High Performance Control of a Permanent Magnet Wind Power Generation System Using an Adaptive Sliding Observer
Zhuang Xu, Pengyao Ge and Dianguo Xu
Senior Member , IEEE Harbin Institute of Technology
150001Harbin, China [email protected] Abstract -- The paper proposes an adaptive sliding observer
to accurately estimate the rotor position for a direct-drive permanent magnet synchronous generator (PMSG). Without any position sensor, the generator torque and flux responses retain high performance both in the steady state and during transients. The problem on the shaft encoder installation difficulty, reliability and disturbance in encoder cable is remedied and the reliability of the PMSG generating system is increased. The grid voltage orientation control is implemented into the grid converter in which nearly sinusoidal input current and fast dc link voltage regulation can be achieved.
Experimental results verify the effectiveness of the proposed observer for PMSG and high performance power production can be obtained.
Index Terms-- Permanent Magnet, Wind Power, Sensorless, Sliding observer.
I. INTRODUCTION
A direct-drive wind generation system employing variable- speed multi-pole Permanent Magnet Synchronous Generator (PMSG) is illustrated in Figure 1. The generator side PWM converter controls the generator torque according to the power reference of the generator, while the grid side inverter controls the dc-link voltage and the active and reactive power flows to the utility grid. The direct-drive PMSG wind generation system eliminates the gearbox, increases the reliability and reduces the system cost. Moreover, the PMSG generation system has very low audible noise and high motor load-ability and produces power energy continuously without any concern of encoder failure with the integration of rotor position observer.
In the applications related to the PMSG wind power systems, Field Oriented Control (FOC) technique of ac machines appears to dominate the thoughts of the academia and industry. The FOC has been used in permanent magnet synchronous motor control [1]–[3], and it was extended naturally to the PMSG wind power application [3]. The rotor position is essential for the coordinate transformation in the
This project is funded by National Natural Science Foundation of China (50777013)
Fig. 1. A direct drive variable speed PMSG wind generator system connected to the power grid.
fields decoupling. Therefore, encoder-based position sensing methods or observers are required for both generating and motoring with FOC.
To maximize the capture of the wind power, maximum power point tracking strategies need to be implemented to control the generator. The optimum active power setting or torque reference can be calculated according to the actual generator speed without any wind speed sensor. The generator operates in torque control mode and the speed loop is not used. Considering the encoder installation difficulty, reliability and disturbance in encoder cable, the reliability of the PMSG generating system has to be increased. Rotor position or speed identification methods should be used. A solution is to adopt a closed-loop observer which makes use of error correction term in order to increase its robustness and improve dynamic performance. The accuracy of the observer is crucial in very low speed range. This paper proposes an adaptive sliding observer that is supported by the theory of variable structure systems [4-7]. The sliding-mode method possesses robustness over a specified magnitude range of system parameter uncertainties and disturbances.
II. MODELING OF PM SYNCHRONOUS GENERATORS The circuit equations of PM synchronous generators in the rotor d–q rotating coordinate are given by (1)
d d re q d 0
re d q q re f
q
v R pL L i
L R pL i v
ω
ω ω λ
+ −
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
= +
⎢ ⎥ ⎢⎣ + ⎥ ⎢ ⎥ ⎢⎦ ⎣ ⎦ ⎣ ⎥⎦
⎣ ⎦ (1)
For the surface-mounted PMSG, the equation (2) can be derived by transforming (1) from d-q to α β− coordinate.
0 sin
0 cos
s re
re f
s re
v R pL i
v R pL i
α α
β β
ω λ θ θ
+ −
⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤
= +
⎢ ⎥ ⎢⎣ + ⎥⎦⎢ ⎥ ⎢⎣ ⎥⎦
⎣ ⎦ ⎣ ⎦ (2)
The EMF is expressed by (3).
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sin cos
re re f
re
e e e
α β
ω λ θ θ
⎡ ⎤ ⎡− ⎤
=⎢ ⎥= ⎢ ⎥
⎣ ⎦
⎣ ⎦ (3) Using (2), a current model of the IPM synchronous motors can be described by equation (4).
