4th International Conference on Power Engineering, Energy and Electrical Drives Istanbul, Turkey, 13-17 May 2013

Maximum Power Point Tracker for Wind Energy

Generation Systems using Matrix Converters
L. P. Afonso S. F. Pinto J. F. Silva
INESC-ID Lisboa
AC Energia, DEEC, Instituto Superior Técnico, Technical University of Lisbon,
Lisboa, Portugal
Email: soniafp@ist.utl.pt , fernandos@alfa.ist.utl.pt

Abstract— This paper presents an optimum speed control system

for wind energy generation systems equipped with Doubly Fed
Induction Generators (DFIG) and Matrix Converters (MC). The
MC controls the rotor currents and guarantees the optimum
speed of the generator necessary to extract the maximum energy
from the wind (MPPT-Maximum Power Point Tracking). The
Stator Flux Oriented Control (SFOC), together with the Sliding
Mode Control (SMC) and the MC Space Vector representation
technique guarantee nearly unitary power factor in the
connection to the grid.
The results show the proposed method presents good results for
variable wind speed below the nominal power, and is more
accurate in terms of MPPT than the torque control technique.
Keywords- Doubly Fed Induction Generator, Wind Turbine,
Matrix Converter, Maximum Power Point Tracking, Optimum
Speed Tracking.
Figure 1. Global model of the system.
I. INTRODUCTION
Using the Stator Flux Oriented Control (SFOC) [8], a
Nowadays, despite the global economic crisis, wind power
reference frame synchronous with the d component of the
is one the most promising renewable energies and has reached
stator flux is chosen. Then, taking into account the real speed
a record of 250GW installed power[1].
and the optimum one, a reference torque which guarantees the
The DFIG configuration is known to have some advantages optimum speed is generated. Using the space vector
mainly due to the fact that the power processed by the representation and the sliding mode control, the MC applies the
converter is only the excitation power, which is around 25% of rotor currents necessary to satisfy the established reference
the total system power. As a result, it is possible to generate torque or the machine speed to extract the maximum energy
active power through the rotor and the stator and control the available at each wind speed.
reactive power in the connection to the grid [2].
The most commonly used three-phase AC-AC converter is II. WIND TURBINE AND DFIG
the rectifier-inverter pair structure [3], [4], and thus, the most
well-known and well-established. However, in the intermediate A. Wind Turbine Model
DC-link large storage components, electrolytic capacitors are With the wind turbine, it is possible to extract the power P
used, decreasing the overall lifetime of the system and (1) from the wind, where Cp (2) is the performance coefficient
increasing the total weight and size of the converter, the losses or power coefficient, ρ is the air density [kg/m3], A is the swept
and the global system costs [5]. Due to these problems the area [m3] by the wind turbine and u is wind speed [m/s].
matrix converter may become a competitive solution for these
applications, as it is a bidirectional direct AC-AC converter, 1
able to establish a desired output frequency, output voltage and P= C p (β, λ ) ρ A u 3 (1)
2
input displacement factor [6], [7]. Furthermore, it is a compact
converter without dc-link or large energy storage elements [6], The power coefficient Cp (2) depends on the pitch angle ȕ
as shown in Fig. 1. and the tip-speed ratio Ȝ (3).
The system is based on a wind turbine model that defines
the DFIG optimum speed according to the wind speed. 12.5
§ 116 · − λi
C p = 0.22 ¨¨ − 0.4β − 5 ¸¸ e (2)
© λi ¹

978-1-4673-6392-1/13/$31.00 ©2013 IEEE

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4th International Conference on Power Engineering, Energy and Electrical Drives Istanbul, Turkey, 13-17 May 2013

−1 From the induction machine equations in the defined dq

§ 1 0.035 ·¸ ω R coordinates system (7) [8] and considering that the stator
λi = ¨ − λ= t (3)
¨ λ + 0.08β β3 + 1 ¸ u resistance can be neglected (it is much smaller than the stator
© ¹
reactance rs<<ωsLs), then (7) can be simplified to (8) using (5):
The mechanical torque Tm extracted from the turbine rotor
is given by (4): ­ dψ sd
°°u sd = rs isd + dt − ωs ψ sq
Tm = P ωm (4) ® (7)
°u = r i + dψ sq + ω ψ
°¯ sq s sq dt
s sd
B. Wind Turbine Optimum Speed
Under the nominal wind turbine power, ȕ is usually set to
zero, and from (1), (2) and (3) the turbine power is given by ­ dψ sd
°u sd ≈
(5): ® dt (8)
°u sq ≈ −ωs ψ sd
¯
12.5
ª º −
« » 1 In steady-state the angular frequency of the reference frame
« » u
− 0.035 synchronous with the stator is the same as the angular
0.22 3« 116
P= ρ Au − 5» e ωt R
(5) frequency ωs of the stator voltages, which is constant [8]. In the
2 « 1 » defined coordinate system the grid voltages applied to the
« u » stator of the induction generator are as defined in (9).
« »
¬« ωt R − 0.035 ¼»
u sd = 0 ∧ u sq = − 3 Vef (9)
From (5) the turbine power only depends on the turbine
speed ωt and on the wind speed u. In order to find the optimum From (8) and (9) it is possible to determine the flux Ψs (10)
speed according to a certain wind speed it is necessary to [8]:
maximize P (6).

