Detailed Dynamic Modeling of Permanent Magnet Synchronous Machine based Wind Turbine for Power System Dynamic Analysis

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Detailed Dynamic Modeling of Permanent Magnet

Synchronous Machine based Wind Turbine for Power

System Dynamic Analysis

M. A. Tabrizi, IEEE Graduate student member , G. Radman, IEEE Senior member ,

Electrical and Computer Engineering Department

Tennessee Technological University

Cookeville, TN, USA

Abstract — Wind energy system as a renewable energy source

has been developed remarkably during the last decade to

supplement large scale power systems, micro grids and smart

grids. While Considerable research efforts have been directed to

wind energy system modeling and analysis, comprehensive modeldevelopments for grid integration studies have been relatively

sparse. This paper presents a detailed dynamic modeling for wind

energy conversion system based on Permanent Magnet

Synchronous Generator. The wind turbine system is equipped

with maximum power point tracking module and pitch angle

control. The control system aims to regulate the generator shaft

speed, DC link voltage, and output reactive power while

minimizing generator losses. The overall system is simulated in

MATLAB. The simulation result indicates the presented model

adequately represents the details of the system performance while

meeting all control objectives.

I. I NTRODUCTION

Wind energy conversion system converts the energy storedin wind to the mechanical energy ready to be used bygenerator. Wind turbine may operate with either fixed speed(approximate speed deviation within 1% of nominal speeddetermined by power system) or variable speed. Since fixeds peed wind turbine’s generator, which is an induction machine,is directly connected to the grid, the generator shaft speed isalmost fixed to the grid frequency. Thus, storing power fluctuations caused by wind variations and drive train is not possible. Therefore, fixed-speed wind turbines have a negativeimpact on the power quality of the grid [1]. Variable speedwind turbines are connected to the grid through power electronic devices making it possible to control the shaft speed.As a result, the power variations caused by wind turbulence can

be absorbed by changing the shaft speed. Hence, in comparisonwith fixed-speed wind turbine, variable speed wind turbinesresult in a better grid power quality [2].

Permanent Magnet Synchronous Generator (PMSG) is oneof the most popular kinds of synchronous generators used invariable speed wind system applications [3] [4]. No need for external excitation system, no copper losses in the rotor circuits, relatively maintenance-free operation, high efficiency,high reliability and high power density are the main reasonscontributing to their popularity [5]. Different types of PMSGs

such as Interior Permanent Magnet Synchronous Generator (IPMSG) and Surface-mounted Permanent MagnetSynchronous Generator (SPMSG) have been utilized in power systems applications and electric machine industries [6]. In

SPMSG, the magnets are mounted on surface of the rotor anddue to their high reluctance, the equivalent air gap is large andconsidered as uniform making the saliency effect negligible. Asa result, the quadrature-axis synchronous inductance is equal todirect-axis one and only magnet torque is produced. Incomparison with SPMSG, IPMSG has gained more attention inwind energy industry. Due to non-uniform equivalent air gap,the saliency effect in IPMSG is significant. The quadrature-axissynchronous inductance is usually larger than its direct-axissynchronous inductance providing both magnetic andreluctance torque. Therefore, IPMSG offers higher efficiencyin addition to higher controllability meaning that the output power could be kept constant within a wider speed range byutilizing flux weakening regime; magnets are covered and

protected providing more robust and solid structure [7], [8], [9].This paper presents a detailed dynamic modeling of a

IPMSG based wind turbine as shown in Fig. 1. The turbine isconnected to the grid through a back-to-back converter and isassumed to be equipped with maximum power point trackingmodule and pitch angle control. The control system aims tocontrol generator shaft speed, DC link voltage, output reactive power and minimize the generator losses. While the presentedmodel is detailed enough to capture all performance objectives,this model can be integrated to power system model for the purpose of transient and dynamic stability studies.

Series filter &

transformer

IPMSG

Gear Box

(GB)

≈

= =

MSC GSC

≈

= V t

R fw+jX fw

I t V dc

I 1 I 2

V s

I s

P t ,T t Grid

Control

System

( M 1) ( M 2)

Measurements Measurements

V

Fig. 1. IPMSG based wind turbine system connected to the system.

Center for Energy System Research, Cookeville, TN.



The remainder of the paper is organized as follow: SectionII, III and IV illustrate the wind power, aerodynamic power control and maximum power point tracking, respectively.Section V and Section VI present a comprehensive formulationfor IPMSG and its control system modeling. Section VII provides the model performance test results and Section VIIIconcludes the paper.

It should be noted that all the symbols and notations used in

this paper are standard symbols for corresponding parametersand their definitions may be found in [10] [11].