/ 0 1/ 0
0 / 0 1/ 1/
s s
s
s s
i e v
i R L L
i e L v
i R L L
α α α
α
β β β
β
⎡ ⎤ ⎡− ⎤⎡ ⎤ ⎡− ⎤⎡ ⎤ ⎡ ⎤
= + +
⎢ ⎥ ⎢⎣ − ⎥⎦⎢ ⎥⎣ ⎦ ⎢⎣ − ⎥⎦⎢ ⎥⎣ ⎦ ⎢ ⎥⎣ ⎦
⎣ ⎦
(4)
Differentiating (3) gives the dynamics of the back EMF based on the assumption that the electrical system’s time constant is smaller enough than the mechanical one; the rotor speed can be considered as a constant during the sampling period, i.e. . Under velocity control, the EMF dynamics are expressed as (5).
re re
e e
e e
α β
β α
= −
=
ω ω
(5) From the new model of (4) and (5), the PM synchronous generator can be described by state space equations as (6).
11 12 1
0 22 0
A A B
A i i
e e v
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥
⎢ ⎥ ⎢⎣ ⎥⎦ ⎣ ⎦ ⎣ ⎦
⎣ ⎦= ⋅ + (6) where i=[ , ]i iα β T and e [ e , e ]= α β T
11 12 22 1
1
1 1 0 0 1
0 1
1 0
s s re
s
A ( R / L ) I A ( / L ) I
A J
B ( / L ) I I
J
= − ⋅
= − ⋅
= ω
= ⋅
⎡ ⎤
= ⎢ ⎥
⎣ ⎦
−
⎡ ⎤
= ⎢ ⎥
⎣ ⎦
III. AN ADAPTIVE OBSERVER FOR SENSOR-LESS CONTROL OF PMSG
A. The design of the observer
The observer is designed based on the stator current model (4). With correction gains multiplied by switching functions, the adaptive observer is in the form of (7) and the overall structure is shown in Figure 2.
11 12 1
22 0
0
ˆA A B
ˆA
ˆ i
i
ˆe ˆe v K sgn( s )
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎢ ⎥ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⋅ +
⎢ ⎥ ⎢⎣ ⎥⎦ ⎣ ⎦ ⎣ ⎦
⎣ ⎦ + (7)
Where ˆi
ˆi ˆiα
β
= ⎢ ⎥⎡ ⎤
⎢ ⎥⎣ ⎦, ˆe
ˆe ˆeα
β
= ⎢ ⎥⎡ ⎤
⎣ ⎦ and
i ˆi i ˆi
s α α
β β
⎡ − ⎤
= ⎢ ⎥
−
⎢ ⎥
⎣ ⎦
ˆA11= −( R / L ) Is ⋅ Aˆ22= ωˆreJ
1
0
⎡ ⎤B
⎢ ⎥⎣ ⎦
11 12
0 22
ˆA A
ˆA
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎣ ⎦
i ˆe
⎡ ⎤⎢ ⎥
1 ⎣ ⎦
S [I 0]
11 12
0 22
ˆA A ˆA
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎣ ⎦
[0 I] tan-1
sgn(s ˆs) K i i−
vs
iS
ˆS i
θˆ
Fig.2. The overall structure of the adaptive observer.
i e
K K K
⎡ ⎤
= ⎢ ⎥
⎣ ⎦
1 2
0 0
i i
i
K k
k
⎡ ⎤
= ⎢ ⎥
⎣ ⎦
The observer switching manifold is defined as
1 2
s i ˆi
s i ˆi
s α α
β β
⎡ − ⎤
=⎡ ⎤⎢ ⎥⎣ ⎦ ⎣= ⎢⎢ − ⎥⎥⎦. The sliding surface is given by s=0. The
error components areiα = − ,iα ˆiα iβ= − iβ ˆiβ eα =eα− and eˆα eβ = − eβ ˆeβ
Then, by subtracting (7) from (6), the error dynamics for the current estimation are given as follows.
