A comparative study of current control schemes for a direct-driven PMSG wind energy generation system

Electric Power Systems Research 143 (2017) 197–205

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Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr

A comparative study of current control schemes for a direct-driven PMSG wind energy generation system E.G. Shehata Electrical Engineering Department, Faculty of Engineering, Minia University, El-Minia, Egypt

a r t i c l e

i n f o

Article history: Received 2 August 2016 Received in revised form 9 October 2016 Accepted 14 October 2016 Keywords: Wind energy PMSG Sliding mode controller Model predictive controller PI-controller

a b s t r a c t Because of many advantages of a direct-driven permanent magnet synchronous generator (PMSG), it becomes one of the most attractive variable speed wind energy generation type. Different control strategies have been studied for controlling the machine side converter of the PMSG. This paper presents a comparative study of three current control schemes for machine side converter (MSC) based on rotor flux oriented control. Firstly, an integral sliding mode controller (ISMC) is designed to regulate the stator current in the synchronous reference frame. The proposed ISMC has two integral switching functions for stator direct and quadrature currents control. Secondly, a finite control set model predictive control (FCS-MPC) is designed to regulate the stator current and replace ISMC. The proposed MPC takes the discrete states of the voltage source inverter into account and the future converter behavior is predicted for each sampling period. Then the switching action that minimizes a predefined cost function is selected to be applied in the next sampling instant. Neither axes transformation nor modulation technique of the stator voltage is required. Finally, to evaluate the performance of the proposed controllers, their performance is compared to the conventional PI-controller under the same operating conditions. The feasibility and effectiveness of the proposed schemes have been demonstrated through computer simulations. The results show the advantages and drawbacks of each controller. © 2016 Elsevier B.V. All rights reserved.

1. Introduction During the last years, the generation of electric power from renewable energy sources has increased potentially to reduce the consumed amount of fossil fuel and pollution level. The wind energy generation system are sharing with a significant amount of renewable energy and this amount will increase in the future [1,2]. Variable speed wind energy generation systems are the preferred technology in the market today [1]. Because of its advantages, doubly fed induction generators (DFIGs) driven by variable speed wind turbines are the dominant type worldwide [3,4]. Controlled active and reactive power, partial-scale power converter, and a certain ride through capability are considered the main merits of this generation system [4]. However, with the development of larger wind power capacity, increased power density, the need for higher reliability and poor low-voltage ride-through capability of the partial scale converters, different types of generation systems have been developed recently. Due to its advantages, a direct-driven PMSG has received high attention in the recent years [5]. PMSGs have

E-mail address: [email protected] http://dx.doi.org/10.1016/j.epsr.2016.10.039 0378-7796/© 2016 Elsevier B.V. All rights reserved.

high power density, low maintenance cost as a result of absence of gearbox and slip rings, and full scale power converters with high grid capability [5,6]. The direct-driven PMSG wind energy system consists of a PMSG coupled directly to the wind turbine. The stator of the PMSG is connected to the utility grid through a full scale back-to-back converters. Vector control/field oriented control and direct power control schemes are the dominated controller for the machine and grid side converters. For MSC, the PMSG stator current is controlled in the synchronous reference frame based vector control/field oriented control using traditional PI-controllers [6–14]. However, the classical PI-controller has poor performance especially during transient operation and do not consider the discrete operation of the converters [9]. As a result of fast and powerful microprocessor, the implementation of advanced control techniques is possible. Fuzzy logic control, neural networks, sliding mode control, model predictive control can now be applied for power converters. Sliding mode controller (SMC) is considered one of the most attractive nonlinear controllers. It is an effective and high frequency switching control strategy for nonlinear systems with uncertainties [15–17]. Fast dynamic response and robustness are the main advantages of the sliding mode controller. In Ref. [15], PMSG torque and the pitch angle of the wind energy system are controlled using slid-

