Modelling a 2.5 MW direct driven wind turbine with permanent magnet generator

Report for the final examination project of the Nordic PhD course on Wind Power by Thomas P. Fuglseth

Department of Electrical Power Engineering Norwegian University of Science and Technology NO-7491 TRONDHEIM, NORWAY

IntroductionThis report is the conclusion of the post-course work following the Nordic PhD-course on Wind Power, which was held on Smla 5th 11th June 2005. The goal of the project was to construct a Simulink model of a 2.5 MW wind turbine with direct-driven permanent magnet generator connected to the grid via a back-to-back three-phase converter. The model was then to be tested in simulation runs.

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ContentsIntroduction................................................................................................................................ 2 Contents...................................................................................................................................... 3 1 System description and modelling ..................................................................................... 4 1.1 Mechanical dynamics ................................................................................................. 4 1.2 Permanent magnet generator ...................................................................................... 5 1.3 Generator-side converter............................................................................................ 6 1.4 Grid side converter and inductance ............................................................................ 6 1.5 DC-link dynamics ...................................................................................................... 7 1.6 System parameters...................................................................................................... 8 1.7 Wind model................................................................................................................ 9 2 Control strategies.............................................................................................................. 10 2.1 Generator side control.............................................................................................. 10 2.2 Grid side converter control....................................................................................... 12 3 Simulation results ............................................................................................................. 13 Conclusions and further work .................................................................................................. 19 References ................................................................................................................................ 20

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1 System description and modellingThis chapter sums up the equations which were implemented in the simulink model. The system in its entirety consists of the following components. A wind turbine generates torque from wind pressure. The torque is transferred via the generator shaft to the rotor of the generator. The generator produces an electrical torque, and the difference between the mechanical torque from the wind turbine and the electrical torque from the generator determines whether the mechanical system accelerates, decelerates or remains at constant speed. The generator is connected to a three-phase converter which rectifies the current from the generator to charge a DC- link capacitor. The DC- link feeds a second three-phase converter which is connected to the grid through a transformer.

Transformer inductance

Back-to-back converter

PM generator

~V3Grid

+

=

Vdc_

=

~

figure 1 System overview

1.1 Mechanical dynamicsIt is not a part of this exercise to model the wind turbine itself and the pitching action of the blades. Instead it is assumed that the turbine is run at variable speed to ensure that the tipspeed ratio remains at a constant optimal point ?opt=7.5 with a power coefficient Cp =0.474. The mechanical torque delivered by the wind turbine is given by: M t = 1 air Rwt2 C p ( ) v3 (2.1) 2 3 Here, ?air is the density of air, in kg/m . Rwt is the radius of the wind turbine rotor in meters and v 8 is the wind speed far away from the turbine. That is, the wind speed before it is slowed down by the wind turbine. Cp (?) is the coefficient of power as a function of the tip-speed ratio ?. Normally this will also be a function of the rotor pitch, but pitching dynamics are not modelled. The tip speed ratio ? is calculated as: R = mek wt v This value is fed to a look- up table to find values of Cp (?). The turbine is connected to the rotor of the generator via a shaft. The turbine, shaft and generator are modelled as a single rotating mass: dmek 1 = (M e + M t ) dt J total

(2.2)

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where Me is the electromechanical torque developed by the electrical machine (when the electrical machine is operated as a generator Me will be negative) and Jtotal is the moment of inertia of the entire mechanical system. In p.u. values equation (2.2) becomes:dn 1 = ( me + mt ) (2.3) dt Tm For more accurate results, the mechanical dynamics could be modelled as a two- mass-system with a spring and damper between them, but this was not done due to constraints in time and in available information.

