Deadbeat–fuzzy controller for the power control of a Doubly Fed Induction Generator based wind power system

Accepted Manuscript Deadbeat–fuzzy controller for the power control of a Doubly Fed Induction Generator based wind power system C.M. Rocha-Osorio, J.S. Solís-Chaves, Lucas L. Rodrigues, J.L. Azcue Puma, A.J. Sguarezi Filho

PII: DOI: Reference:

S0019-0578(18)30480-4 https://doi.org/10.1016/j.isatra.2018.11.038 ISATRA 2986

To appear in:

ISA Transactions

Received date : 21 March 2018 Revised date : 20 October 2018 Accepted date : 27 November 2018 Please cite this article as: C.M. Rocha-Osorio, J.S. Solís-Chaves, L.L. Rodrigues et al. Deadbeat–fuzzy controller for the power control of a Doubly Fed Induction Generator based wind power system. ISA Transactions (2018), https://doi.org/10.1016/j.isatra.2018.11.038 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Highlights •

• •

A fuzzy compensator along with the deadbeat controller in order to obtain a satisfactory response in the rotor side control of the DFIG based wind system in terms of steady-state error. The application of the deadbeat fuzzy controller is a valid alternative to the DFIG Stator Power Control. Experimental test bench using a 3 kW DFIG and a DSP TMS320F28335 card.

*Title page showing Author Details

Deadbeat–fuzzy controller for the power control of a Doubly Fed Induction Generator based wind power system C.M. Rocha-Osorioa,∗, J.S. Sol´ıs-Chavesa , Lucas L. Rodriguesa , J.L. Azcue Pumaa , A.J. Sguarezi Filhoa a

Federal University of ABC - Santo Andr´e , SP - Brazil

Abstract This paper proposes a Fuzzy Logic Controller for improvement of the steadystate response of a Doubly Fed Induction Generator used in a wind energy system, and governed by means of a Deadbeat Power Controller. The generator mathematical model is consistent with the Stator Flux-oriented strategy in the synchronous reference frame. Different simulation scenarios were developed in Matlab/Simulink to evaluate the dynamic and the steady-state responses. In order to obtain experimental results, the simulated scenarios were repeated by means of a test bench and a Digital Signal Processor board. These results demonstrate that the response still follows the power references imposed, despite the fact that the generator parameters (Rr , Ls and Lm ) were varied in a 30 %. A lower steady-state error is also achieved when compared with a Deadbeat and a classical PI controller. All the aforementioned evidence the proper application of this Fuzzy Controller in a wind power system based on a Doubly Fed Induction Generator. Keywords: Doubly fed induction generator (DFIG), fuzzy control, deadbeat controller, power control, wind generation, renewable energy.

∗

Corresponding author Email addresses: [email protected] (C.M. Rocha-Osorio), [email protected] (J.S. Sol´ıs-Chaves), [email protected] (Lucas L. Rodrigues), [email protected] (J.L. Azcue Puma), [email protected] (A.J. Sguarezi Filho )

Preprint submitted to ISA Transactions

20 de Outubro de 2018

1

2

Nomenclature Acronyms DFIG Doubly Fed Induction Generator SFOC DSP Digital Signal Processor SVM FLC Fuzzy Logic Controller RSC RE Renewable Energy SCIM Rs , Rr Ls , L r , L m σ ωs , ω r , ω m θs , θr , θm P, Q ⃗ ⃗v ,⃗i, ψ v, i, ψ p T, k e, ∆e

3

4

5

6 7 8 9 10 11 12 13 14 15 16 17 18 19

s, r α, β d, q ref

Stator Field Oriented Control Space Vector Modulation Rotor Side Converter Squirrel Cage Induction Machine

Symbols stator and rotor resistances; stator, rotor, and mutual inductances; total leakage factor; synchronous, slip, and rotor angular sfrequency; stator flux, slip, and rotor angles; Active and reactive power; voltage, current, and flux space vectors; voltage, current, and flux components; number of poles; sampling period and sampling time ; error and error variation ;

Subscripts stator and rotor; direct-and quadrature-axis expressed at stationary reference frame; direct-and quadrature-axis expressed at synchronous reference frame; reference value.

1. Introduction Wind Energy (WE) has experienced an enormous growth since the year 2000, increasing its percentage in the global energy matrix in countries such as China, USA, India, South-Africa, Brazil, and the European Union. The WE global total cumulative generated at the end of 2017 was 539.12 GW [1]. This type of renewable energy represents a clean generation alternative and can contribute to resolving the environmental crisis because, during its operation stage, WE is free of greenhouse gas emissions. Besides, according to the International Energy Agency, an estimated average carbon dioxide reduction of 600 g/kWh is considered to be obtained from WE [2]. This is possible because of the development of new technologies for generators and power converters [3] which have allowed wind turbines to considerably increase their power levels up to the MW scale. Moreover, a cumulative capacity growth rate increase of up to 8.8 % is expected by the year 2022 [1].