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E.G. Shehata / Electric Power Systems Research 143 (2017) 197–205

ing mode controller without the assumption that the wind speed is measurable. In Ref. [16], a double integral sliding-mode current controller is designed for tracking the optimum dc-side current of the boost converter of the machine sides. High-order sliding mode control is designed in Ref. [17], to avoid the chattering phenomenon of the first order sliding mode controller. However, the chattering phenomena and reaching phase stability problem are still the main problems of the first order SMC. One of the most advanced controller strategies is a model predictive control. The main characteristic of this control type is the use of a model of the system for predicting the future behavior of the controlled variables. This information is used by the controller to obtain the optimal actuation, according to a predefined optimization criterion. A more flexible criterion is used in model predictive control (MPC), expressed as a cost function to be minimized [18,19]. MPCs can be classified into two types, continuous and finite control set. MPC with continuous control set needs a modulator in order to generate the required voltage. This will result in having a fixed switching frequency. In contrast, the finite control set model predictive controller (FCS-MPC) directly generates the switching signals of the converter, do not need a modulator and presents a variable switching frequency. In Ref. [19], the MSC is controlled based on a fixed-frequency model predictive direct torque control while the grid side converter is controlled based on a finite-set model predictive direct power control. In Refs. [20] and [21], a direct model predictive control scheme combining with an instantaneous power calculation for three level back-to-back converter PMSG wind turbine systems is presented. In Refs. [22–24], a two sample ahead prediction horizon FCS-MPC for a four-leg NPC inverter is deigned to achieve high-performance operation with low switching frequency. The results show that the two sample ahead method is favorable where lower switching frequency operation is mandatory. In Ref. [25], a deadbeat algorithm is proposed to control the stator currents of an outer-rotor five-phase BLDC generator driven by wind turbine. The estimation stage uses an extended Kalman filter and generator equations are used to calculate the appropriate voltages for the next modulation period. In this paper, to improve the performance of the direct-driven PMSG wind energy generation system, two current control schemes are designed for MSC control. Firstly, an ISMC is designed to generate the reference stator voltage in the synchronous reference frame based on the stator current error. Two integral switching are selected to describe the switching surface of the two stator current components. A pulse width modulation (PWM) technique is used to generate the switching state of the MSC. For simplicity, the machine parameters are not required. The stability of the proposed ISMC is proven using Lyapunov stability theorem. Secondly, a FCS-MPC is designed to generate directly the switching state of the MSC based on the stator current error and in turn pulse width modulation technique is not required. The proposed MPC takes the discrete states of the converter and discrete model of the PMSG into account. The future converter switching state is predicted for each sampling period. To show the effectiveness of the two schemes, their performance is compared to the classical PI-controller of the MSC performance under the same operating conditions. The performance of the three controllers are tested under fixed and variable wind speed and symmetrical grid fault condition.

2. Wind turbine model The mechanical output power of a variable speed wind turbine is given by [1]: A 3 Pt = cP (, ˇ) v 2

(1)

where cP is a power coefficient,  is an air density (kg/m3 ), A is a turbine swept area (m2 ), v is a wind speed (m/s), ˇ is a blade pitch angle (degree), and  is a tip speed ratio and can be expressed as =

ωt rt

(2)

v

where ␻t is the wind turbine speed (rad/s) and rt is the blade radius in meters. The turbine power coefficient cp (, ˇ) is modeled based on the turbine characteristics is: cP (, ˇ) = c1

c

2

i



− c3 ˇ − c4 e

−c5 i

+ c6 

(3)

The mechanical power as a function of generator speed, for wind speeds and blade pitch angle ˇ = 0◦ , is illustrated in Fig. 1. 3. PMSG dynamic model The dynamic model of the PMSG in the synchronous reference frame can be expressed as [8–10]: Vd = Rs id + ˙ d − ωe

q

(4)

Vq = Rs iq + ˙ q + ωe

d

(5)

where Vd and Vq are the d–q stator voltage components, respectively. id and iq are d–q stator current components, respectively. ␺d and ␺q are the d–q stator linkage flux components, respectively. Rs is the stator resistance. ωe is the electric angular rotor speed (rad/s). The stator flux linkage components can be written as: d

