A Direct Power Controller for Doubly-Fed Induction Generator

Alfeu J. Sguarezi Filho and Ernesto RuppertDSCE-FEEC-UNICAMP

CEP 13083 - 852 , Caixa Postal 6101 , Campinas - SP, Brasilsguarezi@dsce.fee.unicamp.br, ruppert@fee.unicamp.br

Abstract - This paper proposes a direct power control schemefor doubly-fed induction generator for variable speed wind powergeneration. Using a stator flux decoupling and deadbeat control loopsthe algorithm calculates the rotor voltage vector to be supplied to therotor in order to eliminate the active and reactive power errors thatcan achieve high accuracy and fast dynamic power response basedon the estimated stator flux, the active and reactive power and theirerrors. Simulations results are carried out for validation of the digitalcontroller operation.

Keywords - Doubly-fed induction generator, direct powercontrol, wind energy, variable speed constant frequency appli-cations.

I. I NTRODUCTION

The wind energy systems using a doubly-fed inductiongenerator (DFIG) have some advantages due to variable speedoperation and four quadrant active and reactive power capa-bilities compared with fixed speed induction generators. Thestator of DFIG is connected direct to the grid and the rotorlinks the grid by a bi-directional converter. The rotor converteraims to the DFIG active and reactive power control betweenthe stator and ac supply [1], [2].

Control of DFIG wind turbine systems is traditionally basedon either stator-flux-oriented [3] or stator-voltage-oriented [4]vector control. The scheme decouples the rotor current intoactive and reactive power components. Control of the activeand reactive power are achieved with a rotor current controller.Some investigations using PI controllers by using stator-flux-oriented that generates reference currents from active andreactive power errors to the inverter or a cascade PI controllersthat generate a rotor voltage which has been presented by [5],[6]. The problem in the use of PI controller is the tuning ofthe gains and the cross-coupling on DFIG terms in the wholeoperating range. An interesting method to solve these problemshave been presented by [7], [8], [9].

Some investigations by using predictive functional con-troller [10] and internal mode controller [11], [12] havesatisfactory power response when compared with the powerresponse of PI but it is hardly to implement one due to thepredictive functional controller and internal mode controllerformulation. Another possibility to doubly-fed power control

can be made by using fuzzy logic [13], [14]. These strategieshave satisfactory power response although it involves rela-tively complex transformation of voltages, currents and controloutputs among the stationary, the rotor and the synchronousreference frames.

The direct power control was applied to the DFIG powercontrol and it has been presented in [15], [16], [17].Thisscheme calculates the required rotor controlling voltage di-rectly based on the estimated stator flux, active and reactivepower and their errors. In [15] the principles and the imple-mentation of DPC is made with hysteresis controllers andvariable switching frequency. In [16], [17] the principles ofthis method are described in detail and simulations results havebeen presented with variable and constant switching frequencyrespectively. Moreover, the conventional DPC complicates theAC filter design because of its variable switching frequency.An alternative to direct power control is the power error vectorcontrol [18]. This strategy is less complex and obtains similarresults to direct power control.

To improve the power response, to eliminate the torqueripple and to protection of rotor-side converter under grid volt-age sags a proportional control with anti-jamming control wasproposed in [19]. This control has satisfactory power responseand eliminate the rotor current overshoot in voltage sags whenthe loop of torque control is applied, although power and rotorcurrents results were shown only in fixed speed operation.The proportional controller needs to be carefully tuned toensure system stability and adequate response within the wholeoperating range.

The concept of DPC is applied to DFIG under unbalancedgrid voltage conditions by [20], [21]. In [20] the active andreactive powers are made to track references using hysteresiscontrollers. In [21] a notch filter is applied in DPC strategywhich allows the power control. These strategies haver satis-factory active and reactive power response under unbalancedgrid voltage.

