A robust adaptive controller for a DFIG wind turbine with grid voltage and frequency support

Bộ điều khiển thích ứng mạnh mẽ cho tuabin gió DFIG có lưới Hỗ trợ điện áp và tần số. A robust adaptive nonlinear controller is designed for a DoublyFed Induction Generator (DFIG) wind turbine connected to a power grid. The controller main objective is to regulate the generator terminal voltage and rotor speed. A model based control design strategy is adopted. The controller structure and equations are obtained following a backstepping control design method using the DFIG reduced order model. Grid parameters are assumed unknown during the design. Therefore, the controller is provided with an adaptation module that automatically readjusts controller parameters when grid conditions change. Simulations are used to assess the proposed DFIG controller effectiveness.

2010 IEEE International Conference on Control Applications Part of 2010 IEEE Multi-Conference on Systems and Control Yokohama, Japan, September 8-10, 2010 A Robust Adaptive Controller for a DFIG Wind Turbine with Grid Voltage and Frequency Support Francis A Okou, Member, IEEE, Ouassima Akhrif, Member, IEEE, and Sebastien Gauthier Abstract—A robust adaptive nonlinear controller is designed for a Doubly-Fed Induction Generator (DFIG) wind turbine connected to a power grid The controller main objective is to regulate the generator terminal voltage and rotor speed A model based control design strategy is adopted The controller structure and equations are obtained following a backstepping control design method using the DFIG reduced order model Grid parameters are assumed unknown during the design Therefore, the controller is provided with an adaptation module that automatically readjusts controller parameters when grid conditions change Simulations are used to assess the proposed DFIG controller effectiveness T I INTRODUCTION HE increasing penetration of renewable energy generators in general and particularly wind power generators is forcing power utilities to reconsider the way these generators interact with the grid The trend is that wind turbines should remain connected to the grid when severe faults occur while supporting the grid voltage and frequency during normal operations Several researchers are still investigating the best control system for wind generators with these new operating constraints In [1], a combination of proportional–integral (PI) and Lyapunov based auxiliary controller is proposed to stabilize and improve the DFIG post fault behavior In [2], a feedback linearization based nonlinear voltage and slip controller is proposed for a DFIG connected to an infinite bus A direct active and reactive power controller based on stator flux estimation is discussed in [3] The paper proposes a basic hysteresis controller In [4], the well-known vector control is proposed to regulate the active power and reactive power generated by a DFIG In [5], a DFIG control strategy that is very similar to the conventional AVR/PSS for synchronous generator is utilized to support the power grid voltage and frequency The main drawback of the aforementioned works is their inability to cope with large changes in grid parameters This paper proposes a robust adaptive control strategy that makes possible for a wind generator to adequately support the grid voltage and frequency A doubly-fed induction Manuscript received February 15, 2010 This work was supported in part by the Academic Research Program (ARP) F A Okou and S Gauthier are with the Royal Military College of Canada, Po Box 17000 Station Forces, Kingston, Ontario, Canada (phone: 1-613-5416000 x6630; e-mail: aime.okou@ rmc.ca) O Akhrif is with Ecole de Technologie Superieure, (e-mail: ouassima.akhrif@etsmtl.ca) 978-1-4244-5363-4/10/$26.00 ©2010 IEEE machine is used to converter the mechanical power into an electric energy A fast response wind generator is proposed to be able to easily follow changes that occur in the power grid The controller adaptive feature helps to considerably