20

2

21 22 23 24 25 26 27 28 29 30 31

In WE Systems, three well-defined control levels [4, 5, 6] are present:–The first level corresponds to the energy control between the generator and the power grid, and it is based on the behaviour of the mechanical power income from the renewable source. The second level controls the interaction between aerodynamics and the wind turbine mechanical systems. This level will be responsible for generating the control input signals for the first level, such as the power references for the electric control of the wind turbine. Finally, the third control level aims to command the integration between the electrical network and the wind turbine. The three control levels are represented in Fig.1. In this paper, a first level control is studied with a focus on the Rotor Side Converter (RSC).

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

Fuzzy Logic Controllers (FLC) applied to the control of electrical machines have already been used in different and innovative ways, along the time [7, 8], mainly because of the Fuzzy Logic being a powerful tool to resolve some control problems through the emulation of human thoughts when a plant with well-known behaviour is present, as in the case of an electric generator. Literature reports some interesting background papers from the nineties. Among them, the following could be cited: A Deadbeat–fuzzy control algorithm was used as a method to regulate the phase current in a Squirrel Cage Induction Machine (SCIM), validating it with simulated and experimental results; but keeping in mind that the hysteresis band had to be small in order to obtain a satisfactory performance. This causes an increase in that the switching losses, being necessary the addition of a high-frequency resonant inverter, thus increasing the cost of the whole system [9]. The Deadbeat–fuzzy current regulator and the Fuzzy Speed Controller (FSC) described in [10] demonstrate a superior performance when they are applied to an Induction Machine drive, obtaining a stable and a faster step response. On the other hand, the Deadbeat controller in [10] is robust to the parameter variations. However, the deterministic component in the current loop is difficult to be obtained, because it heavily relies on many system parameter variations, feedback signal errors, and load disturbances. Furthermore, the deterministic component is a highly nonlinear term with a huge difficulty to be modeled by means of analytical equations. Therefore, the FLC must be employed for this task [10]. In [11], the same authors employed the Deadbeat–fuzzy current regulator and expanded the experimental results range by doubling the values of the stator and rotor resistances, testing the speed tracking, as well as the transient and the steady-state responses. In the present study, considering 3

59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78

that the Deadbeat controller performance depends on the model accuracy and the parameters evaluation, the FLC can compensate for these uncertainties, improving the DFIG steady-state response in a way similar to the one proposed for the SCIM; though with an intriguing dynamic response due to the presence of the rotor windings. In this context, the control of the RSC in a Back–to–Back topology, through a Deadbeat–FLC, can be an alternative to the use of the classical PI controller for the power control in a WE System. In the last ten years, researchers have expanded the scope of FLC to DFIG’s WE applications and to three-phase Controlled Voltage Sources; however, a power control for a DFIG-based WE system, using a Deadbeat–FLC, has not been reported yet. Nevertheless, some interesting researches can be cited to demonstrate the FLC’s vast field of action in renewable energy applications. For instance, a kind of FLC called Fuzzy-PI Controller, was proposed with success, mainly, for controlling the active and the reactive powers in WE systems [12, 13]. Direct Torque Control strategies were depicted in [14] and [15], using different types of generators for testing the projected FLC algorithms. In [16], a Deadbeat–FLC was used for controlling a Voltage Source (VSR) PWM Rectifier. In 2011 the same VSR prototype was reported again with fully simulation results and an extended description of the experimental setup developed, as well as the results obtained with this small-scale prototype [17].

79

4

Wind Turbine Rotor

Network Stator Circuit Breaker

Qs Ps

Transformer

Ps + Pr

Wind

DFIG

Qs

wm < ws Pr wm > ws Rotor Filter

Wm

BACK TO BACK CONVERTER ROTOR SIDE VSC

GRID SIDE VSC

AC

AC

vwind

Grid Filter

DC

DC DC BUS

ir

b Speed Power Control Control Control Level II

Pg

Pg ,ref

conv g ,ref

P

SVM

VCC SVM

Grid Side Rotor Side Control Control Control Level I

Qgconv ,ref

ig Pg ,Qg

VDC ,ref

Grid Operator Control Level III

Figure 1: Wind power system based on DFIG. 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

Both of the above-mentioned papers present a Deadbeat controller with overshoot and delay control. The FLC is used to obtain a good trade-off between the current overshoot and phase delay, when reference signals undergo a change in their values [17]. Elkhadiri et al. in 2018 present another FLC to govern a DFIG wind turbine. The control was performed by way of the rotor flux oriented vector technique [18]. The FLC uses the rotor current error and the evolution of this error to estimate the control signal Ku . Kalaivani et al. in 2017 propose a FLC for a DFIG powered by a Back–to–Back converter and operating under fault conditions. The error and the change in the error of the rotor voltage signals are the inputs for the FLC based on the Mamdani inference [19]. An interesting alternative with a FLC for a RSC in a DFIG-based WE system operating with constant power control strategy was presented by Mahalakshmi et al. in 2016. This time, the FLC was employed to calculate the pulses of the RSC thyristors, by using the mechanical and the synchronous speed signals. The proposed controller was also tested in an experimental setup [20].