II. WIND POWER

Power extracted from the wind by wind turbine blades isgiven by:

3),(2

1w pt V C A P (1)

According to Betz Law, the highest possible value for power coefficient, C P , is proved to be 0.5926 [10]. Furthermore, the power coefficient will be even smaller than Betz limit and can be expressed as a function of tip speed ratio and wind turbine pitch angle. For 3-blade wind turbine, the power coefficient isgiven by [12]:

)17.0(2)6.5022.0(5.0),(

eC p (2)

where for 3-blade wind turbine the tip speed ratio is defined as:

W

t

V

R

(3)

As seen from above equations, at a given wind speed, theamount of power captured by wind turbine blades is onlycontrolled by turbine shaft angular speed and turbine blades pitch angle.

III. AERODYNAMIC POWER CONTROL

Fig. 2 shows an ideal wind turbine output power vs. windspeed curve. At any wind speed higher than rated wind speed,it is necessary to limit the input power to the wind turbine to prevent turbine mechanical and electrical damages, i.e.aerodynamic power control. There are three major ways of performing the aerodynamic power control namely stall, activestall and pitch control [10]. For newer larger wind turbines, pitch control is the most common method of controlling theaerodynamic power at wind speed higher than rated windspeed. Below rated wind speed, the turbine should use a pitchangle that maximizes the energy capture from the wind i.e. pitch angle equal to zero. Above rated wind speed, the pitchangle is controlled in such a way that the angle of attack isdecreased and aerodynamic power remains fixed at its ratedvalue. The dynamics of the pitch control is moderately fast andcan have significant impact on dynamic studies. The pitchcontrol block diagram is shown in Fig. 3.

In this model, the blade position actuators are rate limitedand there is a time constant associated with the translation of blade angle to mechanical output. The pitch control does notdifferentiate between shaft acceleration due to increase in windspeed or due to system faults. In Fig. 3, for power levels aboverated, the rotor speed will be controlled primarily by the pitch

control, with the speed being allowed to rise above thereference transiently.

Fig. 2. Typical variable-speed wind turbine output power vs. wind speed

curve (ideal case)

∑Delay Block

& Rate limit

P-I +-

P-I +-

Rotor speed

maximum

speed

Output

power

Maximum

power

cmd

Blades

pitch

Fig. 3. Turbine pitch control block diagram

Fig. 4. Power Coefficient variations as a function of Tip Speed Ratio and

pitch angle for a typical wind turbine

IV. MAXIMUM POWER POINT TRACKING

Fig. 4 shows the power coefficient as a function of tip speedratio and pitch angle for a typical 3-blade wind turbine. Asdiscussed, at any wind speed below the rated wind speed, the pitch angle is kept at zero. Therefore, below the rated windspeed, the only control variable is the tip speed ratio which isitself a function of turbine shaft angular speed. As seen fromFig. 4, for each power coefficient curve at any given pitchangle, there is unique tip speed ratio ( λ

mppt ) which maximizes

the power coefficient and consequently the captured power from wind. This tip speed ratio corresponds to a unique turbineshaft angular speed (ωmppt ). For a wind speed below the ratedwind speed, maximum power point tracking is guaranteed if thetip speed ratio is equal to:

)6.5022.0(17.0

1 2 mppt

(4)

0 7 18 2425

0

20

40

60

80

100

120

Wind speed (m/s)

p o w e r ( % )

rated power at rated wind

speed

cut-in windspeed

cut-outwind speed

05

10 1520

2530

00.5

1

1.5

2-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

Tip Speed Ratio ()Pitch Angle ()

P o w e r C o f f i c i e n t ( C P

)



V. IPMSG BASED WIND SYSTEM MODELING

In following subsections, IPMSG dynamic model is presented based on synchronously rotating qd reference frametheory in which the quadrature axis (q-axis) leads the directaxis (d-axis) by 900 [13] i.e. the voltage or current space vector in abc reference is projected into q-axis and d-axis as follows:

)()(

t jd q e jf f f

(5)

Dynamic models are given in terms of per unit quantities.Turbine rated power and magnitude of rated phase voltage atIPMSG terminal are chosen as base power and base voltage for per unitizing of the dynamic model of IPMSG.

A. IPMSG and MSC Dynamic model

In contrast to conventional synchronous generators, IPMSGdoes not have any damper winding located on its rotor. Inaddition, the magnetic field is provided by permanent magnetswhich are poor electrical conductors. As a result, the dynamicof the rotor can be neglected. The IPMSG dynamic model inrotor reference frame (Park’s transformation) is given by:

sq sqr sd b

sd

sd s sd I X pI

X

I RV

mr sd sd r sqb

sq sq s sq I X pI

X I RV

sqm sd sq sd sqe I I I X X T )(

(6)

The electrical torque expression is positive for generator action with positive direction of stator current assumed out of the stator terminals. The torque and generator shaft speed arerelated by the following dynamic equation:

et r T T p H 2 (7)

where H is the turbine and generator shaft inertia constant in

second.