compensate for very large variation into the grid parameters A model based design method is proposed to find the controller equations A backstepping control design method is used to derive the controller structure The proposed controller main advantages are that it considerably decreases the wind generator time response The active power and reactive power generated are automatically readjusted to maintain the grid frequency and voltage at their nominal values The rest of this paper is organized as follows: The doublyfed induction machine model is presented in the section II The proposed controller is designed in the section III Section IV presents the simulation results The paper ends with a conclusion II DFIG STATE SPACE MODEL A doubly-fed induction machine is represented in d/q reference frame by the following equations [6]: v ds = R s i ds − ωs Ψ qs + v qs = R si qs + ω s Ψds + dΨ ds dt dΨqs dt dΨdr v dr = R r i dr − ω sl Ψqr + dt dΨqr v qr = R r i qr + ω sl Ψdr + dt where the variables vds , vqs , (1.a) (1.b) (1.c) (1.d) vdr , vqr represent the d/q reference frame components for the machine stator and rotor voltages Variables i ds , i qs , i dr , i qr stand for the machine stator and rotor current in the d/q reference frame The machine fluxes in the d/q reference frame are presented by the variables Ψ ds , Ψ qs , Ψ dr , Ψ qr Finally, variables ωs and ωsl are stator and rotor current frequencies in rad/s, respectively Parameters R s and R r represent rotor and stator resistances The fluxes and currents are related by the following algebraic relationships 572 (2.a) Ψds = L si ds + L mi dr Ψqs = L si qs + L mi qr (2.b) (2.c) Ψdr = L r i dr + L mi ds Ψqr = L r i qr + L m i qs (2.c) where parameters Ls , L r , and L m represent the stator inductance, the rotor inductance and the mutual inductance The electric torque generated by the DFIG has the following expression in terms of the d/q reference frame stator flux and current Te = p(Ψdsi qs − Ψqsi ds ) (3) The parameter p is the number of pole pairs Note that the rotor current frequency can be expressed in term of the stator current frequency and the rotor speed in rad/s as follows: (4) ω sl = ω s − ω r When the d-axis for the park transformation used to find the DFIG dynamics aligned with rotor flux axis, we have the follow relationships: V (5) Ψqs = , Ψ ds = Ψ s = s ωs The parameter Vs stands for the grid voltage As the consequence, rotor flux components, in the d/q reference frame, have the following expression in terms of the rotor current components and the grid voltage Ψ dr = L m Vs + σL r idr , Ψqr = σL r i qr Ls ωs vqr = R r iqr + ωsl ( di dr dt diqr L m Vs + σL r idr ) + σL r Ls ωs dt Parameters J and B represent the generator inertia and the mechanical friction coefficient, respectively Tm is the mechanical torque supplies by the blades The paper objective is to propose a control system for this wind power generator to regulate its terminal voltage and the rotor speed The rotor speed is related to the grid frequency The following figure represents the wind turbine connected to a grid AC/DC – DC/AC converters are used to generate the rotor voltage Indeed, the DFIG controller generates the reference signals for the rotor-side converter That converter generates the rotor voltages from the capacitor DC voltage This DC voltage is generated by the grid side AC/DC converter from the grid voltage This paper is concerned with the design of the rotor side converter The grid-side converter controller which regulates the capacitor DC voltage is not treated for the sake of brevity The DC voltage across the capacitor is assumed constant, therefore (6) where σ = − L2m (L s L r ) It is common to neglect the stator resistance and to assume a constant grid voltage magnitude for the design of the wind power generator controller These assumptions lead to a reduced order model We get the following equations at the stator: dΨds ≈0 (7.a) v ds = R si ds − ω s Ψqs + dt dΨ qs (7.b) vqs = R s iqs + ωs Ψ ds + ≈ ωs Ψ ds = Vs dt Next equations which