dP 3 Vef
=0 (6) ψs = ≈ const (10)
dωt ωs

From (5) and (6) the optimum speed can be calculated as a From (6) and (10) is obtained the rotor q component
function of the wind speed and the blades length, considering *
reference current irq that guarantees the maximum output
that the air density is ρ = 1.225kg / m3 . power:

6.32497 * Ls ω s
ωTopt = u (7) irq =− T* (11)
R 3 Vef p M
This optimum speed will be further used as a reference for
the speed controller. This value depends on the reference torque T * .

C. Rotor irq Reference Current D. Rotor ird Reference Current

In the Stator Flux Oriented Control, the dq reference frame The d component of the reference current ird *
is used to
is synchronous and coincident with the d component of the
stator flux (5). Consequently the q component of the stator flux control the power factor at the point of connection to the grid,
is zero. guaranteeing nearly unitary power factor. Then the reactive
power Qgrid in the connection to the grid must be zero.
ψ sd = ψ s ∧ ψ sq = 0 (5)

Using this dq coordinate system to describe the dynamic

behaviour of the induction machine model it is possible to
obtain the q component of the rotor current irq (6) as a
function of the electromagnetic torque Tem of the induction
generator [3], [4]:

Ls
irq = − Tem (6) Figure 2. Reactive power flow.
p M ψs

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The reactive power at the matrix converter input Qr may be ­ .

set to zero by controlling the matrix converter input currents. °S rα (erα , t ) S rα (erα , t ) < 0
The d component of the rotor current ird is used to set the stator ® (18)
.
reactive power Qs to zero. The value of Qs is given by (12):
¯ r β r β ( ) (
° S e , t S rβ e , t < 0
rβ )
Qs =u sq ids − uds iqs (12) From (17) and (18), the criteria to choose the voltage space
vectors are [9]:
From the induction machine equations [8], the stator flux d
.
and q component are given by (13). ( ) ( )
- If S rαβ erαβ , t > 0 Ÿ S rαβ erαβ , t < 0 , then the chosen

­°ψ sd = Ls isd + M ird voltage vector must increase the current irαβ ;
® (13) .
°̄ψ sq = Ls isq + M irq ( ) (
- If S rαβ erαβ , t < 0 Ÿ S rαβ erαβ , t > 0 , then the chosen)
Taking into account (5) and considering (9), the reactive voltage vector must decrease the current irαβ ;
power Qs is then given by (14):
( )
- If S rαβ erαβ , t = 0 the chosen vector must guarantee that

3 Vef the current irαβ is kept nearly constant;

Qs = (ψ s − M ird ) (14)
Ls B. MC Input Power Factor Control
Setting the reactive power Qs to zero, then the reference To control the MC input power factor it is necessary to
* control the reactive power Qr at the filter entrance.
current ird is given by (15).
Qr = vq id − vd iq (19)

* 1 3 Vef A zero vq component results by choosing a reference frame

ird = (15)
M ωs synchronized with the grid voltage.

III. MATRIX CONVERTER INPUT AND OUTPUT CURRENTS vd = 3 Vef ∧ vq = 0 (20)

CONTROL
Therefore, the reactive power only depends on voltage vd
A. Matrix Converter Output Currents Control and current iq. Knowing that the unitary power factor implies
Qr=0, iq must be zero to satisfy this condition, so the reference
* *
To follow the reference ird and irq rotor currents, which current iq* is set to zero in order to generate a near unity power
depend directly on the voltages applied to the MC output, urd factor at the input of MC.
and urq, the sliding mode control technique is used [9], [10].
*
Applying the inverse Park transformation to ird *
and irq iq* = 0 (21)
the reference currents ir*α and ir*β are obtained in αβ In this case the sliding surface is given by:
coordinates and the sliding surfaces are then given by (17) [9,
10]: (
S q = k q iq* − iq ) (22)

(
­°S rα (erα , t ) = kα ir*α − irα ) And the stability condition is:
®
( ) ( *
°̄S rβ erβ , t = kβ irβ − irβ ) (17)
( ) (
.
S q eiq , t S iq eiq , t < 0 ) (23)