1.2 Permanent magnet generatorThe generator model is implemented entirely in dq-coordinates. That is to say, there are no AC-states in the model. The generator is modelled with dc voltages and currents in a rotorfixed rotating coordinate system with the d-axis being in the direction of the flux from the permanent magnets. This model is a scaled per-unit model, and essentially the same as that shown in [Nilsen, 2005]. The equations for the d- and q-axis voltages are as follows: 1 d d ud s = rs id s + n q (2.4) n dt 1 d q uq s = rs i qs + + n d (2.5) n dt Here, id , iq , vd and v q are the p.u. d- and q-axis currents and voltages respectively, rs is the p.u. stator resistance. ? n is the basic electrical frequency of the generator, n is the p.u. frequency og the generator and ? d and ? q are the d- and q-axis fluxes respectively. The d- and q-axis fluxes are given as follows: d = xd id s + m

(2.6)

q = xqs i q (2.7) We see that by keeping the q-axis current equal to zero, we can orient the flux entirely in the d-axis direction. More on this will follow later.Next, (2.6) and (2.7) are inserted into (2.4) and (2.5) to get the following: x di s ud s = rs id s + d d n xq iq s n dt xq diqs uq s = rs i qs + + n ( xd id s + m ) n dt Transforming these equations a bit we get: did s n = (ud s rs id s + xq n iq s ) dt xd s diq = n ( uq s rs iq s n ( xd id s + m ) ) dt xq The p.u. torque developed is given by: e = d iqs q id s

(2.8) (2.9)

(2.10) (2.11)

(2.12) 5

These equations are implemented in the simulink diagram shown below.rs 1 s I_d -Cpsim id 1 id 1 ud Wn/Xd wn/xd xd Gain2 psid 2 psid*iq Me

n*psid 3 n n psiq*id

psiq n*psiq iq

xq Gain4

2 uq

wn/xq Gain3 rs

1 s I_q

3 iq

figure 2 Simulink model of permanent magnet generator

1.3 Generator-side converterWhen working with a simplified dq- model it is common to neglect switching dynamics, ripple currents and other fast dynamics in the electrical system. In this case, we follow the example of [Nilsen, 2005] and model the generator side converter as a simple time delay. That is, the voltages on the generator clamps are given as: ud s ( t) = uc o n t, d ( t tv) u dc (t ) (2.13) where ucont,d and ucont,q

uq s (t ) = uc o n t, q (t tv ) u dc (t ) are the control signals given as output from the controller.

(2.14)

1.4 Grid side converter and inductanceFor the grid/transformer inductance, the model given in [Molinas et al. 2005] is used: di g ud g = rg i d g + l g d ng lg iq g + ud c (2.15) dt di q g g g uq = rg iq + lg + ng lg iq g + uqc (2.16) dt Here, the superscript g stands for grid, to differ from the generator variables. v d c and v q c are the voltages on the clamps of the front-end converter.

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By transforming these equations we get: did g 1 = ( ud g ud c rg id g + ng iq g ) dt lg

(2.17) (2.18)

1 (uq g u qc rg iq g ng id g ) dt lg Which is the model implemented in the simulink diagram shown in figure 3. =1 ud_gridi_dq Q

diq g

u_dq_grid

P

3 P 4 Q 1

Power calculations 2 uq_grid 1/L Add1 1 si_dq

id

Integrator Gain R

2 iq

3 ud_converter

Gain1 [2x2] Product Gain2

4 uq_converter

5 n_grid

figure 3 grid inductance model

The grid side converter is modelled as a time delay just like the generator side converter.

1.5 DC-link dynamicsThe equations for the DC-link dynamics are taken from the lecture notes given at the Smla course [Norheim, 2005], and are as follows: dudc 1 = ( ( ud s id s + uq s iq s ) (ud c id c +uq c iqc ) ( Pr l, o s s + Pdloss ) ) (2.19) , dt C udc This follows easily from the principle of conservation of energy. The total power flowing into the capacitor bank must equal the sum of the power flowing out of the capacitor and the losses in the conductors. (2.19) can also be written asdud 1 = ( I v I d ) dt C

(2.20)

1 (ud s id s + uq s iqs Prloss ) is the current flowing out of the generator side , udc 1 converter and I d = (ud c id c + uq c iq c Pd , loss ) is the current flowing into the grid side u dc converter. This means that in when the PM machine is operating as a generator these will both be negative.

where I v =

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The losses are modelled as simple resistances, IE s Prloss = r (ids 2 + iq 2 ) , v and c c Pd , loss = rd (id 2 + iq 2 ) This loss model gives a simple approximation to the real losses in the system, but for more accurate simulation a better loss model that includes switching losses and nonlinear losses in the generator would be preferable.

