Non-Adaptive Speed and Position Estimation of Doubly-Fed Induction Generator in Grid-Connected Operations

The nonadaptive speed and position estimation scheme for a doubly-fed induction generator (DFIG) is presented in this article. The observer structure is based on the extension of the mathematical model of DFIG to the introduced H vector. Based on the defined H vector, the nonadaptive position and speed estimation is proposed. The Lyapunov method is extended to the practical stability theorem to stabilize the structure. The classic stator field-oriented control to active and reactive power control is used in the sensorless control system. The performance of the proposed algorithm of the speed and position observer is validated by simulation and experimental results using the 2 kW generator. The comparison tests to the classical model reference adaptive system based observer are provided. Finally, the article contains a summary in which the non-adaptive speed estimation is discussed in context to the adaptive reconstruction of the rotor speed and position.

Non-Adaptive Speed and Position Estimation of Doubly-Fed Induction Generator in Grid-Connected Operations Marcin Morawiec , Senior Member, IEEE, and Krzysztof Blecharz Abstract-The nonadaptive speed and position estimation scheme for a doubly-fed induction generator (DFIG) is presented in this article.The observer structure is based on the extension of the mathematical model of DFIG to the introduced H vector.Based on the defined H vector, the nonadaptive position and speed estimation is proposed.The Lyapunov method is extended to the practical stability theorem to stabilize the structure.The classic stator field-oriented control to active and reactive power control is used in the sensorless control system.The performance of the proposed algorithm of the speed and position observer is validated by simulation and experimental results using the 2 kW generator.The comparison tests to the classical model reference adaptive system based observer are provided.Finally, the article contains a summary in which the non-adaptive speed estimation is discussed in context to the adaptive reconstruction of the rotor speed and position.