Machine Side Converter (MSC) can be modeled usingaverage model i.e. controllable voltage source:

2;

211

dcd sd

dcq sq

V M V

V M V (8)

where M 1 is a modulation index and is decided by controlloops.

B. DC link dynamic model

Using KCL, the dynamic equation for DC link of the back-to-back converter in per unit is given by:

211 )(2

11

I I M I M pV B sd d sqqdccb

( 9)

C. GSC and R-L series filter dynamic model

Similar to MSC, Grid Side Converter (GSC) can bemodeled using average model i.e. controllable voltage source:

2;

222

dcd d

dcqq

V M V

V M V (10)

where M 2 is a modulation index and is decided by GSCassociated control loops. Using KVL, the dynamic equationassociated with the series R-L filter in synchronously rotatingreference frame is given by:

d fweqb

fwq fwqtq I X pI

X I RV V

q fwed b

fw

d fwd td I X pI

X I RV V

(11)

Active and reactive power flow equation at IPMSM PointOf Interconnection (POI) bus in synchronously rotatingreference frame is given by:

qtd d tq gwd td qtq gw I V I V Q I V I V P ; (12)

Using Phase Lock Loop (PLL), the q-axis and d-axiscomponents of the POI voltage can be obtained such that the q-axis component is aligned with the POI bus voltage i.e. V tq=V and V td =0 making the decouple control of active and reactive power possible.

VI. IPMSG CONTROL SYSTEM DESIGN

As mentioned, in this paper four control objectives areconsidered. In following sub-sections each of these controlobjectives is explained in details and controller parameters aredesigned based on well-known pole-placement strategy.

MSC is assigned with two tasks namely: maximum power point tacking and generator loss minimization. To meet thesetwo control objectives, two control loops based on the model provided in previous sections is developed. The stator referencecurrent components (d and q-axis) are calculated using thesetwo control loops such that the two objectives are met. Stator reference current components (q and d – axis) generated by theseouter-level loops are fed to inner-level current control loopswhich consequently generate the reference voltages for

Voltage-Source Converter switching.

In order to extract the maximum power from wind power atany given pitch angle and wind speed below the rated speed,the generator electrical desirable (reference) rotor speed inrad/sec is given by:

R

V n

Pole MPPT w gb MPPT r

2, (13)

For generator rotor speed to follow the above referencevalue, a Proportional-Integrator ( PI ) controller is used. Thiscontroller is designed using per unitized generator torque-speedequation:

et r

T T p H 2 (14)

The closed loop transfer function for the speed control loopis given by:

)2

()S2

(2H

1)(

2,

H

k

H

k S

k S k sG

ω

iω

p

i p

MPPT r

r

(15)



Comparing the denominator of above 2nd order transfer function with Butterworth polynomial, the PI gains arecalculated:

020 22;2

H k H k pi (16)

Neglecting all other generator losses except stator copper losses, total generator losses are given by:

)(22

sq sd sloss I I R P (17)

The second objective is to minimize the stator copper lossessubject to the fact that the generator electrical torque must bemaximized for any given wind speed below the rated speed i.e.:

MPPT t sqm sd sq sd sqe T I I I X X T )( (18)

This minimization problem subject to single equalityconstraint can be formulated as a Lagrange Multiplier optimization problem. Implementing the necessary andsufficient condition for this optimization problem results in thefollowing 4th order polynomial:

001

,12

,23

,34

,4 a I a I a I a I a ref sd ref sd ref sd ref sd (19)

where

23

34 )(3;)( sd sqm sd sq X X a X X a

31

22 ;)(3 m sd sqm a X X a

20 )( MPPT sd sq T X X a

(20)

In order to satisfy both control objectives, i.e. maximum power point tacking and loss minimization, (14) and (20) are

combined. Therefore definition of a0 is replaced by:

2

0)()(

t sd sqT X X a (21)

The quadrature component of stator current reference is given by:

m sd sd sq

t ref sq I X X

T I

)(, (22)

While outer loops provide the inner current control loopswith current reference values, the inner current loops generatethe desirable modulation indices used to produce switchingsignals for converter. MSC current control loops are shown inFig. 5.