1.6 System parametersThe p.u. values used in the simulation are based on a 2.5 MW wind turbine. The turbine itself is 80 meters in diameter, and with a moment of inertia J wt = 9.0 106 kgm2 [DAWE 2004]. As mentioned earlier, the pitching dynamics are not modelled. It is assumed that the blades are pitched to maintain optimum energy capture. The objective of the generator control is to keep the turbine running at a rotational speed corresponding to a tip-speed ratio which gives us the optimal power coefficient Cp,opt=0.474. In reality this is similar to controlling a fixed-pitch turbine. The Cp -table data are taken from [DAWE, 2004]. The generator is a permanent magnet generator with surface mounted magnets and symmetrical rotor. In other words, the reactance is the same in both the d- and q-axis. The p.u. values that have been used in this simulation are: rs = 0.02 xd = xq = 0.5 These are approximate values that are reasonably representative of a synchronous PM machine in the 2.5 MW range. If we assume that 1 p.u. speed and no load torque requires 1 p.u. stator voltage, we get a p.u. flux of m = 0.865 Physically, the rotor of the generator is assumed to consist of three cylinder sections: a center section with a radius of 0.25 meters and a length of 0.5 meters, a disc section with an outer radius of 2.1 meters and a length of 0.4 meters and an outer cylinder with an outer radius of 2.5 meters and a length of 1 meter. For the purpose of calculating moment of inertia, the rotor is assumed to consist of homogenous steel. Using the formula for moment of inertia for a cylinder 4 4 J = 1 steel l ( router rinner ) 2 with a density ?steel of 7850 kg/m3 we get a total moment of inertia Jtotal = 9.193 106 kgm2 . The per unit mechanical time constant is J Tm = total m e k, n Sn where ? mek,n is the nominal rotational frequency in rad/sec, which in this case equals mek ,n = 16rpm = 1.676 rad s 30 and Sn is the nominal power, in this case 2,5 MW. This gives us a time constant of 9.193 106 1.676 Tm = = 10.3s 2.5 10 6

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The DC- link capacitor is dimensioned using an empirical rule that is commonly used in literature:= dc (2.21) 2 Sn Here, C is the capacitance of the DC- link capacitor, Udc is the nominal DC-link voltage and Sn is the nominal power of the generator. The time constant t dc is generally selected to be a proportion of a full cycle for the grid voltage. A full cycle for a 50 Hz grid is 20 ms, and a time constant of 6-8 ms seems common. (2.21) gives us: 2 S n dc C= U dcrated 2 , Next, we know that the per unit capacitive reactance is: I dcrated , xc = (2.22) C Udcrated , C U dc , rated2

If we insert the expression for C into (2.22) and take into account that I dcrated = Sn / Vdcrated we , , get: Sn U dc , rated 1 xc = = 2 Sn dc Udcrated 2 dc , U dc , rated 2 We see that the per unit capacitive reactance is independent of the physical rated voltages and currents. If we chose the DC- link time constant to be 7.2 ms (0.36 cycles) we get an x c value of: 1 xc = = 0.221 2 (2 50 Hz ) 7.2 10 3 s In testing, this proved to be a bit too small. In the end, x c was set to be 0.5, which yielded better results.

1.7 Wind modelThe wind model used is one developed at the Ris National Laboratory in Denmark. The simulink implementation is designed at the University of Aalborg, and is available from http://www.iet.aau.dk/ as part of the Wind Turbine Blockset beta, a toolbox for wind turbine simulation in Simulink.