1 1
2 2
1 1
i s
i s
s i / L e k sgn( i )
s i / L e k sgn( i )
s α α α
β β
β
⎡ ⎤ ⎡− ⋅ − ⋅ ⎤
⎡ ⎤ ⎢ ⎥
=⎢ ⎥⎣ ⎦=⎢ ⎥ ⎢⎣ ⎦=⎢⎣− ⋅ − ⋅ ⎥⎥⎦
(8)
When the error dynamics reach the sliding-mode surfaces, s1ands2, the error between the actual and the estimated currents will decay to zero. Through the stability analysis, the observer gain K can be determined by the sliding-mode reaching conditions, which is given by the condition:s sT⋅<0.
Boths s1⋅ <1 0ands s2⋅ <2 0are the sufficient conditions to satisfy the reaching of sliding-mode. These conditions guarantee that in finite time S will equals zero and the states stays on the switching surface. Thereafter, the gains of the reduced-order observer can be finalized when the states have reached the sliding manifold and are forced to stay on the surface, i.e. s s= = 0.
The time derivatives of the switching functions are:
1 1 1
2 2 2
i s
i s
i e / L k i s s i i
s s i i i e / L k i
α α α
α α
β β β
β β
⎡ ⋅ ⎤ ⎡− ⋅ − ⋅ ⎤
⋅
⎡ ⎤ ⎢= ⎥=⎢ ⎥
⎢ ⋅ ⎥ − ⋅ − ⋅
⎢ ⋅ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦
(9)
In practice, the extended EMF eαand eβare bounded.
η and1 η are considered to be two known positive parameters 2 satisfying eα < η and 1 eβ < η .2 k1iand k2ihave to be large enough to ensure robustness of the observer to disturbance,
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parameter mismatches and speed estimation error.
If k1i > η +( 1 ˆe ) / Lα sand k2i > η +( 2 ˆe ) / Lβ sthe reaching conditions are satisfied. The sliding motion occurs and the trajectories remain on the sliding surfaces. Then iα and
iβdecay to zero, which means that the estimated currents will converge to their actual values. After sliding take places and the rotor speed is estimated accurately enough, the equivalent control information of the two discontinuous high switching control components is extracted in (10).
1 2
1 0
0 1
i
s i
eq s
e k sgn( i ) / L
/ L e k sgn( i )
α α β β
⎡ ⋅ ⎤ ⎡− ⎤⎡ ⎤
=
⎢ ⋅ ⎥ ⎢ − ⎥⎢ ⎥
⎢ ⎥ ⎣ ⎦ ⎣ ⎦
⎣ ⎦ (10)
The EMF estimation gain matrix Ke is set as
1 1 2
1 2 2
ˆ ˆ
i e i
e s re s
i i e
s re s
k L k L k
K k L k L k
ω ω
⎡− ⎤
= ⎢ ⎥
− −
⎣ ⎦.
1
ke and k2e are positive observer gains which are determined by Lyapunov stability analysis. The rotor angle is observed using phase angle difference between ˆeαandˆeβ. i.e.
1 re
ˆ tan ( ˆe ) ˆe
− α
β
θ = − (11)
B. Convergence and Stability
After sliding motion happens and the rotor speed is estimated accurately enough, with the choice of EMF estimation gain matrix Kein (11), the extended EMF error dynamics is given by
(12) Substituting from (10) yields
(13) The extended EMF errors converge exponentially to zeros.
1e
k and k2eare positive gains and chosen for the desire system dynamics. Complete robustness is achieved during the whole transients and the system will converge asymptotically to the origin with time constant of 1/k. The convergence rates of error dynamics are determined by k1eandk2e. The choices of the observer gains do affect the drive system performance.
If k1i andk2i are much larger than their thresholds, the estimated states would generate chattering, especially in very low speed region. The selection ofk1eandk2e should also be careful to make the system converge faster.k1i,k2i,k1eandk2ehave been scheduled according to the speed variation.
The sliding observer developed in last Section assures the estimation of the rotor angle of the PM generator. However, the fast switching may generate unexpected chattering. The high frequency components of the chattering are undesirable because they may excite un-modeled high frequency system dynamics and even result in unforeseen instability. To alleviate the problem, the discontinuous part of the observer could be smoothed out by introducing a boundary layer around the switching surface. This results in replacing the switching function by a continuous function around the sliding surface neighborhood. In contrast to the adoption of a boundary layer to eliminate chattering, the observer gain can be modified as a function of s to eliminate the sliding chattering. This technique does not reduce accuracy and robustness. Fuzzy logic rules can be adopted for the sliding- mode gain function.