represent the rotor dynamics become the reduced model for the DFIG Note that the model depends on the grid voltage Vs and frequency ωs v dr = R r i dr − ω sl σL r i qr + σL r Vs = ωs Ls ids + ωs L m iqr (9.b) The rotor speed dynamics are represented by the following equation: J d (10) ω r = −Bω r + Tm + Te p dt (8.a) (8.b) The DFIG electrical torque and terminal voltage could be written in terms of the reduced order model state variables as follows: L (9.a) Te = −pVs m iqr Ls Fig A doubly-fed induction machine based wind power generator III ROBUST ADAPTIVE CONTROLLER DESIGN This section presents how the DFIG controller structure and equations are obtained A model based design method is proposed and it is based on the DFIG model presented in the previous section The controller is an adaptive voltage and speed regulator which guarantees that the active power and reactive power delivered by the generator are automatically adjusted to support the grid when a change occurs A systematic method to find the controller equations is now presented A Rotor Speed Regulator Equations (10) and (8.b) are considered for the design of this rotor speed regulator For the sake of clarity, these equations are repeated below d (11.a) ωɶ r = −θ1ω r + θ2 − θ3iqr dt d (11.b) iqr = −θ4 − θ5idr − θ6 iqr + θ7 Vqr dt 573 Parameters that appear in this model have the following expressions: ϕ0 = −αɺˆ v1 − (αˆ + θ1 = p p p2 Lm B , θ2 = Tm , θ3 = Vs , θ5 = ωsl J J J Ls ϕ = −(αˆ + θ4 = ωsl L m Vs R , θ6 = r , θ7 = σL r Ls ωs σL r σL r Note that the difference between the variable ω r to be is denoted by regulated and its reference ω ref r ref ~ ω r = ω r − ω r It is assumed during the design that the grid voltage Vs and frequency ωs are unknown or they could change As a consequence, parameters θ2, θ3 , θ4 and θ5 are assumed unknown However, θ1, θ6 and θ7 are assumed perfectly known and they are equal to their nominal values, θ1N, θ6N and θ7N respectively The design objective is to find Vqr expression in such a way that the system described by equations (11.a, 11.b) is stable and the variable ωɶ r is maintained at zero Since, Vqr doesn’t influence ωɶ r directly but does it via the variable iqr, the value of iqr that will maintain ωɶ r at zero is find first using equation (11.a) It has the following expression k (12.a) i*qr = αˆ v1 + v12 ωɶ r k v1 = k1ωɶ r + θˆ − θ1N ω r + (1 + ω 2r )ωɶ r (12.b) where αˆ is an estimation of α = θ3 The way that estimation is obtained will be discussed later on The terms k k 4v12 ωɶ r and (1 + ω 2r )ωɶ r appear in equation (12.a) and (12.b) respectively to compensate for the likely difference between α3 , θ1 and θ2 ; and their corresponding estimation or nominal values αˆ , θ1N and θˆ The term θˆ − θ1N ω r is included to cancel the corresponding term in equation (11.a) The parameter k1 is a positive gain selected by the designer to stabilize the system The parameter k is also a gain Its role will be explained later on Next, the error between the signal iqr and its reference i*qr is defined and it is equal to ~ iqr = i qr − i*qr (13) The control signal Vqr expression will be found in such a way that its drives this error signal to zero That expression is obtained from the error dynamics that have the following form: d ɶ (14) iqr = ϕ0 + ∑ ϕi θi + θ7 Vqr dt i =1 where the nonlinear functions that appear in the above equation have the following expressions: k ɺ v1ωɶ r )θˆ 2 k k k k v1ωɶ r )(−θ1N + k1 + (1 + ω 2r ) + ω r ωɶ r ) − v12 4 ϕ1 = −ϕω r ; ϕ2 = ϕ ; ϕ3 = −ϕiqr ; φ = −1 ; ϕ5 = −idr ; ϕ6 = −i qr The control input equation that will stabilize the system while maintaining the error signal at zero has the following form: k Vqr = α 7N v − v 22 ɶiqr (15.a) v = {−ϕ0 − ϕ1θ1N − ∑ ϕi θˆ i − ϕ6 θ6N } − k ɶiqr + θˆ 3ωɶ r i= (15.b) k 2 ɶ − ( ∑ ϕi + ωɶ r )iqr i =1 The parameter k2 is a positive gain that needs to be selected by the designer to stabilize the system The term θˆ ωɶ is r added for stability reason too The terms in bracket are added to cancel their corresponding terms in equation (14) The last terms of equation (15.b) are incorporated to compensate for the possible difference between the estimated parameters or the nominal values and their corresponding parameters The system closed loop dynamics have the following form: αɶ d k ωɶ r = −k1ωɶ r − θ3 ɶiqr − θɶ1ω r + θɶ − v1 ωɶ r + v1 dt α3 α3 k − (1 + ω 2r )ωɶ r (16.a) d ɶ k iqr = −k ɶiqr + θˆ 3ωɶ r + ∑ ϕi θɶ i − ( ∑ ϕi2 + ωɶ r2 )iɶqr dt i =1 i =1 (16.b) αɶ k 2ɶ − v iqr − v α7 α7 These equations are necessary to study the system stability The tilde parameters represent the difference between any estimated or nominal parameter and its actual value The speed regulator part of the controller is illustrated at Figure This regulator consists indeed of a ωr-controller which generates the reference for the iqr-controller Next section discusses the adaptation laws used to update the estimated parameters that appear in the controller equations The adaptation laws are derived from a stability study that involves the closed loop system describes by equations (16.a, 16.b) and the adaptation module The proof of stability guarantees that the two dynamics could coexist without any instability problem The following candidate Lyapunov function is used for this purpose 574 6 ɶ2 θɶ αɶ 32 (ω r + iqr ) + ∑ i + 2 i = βi β3' Voltage controller Ωɶ v ωref r − ωɶ r + Eq 22 Eq 12 + ɶi dr − i*dr vdr idr controller ɶiqr i*qr ωr controller Eq 23 − + iqr ωr Eq 15 vqr Rotor Side Converter + Gate signals idr Ω v controller − The system stability is now discussed The Lyapunov function derivative has the following form: ɺɶ ɺ = −(k ωɶ + k ɶi ) + θɶ (ωɶ + ϕ ɶi + θ2 ) − k (ω + ɶi ϕ2 ) V r qr r qr r qr β2 ɺ ɺ θɶ θɶ k +θɶ (−ωɶ r ɶiqr + ϕ3 ɶiqr + ) − (ω r2 + ϕ22 )iɶqr + θɶ (ϕ4 ɶiqr + ) β3 β4 ɺɶ θ αɶɺ k k 2 − ϕ24 ɶiqr + θɶ (ϕ5 ɶiqr + ) − ϕ52 ɶiqr + αɶ ( + v1ωɶ r ) β5 β3' V ref −t + Vt ∫ Ωv the estimated parameter θˆ i are represented by θ im and θ iM The positive real number ρi is selected by the designer (17) 2-Level PWM V= −1 k 2 ɶ k 2 (αɶ v ɶiqr + ɶiqr v ) − θ1 (ω r ωɶ r − ϕ1ɶiqr ) − (ω r2 ωɶ r2 + ɶiqr ϕ1 ) α7 4 k 2 k 2 θɶ ϕ6 ɶiqr − ɶiqr ϕ6 − v1 ωɶ r 4 iqr controller speed controller Fig The controller interne structure The function V is positive definite since coefficients βi and β are positive real numbers The derivative of the candidate ' Lyapunov function has the expression given at the end of this paragraph (equation 18.a) The adaptation laws are selected in such a way that the Lyapunov function derivative is negative semi-definite if αɶ , θɶ1 , and θɶ vanish The resulting adaptation laws have the following expression ɺ θˆ = β2 Pr oj  ϕ2 ɶiqr + ωɶ r , θˆ  ,   ɺˆ θ3 = β3 Pr oj  ϕ3 ɶiqr − ωɶ r ɶiqr , θˆ  ,   ɺˆ θi = βi Pr oj  ϕi ɶiqr , θˆ i  , i=4, 5,   ( ( ( ) ) ) ɶ r ) , αˆ  ; αˆ = β3' Pr oj ( v1ω The projection function involves in the adaptation laws expressions is defined as follows: Proj y, θˆ i = y , if {θim ≤ θˆ i ≤ θiM } or {θˆ i ≤ θim and y ≥ 0} or {θ ≤ θˆ and y ≤ 0} [ ] [ ] Proj y, θˆ i = y[1 − [ ] iM θ im θ im i − θˆ i2 − (θ im − ρ i ) ] , {θˆ i ≤ θ im and y < 0} 2 θ i2 − θˆ iM Proj y, θˆ i = y[1 − ] , {θ iM ≤ θˆ i and y > 0} (θ iM + ρ i ) − θ iM (18.a) Substituting the adaptive laws in equation (18.a) and considering the fact that the projection function has the following properties Proj(y, θˆ ) ≤ y , θɶ Proj(y, θˆ ) ≥ θɶ y (18.b) On can show that, the Lyapunov function derivative satisfies the following inequalities, ɺ ≤ −(k ωɶ + k ɶi ) + ( αɶ + θɶ + θɶ ) V r qr k α7 (18.c) γ2 ≤ −2 min(k1 , k )V + k considering the fact that the following inequality is true k y2 xy ≤ x + , ∀x, y ∈ R , k ∈ R + (18.d) k The parameter γ depends on the errors αɶ i and θɶ j defined before Integrating (18.c), yields; t γ2 V ≤ V(0)e −2 min(k1 ,k ) t + ∫ e −2 min(k1 ,k )(t − τ ) dτ k (18.e) As consequence, when the time t goes to infinity, the following inequality is true: γ2 ωɶ 2r ≤ (18.f) k min(k1 , k ) The variable ωɶ r can therefore be made arbitrary small independently to the presence of the disturbance γ increasing the gain k The system is said to