= Ld id +

q

= Lq iq

(6)

pm

(7)

where Ld and Lq are the d-axis and q-axis stator inductance, respectively. pm is the permanent magnet flux linkage. The generator electromagnetic torque can be written in terms of the magnetizing current and flux linkage components as follows:



Te = 1.5P d iq −

q id



= 1.5P



pm iq





+ Ld − Lq id iq



(8)

For multi-poles PMSGs with surface mounted permanent

magnets, the direct and quadrature axis inductances are equal

Ls = Lq = Ld . The electromagnetic torque of the generator is given by

Te = 1.5P

pm iq

(9)

where P is the number of pole pairs. It is shown that the generator torque can be controlled by controlling the stator q-axis current component only. For maximum output torque and high efficiency, the d-axis current is set to zero (is = iq ). 4. Conventional vector control of MSC The MSC of the PMSG is controlled to regulate the stator current in the synchronous reference frame [8–11]. In addition, the generator torque and stator reactive power are controlled based on stator current control. So, the controller of the MSC normally has a fast inner current loop, controlling the stator d–q-axes current, combined with an outer slower loop for torque and/or stator reactive power control. For high power machines, the stator winding resistance is much smaller than the synchronous reactance. Based on Eqs. (4) and (5), the reference value of the stator voltage in the synchronous reference frame can be estimated as:  comp Vd∗ = Ld i˙ d − ωe Lq iq = Vd − Vd

Vq∗ = Lq i˙ q + ωe

pm

=



comp Vq + Vq

(10) (11)

E.G. Shehata / Electric Power Systems Research 143 (2017) 197–205

199

Turbine output power (pu of nominal mechanical power)

Turbine Power Characteristics (Pitch angle beta = 0 deg) 1.2

13 m/s

1 12 m/s 0.8 Max. power at base wind speed (11 m/s) and beta = 0 deg 11 m/s 0.6 10 m/s 0.4

9 m/s 8 m/s

0.2

7 m/s 6 m/s

1.2 pu

0

-0.2 0

0.2

0.4

0.6 0.8 1 Turbine speed (pu of nominal generator speed)

1.2

1.4

Fig. 1. Turbine and tracking characteristic with pitch angle = 0◦ .

axis currents. In this section, the sliding mode controller is designed for interior PMSG and can be applied to surface mounted type. The integral switching functions can be expressed as [16,17]:



Sid =

p Kid eid

i + Kid

eid dt

(12)

eiq dt

(13)

 p

i Siq = Kiq eiq + Kiq

p

Fig. 2. Vector control of MSC.

The first term in Eqs. (10) and (11) shows that stator d- and q-axis currents can be regulated by controlling the stator d- and q-axis voltage, respectively, while the second term of the two equations represents the compensation terms. The block diagram of the vector control of the MSC is shown in Fig. 2. A maximum power point tracking is implemented to generate the reference electromagnetic power/torque based on generator speed. Below rated speed, the pitch angle (ˇ) is kept to the optimal value which is zero and the turbine speed is changed in such a way that the tip-speed ratio () is optimal. The reference generated power/torque is estimated based on the measured generator speed (ωr ) where a P − ωr3 characteristic is implemented in the MPPT block [26]. The reference value of the stator q-axis current is estimated from the reference electromagnetic torque while the reference value of the stator d-axis current is set to zero. The estimated reference voltage in d–q frame is transformed to the abc frame using rotor position angle then sinusoidal pulse width modulation (SPWM) generates the switching signals of the MSC. In spite of its simplicity, conventional PI-controller has some problems such as difficult to tune the parameters and inefficient to handle system nonlinearity. In addition, the compensation term needs the generator parameters and rotor speed measurement. 5. Integral sliding mode control of MSC In this section, an integral sliding mode controller is designed to replace PI-controllers. Two integral switching functions are selected to describe the switching surface of the stator d- and q-

p

i , K , and K i are positive where eid = id∗ − id , eiq = iq∗ − iq , Kid , Kid iq iq gains. Using Eqs. (4) and (5), the time derivative of the switching functions can be presented as:



p i eid + Kid S˙ id = Kid

 p i S˙ iq = Kiq eiq + Kiq

Vd − Rs id − ωe i˙ d∗ − Ld

Vq − Rs iq + ωe i˙ q∗ − Lq



q

(14)

 d

(15)