97

5

98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135

Similar to fuzzy controllers applied in power electronics, another nonmodel based controller (free model controller) is proposed in [21] but applied to an autonomous humanoid robot, demonstrating fast and robust convergence to cope with external disturbances. This type of controllers could also be applied to the wind power system (which have a multi-input, multioutput, highly coupled, nonlinear and drifting system response) and it could be the future research direction of this project. The fuzzification, decision making, rule base, and defuzzification proposed here, need the rotor current error and the rotor current error variation to obtain the new rotor voltages that will be applied in the next control step (as can be observed in Fig. 3b), the stator-flux oriented equations for the DFIG are modeled in the synchronous reference frame, thus easing the complexity of the mathematical dynamic expressions of the generator and the Deadbeat-FLC. Therefore, the steady-state error percentage is null in the power responses, even when a random wind speed profile is considered, and a variation of 30 % in the DFIG’s parameters is programed in the DSP algorithm. A whole description of the Deadbeat–FLC is depicted in the sections below. The Deadbeat–FLC allows a better tracking of the reference signals, and its superior performance is similar to other novel DFIG controllers proposed in the literature, for instance: A DPC-Deadbeat proposed in [22] and probed for normal operation, during a balanced voltage sag conditions and a parametric variation up to 7 %. Similarly, in [23] a robust finite control is depicted, but in that case is used a cost function to minimize error due to parametric variations. To evidence the superior performance of the Deadbeat–FLC presented in this study, comparative simulation results were carried out via Matlab/Simulink. The two controllers chosen for comparing are a Deadbeat Power Controller proposed by Sguarezi [24], and a classical DFIG PI Controller adjusted as reported in [25], both working under the same operating conditions than the Deadbeat-FLC herein proposed. For obtaining the experimental results, a 3 kW DFIG-based wind energy system, controlled via a TMS320F28335 DSP system from Texas Instruments, was used. In order to explain the Deadbeat–FLC, the present paper is organized as follows: First, a fully description of the DFIG power stator control strategy is resumed in Section 2.1. Next, Section 2.2, depicts the Deadbeat Controller theory. Then, the compensation technique applied using the FLC is thoroughly explained in Section 2.3. Later, the Simulation Results are presented in Section 3. After that, in Section 4, the experimental results 6

137

are presented and compared with the same scenarios simulated via Simulink. Finally, some conclusions for this work are enlisted in Section 5.

138

2. Power control of Doubly Fed Induction Generator

136

139 140 141

Rs

isd

Fig. 2 shows the equivalent circuit of a DFIG in dq reference frame. Using the stator-flux-oriented vector control (SFOC) [5], the generator voltages fluxes and currents can be expressed as follows:

Lss R

wr 644474448 Lsr (ws - wm ) yrq

Lss

ws ysq

Rss L

-+

Rss L

+ + -

Lss R

Rs

isq

ird

Lss R

L R ss

Lm

Rss L

+-

Rss L

+ +

d yrd dt

wr 644474448 Lsr (ws - wm ) yrd Rr

Lss

ws ysd

+

d ysd dt

vsd

+-

Rr

+

vrd -

dt

-

-

R L ss

Lm

irq +

d yrq dt

-

Figure 2: Equivalent circuit of a DFIG in dq reference frame.

142

dψsd ⃗vs,dq = Rs⃗is,dq + + jωs ψsd dt

(1)

⃗r,dq dψ ⃗r,dq ⃗vr,dq = Rr⃗ir,dq + + jωr ψ dt

(2)

⃗s,dq = ψsd = Ls⃗is,dq + Lm⃗ir,dq ψ

(3)

⃗r,dq = Lr⃗ir,dq + Lm⃗is,dq ψ

(4)

⃗r,dq ⃗is,dq = ψsd − Lm ψ σLs σLs Lr

(5)

⃗ ⃗ir,dq = ψr,dq − Lm ψsd σLr σLs Lr

(6)

7

Lss R

+

d ysq

vsq

-+

vrq

143 144 145

146 147 148

In steady state, and neglecting the voltage drop in stator resistance, the stator flux is proportional to the network voltage vs . Thus, the stator voltage components are: vsd = 0

(7)

vsq = vs ≈ ωs ψs

(8)

2.1. Power control using deadbeat–fuzzy controllers According to (5), (7) and (8) the active and reactive powers of the stator are calculated as follows : 3 Lm Ps = − vs irq 2 Ls ) ( 3 ψs Lm − ird Qs = vs 2 Ls Ls

149 150 151 152 153 154 155

(9) (10)

In (9) and (10), it can be realised that the active Ps and reactive Qs power can be controlled independently, as well as proportionally, according to the rotor current components irq and ird , respectively. Fig. 3(a) illustrates the rotor side power control with deadbeat–fuzzy controllers. This control scheme consists of an estimation stage (please see the box entitled as Estimator), where the active and reactive power are estimated from stator voltages and currents using the following equations: Ps =