The gains kα and kβ should be greater than zero with a limit According to (22), (23) and [9] the criteria to choose the
value imposed by the semiconductors maximum switching current state space vectors are:
frequency, as these gains affect directly the switching
frequency [10]. ( )
- If S q eiq , t < 0 the chosen vector must guarantee
.
To guarantee the right choice of the voltage state space
vectors, defined by the SVM [7], [9], [10] the stability
( )
S iq eiq , t > 0 , increasing the current iq;
conditions have to be satisfied. ( )
- If S q eiq , t > 0 the chosen vector must guarantee
.
( )
S iq eiq , t < 0 , decreasing the current iq.

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IV. WIND TURBINE SPEED AND TORQUE CONTROL As a second order system, the transfer function (27) can be
written in the canonical form (28):
A. Wind turbine speed control
From (7) the generator speed reference ω*m is given by ωm ω02
= (28)
(24), where G is the gear ratio. ω*m s 2 + 2ξω0 s + ω02

6.32497 Comparing (27) and (28), the gain Tp is obtained:

ω*m = G u (24)
R
1 4ξ 2Td
Tp = = (29)
Since the MC is set to follow a reference current it is ir* ωo K d Td Kd
necessary to design a control system able to generate this
reference in order to reach the reference speed ω*m as soon as B. Wind turbine torque control
possible. Knowing ωTopt (7) and replacing it in (4) results in the
To design the speed controller it is assumed that the matrix relation between the turbine torque (TMPPT) and the maximum
converter may be modeled as a first order system with a pole electrical power (30).
− 1 Td dependent on the switching period. The DFIG may also
be modeled as a first order system with one pole − K d J Pmax
TMPPT = (30)
dependent on the generator inertia. The block diagram of the G ωTopt
controlled system is shown in Fig. 3.
The maximum power is obtained from (6) and (5) using (7).
Tm Then, the reference torque is given by (31):
* Tem ωm
ω*m T 1 1 Kd
C (s )
sTd + 1 1 Kd s +1
0.843213 R 2 u 3
T* = (31)
G ωTopt
Figure 3. Block diagram of the system.
The reference torque T* will be further used to define the
The compensator will produce a reference torque T* that *
* reference current irq , that will be used to guarantee the torque
will establish the reference output current irq (11). Using the
based maximum power point tracking.
sliding mode control strategy, the state space vectors which
*
guarantee the desired irq and consequently the desired speed V. RESULTS

ω*m will be applied to the matrix converter. In order to evaluate the performance of these MPPT
techniques the whole model of the system was designed using
A proportional-integral controller (PI) is then used [9] to Matlab / Simulink.
guarantee a zero static error. A 1.5MW based Nordex S77 turbine, with R=36.5m and
gear ratio G=104 was used. The induction generator
sTz + 1 K parameters are those presented in Table I.
C (s ) = = Kp + i (25)
sT p s
TABLE I. DFIG PARAMETERS
In order to cancel the low frequency pole of the system at Vef (V) P (MW) rs (Ω) rr (Ω) Ls (mH) Lr (mH) M (mH) p
− K d J , the PI compensator zero Tz is given by (26): 690 1.5 2.2 1.8 3.02 2.95 2.9 2

Kd
Tz = (26) A wind speed chart was established according to some real
J speed records. These records are discrete and correspond to the
From Fig 3, (25) and (26) the closed-loop transfer function average speed per hour. However, due to CPU processing
of the system is given by (27): speed limitations, the simulation was reduced to 48 seconds,
based on 48 hours of wind records. Once the simulation time
1 had to be reduced, the moment of inertia J of the whole system
had to be reduced to acceptable values that allow the turbine
ωm T p K d Td speed to change within this time scale.
= (27)
ωm s 2 + 1 s +
* 1
From the total equivalent moment of inertia J of the
Td T p K d Td turbine, the damping ratio ξ of the compensator, and the delay
Td the speed controller was sized and the resultant parameters
are presented in Table II.

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TABLE II. SPEED CONTROLLER PARAMETERS Both control methods set a certain reference torque Tref and
J (kg m ) 2
ξ Td (ms) Tz (ms) Tp (ms) the MC is controlled in order to follow this reference. As Fig. 5
shows the reference torque is followed by the system in both
145.6 2 2 1 3.02 40
situations.
As Fig. 6 shows, the grid voltage and current are in
Using the same wind chart the numerical results obtained opposite phase guaranteeing Qs ≈ 0 thus supplying active
for both control systems are presented in Tables III to V. The
results of Energy Ratios (Table IV) and Energy Losses (Table power to the electric network. These results were obtained for
V) were calculated according to the formulas presented in the both control methods, keeping the power factor of the whole
Appendix. system above 0.99 during all the simulations.