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2 Control strategiesThis chapter details the strategies used in the control systems of the generator and the grid side converter.

2.1 Generator side controlThe generator side control is based on the stator voltage equations (2.10) and (2.11). To simplify the control design we divide the control system into two parts: a decoupling controller and a PI controller. In other words, we give the input voltages the form: ud s = udI s + u dII s (2.1)

uq s = uqI s + uqII s We see that if we give udIIs and uqIIs the following form: udII s = n x iqs q

(2.2) (2.3)

uqII s = n xd id + n m (2.4) then we get the following first order systems to control: did s r = n s i d s + n udI s (2.5) dt xd xd diqs r = n s iq s + n uqI s (2.6) dt xq xq Equations (2.5) and (2.6) represent two first order systems with one control variable each. In short, we use the control system to decouple the d- and q-axis equations from each other, as well as removing the speed-dependent terms. The d- and q-axis currents can now be controlled with two ordinary PI controllers. The decoupling controller is dependent on measurements of the currents as well as knowledge of the impedances x d and x q and the flux generated by the permanent magnets. However, by using PI controllers, any errors introduced by minor inaccuracies in the assumed values of ? m , x d and xq are compensated for by the integrating controller.As shown in [Nilsen, 2005], the system defined by (2.5) and (2.6) can be represented by the following simplified transfer functions: i (s) n Td h0 d ( s ) = d = (2.7) udI ( s ) xd (1 + Td s ) (1 + Tsum s) iq (s ) n Tq h0q ( s) = = (2.8) u qI ( s ) xq (1 + Tq s) (1 + Tsum s ) x xd and Tq = q n rs n rs Tsum is the sum of all the smaller time constants in the system such as filtering of current measurements and linearized switching delays. where Td = The current controllers are PI controllers with limited output and anti- wind-up features. Antiwind-up is implemented by disabling the integrator while the output from the controller is equal to the upper or lower limit. The general transfer function of the PI controller is as follows:

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1 + Ti s (2.9) Ti s Controller design follows the guidelines given in [Nilsen, 2005]. Since the generator has a symmetrical rotor with x d =x q the controller design for both axes will be the same. We start out with what is termed the numerical optimum design, which gives us: xd K p, d = Ti, d = Td 2 n Tsum When numbers are inserted, we get: K p, d = 5.65 Ti, d = 0.085 h pi ( s ) = K p

During testing, it was revealed that these controller values gave a controller that was too aggressive, leading to rapid saturation and overshoot. After some testing and tuning, the following current controller parameters were selected: K p ,d = 2.0 Ti, d = 0.05 which yielded better performance. The speed controller is quite interesting compared to most regular motor drives, as what we are really interested in controlling is the tip speed ratio. While this works quite well in simulations, tip speed ratio can be difficult to control in a real turbine. The rotational speed of the turbine can be measured with a high degree of accuracy, but wind speed measurements (which are necessary to calculate the tip speed ratio) are easily affected by turbulence and other factors. In this simulation we assume perfect wind speed measurements, but in reality some form of estimation based on measurements and modelling will probably have to be used. This will again lead to some loss of production, as the controller is unable to perfectly optimize the production. For speed control, the system was tested with a PI-type controller designed using the symmetrical optimum strategy, again taken from [Nilsen, 2005]. This method gives the following controller parameters: Tm K p, n = Ti, n = 4 Ts u m, n 2 Ts u m, n m Tm is the mechanical time constant of the generator and turbine, as detailed in section 1.1. Tsum,n is the sum of the smaller time constants: Ts u m, n = Te q, i + Tf i, n . Tfi,n is the speed filter time constant, while Teq,i is the equivalent time constant for the electrical system. In other words, the time constant we find if we represent the electrical dynamics as a single first-order system: Te q,i = 2 (Tv + Tf , i ) . This method gave the following controller parameters:K p, n = 39780 Ti, n = 0,0012

This illustrates that while methods for obtaining controller parameters are a useful tool for initial controller design, they are not perfect. In this case the high proportional amplification meant that the controller for all practical purposes worked like a hysteresis controller. The result was a lot of overshoot and a high degree of oscillatory behaviour of the system, and the controller had to be tuned manually. Unfortunately none of the PI controller designs gave satisfactory behaviour. Generally the system dynamics were either too slow, or unstable.

