power angle control of grid-connected voltage source

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Titel: Power Angle Control of Grid-Connected Voltage

Forfattare:

Source Converter in a Wind Energy Application

Jan Svensson

RAPPORT INOM OMRAdET VINDENERGI

Rapportnummer: VUMD-95/12

Projektledare: Ola Carlsson

Projektnummer: 506 258-8 Elmaskin CTH

Projekthandlaggarepa NUTEK: Hans Ohlsson

Postadress Besoksadress Telefon Telefax Telex

Power Angle Control of Grid-Connected Voltage Source Converter in a Wind Energy Application

Jan Svensson

Technical Report No. 218 L

Submitted to the School of Electrical and Computer Engineering Chalmers University of Technology

in partial fulfilment for the degree of Licenciate of Engineering

Department of Electric Power Engineering Division of

Electrical Machines and Power Electronics

Goteborg, Sweden 1995

Chalmers Tekniska Hogskola Institutionen for Elkraftteknik

Avdelningen forElmaskinteknik och Kraftelektronik

S-41296 Goteborg, Sweden

ISBN 91-7197-228-5

Chalmers Bibliotek Reproservice

Goteborg, November 1995

DISCLAIMER

Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.

Abstract

In this thesis, the connection of a voltage source converter to the grid in a wind energy application is examined. The possibility of using a cheap control system without grid current measurements, is investigated. The control method is based on controlling the voltage angle of the inverter, which governs the active power flow.

The highest frequency of the power variation, coming from wind turbine, is approximately 5 Hz. Since the proposed control method easily can handle such power variations it is very well suited for wind turbine applications.

The characteristics of the system depend on the DC-link capacitor, the grid filter inductance and resistance. Large values of the resistance damp the system well but increase the energy losses. A high inductance leads to a reduced harmonic level on the grid but makes the system slower. By using feedforward of the generator/rectifier current signal, the performance is increased compared to an ordinary Pi-control.

Combining the Linear Quadratic (LQ) control method with Kalman filtered input signals, a robust control method with a good performance is obtained. The LQ controller controls both the phase displacement angle and the modulation index, resulting in higher bandwidth, and the typical power angle resonance at the grid frequency disappears.

Acknowledgements

I would like to thank my supervisor, Dr Ola Carlson, for his support in this project and my examiner Professor Kjeld Thorborg for a good start Furthermore, I would also like to thank my colleagues Torbjom Thiringer and Michael Lindgren for many interesting discussions and of course the rest of the crew at the department.

The financial support for this thesis is given by the Swedish National Board for Industrial and Technical Development (NUTEK), through the wind power consortium and it is gratefully acknowledged.

Table of contents

Table of contents

List of symbols.................................................................................................................. 11 Introduction.................................................................................................................... 4

2 Properties of the voltage source converter.................................................................... 62.1 Main circuit.............................................................................................................62.2 Control of output voltage.......................................................................................8

2.2.1 Pulse width modulation (PWM) by means of the suboscillation method.....82.2.2 Adding zero components to the sinusoidal reference curve......................... 82.2.3 On-line zero components...............................................................................9

2.3 The linearized switching state space vector........................................................... 9

3 Harmonic distortion and filtering..................................................... ..............................113.1 Background............................................................................................................113.2 Voltage harmonics of the converter...................................................................... 113.3 Voltage harmonics on the grid induced by converter...........................................14

3.3.1 The electrical system of a wind power plant with variable speed.................143.3.2 The equivalent circuit of the plant................................................................. 153.3.3 The parameter of the equivalent circuit..........................................................173.3.4 Different types of grid filters......................................................................... 173.3.5 The L-filter.................................................................................................... 183.3.6 The LC-filter...................................................................................................203.3.7 The LC-filter with a damping link................................................................213.3.8 Comparison between the grid filters............................................................. 29

4 Power angle control of the converter............................................................................. 304.1 The system configuration...................................................................................... 304.2 The basic idea of power angle control...................................................................324.3 Non-linear model of the system............................................................................ 324.4 The reduced non-linear model............................................................................... 354.5 Linearization of the system....................................................................................36

4.5.1 Operating points.............................................................................................364.5.2 Linearization of the inverter.......................................................................... 364.5.3 The state equation of the DC-link................................................................. 384.5.4 The state equation for the linearized system.................................................384.5.5 The step response of the DC-link current.....................................................40

4.6 Regulators.............................................................................................................. .444.6.1 The Pi-regulator............................................................................................ 444.6.2 The PIEF-regulator....................................................................................... 45