I. INTRODUCTION
T HE doubly-fed induction generator (DFIG) is one of the most used constructions in high-power wind power plants due to a few advantageous features.A generator of this type enables the wind turbine to operate a wide range of changes in the rotational speed of the shaft to improve the efficiency of energy production from wind.The power of the rotor-side converter is only a fraction of the power of the generator.Other advantages of the system are high precision control of active and reactive power and good quality power transferred to the power grid.A high-performance generator system has to use advanced control methods.Generator control techniques in different are generally reported based on known vector control methods, such as field-oriented control [1], [2], [3], direct power [4], [5] or torque control [6], and nonlinear control [7], [8], [9].In all the above-mentioned control techniques, information about the generator shaft position and its angular velocity is necessary for the implementation of the control system.Measurement of the generator rotor position with the use of the most popular solution in the form of a mechanical rotary motion sensor causes technical difficulties in its assembly and significantly reduces the reliability of the entire system.Therefore, in advanced electric drive control systems, a tendency can be observed to eliminate the speed sensor and replace it with algorithmic solutions.This approach, known as sensorless, can significantly increase the reliability of the control system and reduce the wiring of the system.Another important aspect of the application of sensorless algorithms is their use for diagnostics of generator operation.The least complicated technique is the algorithms operating in an open calculation loop [10], [11], where the rotor position angle is determined based on the measured and calculated value of the stator or rotor current in a specific coordinate system.These algorithms are sensitive to changes in machine parameters and measurement noise due to no error correction mechanism.
The model reference adaptive system (MRAS) algorithm is the most numerous group of solutions for estimating the shaft angle and selected generator state variables.Depending on the adopted mathematical model of the generator and the working variables, the MRAS algorithm takes various variants, the most important being stator flux (SF-MRAS) [12], rotor or stator current (RC-MRAS) [13], and back electromotive force (EMF-MRAS) model [14].In general, the operation of MRAS algorithms, depending on the choice of the reference model structure and the error correction mechanism, is sensitive to the operation of the generator system near a low slip frequency.A different solution is shown in [15], [16], where rotor angle estimation is realized by the applications of the second-order generalized integrator and frequency-locked loop structures to filtered rotor current in the sensorless generator control system connected to the power grid.The correctness of the algorithm operation is related to the limited dynamics of the machine shaft speed change [17].
In [18] and [19], the authors present a method for estimating velocity and angle using a high-order sliding mode observer and a phase-locked loop.The operation of this algorithm is based on accurate, current measurements.Considering the assumed sliding plane, the error of the observer's correction oscillates around minimal values, which may affect the estimation accuracy.Another algorithm approach is the high-frequency method [20], [21], whereby a high-frequency current or voltage signal is injected on the rotor side, and then the stator side signal is measured.Rotor angle position and speed are estimated from the signals' phase shift and frequency.Its effectiveness and accuracy are influenced by the impedance value on the power grid side.An interesting approach is using frequency methods based on fast Fourier transform (FFT) analysis to identify and extract a spectral frequency in the generator's electrical signals that have a known relationship to rotor speed to estimate rotor position [22].These algorithms require the use of fast digital signal processor (DSP) processors and are characterized by a decrease in estimation accuracy along with a decrease in the load on the generator system [23].
The next very interesting group is variable state observers based on the Luenberger theory and generator vector models.These advanced estimation algorithm solutions generally are divided into two groups, the full-order [24], [25] and reducedorder [26], [27], with different technic approaches.In [28] the full-order observer is proposed where the rotor position estimation error is regarded as an additional machine parameter, and it is tracked by the proposed adaptive law that makes compensation for the projection error from the rotor reference frame to the stator reference frame.A full-order observer is also shown in [29], [30], and [31], but the backstepping technique was used to improve performance.The most important task for the variable state observer is his stability in a wide range of machine rotor speeds both below and above the synchronous speed and variationist in generator parameters.
The main contribution of this article is to propose and extend the non-adaptive estimation scheme of position and the rotor speed of the DFIG system, which was first time proposed in [32].The proposed observer is based on the DFIG mathematical model derived for the stator and rotor current vectors and stabilized using the Lyapunov method.The proposed solution brings benefits, in particular during the uncertainty of parameters of DFIG, disturbances in the measurement of the generator stator and rotor currents, and the generator operating states from the sub-and super-synchronous speed range.The order of observer structure is decreasing in comparison to adaptive structure, from r = 6 to r = 5.The problem with tuning the observer does not exist.The observer structure presented in [32] is extended to the stability analysis and uncertainties of chosen parameters.The simple adaptation mechanism is proposed to adjust the introduced in [32] tuning gain, which limits the overshoots of estimation errors during dynamic states of dynamic states the DFIG shown in Sections III and IV.
Presented simulation and experimental tests for a 2 kW generator show the state variables' small value of estimation errors and excellent performance in all working ranges.

II. MATHEMATICAL MODEL OF DOUBLY-FED GENERATOR
The mathematical model of the DFIG system can be determined in different reference frames (stationary or rotating frames).The differential equations for the rotor and stator current vector components were presented in [9] and [30] and the electromechanical equation for the rotor angular speed where where i s , i r , u r , u s are the currents and voltages of rotor and stator vectors, respectively, T e and T L are the electromagnetic and load torque, f r is the coefficient of viscous friction.It is assumed that all machine parameters are known; the stator current vector components and the stator voltage vector components can be measured; the rotor voltage vector components are the control variables; rotor current vector components, as well as the rotor angular speed ω r , are estimated by the speed observer structure.The generator is connected to the ac grid from the stator side and to the voltage source converter VSC from the rotor side.