be ultimately bounded It is therefore stable It will guarantee that estimated parameters remain and converge inside the predefined domain θ + ρ θ − ρ Predefined lower and upper bounds of [ im i iM i ] B Output Voltage Regulator The output voltage regulator design follows the same scheme presented previously for the rotor speed regulator The design uses equations (8.a), (9.b) and equation (19) given below 575 t Ω v = ∫ (Vs − Vsref )dτ Vsref (19) is the terminal voltage reference An integrator is added to the system to be controlled to guarantee a zero steady state error despite the presence of perturbations that haven’t been modeled The augmented system state space representation has the following form, therefore: d (20.a) Ω v = θ8idr + θ9 − Vsref dt d (20.b) idr = −θ10 idr + θ11iqr + θ12 Vdr dt where the parameters appearing in the model have the following expressions: R θ8 = ωs L m , θ9 = ωs Ls ids , θ10 = ωsl , θ11 = r ; σL r θ12 = σL r Since the grid parameters are assumed unknown, parameters θ9, and θ10 are unknown The design scheme gives the structure illustrated at Figure inside the box labeled Voltage controller That regulator consists of two subcontroller named Ωv-controller and the idr-controller The first sub-controller is used to synthesize the second subcontroller reference The errors signals involve in the diagram given at Figure are defined as follows: ɶ = Ω − Ωref = Ω (21.a) Ω v v v v ~ * (21.b) idr = i dr − i dr k k ɶ k ϕ = {(k + )(α8N − v3Ω V ) + v3 } , ϕ8 = ϕi dr ; ϕ9 = ϕ 4 k ɶ ɺˆ ref ϕ0 = (α8N − v3Ω V )θ9 − ϕVs , ϕ10 = −i dr ; ϕ11 = −iqr The same stability analysis is performed to obtain the reactive power regulator adaptation laws The following candidate Lyapunov function is used for this purpose 1 10 θɶ 2 V = (Ω2v + ɶidr )+ ∑ i (25) 2 i =9 βi Parameters βi are positive gains selected by the designer The adaptation laws have the following expressions: ɺ θˆ10 = β10 Pr oj  ϕ10 ɶidr , θˆ10  , i=10,   ɺˆ ɶ , θˆ  θ9 = β9 Pr oj  ϕ9 ɶidr + Ω V 9  Next figure illustrates the system in closed loop and the module involved in the implementation A tachometer is needed to obtain the rotor angular position θr from its ( DFIG (23.a) Tachometer AC/DC & DC/AC converters ωr ∫u V-ω controller θr θ2 − θ1 − + + θs abc / dq ∫u ωs 10 i =9 idr ,iqr ir,abc PLL Vs,abc π Fig The DFIG control system IV SIMULATION AND RESULTS (23.b) The nonlinear functions The parameter α12 = θ12 appearing in equation (23.b) have the following expressions Vt ,ωr The proposed controller is tested in the next section Preliminary results are also shown Simulation results for two common power system contingencies are presented v = −ϕ0 − ϕ8 θ8N − ∑ ϕi θˆ i − ϕ11θ11N − k ɶidr − θ8N Ω V k 11 ɶ − ( ∑ ϕi2 + ΩV )idr i =8 Grid side Ωr v k 2ɶ v idr ) position are used by the abc/dq module that gives the rotor daxis and q-axis currents The rotor currents along with the active power and reactive power are variables needed to implement the control and adaptation laws It has the following expression: k ɶ k ɶ ref ˆ ɶ i*dr = α8 v3 − v32 Ω V , v1 = − k 3Ω V + Vs − θ9 − Ω V 4 (22) Parameters k and k are positive gains selected by the Vqr = α12 v − ) speed A phase lock loop (PLL) module is required to obtained the stator flux angular position θs These angular The d-axis rotor current reference i*dr is obtained using ɶ to zero equation (20.a) It is such that it drives the error Ω designer to stabilize the system and α8 = θ8 The control input expression is selected in such a way that it steers the error signal ɶidr to zero It has the following expression: ( The power system illustrated at the following figure is used to assess the proposed controller performances The system consists of a 13 Kw wind power generator supplying a 340 V, 60Hz power grid The controller objective is to maintain the generator terminal voltage and rotor speed at 576 Ls = 50mH , L r = 50mH Controller gains are set to 10 except the gain k which is equal to The controller capability to stabilize the DFIG signals, when the demanded active power changes