If s is large, then k is large.
If s is small, then k is small.
If 0s sΔ > and sΔ is large, then k is large.
If 0s sΔ > and sΔ is small, then k is small.
If s sΔ < and s0 Δ is large, then k is small.
If s sΔ < and s0 Δ is large, then k is small.
To satisfy the above rules, the sliding gains are designed as follows:
(14) n=1,2; c, σ and n σ are positive tuning parameters. Δn
IV. EXPERIMENTAL RESULTS
As illustrated by the photos in the fig.3 and fig.4, an experimental platform has been set up to test the performance under the field oriented control with the adaptive sliding observer without any shaft encoder. The active power reference is provided by tracking the maximum wind power point. The torque component of the generator current is regulated with the decoupling control. The grid voltage orientation control has been implemented to keep the dc-link voltage constant and control the active and reactive power separately. The controller board consists of TMS320F2812 DSP and EPM570CPLD from Altera. The DSP takes care of control strategy, position and stator flux estimation algorithms.
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Fig. 3. Experimental platform for the testing of the 10 kW permanent magnet synchronous generator
Fig.4. The control cabinet with prime mover variable speed drive inside and two 750kW power modules
The CPLD handles the logics of PWM driving signals and protects the IGBT power module. An induction motor fed by a variable-speed drive worked as the prime mover to emulate the wind turbine. The PMSG was coupled with the induction motor through a gear box with the transformation ratio 1440:200. A torque meter was installed on the shaft. The parameters for the PMSG and induction motor were listed in Table I.
Firstly, the steady state performance was tested without any position sensor and the estimated rotor angle was obtained from the observer. Fig. 5 demonstrates the high performance control with the proposed adaptive sliding observer when the generator was driven at 120 rpm. The d- axis current was regulated at zero for the optimum torque control while the q-axis current command was set to - 0.2(p.u.). It can be noted that both controlled currents tracked the references very well. It can be found that the current waveform maintains perfect sinusoidal. The bottom subfigure of fig.5 recorded the estimated rotor angle with which the generator’s torque and flux were decoupled and controlled using field oriented control method. The chattering problem had been remedied effectively. It can be seen from the figure that the dc-link voltage was stable enough and the estimated rotor position was very smooth. The estimation error between the estimated and measured rotor position was examined and shown in fig.6. With the adaptive observer, the
estimation accuracy was very high. The pulse jump in the figure was due to the period jump from the present period to the next one.
The grid voltage orientation had been implemented to keep the dc-link voltage constant and control the active and reactive power separately. During the experiments, the generator side operation was launched and operating frequency reached to 26.6 Hz after the grid side converter was started. The DC-link voltage was kept constant at 600V.
It can be found in the Fig.7 that full power generation of the 10kW PMSG generation system was recorded. The reactive power was set to zero and the unity power factor had been achieved. The waveforms of the generator current (26.6Hz), grid voltage (50Hz) and grid current exhibit very good sinusoidal forms.
Fig 5. Test of sensor-less steady state operation at 120rpm for the 10kW PMSG wind generator system.
Fig.6 The estimation error between the measured and estimated.
Fig 7. Full power generation of the 10kW PMSG generation system.
From the top to bottom, grid current (green), dc-link voltage(yellow), grid voltage (pink) and the generator current(blue).
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APPENDIX Table I
PMSG
Parameters Values Stator resistance 0.45Ω
Inductance 0.005Η
Permanent magnet flux 0.8Wb
Rated speed 200rpm
Pole pairs 8Pairs
Start torque <6.5Nm
Rated current 43.78A
Rated line voltage 230V
Rated power 10kW
Induction motor
Rated phase voltage 220V
Rated current 30.3A
Rated speed 1460rpm
Rated frequency 50Hz
Power rating 15KW
Torque meter
Measuring range 1000Nm
Tooth number 1440
Speed range 3000rpm
ACKNOWLEDGMENT
The authors gratefully acknowledge the contributions of Zhou Weilai, Sun Jinghua and Liu Jia who are with JZE for their work with this project.
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