A. Speed Observer Structure and Stabilizing Functions
Considering the model ( 1)-( 4) in (xy) rotating reference frame for ω a = 0 and coordinate changes within the IM model Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
following mathematical model of the machine is achieved where The speed observer structure is based on the mathematical model ( 5)- (9).The observer for this model has the following form: where estimated state variables are marked by "^" and The new input variables v rx , v ry and v Hx , v Hy , v θ are introduced to the observer structure, which can be named as the stabilizing functions and allow the observer convergence to the real value of the machine.The stabilizing functions will be obtained by using the Lyapunov theorem.The estimation errors in the observer system can be defined as follows: The rotor speed in the DFIG system depends on the wind speed of the turbine, which is assumed to be constant.Therefore, the derivative of the rotor speed, which occurs in ( 14) and ( 15) can be approximated, considering the time ΔT between calculating the rotor speed dω r /dτ ≈ Δω r /ΔT .This simplification does not significantly influence the accuracy of the speed observer because the rotor speed is treated as the parameter whose value in the DFIG system is constant.
Considering the defined errors in (19) as well as ( 5)-( 9) and ( 12)-( 16) the model of estimation errors has the following form: where in ( 20) and ( 21) the stator and rotor current components are treated as measured values [as the inputs to the observer system ( 12)-( 16)].
The speed observer ( 12)-( 16) will be asymptotically stable if the Lyapunov function is positively defined The derivatives of the function (25) by using ( 20)-( 24) has the following form: The derivative of the Lyapunov function is negatively defined if the stabilizing functions are determined where (c x , c y , c Hx , c Hy , c θ ) > 0 are the observer tuning gains.

B. Stability Analysis of the Observer System
Substituting ( 27), ( 28), ( 29), (30) to (26) for (c x , c y , c Hx , c Hy , c θ ) = 1.0 is easy to check that the system is asymptotically stable and the derivative of the Lyapunov function has the form Assuming the ideal condition, it can be considered that c x = c y = c xy and c Hx = c Hy = c H and the minimum values of these tuning gains should be at least To satisfy the Lyapunov theorem V < 0, the stabilizing function v θ should be In the sensorless control, the rotor speed is not measured; therefore, the deviation θr in (33) should be replaced by θH , which means the difference between the estimated values of H x and H y , calculated from ( 17) and ( 18) and reconstructed from the observer structure by using ( 14) and ( 15) as follows [30]: where ϑ = H x Ĥy − H y Ĥx Then, since the system ( 5)-( 9) stays in operation domain D, hence if with δ 1 , δ 2 , δ θ > 0 and for ĩrx , y ≤ ε 1 , θr ≤ ε 2 , ωr ≤ ε 3 and ε 1 , 2 , 3 << 1 are sufficient small reals, the derivative of the Lyapunov function takes the following form: where The condition (39) implies the convergence of vector values îr to i r and H to H, hence θr tend to real value θ r in finite time, noted as t 2 .
Remark 1: The rotor speed value can be determined from the following dependence [32] where are the rotor flux components that can be calculated from the above dependencies.It is worth noticing that if (39) is satisfied, then the vectors of estimated values tend to their real in finite time, noted as t 2 and ψ2 rx + ψ2 ry = 0 the rotor flux vector components defined in (42) and (43), tend to real values in finite time, noted as t 3, and the same the rotor speed estimated from (40), converges exponentially to its real value ω r .For (c x , c y , c Hx , c Hy , c θ , c f ) > 0, the observer system is practically stable [33].
Remark 2: The form of (40) results from ( 17) and ( 18) by using ( 42) and (43).The value of s ω defined in (41) depends on the mutual position of the vectors H and ψ r .For the perpendicular case, the s ω = 0 because s ω = cos( H, ψ) this is the scalar product of the vectors.In the rotor speed dependence, (40) the value of c f can be determined by using a simple adaptation mechanism where s ω f means the filtrated signal of s ω by using LPF with 1/T = 0.005 and s ω ref is the reference value of the scalar product, which in the ideal condition could be equal to zero and it is assumed γ = 10/T = 0.05 p.u.In Fig. 1, in the DFIG system, the active reference power is set to −0.1 p.u. and reactive power −0.6 p.u.The rotor speed crosses from the sub to the supersynchronous range.The value of c f coefficient is estimated from (44) and limited according to (45).For the subsynchronous speed range, the cf value is about 1.0 up to 3 (depending on the generator load).If the rotor speed is increased, the value of c f is not constant, and during the dynamic state is about c f = 20 p.u.After that, it settles to about 10-11 p.u.The same behavior is visible in the experimental tests.
The maximum value of c f for ( ψ2 rx + ψ2 ry ) ≈ 1, |ω r max | = 1.2 p.u. and |s ωref | = 0.005 can be determined and c f should be chosen 0 ≤ c f ≤ c f max .The observer scheme with the folwchart diagram is presented in Fig. 2. Remark 3: As aforementioned, the observer structure will be stable for the tuning gains determined from (32) and (38).The Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.observer tuning the speed of observer convergences.It is worth noticing that for c θ = 0, the observer structure is unstable due to the open integrator (16).The tuning gains can be specified from the model of observer errors linearized near an equilibrium point in the (d-q) reference frame connected to the stator flux vector.The general form of the linearized system is where A and B are the Jacobian matrices; Δx(t) = [ ĩrd , ĩrq , Hx , Hy , θr ] T ; Δu(t) is treated as known control inputs.
By using the observer errors model ( 20), ( 21), ( 22), ( 23) and for (dq), the matrix A can be determined as follows: (47) where a 1 = L s /w σ and values of ω * r are determined for the chosen working points and ω ψ = 1.0 p.u.
It is worth noticing that the position error equation is omitted in the linearized system because the rotor speed is treated as the parameters, and due to the open integrator form, the value of c θ should always be c θ ࣔ 0. For c θ = 0, it is easy to check that the one pole of the system matrix A has equal to zero.It means that the observer system has undamped oscillations.Therefore, it can be assumed that c θ ࣔ 0 and to analyze the reduced linearized system order with n = 4.The spectrum of the matrix of the linearized observer system for different tuning gains value c xy from 0.1 to 25 is presented in Fig. 3(a).The values of poles are close to zero for c θ = 0.In Fig. 3(b) the value of c H is changed from 1 to 5 p.u., but c θ = 0.1 p.u.
The next section confirms the theoretical investigations by using the simulation results in the 2 kW DFIG system.