3 (vsα isα + vsβ isβ ) 2

(11)

3 (vsβ isα − vsα isβ ) (12) 2 Furthermore, the stator position (13) and stator angular frequency (14) were estimated by way of the stator magnetic flux components from (15). ( ) ψsβ −1 θs = tan (13) ψsα Qs =

156 157

ωs =

(vsβ − Rs isβ ) ψsα − (vsα − Rs isα ) ψsβ (ψsα )2 + (ψsβ )2 8

(14)

(a)

Estimator 3 Ps = (vsaisa + vsbisb ) 2 3 Qs = (vsbisa - vsaisb ) 2 r r r ys ,ab = ò vs ,ab - Rs is ,ab dt

(

)

Qs ,ref

qs

ab r is ,ab

ab abc

ωm

1 Lm

ys Network

BRIM DFIG

ird ,ref

r ir ,dq

irq ,ref

Deadbeat-fuzzy C Control (See Fig. 3(b)) C r Load is ,dq wr

-

eird z

ird ,ref

ird r is ,dq

irq ,ref i rq

wr

-1

+ Deird

Deadbeat Controller

r ¢ vrdq

z -1

+ Deirq

r vr ,dq

158

159

∫ ( ) ⃗vs,αβ − Rs⃗is,αβ dt

The mechanical rotor position, θm , can be calculated from: ∫ p θm = ωm dt 2

160 161 162

(15)

(16)

and the slip position θr , is calculated using (13) and (16): θr = θs − θm

(17)

In addition, another stage of the control scheme regards the proposed controller. In Fig. 3(b), the deadbeat control loop compensated with a Mamdani type fuzzy logic controller (FLC) is detailed. It can be noticed 9

r + vr ,dq +

Dvrq

irq irq ,ref - ei rq

Figure 3: Rotor side control using deadbeat–fuzzy controllers. (a) General scheme, (b) deadbeat–fuzzy loop.

⃗s,αβ = ψ

FLC ird Dvrd

-

DC

Bus r DC abr ir ,abr abc qs - qm ab abr S q abc qr s r r r ir ,ab vr ,abr abr r ab ir ,dq SVM dq dq

+

2 Ls 3 vs Lm

+

AC

qm

-

+ -

ò

dq

r is ,ab

r vs ,ab

+

r is ,dq

2 Ls 3 vs Lm

(b) -

-

sa

Ps ,ref

+

æy ö qs = tan-1 ççç sb ÷÷÷ è y ø÷

FLC irq

163 164 165 166

167 168 169 170 171

that the deadbeat controller has the role of generating the rotor voltage ′ ⃗vr,dq , and the fuzzy control aims at compensating the steady–state error by providing the rotor voltage ∆⃗vr,dq . The deadbeat controller and the fuzzy controller are discussed in sections 2.2 and 2.3. 2.2. Deadbeat Controller Using the equations of a discretized continuous linear system, the Deadbeat controller calculates the input u(k), to ensure that the output x(k) reaches the reference values xref . A continuous linear system is represented as follows: x¯˙ = A¯ x + B u¯ + Gw¯ y¯ = C x¯

172 173 174

175

where A, B and G are matrices n × n, w(k) ¯ is the perturbation vector, and C is a matrix identity. thus, discretizing (18) : x¯(k + 1) = Ad x¯(k) + Bd u¯(k) + Gd w(k) ¯ (19) ∫τ ∫τ where, Ad = eAT ∼ = I+AT , Bd = eAT Bdτ ∼ = BT , Gd = eAT Gdτ ∼ = GT 0

176 177 178

(18)

0

, T (1/fm = 1/10000Hz) is the sampling period and k the sampling time. In order to guarantee a null error, the input is calculated in the following way [26] u¯(k) = F (¯ xref − x¯)

179 180 181

(20)

where, x¯ref represents the reference, while F is the gain matrix. Substituting (20) in (19), and considering that x¯ref = x¯(k + 1), the input u¯(k) can be written using the following equation [24]: F

182

z }| { [ ] u¯(k) = Bd−1 Ad A−1 ¯ref − x¯(k) − A−1 ¯ d x d Gd w(k)

(21)

The block diagram of the deadbeat controller can be observed in Fig. 4

10

Gain Matrix

x ref (k )

+

x (k ) Plant u (k ) F Model

-

Figure 4: The block diagram of a deadbeat controller

183 184 185

By applying the deadbeat control to the control of the DFIG powers, the voltage of the rotor expressed in (2) is rewritten in terms of the currents, as follows: ⃗r,dq ψ

⃗r,dq ψ

⃗vr,dq 186

187 188

190

(22)

In other words, the rotor voltage components can be expressed as [27]: vrd = Rr ird +

d (Lr ird + Lm isd ) − ωr (Lr irq + Lm isq ) dt

(23)

vrq = Rr irq +

d (Lr irq + Lm isq ) + ωr (Lr ird + Lm isd ) dt

(24)

but considering that isd = Lψss − LLms ird , isq = − LLms irq and substituting them in the derivative of (23) and (24):

vrd

189

}| { z }| { z d = Rr⃗ir,dq + (Lr⃗ir,dq + Lm⃗is,dq ) + jωr (Lr⃗ir,dq + Lm⃗is,dq ) dt