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In the end, after a lot of testing and trial and error, it was decided to go for a limited PID-type controller. The transfer function of a PID controller is 1 + Ti s + Ti T s2 d hPID (s ) = K p (2.10) Ti s The structure of a PID controller with limited derivative control is illustrated in figure 4. The limitation in derivative is implemented with a rate limitation simulink block. The purpose of this limitation is to avoid controller saturation caused by high frequency signals. TdKp

d dt

+1 Ti

+ +

figure 4 PID controller

Controller parameters were selected mostly through manual tuning to be: K p ,n = 1 Ti, n = 0.8 These were found to give satisfactory performance.

Td , n = 0.01

2.2 Grid side converter controlThe controller structure for the grid side converter is taken from [Norheim, 2005]. The original design consists of two PI controllers, one for the amplitude and one for the phase angle of the converter voltage. That is: uc = m2 udc e j (2.11) The amplitude m2 controls the reactive power to the grid, while the phase angle a controls the DC-link voltage. In this case, the system was difficult to stabilize with only PI controllers. In particular, m2 was prone to oscillatory behaviour. A relatively common occurrence was that m2 would oscillate, while a steadily dropped. This would continue up to a point where a would drop sharply, causing high grid currents, and making the DC-link voltage to drop almost immediately to zero. Naturally, this would not happen in a physical system. This behaviour was only possible because of the simplifications done in modelling the system. However, it should still be avoided. Eventually, a solution with a PI controller for a and a PID controller for m2 was settled upon, and was found to give satisfactory performance. The controller parameters were: K p, m2 = 0.25 Ti, m2 = 0.05 Td , m2 = 0.15K p, = 1 Ti, = 0.5

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3 Simulation resultsThe results in this chapter are based on a 150 second simulation with a mean wind speed of 6 m/s. Simulation over a longer time frame would have been better, but simulation time was limited by the amount of memory on the PC used to run the simulation. The first illustration, figure 5, shows the wind speed time series, and the corresponding generator speed:v 7.5w ind

7 [m/s]

6.5

6

5.5

0

50

100

150

n 0.9 0.85 0.8 0.75 0.7 0.65 0

[p.u.]

50 t [s]

100

150

figure 5 wind speed and corresponding rotational speed

As we can see, the speed controller does a good job of controlling the speed depending on the wind speed. The next illustration, figure 6, shows the corresponding tip speed ratio:tip speed ratio 8 7.8 7.6 7.4 7.2 7

0

50 t [s]figure 6 tip speed ratio

100

150

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As we can see, the controller managed to keep the tip speed ratio within an acceptable region of the desired tip speed ratio of 7.5. Next, we look at the generator currents and voltages. Figure 7 shows the generator dq-axis currents, while figure 8 shows the corresponding voltages.id 0.4 0.3 [p.u.] 0.2 0.1 0 -0.1 0 50 iq 0.5 0 [p.u.] -0.5 iq iqref 100 150

-1

0

50 t [s]

100

150

figure 7 d- and q-axis generator currentsud 2 1 [p.u.] 0 -1 -2

0

50 uq

100

150

4 2 [p.u.] 0 -2 0

50 t [s]

100

150

figure 8 d- and q-axis generator voltages

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In figure 7, the d-axis reference is constantly equal to zero. While the d-axis current stays reasonably close to zero the entire time, the controller performance probably could have been a bit better. The q-axis reference depends on the commanded torque from the speed controller, and is shown in red in the lower plot of figure 7. The obtained q-axis current is shown in blue. In general, the obtained current follows the commanded current quite well, but we see that it overshoots the reference most of the time, so perhaps a slightly less aggressive controller would have been better. On the grid side, we get the currents shown in figure 9.id,grid 0 -0.2 [p.u.] -0.4 -0.6 -0.8