IV. SIMULATION RESULTS
The simulation model of the DFIG system contains the doubly-fed machine model, wind emulators, and the VSI model with the space vector modulation (SVM).
The sample period for the control system is 150 μs and for the SVM is 300 μs.The system parameters are given in Table II.In Fig. 4(a) the active power (s p ) is changed from −0.1 to −0.35 after 100 ms (reactive power (s q ) is set to −0.6), and after 400 ms, the active and reactive powers are changed togethers p to 0.35, s q to 0.2 p.u.As a result, the estimation error of rotor speed is smaller than 0.01 in the steady-state and about 0.015 in the transient state.On the other hand, the rotor position error is about 0.012 in the steady state and approximately 0.017 in the transient state.The components of rotor current i rd , i rq are shown.
In Fig. 4(b) the active power s p is set to 0.02 p.u. and reactive power s q is set to −0.6 p.u.The rotor speed of the DFIG is changed from the subsynchronous to supersynchronous mode.The estimated speed error is increased during the rotor speed crossing through the synchronous speed (1.0 p.u.).The error of rotor position is almost constant and at about 0.01 p.u.
In Fig. 5(a) the rotor current estimation errors ĩrx,yS , rotor flux "x" component ψ rxS , measured and estimated rotor position, estimated rotor speed and rotor speed and position errors, and rotor current vector components for the DFIG steady-state are presented.The rotor speed and position errors are about 0.015 p.u.In Fig. 5(b) the DFIG parameters uncertainties test is presented.After 300 ms the stator resistance is changed twice.The rotor position error is changed from −0.012 to 0.012 rad; the rotor speed error is below 0.01 p.u. in the stationary state; after 600 ms, the nominal value of magnetizing inductance is twice (together with stator and rotor inductance).This change has resulted in a reduction of the errors of rotor speed and position below 0.01.Changes in the nominal value of stator resistance and the magnetizing inductance do not affect the stability of the proposed observer structure.
In Fig. 6, the DFIG turbulences are modeled.The turbulences are turned on in the control system.It is visible that the estimated rotor speed has oscillated; however, the estimated speed and position errors are below 0.02. in the steady state.The proposed observer is stable during these simulated conditions.
The theoretical issues are confirmed using experimental investigations presented in the following section.