[ ( )] d ψs Lm = Rr ird + Lr ird + Lm − ird − ωr (Lr irq + Lm isq ) dt Ls Ls

(25)

[ ( )] d Lm vrq = Rr irq + Lr irq + Lm − ird + ωr (Lr ird + Lm isd ) (26) dt Ls ( ) and taking in-to account that dtd Lψss = 0 because of SFOC. By doing so, the voltage rotor component can be defined as ( ) Ls Lr − L2m dird − ωr Lm isq (27) vrd = Rr ird − Lr ωr irq + Ls dt 11

(

vrq = Rr irq + Lr ωr ird + 191 192

Ls Lr − L2m Ls

)

dirq + ωr Lm isd dt

From the rotor components expressed in (27) and (28) the rotor voltage vector is: ( ) L2m d⃗ir,dq ⃗ ⃗ ⃗vr,dq = (Rr + jLr ωr ) ir,dq + jLm ωr is,dq + Lr − (29) Ls dt

193

thus, writing (29) in state–space form:

194

d⃗ir,dq = H⃗ir,dq + K⃗vr,dq + L⃗is,dq dt rewriting (30), the terms H, K, and L are defined as follow: [

dird dt dirq dt

]

=

z[

H

−Rr σLr −ωr σ

+ 195 196 197 198

(28)

}| z[

ωr σ −Rr σLr

]{ [

0

}|

ird irq

]

L

−ωr Lm σLr

ωr Lm σLr

0

+

z[

]{ [

K

1 σLr

0

isd isq

}|

0 1 σLr

]{ [

vrd vrq

]

(30)

] (31)

Expressing (31) in form of a discretized differential equation where T is a sampling period at time k, and renaming x¯ = ⃗ir,dq , Ad = H, Bd = K, u¯ = ⃗vr,dq , Gd = L and w ¯ = ⃗is,dq . The vector for the rotor current can be estimated: ⃗ir,dq (k + 1) = Ad⃗ir,dq (k) + Bd⃗vr,dq (k) + Gd⃗is,dq (k)

(32)

and its components can be written according to:

199

z[

⃗ir,dq ref

A

B

d }|d }| ]{ z[ ]{ [ ] z[ T }| ]{ [ ] Rr T ωr T 0 1 − σLr ird (k + 1) i (k) vrd (k) rd σ σL r = + −ωr T T irq (k + 1) irq (k) vrq (k) 1 − RσLr Tr 0 σL σ r

+

z[

Gd

0

}|

−ωr Lm T σLr

ωr Lm T σLr

0

12

]{ [

isd (k) isq (k)

]

(33)

200 201

At last, the rotor voltage components can be calculated using (21) and (33):

vrd (k) = σLr

ird,ref − ird (k) + Rr ird (k) − Lr ωr irq (k)) − Lm ωr isq (k) (34) T

vrq (k) = σLr 202

irq,ref − irq (k) + Rr irq (k) + Lr ωr ird (k) + Lm ωr isd (k) T

where the rotor current references from (9) and (10) can be expressed as: ird (k + 1) = ird,ref = −

2Qs,ref Ls ψs + 3vs Lm Lm

(36)

2Ps,ref Ls 3vs Lm

(37)

irq (k + 1) = irq,ref = − 203 204 205 206

(35)

Regarding (34) and (35), it can be perceived that the Deadbeat controller can adjust the rotor voltage employing the rotor current (⃗ir,dq ), its references (⃗ir,dqref ), the stator current (⃗is,dq ), and the angular speed (ωr ). In Fig. 5, the input–output structure of the deadbeat controller is illustrated. r is ,dq r ir ,dq r ir ,dqref wr

Deadbeat Controller ird ,ref - ird (k ) sLr + Rr ird (k ) - Lr wr irq (k ) - Lm wr isq (k ) T sLr

irq ,ref - irq (k ) T

r vr ,dq

+ Rr irq (k ) + Lr wr ird (k ) + Lm wr isd (k )

Figure 5: Input–output structure of the Deadbeat controller.

207 208 209 210 211 212 213 214

2.3. Design of fuzzy logic controller Fuzzy logic controllers (FLCs) are widely used in various systems, being one of their main advantages the offering of solutions based on experience. This allows a performance which is equivalent to a human operator, without using the mathematical model of the system. For this reason, this type of controller is used in complex systems in which the mathematical model is unable to respond with accuracy (e.g. parametric variation and external disturbances). 13

215 216 217 218

In order to compensate the deadbeat controller steady state error, two FLCs are used to estimate the rotor voltage components ∆vrd and ∆vrq . The input of each controller are the rotor current error eird and eirq and its variations ∆eird and ∆eirq , where: ird,ref (k+1)

eird =

(38)

z}|{ z }| { irq,ref − irq

(39)

irq,ref (k+1)

eirq =

ird (k)

z }| { z}|{ ird,ref − ird

∆eird =

irq (k)

eird − z −1 eird T

(40)

eirq − z −1 eirq (41) T where T denotes the sample time. The operation of the FLC can be summarized in three main stages, as presented in Fig. 6. ∆eirq =