This paper proposes a direct power control scheme that canachieve high accuracy and fast dynamic power response byusing a stator flux decoupling and deadbeat control loops. Thescheme is derived using a discrete state-space o DFIG model.

https://doi.org/10.24084/repqj08.219 58 RE&PQJ, Vol.1, No.8, April 2010

The power control aims the active and reactive power controlcalculates the rotor voltages required to guarantee active andreactive power reach their desired reference values based onthe estimated stator flux, the active and reactive power, andtheir errors. Simulation results are carried out for validationthe proposed controller.

II. DFIG M ODEL AND DPC

Doubly-fed induction generator model in synchronous ref-erence frame is given by [22]~v1dq = R1~i1dq + d~�1dqdt + j!1~�1dq (1)~v2dq = R2~i2dq + d~�2dqdt + j (!1 � PP!me )~�2dq (2)

the relationship between fluxes and currents~�1dq = L1~i1dq + LM~i2dq (3)~�2dq = LM~i1dq + L2~i2dq (4)

and generator active and reactive power areP = 32 (v1di1d + v1qi1q) (5)Q = 32 (v1qi1d � v1di1q) (6)

The subscripts1 and 2 represent the stator and rotorparameters respectively,!1 is the synchronous speed,!me is the machine speed,R1 and R2 are the estator and rotorwindings per phase electrical resistance,L1 , L2 and Lmare the proper and mutual inductances of the stator and rotorwindings,~v is the voltage vector ,PP is the machine numberof pair of poles.

DPC aims independent stator activeP and reactiveQ powercontrol by means a rotor flux regulation. For this purpose,PandQ are represented as functions of each individual rotorflux. Using stator flux oriented control, that decouplesdqaxis, (3) and (4) the relationship between stator currents androtor fluxes is given byi2d = L1L1L2 � L2M �2d � LML1 L1L1L2 � L2M �1 (7)i2q = LML1L2 � L2M �2q (8)

and the active (5) and reactive (6) power can be computed byusing Equations (7) and (8)P = � 3v1LM2�L1L2�2q (9)Q = 3v1LM2�L1L2 ���2d + L2LM �1� (10)

Equation (2) indicates that the rotor flux change is directlycontrolled by the applied rotor voltage when rotor resistance

is neglected. The components of rotor flux are�2d = �2 os Æand�2q = �2 sin Æ that can be seen in Figure 1. Thus, rotorfluxes will reflect on stator active and reactive power as canbe seen in (9) and (10). Consequently this principle can beused on stator active and reactive power control on rotor sidein the DFIG with stator connected to the grid.

Figure 1. Stator and rotor flux vectors in the synchronous dq frames.

I II. D IRECT POWER CONTROL STRATEGY

Since the applied rotor voltage is also constant during apower control period for voltage-fed PWM and neglecting therotor resistance, then (2) can be discretized atkth samplinginstant in synchronous referential frame using the stator fluxposition and active (9) and reactive (10) power. Thus, theequations that contains only the rotor voltages and power aregiven byQ(k + 1) = v2d(k)TA +Q(k) + !slP (k)T (11)

andP (k+1) = v2q(k)TA +P (k)�!slQ(k)T�!sl L2LM A�1 (12)

where !sl = !1 � PP!me , � = 1 � L2ML1L2 and A =� 2�L1L23v1LM .In space state form (11) and (12) become�qp (k + 1) = Ad �qp(k) +Bd�v2(k) +Gd��(k) (13)�Q(k + 1)P (k + 1)� = � 1 !slT�!slT 1 � �Q(k)P (k)�++ �TA 00 TA� �v2d(k)v2q(k)�++" 0 !slL2TLM A�!slL2TLM A 0 # ��1(k)0 �

(14)

From now it will be assumed that the mechanical timeconstant is much greater then the electrical time constants.Thus !me = onstant is a valid approximation for eachsample period [23], [24], [25].

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A. Decoupling Control

The digital control technique allows to calculate requiredinput to guarantee that the output can achieve high accuracyand fast dynamic response using a discrete equation of thecontinuous linear system [26], [27], [28].