V. EXPERIMENTAL RESULTS
Experimental results were carried out in the 2 kW generator system (Table II in Appendix).The sensorless control system with the Field-oriented control from Fig. 5 was implemented in a DSP board with a Sharc ADSP21363 floating-point signal processor and Altera Cyclone II FPGA.The PWM switching frequency was 3.3 kHz.The code of the algorithm was not The experimental investigations are divided into the following scenarios: steady-state of the DFIG, nominal power changes of active and reactive, and nominal parameters uncertainties.In the figures, the selected variables are presented as follows.
1) s p , s q active and reactive power respectively.
3) Estimated θr and measured θ rM value of the rotor position and their error θr .4) Values of the signals s ω and θH .5) Stator and rotor vector components i sxS , îrx,yS .6) Error of the estimated rotor vector component ĩrxS .7) Rotor flux vector components ψrx,yS .

A. Steady State of the DFIG System
In this scenario, the DFIG is in the steady state in which the active power is set to −0.1 p.u. and reactive power is set to −0.6 p.u. the rotor speed is about 0.72 p.u.In Fig. 8(a) the estimated rotor speed, rotor current vector components îrx,yS , and rotor flux vector components ψrx,yS are shown.The current vector components have visible higher harmonics resulting from the machine's design (slots harmonics [30]).In Fig. 8(b), the rotor speed is about 0.82 p.u. and the measured and estimated rotor position, stator vector component i sxS , and the error of the estimated rotor vector component ĩrxS are shown.The average value of error ĩrxS is smaller than 0.02 p.u.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.In Fig. 9(a) the chosen variables in the steady state of the DFIG are presented.For the subsynchronous range, the rotation speed is about 0.75 p.u. for the supersynchronous range, which is about 1.2 p.u.The estimated and measured rotor position values are compared, and the values of filtrated scalar product s ωf and stabilizing function θH are shown.In Fig. 9(a) the value of s ωf is almost equal to zero, and the value of c f is determined using the adaptive term (44) for s ωref = 0.001 p.u.The estimation of position error is then about −0.03 rad.
In the supersynchronous case presented in Fig. 9

B. Dynamic States of DFIG
In Fig. 10(a) the active power is changed to −0.35 p.u., and reactive power is maintained to a constant value of −0.6 p.u.The estimated rotor speed error in the transient state is 0.05 p.u. in the steady state is smaller than 0.02 p.u.The estimated rotor  position error is about 0.1 rad. in the transient state and, after this, is decreased to about 0.04 p.u.In Fig. 10(b), after 300 ms, the reactive power is changed from −0.6 to −0.4 p.u.The rotor speed error is constant and at about 0.02 p.u. however, the rotor position error is increased to 0.07 rad.This error occurs after the value of the reactive power is decreased to −0.4 p.u.This test confirms that the reactive power value influences the estimation errors of the speed observer structure.
In Fig. 11 the rotation speed of the generator rotor increases from 0.72 to 1.25 p.u. as a result of increasing the external driving torque of the generator.The generator power control system works correctly, the values of active and reactive power set in the control system are kept almost constant around reference values.The time waveforms presented in Fig. 11 refer to the unfavorable operating point with an underloaded generator.The references to active and reactive powers are −0.1 and −0.6 p.u., respectively.The sensorless control system works stable, crossing through the synchronous rotor speed.The rotor position error is smaller than 0.15 rad. in the transient states and, after reaching the synchronous speed, is decayed to the constant of about −0.05 p.u.The value is maintained, and it results from c fmax and s ω .It means that the vectors H ψ r are not perpendicular s ω = 0. Hence, to minimize the position error, the value of c f should not be limited to c fmax .On the other hand, the term (−c f s ω ) is introduced to (40) and without its bounded; the rotor speed value may not tend to the actual value though this θr does not tend to real value θ r in finite time, noted as t 2, and the observer system can be unstable.
In Fig. 11(b), the value of c f is set to constant value c f = 11 p.u.The constant value of c f was presented in [32].For this case, the maximum value of the estimated error of rotor position is about 0.15 rad in the supersynchronous speed range.The sensorless control system is stable.In Fig. 12, the DFIG crosses from ω r = 0.75 to ω r = 1.2 p.u.The cf value is estimated adaptively using (44) and bounded to c fmax .Then the value of s ω is changed from − s ω to +s ω, which is visible in Fig. 12. Therefore sgn(s ωf ) is added to (44).The estimation error of the rotor during the transient state is about 0.12 p.u.; however, in the steady state is smaller than 0.05 p.u.In Fig. 12(b) the chosen variables are presented for the MRAS observer presented in The active power is set to −0.1 and the reactive power to −0.6 p.u.The MRAS system is stable; however, the estimated position and speed oscillate.The average rotor position is about 0.12 [rad].
The coefficient c f in (40) has not a physical meaning.However, this value influences the estimated rotor speed and position.This minimizes the position and rotor speed errors, particularly in the crossing through the synchronous speed visible in Fig. 12.