219 220 221

crisp inputs NB

NM

NS

ZE

PS

PM

eird

PB

1

0.5

Deird eirq

Deirq

0

Input membership functions -20

-10

0

10

20

FUZZIFICATION fuzzy inputs

(See Fig. 7)

Fuzzy Rules

FUZZY PROCESSING NB

NM

NS

ZE

PS

PM

PB

1

fuzzy outputs

0.5

0

Output membership functions -20

-10

(See Fig. 7)

0

10

20

DEFUZZIFICATION

Dvrd Dvrq

if ei rd is NB and Dei rd is NB then Dvrd is NB M

(See Table 1)

crisp outputs (control outputs) Figure 6: Operation of the FLC.

222 223

1) Fuzzification: the crisp inputs (in this case eird , eirq , ∆eird , and ∆eirq ) are transformed into fuzzy inputs using the input membership sets presented 14

224 225 226 227 228 229 230 231 232 233 234 235 236 237

in Fig. 7. Each FLC has 5 triangular membership functions and 2 trapezoidal membership functions for each linguistic variable. 2) Fuzzy processing: the fuzzy inputs are processed according to the base rules presented in Table 1 [28]. In order to obtain the fuzzy outputs, 49 fuzzy rules and a Mamdani type fuzzy inference were used; owing to the simplicity in the formulation of the fuzzy rules, when compared with the Sugeno type fuzzy inference; whereas the Sugeno type fuzzy inference requires a detailed mathematical analysis for adjusting the several parameters. 3) Defuzzification: the fuzzy outputs are transformed into crisp outputs using the traditional Center of Gravity (COG) method, which calculates the gravity center of the area under the membership function and the output membership sets illustrated in Fig. 7. The control outputs ∆vrd and ∆vrq , considering the COG method can be written by means of (42) and (43), respectively:

∆vrd =

n ∑

µRi ∆vrd Ri

i=1 n ∑

∆vrd Ri

i=1 n ∑

∆vrq Ri

(42)

i=1

∆vrq =

n ∑

µRi ∆vrq Ri (43)

i=1

238 239 240 241 242 243 244 245 246

where, n is the number of rules, µ is the membership degree, and Ri is the rule evaluated. The FLCs setting is based on the assumption that the generator parameters are not error-prone, and that the errors that may occur are reflected in the current variations [29]. Therefore, the fuzzy controller was adjusted so that those variations would be compensated, thus improving its performance. To that end, the error shown in (38) and (39), and the error variation shown in (40) and (41) are adjusted in such a way that the proposal achieves a satisfactory performance even in conditions of parametric variation.

15

NM

NS

ZE

PS

PM

Mamdani Inference / 49 Rules

PB

eird

Fuzzy Rules

Dvrd

0.5

NB NB NB NB NM NS NS ZE 0

-10

-20 1

NB

0 eird

NM

NS

ZE

10 PS

PM

NM NB NM NM NM NS ZE PS

20

NS NB NM NS NS ZE PS PM

PB

Deird ZE NB NM NS ZE PS PM PB

PS NM NS ZE PS PS PM PB

0.5

Degree of membership

1

0

0 Deird

10

1

NB

NM

NS

ZE

PS

PM

0 -10

0 eirq NS

ZE

10 PS

PM

eirq

NM NB NM NM NM NS ZE PS

3

NS NB NM NS NS ZE PS PM

PB

Deirq ZE NB NM NS ZE PS PM PB

PS NM NS ZE PS PS PM PB

0.5

0

0 Deirq

125

250

PS

PM

PB

Output Membership Sets 1

NB

NM

NS

ZE

PS

PM

PB

0.5

0

-180

-125

0 Dvrq

125

180

Active Power Control

PB ZE PS PS PM PB PB PB -1.5

0 Dvrd

NS ZE

NM

PM NS ZE PS PM PM PM PB -3

PB

Negative Negative Negative Zero Positive Positive Positive Big Médium Small Big Small Medium

NB NM NS ZE PS PM PB

NB NB NB NB NM NS NS ZE

NM

PM

Reactive Power Control

Mamdani Inference / 49 Rules Fuzzy Rules

NB

PS

Linguistic Terms

Dvrq

-3

-125

NB

0.5

1

ZE

0

-250

20

PB

NS

0.5

PB ZE PS PS PM PB PB PB -10

-20

NM

NB

PM NS ZE PS PM PM PM PB

Input Membership Sets

Degree of membership

Output Membership Sets

NB NM NS ZE PS PM PB Degree of membership

NB

Degree of membership

Degree of membership

Degree of membership

Input Membership Sets 1

3

1.5

Figure 7: Fuzzy logic controller.