The switching function can be expressed in the state space form as:

˙ Sid S˙ iq

=

   Si

i Kid eid i Kiq eiq

+

p Kid i˙ d∗

+

p Kiq i˙ q∗



⎡



⎢

p

+⎣



C

p





Kid

Kiq



Rs id + ωe Ld Rs iq − ωe Lq



q

d

⎤ ⎡ ⎥ ⎢ ⎦−⎣  

m

p

Kid

0

Ld

p

0

Kiq



Lq

⎤

⎥ Vd (16) ⎦ Vq    U

D

S˙ i = C + m − DU

(17)

However, the machine parameters change under operating conditions such as heating and saturation. In addition, the stator currents and rotor speed changes within bounded values so it can be added to the system uncertainties. The previous equation can be rewritten as S˙ i = C + (m + m) − (D + D)U = C − DU + W

(18)

where m and D are the uncertainties matrices. The lumped uncertainties matrix (W ) can be expressed as −W = m + m − DU

(19)

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E.G. Shehata / Electric Power Systems Research 143 (2017) 197–205

The output voltages in the stationary reference frame (˛–ˇ) can be expressed in terms of the switching states as



V˛

⎡



Vˇ

=

2 3

⎢ ⎣

1

1 2 √ 3 2

0

⎤

1 2 √ 3 − 2

−

−

⎥ ⎦ SVdc

(24)

Also, the d–q representation of output voltages can be expressed in terms of the switching states as Vdq = C1 C2 SVdc



Fig. 3. Block diagram of the ISMC for MSC.

When the trajectories of stator currents coincide with the sliding surfaces S˙ i = Si = 0

(20)

Using Eqs. (18) and (20), the control effort (stator voltage reference values) of the ISMC can be expressed as U ∗ = D−1 (C + i sat(Si ))



where D−1 =





Ld 0



0 , i = Lq



(21)



id , i ≥ |W |, sat(Si ) = iq

Si , |Si |+1

and

Vd∗ . Vq∗ Sat function is selected instead of sign function to reduce chattering effect. However, the designed ISMC has instability problem during reach phase. To overcome this problem, two terms are added to the control effort equation to become U∗ =

U∗ = D

 −1

C + i sat(Si ) + K c Si + K t i



where i = id

iq

T



c , K c = Kid



c Kiq



t , K t = Kid

t Kiq

T

c , Kc , , Kid iq

t , K t are positive gains. The stability of the proposed ISMC is Kid iq proved using Lyapunov stability theory as shown in Appendix A. Fig. 3 shows the block diagram of the proposed ISMC for MSC control. The stator reference voltages are estimated using the stator current values without speed signal. As in the case of PI-controller, the estimated reference voltage in d–q frame is transformed to the abc frame using rotor position angle then SPWM generates the switching signals of the MSC.

Va

⎤

⎡

2

⎢ ⎥ 1⎣ ⎣ Vb ⎦ = 3 −1 Vc where S = [ Sa

−1 2

−1

Sb

⎤

−1 ⎦ SVdc

−1 −1

2 T

Sc ] and Si = {

⎡

sin  cos 

(23)

2 3

and C2 =

⎣

1

⎤

  ⎢ ⎥ id (k) =⎣ ⎦ −ωe Lq Ts Rs Ts iq (k + 1) iq (k) 1−

id (k + 1)



1−

ωe Lq Ts Ld

Rs Ts Ld

Lq

 +

0 −ωe pm Ts Lq



⎡T

Lq

⎤

  ⎥ Vd (k) ⎦ Ts Vq (k)

s

0

⎢ Ld +⎣ 0

(26)