C. Parameter Uncertainties of DFIG
The uncertainties test of the parameters in the observer system was carried out to check the behavior of the proposed observer to the variability of chosen nominal parameters.The values of the stator and rotor resistances are changed at the same time.These values were changed two times according to R s,r = 2R s,rN .This test is presented in Fig. 13(a).The active and reactive powers are −0.1 and −0.6 p.u., respectively.The rotor speed is about 0.75 p.u.After 0 s values of the resistances are changed, and after 1s, the nominal values are set.The estimated value of rotor position error is about 0.1 rad.In Fig. 13(b), after 0 s nominal value of the magnetizing inductance L m = 0.75L mN , and after 0.8 s is set to the nominal value.The estimated rotor position error is about 0.07 rad.Changes in the nominal parameters of the generator do not stimulate the observer structure significantly and have the influence of a negative effect on the estimated rotor position value, however, for the assumed range of the parameters changes, the observer structure is stable.

VI. DISCUSSION
The performance of the proposed non-adaptive speed estimation approach is shown in Sections IV and V.The proposed sensorless control system works stable even if the nominal parameters of the generator are changed.This test has been presented in Fig. 13.The error of rotor position is increased up to 0.1 rad; for the nominal parameters is decreased up to 0.04 p.u. (for R s,r = 2R s,rN ); almost the same position error value occurs during the changes of the nominal value of the magnetizing inductance L m = 0.75L mN [see Fig. 13(b)].The average error of estimated rotor speed is always below 0.025 p.u.; and higher in case of uncertainties of L m .In Fig. 10 the responses of the stator's active and reactive power are shown.As shown in Fig. 10(b), the observer structure is sensitive to reactive power changes.In this case, the estimated position errors changed their value from positive to negative; however, their average value does not change and is kept to about 0.05 rad.In Figs.11 and 12 the rotor speed of DFIG is changed from subsynchronous to supersynchronous.The rotor speed is estimated non-adaptively; however, the additional stabilizing function is introduced to the estimation law (40) and particularly described in Section III.The reinforcement of this function marked as c f can be determined as the constant or by an adaptive approach (44).Both cases are shown in Fig. 11.After changing the rotor speed, for c f = 11 [see Fig. 11(b)], the estimated rotor position has a constant value of about 0.15 p.u.While c f is adaptively estimated, the position error is minimized almost to 0.03−0.05p.u., which is visible in Fig. 12(a).
The performance of the proposed observer structure was compared to the well-known [34] SFMO-MRAS structure.In the MRAS [34], the reinforcement in the integration part of PI controllers was taken 0.9 p.u.; for the error, the part was 1.0 p.u.The active power changes are shown in Fig. 14(a), while changes in the reactive power are in Fig. 14(b).The rotor position error is more oscillating than in the proposed observer structure.Their average value is above 0.18 rad [see Fig. 14(a)].If the reactive stator power is generated after 300 ms, the power is −0.4 p.u., then the average rotor position error is 0.15 p.u. and their value is smaller than in Fig. 14(a).The rotor speed error is smaller than 0.05 p.u. for both cases.
Fig. 15 shows the estimated and measured rotor position waveforms and their error for subsynchronous and subsynchronous operation.The position error for the subsynchronous speed has a higher value (up to 0.27 rad) than the supersynchronous (up to 0.2 rad).In Fig. 12(b), the rotor speed is changed from the subsynchronous to supersynchronous operation for MRAS.It is visible that the rotor position error is changed from −0.15 to −0.2 rad during the changing of rotor speed.As a result, the maximum estimated rotor speed error has a value up to 0.07 p.u.
The comparison of estimation rotor speed and position errors between the proposed non-adaptive speed estimation structure and SFM-MRAS [34] is given in Table I.