Table 1: Fuzzy rules, ∆vrd and ∆vrq [28] ∆eird /eird NB NM NS ZE PS PM PB

247

248 249 250

NB NB NB NB NB NM NS ZE

NM NB NM NM NM NS ZE PS

∆vrd NS NB NM NS NS ZE PS PS

ZE NM NM NS ZE PS PM PM

PS NS NS ZE PS PS PM PB

PM PB NS ZE ZE PS PS PM PM PB PM PB PM PB PB PB

∆eirq /eirq NB NM NS ZE PS PM PB

NB NB NB NB NB NM NS ZE

NM NB NM NM NM NS ZE PS

∆vrq NS NB NM NS NS ZE PS PS

ZE NM NM NS ZE PS PM PM

PS NS NS ZE PS PS PM PB

PM NS ZE PS PM PM PM PB

PB ZE PS PM PB PB PB PB

3. Simulation results In order to validate the control scheme proposed in Fig.3, a simulation model developed in Matlab/Simulink was tested under different power conditions and parameters, according to Tables 2 and 3, respectively.

16

Table 3: Parameters of the DFIG. Parameter Value Rated Stator Voltage Vs,n 220/380 ∆–Y Rated Stator Current Is,n 12 A Rated Power Pn 3 kW Rated Speed 1800 rpm Rated Frequency fs 60 Hz Stator Resistance Rs 1Ω Rotor Resistance Rr 3.1322 Ω Mutual Inductance Lm 0.1917 H Stator Inductance Ls 0,2010 H Rotor Inductance Lr 0.2010 H Number of Poles p 4 Lumped Inertia Constant J 0.05 kgm2 Frequency Modulation, fm 10000 Hz

Table 2: Network conditions Condition 1 2 3

251 252

Fig.8 shows the simulation results of the wind power system when the generator operates under variable speed, according to the rotor speed profile illustrated in Fig. 8 (a). Rotor speed [rad/s]

253

Stator Power Power Time P [W] Q [var] Factor [s] -1,000 -620 0.85 1.1–1.2 -1,500 930 0.85 1.2–1.5 -2,000 0 1 1.5–1.6

(a)

240 220 200 180

Reactive Power [var]

1.2

1.25

1.3

-1200 -1400

(d)

-1600 -1800

1.35 1.4 1.45 1.5 Time [s] 9 i rq,ref Deadbeat 8 Deadbeat-Fuzzy PI 7

Rotor Current [A]

-1000

-2000

(c)

1.15

Ps,ref Deadbeat Deadbeat-Fuzzy PI

500

(e) 0 Qs,ref Deadbeat Deadbeat-Fuzzy PI

-500 1.1

1.55

1.15

1.2

1.25

1.3

1.35 1.4 Time [s]

1.45

1.5

1.55

1.6

1.6

6 5 4 6

1000

Rotor Current [A]

(b)

Active Power [W]

160 1.1

i rd,ref Deadbeat Deadbeat-Fuzzy PI

4 2 0 -2 1.1

1.15

1.2

1.25

1.3

1.35 1.4 Time [s]

1.45

1.5

Figure 8: Response of the deadbeat, deadbeat–fuzzy and PI controllers under rotor speed variation : (a) rotor speed profile, (b) stator active power, (c) stator reactive power, (d) rotor current component irq (e) rotor current component ird .

17

1.55

1.6

254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269

The results obtained were compared with the deadbeat controller considered in [27], in which the rotor voltages were calculated according to (34) and (35) based on the dynamic mathematical model of the generator and without using a cost function. Similarly, the results were compared with the PI controller proposed in [25] whose controller gains were adjusted by the pole-placement technique and without using any cost function. The stator active power is presented in Fig.8 (b); where it can also be observed a similar performance, in terms of steady-state error and rise time, between the deadbeat–fuzzy controller (line red) and the deadbeat (line blue), as well as a better performance when compared with PI controller (green line). Similarly, the above behaviour is presented by observing the rotor current component irq shown in Fig. 8 (d). By comparing the behaviour of the reactive power and the rotor current component shown in Figs. 8 (c) and (e), respectively, a superior performance of the Deadbeat–FLC can be perceived. In Table 4, the percentage of the steady–state error, the settling time (ts ), and the rise time (tr ) are compared for all the alternatives. Table 4: Controllers comparison Parameters Error [%] ts [s] Deadbeat–fuzzy 4.3 0.0011 Deadbeat 17 0.0008 PI 5.3 0.035 Controller

270

271 272 273 274 275 276

tr [s] 0.0008 0.0004 0.048

4. Experimental results The deadbeat–fuzzy controller proposed in Fig. 3 was experimentally validated using a small-scale setup compounded of a 3 kW DFIG coupled to a DC motor that emulates the mechanical system of the wind turbine (rotor blades, shaft). The controller was implemented by way of a Digital Signal Processor (DSP) TMS320F28335, and a signal acquisition board. Fig. 9 presents (a) the laboratory setup, and (b) the connection diagram.