Lq

where k denotes the number of sample. In this section, the FCS-MPC is designed to determine directly the switching state S of the MSC, thus S can be taken as the control inputs. The MSC model (25) is combined with the PMSG model (26), and the combined model can be written as x (k + 1) = A (k) x (k) + m (k) + B (k) u (k)

where x (k+1) =

In this section, MPC is designed to determine directly the switching state of MSC. The concept of the FCS-MPC depends mainly on the topology of the three phase inverter. Since the machine side converter is commonly a three phase two level voltage source inverter, there are six active and two no active combinations of inverter states. The switching states and output voltage in the stationary frame (˛–ˇ) of the three-phase two level inverter are illustrated in Table 1. Compared to PI-controller and ISMC, the stator reference voltage transformation from d–q frame to abc frame and SPWM are eliminated. Using the switching state functions, the phase voltages can be written as

⎡



cos  −sin 



6. FCS-model predictive control of MSC



⎤

1 1 − − √2 √2 ⎦. 3 3 0 − 2 2 The PMSG time continuous model is given in Eqs. (4)–(5). This model includes nonlinear terms and thus needs to be linearized. In order to linearize the PMSG model, the rotor speed is assumed to be constant for one sample time. This is a valid approximation because the mechanical time constant is much greater than the electrical time constant. In addition, The FCS-MPC uses a discrete-time model for the prediction of the currents at a future sample period. For discretization, the forward Euler method with sampling time Ts is applied to the time-continuous model (4) and (5). A discrete-time state-space model of PMSG can be expressed as follows: where C1 =

(22)

T

(25)

⎡

 x (k) =

id (k) iq (k)

⎡T

s

⎢ Ld

B (k) = ⎣

0

id (k+1) iq (k+1)





, A (k) = ⎣

 , m (k) =

−ωe

Ts Lq

ωe Lq Ts ⎤ Ld ⎦,

Rs Ts Ld −ωe Lq Ts Lq

1−

1−

Rs Ts Lq



0 pm Ts

,

Lq

⎤ 0

(27)

⎡

⎥ ⎦ C1 C2 VDC , and u (k) = S

The main objective of a MSC controller is to track the desired stator d- and q-axis current references. A cost function has to be developed to find the optimal control actions. The cost function for the proposed FCS-MPC can be expressed as



J = id∗ (k + 1) − id (k + 1)

2



+ iq∗ (k + 1) − iq (k + 1)

2

(28)

1 if the upper switch of leg i-th is ON For the prediction algorithm, the cost function is evaluated for each of the possible seven voltage vectors, giving seven different 0 if the lower switch of leg i-th is ON

E.G. Shehata / Electric Power Systems Research 143 (2017) 197–205

201

Table 1 Switching states of inverter.

Sa Sb Sc V␣ V␤

V 0

V 1

V 2

V 3

V 4

V 5

V 6

V 7

0 0 0 0 0

1 0 0 2 Vdc /3 0

1 1 0 Vdc /3 √ Vdc / 3

0 1 0 −Vdc√/3 Vdc / 3

0 1 1 −2 Vdc /3 0

0 0 1 −Vdc /3 √ −Vdc / 3

1 0 1 Vdc /3 √ −Vdc / 3

1 1 1 0 0

Table 3 PI- controller gains of RSC. Kp1 Ki1

1.4 136.11

Kp2 Ki2

1.4 136.11

Table 4 Integral sliding mode controller gains. p

Fig. 4. Block diagram of FCS-MPC for MSC. Table 2 Parameters of the simulated PMSG [11]. Rated power Rs (pu) Rated wind turbine speed DC-link voltage (V) Line inductance (pu) Line resistance (pu)

1.5 MW 0.01 1.2 pu 1150 0.3 0.003

Ld = Lq (pu) PM flux linkage (pu) Pole pairs Rated wind speed (m/s) Rated stator voltage Grid frequency (Hz)