VII. CONCLUSION
In the article, the speed and position observer structure which was based on the non-adaptive approach was considered.The non-adaptive dependence (40) was based on the scalar and cross-product of the introduced internal vector H and the rotor flux vector ψr .The scalar product s ω of the two vectors has not a constant value.However, its value depends on the DFIG working points (as mentioned in Section III-B and confirmed by simulation and experimental results).The integration of the estimated rotor speed value determines the rotor position.However, integrating the rotor speed value, especially for an initial condition, introduces the estimation error of the position.Therefore, it was proposed to introduce the stabilizing function to the position differential equation.This stabilizing function was based on the error angle between the estimated value of the vector H and vector ψr (calculated from measured values) multiplied by ωr .It prevents the unstable of the whole observer structure and improves the properties, especially during the "difficult" DFIG working condition while the generator is crossing through the synchronous rotor speed (in which the frequency of the rotor current vector is close to zero).
The stabilizing functions in the proposed observer structure are determined using the Lyapunov approach.Based on the Lyapunov stability theorem of the linearized observer close to the equilibrium points and by the pole placement method, the

Fig. 1 .
Fig. 1.DFIG is crossing from subsynchronous to supersynchronous speed, after 100 ms the value of c f is determined from (44).Simulation results.

Fig. 2 .
Fig. 2. (a) Scheme of the observer structure.(b) Flowchart diagram for the observer structure.

Fig. 4 .
Fig. 4. (a) Active and reactive power changes for the constant rotor speed of about 0.8 p.u.(b) Crossing from subsynchronous to supersynchronous rotor speed from 0.72 to 1.25 p.u.

Fig. 5 .Fig. 6 .
Fig. 5. (a) Steady-state of DFIG and the estimated variables by the speed observer and their errors for subsynchronous mode -index "S" means reference frame connected to the stator.(b) After 300 ms the stator resistance and after 600 ms magnetizing inductance are changed twice.

Fig. 7 .
Fig. 7. Vector control structure for the rotor side converter.
(b), the value of s ω ref = 0.001 p.u.However, s ωf ∼0.0025 p.u. and the value of c f is limited to c fmax = 20 [from(45)].Respectively, the generator's active power was set to −0.3 p.u. and reactive to −0.4 p.u..The filtrated value of s ωf, as well as instantaneous s ω, are presented.
10. (a) Reactive power is equal to −0.6 p.u. and after 400 ms, the active power is changed to −0.35 p.u.(b) Active power is −0.1 p.u. and after 300 ms, the reactive power is changed to −0.4 p.u.

Fig. 11 .
Fig. 11.Characteristic speed observer waveforms for the crossing of generator rotor speed through synchronous speed for (a) value of c f is determined adaptively from (44), and (b) c f = 11 and it is constant.

Fig. 12 .
Fig. 12. DFIG is crossing from subsynchronous to supersynchronous speed; the value of c f is determined from (44).

Fig. 13 .
Fig. 13.Uncertainties of nominal parameters in the speed observer structure for (a) after 0 s R s,r = 2R s,rN and after 1 s, R s,r = R s,rN , (b) after 0 s L m = 0.75L mN and after 0.8 s L m = L mN .

Fig. 14 .
Fig. 14.MRAS [34](a) reactive power is equal to −0.6 p.u. and after 350 ms, the active power is changed to −0.35 p.u. and (b) the active power is −0.1 p.u. and after 300 ms the reactive power is changed to −0.4 p.u.

TABLE I SPEED
AND POSITION ESTIMATION ERRORS