To design the gains of PI controllers in current control

loops, their transfer functions are compared with the 2

nd

order Butterworth Polynomial. The gains are calculated as follow:

200 ;2

b

sd iisd s

b

sd pisd

X K R

X K

200 ;2

b

sqiisq s

b

sq pisq

X K R

X K

( 23)

+-

+-

I sd,ref

I sd

M d1++isd , p K

S

K isd ,i

sq sqr I X

dcV

2

+-

++

- I sq,ref

I sq

M q1++isq , p K

S

K isq ,i

sd sd r I X

mr

dcV

2

Fig. 5. Machine side converter current control loop

Despite of MSC, GSC is locked to the grid frequency. It isassigned the task of DC link voltage control and reactive power (or power factor) control at the POI bus. To regulate the DClink voltage, a PI controller is used. Using the dynamic

equation for DC link voltage, the control loop transfer functionis given by:

c

bidc

c

b pdc

idc pdc

c

b

ref dc

dcdc

B K S

B K S

K S K

BV

V sG

2,

)(

(24)

Comparing the denominator of above 2nd order transfer function with Butterworth polynomial the PI gains arecalculated:

200 ;2

b

cidc

b

c pdc

B K

B K (25)

Assuming the quadrature-axis is aligned with IPMSG

terminal voltage, the direct axis of reference current at POI buscan be calculated using the reference reactive power:

tq

ref g ref d

V

Q I

,, (26)

Neglecting the inverter power loss, the following is proven:

)(2

1222 d d qq I M I M I (27)

In order to implement the DC link control and reactive power control, the quadrature component of terminal referencecurrent is calculated by combining (9) and (27):

2

,211

,

2)(

q

ref d d c

dcb sd d sqq

ref q M

I M B

pV I M I M

I

(28)

Similar to MSC control loops, while GSC outer controlloops provide the inner current control loops with currentreference values, the inner current loops generate the desirablemodulation indices used to produce switching signals for GSC.GCS current control loops are shown in Fig. 6.



+-

+-

I d,ref

I d

M d2++id , p K

S

K id ,i

q fwe I X

dcV

2

td

+

+-

++

+ I q,ref

I q

M q2+

+iq , p K

S

K iq ,i

d fwe I X

tqV

dcV

2

Fig. 6. Grid side converter current control loop

Based on the similar procedure used for MSC, the PI gainsfor GSC current control loops are given by:

fw

b

fwiq pid p R

X K K 0,, 2

20,,

b

fwiqiid i

X K K

(29)

VII. IPMSG MODEL PERFORMANCE TEST

In this section, IPMSG model performance in case of different disturbances is evaluated. Test system is assumed to be an IPMSG connected to the infinite bus through its POI bus.The infinite bus with a known voltage magnitude and anglerepresents a strong power grid. Two different scenarios areconsidered and the model performance is evaluated as follow:

A. pitch angle control activated

As the first scenario, IPMSG based wind turbine responsesto the change in wind speed is considered while pitch control isactivated and turbine is working in unity power factor mode.

The results are shown in Fig. 7. As shown in this figure, the pitch control loop is set such that for a wind speed above 13.5m/s (1.156 pu), the blades are pitched to keep the turbine shaftspeed and output power at the maximum acceptable values andonce the wind speed falls below the maximum acceptablevalue, the pitch angle goes back to zero to let the turbine work at maximum power point tracking mode.

B. pitch angle control deactivated

As the second scenario, IPMSG based wind turbineresponse to the change in wind speed is considered while pitchcontrol is deactivated and turbine is working in unity power factor mode. The results are shown in Fig. 8. As shown in thisfigure, turbine pitch angle is kept at zero for all the wind speedsand turbine works under maximum power point tracking modeeven though speed, output power and current is far more thanthe nominal and acceptable values.

VIII. CONCLUSION

This paper presented a comprehensive dynamic model for Internal Permanent Magnet Synchronous Generator based windsystem. While numerous research efforts have focused uponthe modeling of IPMSM, the developments of the

comprehensive model required to evaluate the technology for the purpose of grid interconnection have been relatively sparse.The presented model includes maximum power point trackingmodule, pitch angle control and average model for grid sideand machine side converters. Control system aims to controlthe generator speed, DC link voltage and reactive power whileminimizing the generator ohmic losses to achieve better efficiency. While this model is detailed enough to capture all performance objective, this model can be integrated to power system model for the purpose of transient stability studies

Simulation results indicate that the model accurately andadequately represents the system performance details and allcontrol objectives are met. However, as an ongoing research

effort, authors believe more challenging performance tests byusing larger test systems and other types of disturbances shouldalso be carried out.

Fig. 7. Model performance test results in case of wind speed change with pitch angle control activated.