The rotor voltages�v2 equations at the(k � 1)th samplinginstant is given by�qp (k) = Ad �qp(k � 1) +Bd�v2(k � 1) +Gd��(k � 1) (15)

BecauseGd��(k) andGd��(k � 1) are approximately equal,hence, combining (13) and (15) yields an expression whichcontains only the inputs�v2 and outputs�qp.�v2 (k) = �v2 (k � 1) +B�1d f[ �qp(k + 1)� �qp(k)℄�Ad[ �qp(k)� �qp(k � 1)℄g (16)

Equation (16) indicates that the input can be calculated byusing the outputs at thek�1, k andk+1th sampling instant.This relationship can be used to calculate the input need toprocedure the reference at each control instant because boththe feedback and the reference atk�1 and k sampling periodinstant are available when (16) is evaluated, either of them canbe used in creation of the input.

The control law that uses the reference�qpref at thek + 1andkth sampling instants to calculate the input�v2ff can beformulated as �v2ff (k) = �v2 (k � 1) +B�1d [( �qpref (k + 1)� �qp(k))�Ad( �qpref (k)� �qp(k � 1))℄ (17)

which means v2dff (k) = v2d (k � 1) +AT [(Qref (k + 1)�Q(k))� (Qref (k)�Q(k � 1))℄�A !sl(Pref (k)� P (k � 1)) (18)

and v2qff (k) = v2q (k � 1) +AT [(Pref (k + 1)� P (k))� (Pref (k)� P (k � 1))℄+A !sl(Qref (k)�Q(k � 1)) (19)

wherePref (k+1), Qref (k+1), Pref (k) andQref (k) are thereferences atk + 1 andk sampling instant.

The total input applied is given by�v2ref (k) = �v2ff (k) + �v2fb (k) (20)

where�v2fb is achieved making (16)� (17), considering thatAd andBd is estimated correctly and treating the modelingerrors as a disturbance to the output control, then it can besimplified by replacing the output with output errors that isgiven by� �qp (k + 1) = Ad� �qp (k) +Bd�v2fb (k) (21)

where � �qp (k) = �qpref (k) � �qp (k). Note that whenrealizing the control,�qpref (k+1) is not available at the time

when �v2ff is calculated. Therefore,�qpref (k) must be usedinstead �qpref (k + 1). In essence, the control reference mustbe delayed one sampling period in the control. The effectof delaying the reference one sampling period will shift theoutput response back one sampling period as well, but theshape of the output response remains the same.

B. Deadbeat Feedback Control

To achieve a fast output response and a null steady stateerror a separate feedback control is made by using (21) andmaking�v2fb (k) = �G � �qp (k). One is given by� �qp (k + 1) = (Ad �BdG )� �qp (k) (22)

From (22) the gains required to null steady state error arefound by (Ad �BdG ) = 0. Thus,G is given byG = A� 1T � j!sl� (23)

The block diagram representing (17) and (22) is shown inFigure 2.

Figure 2. Decoupling and deadbeat control block diagram.

The DPC control scheme uses a digital controller to obtainrotor voltages which should be applied on generator in orderto guarantee active and reactive power reach their desiredreference values, the method directly calculates required rotorcontrol voltage with each switching period, based on theestimated stator flux, the active and reactive power, and theirerrors. The power control block diagram is shown in Figure 3.The converter that is connected to the grid control the voltageof the link DC and one can be controlled by using any controlpresented in [29].

Figure 3. DPC block diagram.

Thus, if the d and q axis voltage components are calculatedaccording (18), (19), (20) and (23) above are applied to thegenerator, then the active and reactive power convergence to

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their respective commanded values will occur. The desired ro-tor voltage in the rotor�� reference frame generates switchingsignals for the rotor side using either space vector modulationthat is given byv2�� = v2dq eÆs�Ær .

Stator currents and voltages, rotor speed and currents aremeasured to stator flux positionÆs and magnitude�1, syn-chronous frequency!1 and slip frequency!sl estimation.

C. Estimation

To digital power control is required to calculate the activeand reactive power values, their errors, the stator flux magni-tude and position, the slip speed and synchronous frequency.