0.7 0.9 40 11 575 V 60

current predictions. For this purpose, the seven values of the objective function J are calculated and denoted as J0 , J1 , J2 , . . .J7 . A simple search function is used to find the minimal value of J and its associated index k. Once this index is found, the switching state at sample k to the MSC and the corresponding voltage are determined through Table 1. However, in order to reduce unnecessary switching, if the index is found to be 0, then the previous states of the MSC are required to determine whether the index 0 or 7 should be used in the control action. When the sampling time progresses to t = k + Ts, the new measurements of id(k + 1), iq(k + 1) currents, rotor speed ωe (k + 1), and electrical rotor angle e(k + 1) are obtained. With all the variables in the objective function being updated, a minimization is performed to find the new minimal value of J and its index k + 1, leading to the control signals for the MSC. The essence of the finite control set method is based on the receding horizon control principle, which uses one-step-ahead prediction and on-line optimization to solve the constrained optimal control problem. Fig. 4 shows the block diagram of the FCS-MPC. Compared to Figs. 2 and 3, FCS-MPC is the simplest one for implementation, where axes transformation of the reference stator voltage and SPWM technique are eliminated. To reduce the computation time of the FCS-MPC, one horizon is assumed during each sample time. Moreover, there are no gains to be tuned. 7. Simulation results and discussions In order to evaluate the performance of the proposed control schemes, several sets of simulations are conducted using Matlab/Sim-Power Systems toolbox. The effectiveness of the PIcontroller, ISMC and FCS-MPC of the machine side converter is examined under different operating conditions. The parameters of the PMSG are given in Table 2. PI-controller gains of the MSC are determined using “Ziegler–Nichols” method while ISMC gains are selected to give optimum performance and given in Tables 3 and 4, respectively. The grid side converter (GSC) is controlled based grid voltage oriented control to regulate the DC-link voltage and unity

p

Kid

0.5

Kiq

2

i Kid c Kid t Kid

20

i Kiq

50

0.5

c Kiq

1

0.1

t Kiq

0.1

id

0.1

iq

0.01

Table 5 PI- controller gains of GSC. KpG1 KiG1

0.83 5

KpG2 KiG2

0.83 5

KpV 1 KiV 1

8 400

power factor of the overall system (Qg = 0). The block diagram of the GSC control is given in Fig. 5 and its PI-controller gains are determined and given in Table 5. The block diagram of the overall proposed system is shown in Fig. 6. In this scheme, wind turbine model estimates the mechanical torque/power based on the wind velocity and rotor speed. PMSG converts the mechanical torque/power into active electric power. The electric power is transferred to the grid through back-to-back converters. The machine and grid side converters are three phase two level inverter. A filter is added to remove the high frequency harmonic components. The wind energy generation system is assumed to be connected to the utility grid at low voltage (575 V) where different grid faults are expected. A DC-chopper or crow bar protection is used to dissipate the power of the generator when grid faults occur. In turn, the wind energy generation system still work even during grid fault. The switching frequency of the SPWM is 1650 Hz. 7.1. Case (1) Variable wind speed with normal grid conditions In this case, the response of the three controllers is tested under wind speed variation. The PMSG is operated in the maximum power point tracking mode where its active power/torque is controlled according to the optimal speed curve depicted in Fig. 1. Initially, the wind speed is assumed to be 8 m/s and then increased to 12 m/s at t = 10 s. In turn, the electromagnetic torque and stator q-axis current change with wind speed variation. The stator d-axis current is assumed to be fixed at zero value to extract maximum torque/power per ampere and avoid demagnetization of the permanent magnet. The simulation results of the PI-controller, ISMC, and FCS-MPC are illustrated in Fig. 7(a–c), respectively. The figure shows the waveforms of grid active power, grid reactive power, DClink voltage, electromagnetic torque, stator d–q-axes current, and generator speed, respectively. All the values are given in per unit except the DC-link voltage is given in volt. It is shown that, the three controllers has high performance under steady state operation. The stator active power changes with rotor speed to track maximum power point. The actual values of the stator active/reactive

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E.G. Shehata / Electric Power Systems Research 143 (2017) 197–205

Fig. 5. Vector control of the GSC.