0 20 40 60 80 100 1201

1.1

1.2

1.3

1.4

1.5

r

- V

w ( p u )

time (sec.)

0 20 40 60 80 100 1201.998

1.999

2

2.001

2.002

2.003

2.004

V d c

( p u )

time (sec.)

0 20 40 60 80 100 120-2

0

2

4

6

8

10

B

( D e g r e e )

time (sec.)

V w

r

0 20 40 60 80 100 1200.9

0.92

0.94

0.96

0.98

1

1.02

P g e n

( p u )

time (sec.)

0 20 40 60 80 100 120-1

-0.5

0

0.5

1

1.5x 10

-21

Q g e n

( p u )

time (sec.)

0 20 40 60 80 100 1200.16

0.17

0.18

0.19

0.2

0.21

0.22

P l o s s

( p u )

time (sec.)



Fig. 8. Model performance test results in case of wind speed change with pitch angle control deactivated.

REFERENCES

[1] A. Petersson, Analysis, Modeling and control of Doubly-Fed InductionGenerators for Wind Turbines, Goteborg, Sweden: Doctoral Dissertation,Chalmers Univ., 2005.

[2] A. Larsson, P. Sorenson and F. Santjer, "Grid impact of variable speedwind turbines," in Proc. of European Wind Energy Conf. and Exhibit.,

Nice, France, 1999.

[3] F. Velenciaga and P. F. Puleston, "High order sliding control for a windenergy conversion system based on a permanent magnet synchronousgenerator," IEEE Trans. Energy Convers., vol. 23, no. 3, pp. 860-867,Sep. 2008.

[4] S. Brabic, N. Celanovic and V. A. Katic, "Permanent magnet synchronousgenerator for wind turbine application," IEEE Trans. Power Electron.,vol. 13, no. 3, pp. 1136-1142, May 2008.

[5] W. Qiao, L. Qu and R. G. Harley, "Control of IPM Synchronous Generator for maximum wind power generation considering magnetic saturation,"

IEEE Transactions on industry applications, vol. 45, no. 3, pp. 1095-1105, May/June 2009.

[6] Y. Chen, P. PilLary and A. Khan, "PM wind generator topologies," IEEETrans. Ind. Appl., vol. 41, no. 6, pp. 1619-1626, Nov./Dec. 2005.

[7] T. M. Jahns, "Flux-weakening regime operation of an interior permanent-magnet synchronous motor drive," IEEE Trans. Ind. Appl., Vols. IA-23,no. 4, pp. 681-689, Jul./Aug. 1987.

[8] J. M. Kim and S. K. Sul, "Speed control of interior permanent magnetsynchronous motor drive for the flux weakening operation," IEEE Trans.Ind. Appl., vol. 33, no. 1, pp. 43-48, Jan./Feb. 1997.

[9] S. Morimoto, M. Sanada and Y. Takeda, "Effetcs and compensation of magnetic saturation in flux-weakening controlled permanent magnetsynchronous motor drives," IEEE Trans. Ind. Appl., vol. 30, no. 6, pp.1632-1637, Nov./Dec. 1994.

[10] J. F. Manwell, J. G. McGowan and A. L. Rogers, Wind Energy ExplainedTheory, Design and Application, West Sussex: John Wiley & Sons Ltd,2002, p. 4.

[11] P. Kundur, Power System Stability and Control, McGraw-Hill Inc., 1994

[12] H. K. Davijani and O. Ojo, "Loss minimization control of grid connectedinterior permanent magnet wind turbine generators," in Press..

[13] A. N. Yazdani and R. Iravani, Voltage Source Converters in power systems, Hoboken, NJ, USA: John Wiley & Sons. Inc., 2010

0 20 40 60 80 100 1201

1.1

1.2

1.3

1.4

1.5

r

- V

w ( p u )

time (sec.)

0 20 40 60 80 100 1201.9996

1.9998

2

2.0002

2.0004

V d c

( p u )

time (sec.)

0 20 40 60 80 100 120-1

-0.5

0

0.5

1

B

( D e g r e e )

time (sec.)

0 20 40 60 80 100 1200.8

1

1.2

1.4

1.6

1.8

2

P g e n

( p u )

time (sec.)

0 20 40 60 80 100 120-1.5

-1

-0.5

0

0.5

1x 10

-22

Q g e n

( p u )

time (sec.)

0 20 40 60 80 100 1200.1

0.2

0.3

0.4

0.5

0.6

P l o s s

( p u )

time (sec.)

Detailed Dynamic Modeling of Permanent Magnet Synchronous Machine based Wind Turbine for Power System Dynamic Analysis

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