TABLE III. ENERGY RESULTS

Torque control Speed control
Available wind energy [Wh] 13668 13668
Turbine delivered energy [Wh] 13490 13653
Generated energy [Wh] 13073 13122
Energy insert into the network [Wh] 12828 12879

TABLE IV. ENERGY RATIOS

Torque control Speed control
Cp (%) 40.662 41.154
Generator efficiency (%) 96.908 96.111 a)
MPPT efficiency (%) 93.856 94.227
Rotor generated energy (%) 1.721 5.283
Stator generated energy (%) 98.279 94.717

TABLE V. ENERGY LOSSES

Torque control Speed control
Global losses (%) 4.905 5.671
Generator losses (%) 3.092 3.889
Converter losses (%) 1.746 1.724
Filter losses (%) 0.0960 0.1030
b)
Figure 5. Electromagnetic torque Tem (blue) and reference torque T* (green)
Fig. 4 shows the generator speed for both control cases. obtained for: a) Torque control technique; b) Speed control technique.

a)

Figure 6. Grid voltage (blue) and stator current (green) in opposition of

phase.

VI. CONCLUSION
Even though both control methods were designed to follow
the maximum power, they present different results.
The fraction of the available wind energy delivered to the
b) network is slightly larger (0.37%) for the speed control case.
Figure 4. Generator speed (rpm) ωm (blue) and ω* (green): a) Torque control
However, in the torque control method the reference torque is
technique; b) Speed control technique. set continuously and within a smaller range of values avoiding
the fast changes of the rotor current and consequently the
generated power when the wind speed changes. Another

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important detail is that when the wind speed drops too fast, the Eturbine − E generated
speed controller forces the system to generate power above the LDFIG = (A7)
nominal value. Eturbine

In spite of delivering higher power to the grid, the speed E L MC

control is very sensible to wind speed variations causing highly LMC = (A8)
E generated
variable rotor currents and generated power. This may be a
negative characteristic to the electrical grid that needs to satisfy
E L filter
the instantaneous load demand. L filter = (A9)
E generated
APPENDICES The MC uses ideal switches in the Simulink schematics.
Thus, the MC losses are the result of the conduction losses and
A. Energy ratios and losses the snubber circuit losses. Since there are no switching losses
The calculation of the energy ratios and the different losses covered by this Simulink block, they are included in the
of the system was done assuming the following equations: snubber circuit. The values used are:
C s = 0.5μF for the snubber capacity of the ideal switch;
Cp – Performance coefficient or power coefficient for the wind
power Rs = 500Ω for the snubber resistance of the ideal switch;
Egenerated – Energy generated by the DFIG [Ws]
Egrotor – Energy generated by the rotor [Ws] Ron = 3.3mΩ for the switches internal resistance.
Egstator – Energy generated by the stator [Ws]
ELfilter – Energy correspondent to the filter losses [Ws] ACKNOWLEDGMENTS
ELMC – Energy correspondent to the MC losses [Ws]
This work was supported by national funds through FCT –
Enetwork – Energy delivered to the network [Ws]
Fundação para a Ciência e a Tecnologia, under project PEst-
Eoptimum – Maximum of available wind energy [Ws]
OE/EEI/UI4064/2011.
Erotor – Fraction of the generated energy by the rotor
Estator – Fraction of the generated energy by the stator REFERENCES
Eturbine – Energy delivered from the turbine to the system [Ws] [1] World Wind Energy Association WWEA, Half Year Report 2012.
Ewind – Total energy in the wind [Ws] [2] MÜLLER, S.; Deicke, M.; Rik, W.; "Doubly fed induction generator
LDFIG – Generator losses systems for wind turbines", IEEE Industry Applications Magazine,
Lfilter – Filter losses vol.8, no.3, pp.26-33, May/Jun 2002.
Lglobal – Global losses of the system [3] Barakati, S.; Modeling and Controller Design of a Wind, Doctoral
LMC – Matrix Converter losses Disseratation, University of Waterloo, Waterloo, Canada, 2008.
ηDFIG – Efficiency of the generator [4] Pena, R.; Clare, J.C.; Asher, G.M.; Doubly Fed Induction Generator
Using Back-to-Back PWM Converters and its Application to Variable-
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E wind POWERENG´07, International Conference on Power Engineering,
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Erotor = (A4) [9] Afonso, L., "Maximum Power Point Tracker of Wind Energy
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Eturbine − Enetwork
Lglobal = (A6)
Eturbine

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