or when a short circuit occurs at the generator terminal, is tested At the time instant t equal 0.5 second, the generator local load decreases to about 30% Before that change, the wind power generator was generating 6150 W Fig.5 shows the generator active power waveform One can note that the generator decreases its active power to about 10% to be able to maintain the same rotor speed Fig The single DFIG – Infinite bus system A c tiv e power deliv ered (w) 6200 5600 5400 5200 5000 0.4 0.6 tim e s 0.8 rotor s peed (rad/s ) 96.4 96.35 1.4 1.38 1.36 1.34 1.32 1.3 0.95 tim e s 1.05 1.1 The rotor speed waveform is illustrated at Fig It takes about 0.3 second to the controller to stabilize the DFIG signals to their steady values The wind power generator is therefore capable to support the power grid frequency when the demanded power changes The second contingency occurs at the time instant equal second It is a 50ms grounded short-circuit at the generator terminals (the stator side) Fig shows the stator voltage waveform and Fig illustrates the generator reactive power profile One can see that these signals return to their pre-fault values quickly after the fault is cleared This fast response should increase the DFIG fault ride-through capability A nonlinear adaptive controller is proposed to increase a wind power generator grid-fault ride-through capability A control system that considerably improves the generator time response is designed The simulation results show that the generator is capable to readjust quickly the generated active power and reactive power to maintain its terminal voltage and the rotor speed constant when a change occur inside the power grid The controller adaptive feature help to reduce the system sensitivity to the power network condition variations Future works will evaluate the proposed controller in a large scale power system environment 96.3 0.5 0.6 0.7 tim e s 0.8 0.9 Fig The rotor speed during the first contingency [1] [2] 400 300 [3] 200 [4] 100 0 95 x 10 V CONCLUSION 5800 96.25 0.4 1.42 Fig The output reactive power during the second contingency 6000 Fig The output active power during the first contingency T erm inale v oltage (V ) Reac tiv e power deliv ered (w) their reference values (340 V, 96.34 rad/s) The machine parameters are: R s = 0.05Ω , R f = 0.38Ω , L m = 47.3mH , tim e s 1.05 1.1 Fig The terminal voltage during the second contingency [5] [6] 577 REFERENCES M Rahimi, M Pamiani, “Transient performance improvement of wind turbines with doubly fed induction generators using nonlinear control strategy,” IEEE Trans Energy Conversion, , To be published W Feng, Z Xiao-Ping, J Ping, and M J H Stering, “Decentralized nonlinear control of wind turbine with doubly fed induction generator,” IEEE Trans Power Systems, vol 23, pp 613–621, May 2008 X Lie, P Cartwright, “Direct active and reactive power control of DFIG for wind energy generation,” IEEE Trans Energy Conversion, vol 21, pp 750–758, Sept 2006 S Muller, M Deicke, and R W De Doncker, “Doubly fed induction generator systems for wind turbines,” IEEE Indus Applications Magazine, vol 8, pp 26–33, June 2002 F M Hughes, O Anaya-Lara, N Jenkins, and G Strbac, “Control of DFIG-based wind generation for power network support,” IEEE Trans Power Systems, vol 4, pp 1958–1966, Nov 2005 P C Krause, O Wasynczuk and S D Sudhoff, Analysis of Electric Machinery and Drive Systems, second edition, USA, WileyInterscience, 2002, ch  ... is proposed and it is based on the DFIG model presented in the previous section The controller is an adaptive voltage and speed regulator which guarantees that the active power and reactive power... generator systems for wind turbines,” IEEE Indus Applications Magazine, vol 8, pp 26–33, June 2002 F M Hughes, O Anaya-Lara, N Jenkins, and G Strbac, “Control of DFIG- based wind generation for. .. gives the rotor daxis and q-axis currents The rotor currents along with the active power and reactive power are variables needed to implement the control and adaptation laws It has the following