The flux estimation using (1) is given by~�1�� = Z �~v1�� �R1~i1��� dt (24)

and the flux position by using equation (24) asÆs = ar tan��1��1�� (25)

The synchronous speed!1 estimation is given by!1 = dÆsdt = (v1� �R1i1�)�1� � (v1� �R1i1�)�1�(�1�)2 + (�1�)2 (26)

and the slip speed estimation by using the rotor speed andsynchronous speed is!sl = !1 � PP!me (27)

The angle in rotor stationary reference frame is given byÆs � Ær = Z !sldt (28)

IV. SIMULATION RESULTS

In the simulation of proposed digital control strategywas used MATLAB/SimPowerSystemsR package. The digitalpower control strategy has aT = 10�4s and the DFIG param-eters are shown in Appendix. Figure 3 shows the schematicof the implemented system. To the power factor (PF) controlthe reactive power reference is given byQref = Pref p1� PF 2PF

Initial studies with various active and reactive power stepsand constant rotor speed at 226.6 rad/s were carried out to testthe dynamic response of the proposed power control strategyare shown in Figure 4. The initial active power and powerfactor references were being -50 kW and FP=+0.85. The activepower and power factor references were step changed from -60 to -100 kW and from PF of 0.85 to -0.85 at 0.25 s and thepower reference were step changed again from -100 to -149.2kW and from PF of -0.85 to 1 at 0.5 s, respectively. The rotorand stator currents and the rotor speed are shown in Figure 5.The dynamic response of both active and reactive power arein few milliseconds, there is no overshoot of either stator/rotoror the active/reactive power and the satisfactory performanceand robustness of the controller can be seen.

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Figure 4. Response of step tests for active and reactive power and rotorcurrents in supersynchronous operation.

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Figure 5. Stator and rotor currents and rotor speed in supersynchronousoperation.

Studies with various power steps and rotor speed werecarried out to further test the proposed power control schemes.During the period 0.25-0.6 s, the rotor speed increased from151.1 rad/s to 226.6 rad/s. The Figure 6 shows the results stepreference tests of active and reactive power. The power steps,i.e., active power and power factor references were changedfrom -60 to -100 kW and from PF of 0.85 to -0.85 at 0.25 s.The rotor current and speed and stator current are shown inFigure 7.

To test the impact of the parameters variations on the systemperformance the rotor resistanceR2 and mutual inductanceLM are increased of 20%. The same tests of step reference

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of active and reactive power with rotor speed variation andwith parameters variation are shown in Figures 8 and 9.Comparing Figures 6 and 8 and Figures 7 and 9, there ishardly any difference, and even with such large inductanceand rotor resistance errors, the system maintains satisfactoryperformance under both steady-state and transient conditions.

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Figure 6. Response of step tests for active and reactive power in severalspeed operation.

V. CONCLUSION

This paper has presented a DPC by using stator fluxdecoupling and deadbeat controllers for doubly fed inductiongenerator that can achieve high accuracy and fast dynamicpower response. The controller uses the DFIG equations tocalculate the required rotor voltages in order to the activeand the reactive power values reach the desired reference.The power control scheme helps the protection of rotor-sideconverter because there is no overshoot in the rotor current.Constant converter switching frequency is achieved that easesthe design of the power converter and the ac harmonic filter.The impact of machine parameters variations is analyzed andfound to be negligible due to the fact that in the deadbeatcontroller formulation the model errors was treated as adisturbance to the output control. The simulations confirm theeffectiveness and the robustness of the DPC during severaloperating conditions and variations of machine parameters.

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Figure 9. Stator and rotor currents with in several speed operation and withparameters variations.

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APPENDIX

Doubly-fed induction generator parametersR1 = 0:02475; R2 = 0:0133 ; Lm = 0:01425 H ; Ll1 = 0:000284 H ;Ll2 = 0:000284 H ; J = 2:6 Kg �m2; PP = 2; PN = 149:2kV A; VN = 575 V .

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