18

a)

b)

COMPUTER Code Composer Studio (CCS)

DSP TEXAS INSTRUMENTS (TMS 320F28335)

INTERFACE BOARD 0-3V

A/D PWM CONVERTER

HALL EFFECT SENSORS

is

INTERFACE BOARD 0-3V / 0-15V

D/A CONVERTER

SCOPE

vs AC DC

ROTOR

DC BUS

Network

STATOR MECHANICAL JOINT

ROTARY BRIM ENCODER

DFIG

DC MOTOR 3kW

Figure 9: Setup of a 3 kW DFIG system (a) Laboratory Setup, (b) Connection Diagram.

277 278

279 280 281 282 283 284 285 286 287 288 289 290 291

The tests, which were performed at constant speed, variable speed, and parametric variations; were implemented as follows: 4.1. Constant rotor speed test In this test, the deadbeat–fuzzy controller was tested under constant rotor speed (ωm = 1700 rpm). In addition, the following rotor current references: ird,ref : 3A to 1A and 1A to 3A; irq,ref : 1A to 3A and 3A to 1A, were used In Figs. 10(a) and (b) the response of the deadbeat–fuzzy controller, according to the conditions previously described, is presented. Besides, the experimental results of the stator current and the stator voltage in phase A were presented in Fig. 10 (c); while, in Fig. 10 (d), an experimental comparison between the deadbeat–fuzzy and the PI controllers is also presented. The experimental results indicated an adequate behaviour of the controller, with a settling time equal to 1.8 ms, without overshooting and a null steadystate error, in spite of the noise caused by the switching of the back to back converter.

19

ird ,ref = 1A

ird

ird ,ref = 3A

ird ,ref = 1A

ird

(a)

(b)

irq

(c)

irq ,ref = 3A

ird = 1A

irq ,ref = 1A irq ,ref = 3A

irq = 3A

irq = 1A

1.8 ms

irq ,ref = 1A 3.5

(d)

Rotor Current [A]

irq

vsA = 311V isA = 4.1A

isA = 3.4A

3 2.5 2 1.5 1 0.5 -0.03

Deadbeat-Fuzzy PI i rq,ref -0.02

-0.01

0 Time [s]

0.01

0.02

0.03

Figure 10: Experimental results under constant rotor speed (a) ird and irq (100 ms/div.), (b) Detail in the rotor current step ird and irq (2 ms/div.), (c) stator current and stator voltage in phase A (10 ms/div.). (d) Experimental comparison between deadbeat–fuzzy and PI controllers.

292 293 294 295 296 297 298 299 300 301

4.1.1. Variable rotor speed test In this test the same rotor current references presented in the constant rotor speed test were considered. However, it was applied a variable rotor speed according to Fig. 11 (purple line), in which, roughly speaking, the speed varied from 2000 rpm to 1670 rpm where the machine synchronous speed was 1800 rpm. The rotor current variation, in the reference frame fixed in the rotor physical structure (ir,αr ), can be observed in the blue line in the same figure. Finally, an invariant behaviour can be perceived in the current components (ird –green line and irq –orange line), despite the change of the rotor speed.

20

ir ,ar

wm = 1800 rpm ird = 1A irq = 3A

Figure 11: Rotor current response considering a variation of rotor speed from 2000 rpm to 1670 rpm (1s/div.). 302 303 304 305 306 307 308 309 310

4.1.2. Parameter variation test In this test, the same rotor current references used in the previous tests were also considered, as well as a constant rotor speed. Nevertheless, a variation of 30 % in the rotor resistance Rr , stator inductance Ls , and magnetization inductance Lm was applied. In Fig.12, it can be observed that the response of the Deadbeat controller compensated with fuzzy, presented a behaviour similar to the constant speed test, thus demonstrating the advantages of the proposed control to counteract the negative effects due to the machine parametric variation.

ird ,ref = 3A

ird

ird ,ref = 1A irq ,ref = 3A irq ,ref = 1A

irq

Figure 12: Rotor current response considering parameters variation of 30% (100ms/div.).

311

312 313

5. Conclusion A Deadbeat-fuzzy Controller designed to decrease the steady state error and improve the robustness of the machine parameter variations has been 21

325

presented in this paper. This one was used for controlling the stator power of a DFIG-based wind generation system. This new controller had been simulated using Matlab/Simulink and tested experimentally under different power conditions, mechanical speed, and a parametric variation of up to 30 % on Rr and Lr , in an experimental manner. The simulation results corroborate that the compensation made by the fuzzy logic controller allows a more precise DFIG power control, reducing the steady-state error that the deadbeat controller by itself cannot avoid. These results were also demonstrated in an experimental way by using a small–scale prototype at the same conditions. Regarding all the above mentioned, the proposed Deadbeat Fuzzy Controller is a suitable solution to the rotor side control in a DFIG-based wind energy system.

326

Acknowledgment

314 315 316 317 318 319 320 321 322 323 324

328

The authors would like to thank to CAPES and FAPESP for their financial support.

329

References

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334 335 336

337 338 339 340

341 342 343

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25