Fig. 6. Block diagram of overall control of MSC and GSC.

power, electromagnetic torque and stator currents (Iq, Id) can track well the reference values. ISMC and FCS-MPC have faster transient response and smoother operation compared to PI-controller

as shown in Fig. 7b and c. FCS-MPC has fast transient response and good tracking of the reference values, however, stator current waveforms in Fig. 7c have high ripples as a result of low

E.G. Shehata / Electric Power Systems Research 143 (2017) 197–205

203

Fig. 7. Simulation results of the PI-controller, ISMC, and FCS-MPC of the MSC during wind speed variations, respectively.

switching frequency and low number of horizon per sample. With larger number of prediction horizon per sample or higher switching frequency, these oscillations may disappear. Table 6 shows a com-

parsion between the three controllers results. It can be deduced that ISMC has superior performance.

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E.G. Shehata / Electric Power Systems Research 143 (2017) 197–205

Fig. 8. Simulation results of the PI-controller, ISMC, and FCS-MPC of the MSC under symmetrical fault condition.

Table 6 Comparsion between the three controllers performance.

Switching frequency (Hz) Vdc ripple Id ripple Iq ripple

PI-controller

ISMC

FCS-MPC

1620 ±2 V (0.17%) ±0.008 pu ±0.01 pu

1620 ±2 V (0.17%) ±0.007 pu ±0.008 pu

400 ±5 V (0.435%) ±0.05 pu ±0.045 pu

FCS-MPC has acceptable performance with low switching losses. 7.2. Case (2) Symmetrical voltage drop In this case, the performance of the three controllers are tested under the event of grid fault. A three-phase symmetrical fault is assumed to occur at the connection point of the generation system. To emulate this fault, the fault was assumed at t = 10 s and was subsequently cleared at t = 10.15 s. The wind speed is assumed constant at 12 m/s during the simulation. During the fault interval, the DC-link chopper resistance absorbs the generated power. The simulation results of the three controllers are shown in Fig. 8. The results show that the PI-controller has high oscillation in the waveforms of the stator currents (Id, Iq) during fault instant and needs long time to decay after fault clearing. Moreover, the generator electromagnetic torque waveform has undesired oscillations [10]. In contrast, the current waveforms of the ISMC are smooth during and after the fault and have small oscillations. The current and torque waveforms have low variation during and after fault. As in the previous case, the low switching frequency of the FCS-MPC causes some oscillations in the current and DC-link voltage waveforms. Comparing the results of the three controllers, it is shown that the proposed ISMC and FCS-MPC improved the performance of the PMSG under abnormal operating conditions. 8. Conclusion Three different control schemes for the machine side converter of PMSG driven by wind turbine based on vector control are presented in this paper. PI-controller, ISMC and FCS-MPC per-

formance is compared under the same operating conditions. The three controllers are tested under normal and symmetrical fault grid conditions. In spite of its simplicity and good performance during steady state, PI-controller has poor performance during wind speed or stator power variations. Moreover, high dip and overshoot appear in the stator current and electromagnetic torque during grid fault. The designed ISMC has fast transient response and smooth operation during wind speed variation and symmetrical grid fault. FCS-MPC is considered the simplest and fastest controller, however, as a result of low switching frequency and low number of prediction horizon per sample, some oscillations appear in the stator currents waveforms. It can be concluded that, ISMC has the superior performance while FCS-MPC is the simplest controller. Appendix A. Lyapunov stability function can be used to ensure the stability of the proposed ASMC as V=

1 T SS 2 i i

(A.1)

Taking the time derivative of (A.1) yields: V˙ = SiT S˙ i

(A.2)

Using (18) and (21), Eq. (A.2) can be expressed as V˙ = SiT (−i sat(Si ) + W )

(A.3)

−i |Si | + WSiT

(A.4)

V˙ =

Because i ≥ W and |Si | ≥ Si , therefore V˙ ≤ 0. References [1] Siegfried Heier, Grid Integration of Wind Energy, Onshore and Offshore Conversion Systems, John Wiley & Sons, Ltd., 2014. [2] Manfred Stiebler, Wind Energy Systems for Electric Power Generation, Springer-Verlag, Berlin, Heidelberg, 2008. [3] Iulian Munteanu, Antoneta Iuliana, Bratcu Nicolaos, Antonio Cutululis, Emil Ceang, Optimal Control of Wind Energy Systems, Springer-Verlag London Limited, 2008.

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