vector-sensitivity study, Industrial

6 Fig. : Estimated and actual rotor positions during rotor speed almost equal to the synchronous speed Fig. : Estimated and actual rotor positions during step change of turbine torque. Electronics, IEEE Transactions on, vol., no., pp.,. [] R. Cardenas, R. Pena, S. Alepuz, and G. Asher, Overview of control systems for the operation of dfigs in wind energy applications, Industrial Electronics, IEEE Transactions on, vol., no., pp., July. [] I. Perez, J. Silva, E. Yuz, and R. Carrasco, Experimental sensorless vector control performance of a dfig based on an extended alman filter, in IECON - th Annual Conference on IEEE Industrial Electronics Society,, pp.. [] W. Leonard, Control of Electrical Drives. Spinger, New Yor, USA,.