0

50 iq,grid

100

150

0.04 0.02 [p.u.] 0 -0.02 -0.04

0

50 t [s]

100

150

figure 9 grid d- and q-axis currents

The q-axis current stays quite well around zero, while the d-axis current varies with the power produced by the wind turbine. Note that the d-axis current is negative because the turbine produces power, in other words we are using a load reference for the converter system. The grid side converter voltages are shown in figure 10. These show pretty much what we would expect, with the d-axis voltage staying nearly constant around 1 p.u. while the q-axis voltage varies with the reactive power required to control the dc- link voltage. Figure 11 shows the active and reactive power while figure 12 shows the commanded voltage vector given by m2 and a.

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ud,grid 1.5 1 [p.u.] 0.5 0 0

50 uq,grid

100

150

0.8 0.6 [p.u.] 0.4 0.2 0

0

50 t [s]

100

150

figure 10 grid side converter voltages

Active power P 0 -0.2 [p.u.] -0.4 -0.6 -0.8

0

50 Reactive Power Q

100

150

0.04 0.02 [p.u.] 0 -0.02 -0.04

0

50 t [s]

100

150

figure 11 active and reactive power

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m2 -0.4

-0.5 [p.u.] -0.6 -0.7 0

50 alpha

100

150

-2

-2.5 [rad] -3 -3.5 0

50 t [s]

100

150

figure 12 m2 and a

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Finally, figure 13 shows the DC-link voltage, as well as the currents Iv and Id .

Udc 2 1.9 [p.u.] 1.8 1.7 1.6 0 50 Iv 0.5 100 150

[p.u.]

0

-0.5 0

50 Id

100

150

0 -0.1 [p.u.] -0.2 -0.3 -0.4 0 50 t [s] 100 150

figure 13 DC-link voltage and current flowing into and out of the DC-link capacitor

While the controller does a reasonable job of controlling the DC-link voltage around the set point of 1.75, performance could be better. A simple way of stabilizing the DC- link voltage is by selecting a bigger capacitor, but there probably is room for some controller improvement as well.

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Conclusions and further workThe simulations show that the model and controller are functioning well. The control strategy works well, altho ugh the DC-link voltage ideally should be controlled more accurately. While the controllers worked very well during initial testing with simple wind conditions (constant or slowly varying wind speed), simulations with more complex wind conditions show that the controllers are not quite perfect yet. While the wind turbine model does a reasonable job of simulating a real turbine, accuracy can be improved in some respects. Implementation of a better loss model would improve the quality of the model greatly. Also, it would be interesting to have data from a real wind turbine to base the model parameters on instead of approximate values. The controller structure should be improved by adding a power limiting controller for when the wind speed increases above nominal wind speed. This should be coupled with a model including pitching dynamics and better modelling of the aerodynamics to improve the controllability of the model. Now that a working wind turbine model is in place, it would be interesting to test it with different controller structures. One area that could be explored is in the use of nonlinear control of the DC-link voltage, as this is a system that is inherently nonlinear in nature. Also, it could be interesting to compare the current solution with a pair of PI-controllers and a decoupling grid for generator control with a multivariable nonlinear controller that includes the cross-coupling terms.

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References[Nilsen, 2005] Roy Nilsen, Elektriske Motordrifter, Department of Electrical Power Engineering, Norwegian University of Science and Technology 2005

[Norheim, 2005] Ian Norheim, Dynamic modelling of turbines (2), Generators, converters and control, Presentation from the Nordic Wind Course, Smla 7th June 2005 [DAWE, 2004] Various authors, DAWE PhD Course/Advanced school 2004, Compendium and course material, Institute of Energy Technology, Aalborg University 2004.

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