4.7 The total linearized system with a regulator........................................................ 464.7.1 The open loop of the linearized system with a Pi-regulator........................464.7.2 The closed loop of the linearized system with a Pi-regulator......................464.7.3 The closed loop of the linearized system with a PIFF-regulator.................47

Table of contents

t

! 4.8 The step responses of the linearized system with different regulators.................. .48] , 4.9 Simulation of the non-linear system.......................................................................53i 4.9.1 Step responses of the generator/rectifier current............................................. 54! 4.10 Parameter variations.............................................................................................. 59, 4.10.1 DC-link voltage fluctuation when using a Pi-controller............................... 59! 4.10.2 DC-link voltage fluctuation when using a PIFF-controller...........................61I 4.10.3 Allowed parameter value variations to maintain the fluctuation below 5%

; ................................................................................................................................. 62' 4.11 Conclusion of the power angle control................................................................. 64t| 5 Power angle control using linear quadratic method....................................................... 65I 5.1 Linear quadratic control.......................................................................................... 65

5.2 LQ-control of the linearized inverter system..........................................................66’ 5.2.1 State space equation of the linear inverter system.......................................... 66

5.2.2 Derivation of the LQ-control with integral error. feedback and feedforward.............................................................................. 67

5.2.3 LQ-controller for the linearized inverter system........................................... 701 5.2.4 Analysis of the linearized inverter system

with LQ-PI and LQ-PIFF controllers.............................................................725.3 The Kalman estimator............................................................................................ 76

i 5.3.1 The non-linear Kalman estimator....................................................................775.3.2 Stability of the system when using an LQ-controller and a Kalman estimator..................................................................................................................................78J 5.4 Simulation of the inverter system........................................................................... 815.4.1 The simulation model..................................................................................... 815.4.2 Step response of load steps ........................................................................82

5.5 Conclusions of inverter system with an LQ-controller.......................................... 86

6 Laboratory set-up and measurements............................................................................. 876.1 Laboratory set-up................................................................................................... 87

6.1.1 Main circuit.....................................................................................................876.1.2 DC-link...........................................................................................................886.1.3 Grid filter........................................................................................................886.1.4 Current and voltage measurement.................................................................. 886.1.5 Control unit.................................................................................................... 88

6.2 Measurements on laboratory set-up........................................................................906.2.1 Measurement 1............................................................................................... 906.2.2 Measurement 2............................................................................................... 94

6.3 Conclusion of the measurements............................................................................99

7 Conclusions and future work..........................................................................................1007.1 Conclusions............................................................................................................ 1007.2 Future work............................................................................................................ 101

I

Table of contents

References.......................................................................................................................... 102

Appendix A. Transformations.......................................................................................... 104A.1 The transformation between a three-phase system and an aP-system................ 104A.2 VSC current and voltage space vectors.................................................................105A.3 The connection between an aP-system and a dq-system.................................... 107A. 4 Voltage and current vectors in the aP-system and the dq-system, respectively.. 108

Appendix B. Normalization.............................................................................................. 109B. l Normalization of power angle controlled inverter................................................ 110

List of symbols

List of symbolssuperscript meaning* reference(ap) vector in the (aP)-reference frame(dq) vector in the (dereference frame

subscriptddchighlown(n)0qssctrfINPVa

P1,2,3,/CLPICLPIFFLQP1FFLQREGMOLPIPIPIFFVM

meaningdirect quantity in the dq-reference frame quantities of the dc-link of the converter high voltage side of transformer low voltage side of transformer rated valuen:th order harmonic component operating pointquadrature quantity in the dq-reference frameseries filtershort-circuittransformerintegral term or currentquantities of the gridproportional termvoltagedirect quantity in the ap-reference framequadrature quantity in the aP-reference framephase quantitiesstate equation matricesstate equation matricesstate equation matricesstate equation matricesstate equation matricesstate equation matricesstate equation matricesstate equation matricesstate equation matrices

variable/fsw/»iijcK

ekkFFkPk,n

meaning unitfrequency Hzswitching frequency for a valve Hzresonance frequency Hzcurrent Athe current flowing into the dc-link Athe current flowing from the dc-link to the converter

Agrid voltage Vnon-dimensional integer feedforward gainproportional gain in a Pi-controller constant for modified Pi-controller non-dimensional integer

1

List of symbols

PsswtuUTHD

«xUkVV

vowcEGmLMPPmQRSS„luuGlow, It

Ghigh,n

XzT,TKCO

6£

¥A,B,C,DE,F,GHkLPQ,R

frequency ratioLaplace operatorswitch state of a valve, (±1)time svoltage Vtotal harmonic voltage Vrelative voltage of convertervoltage drop over a transformer p.u.potental Vmeasurement noisepotental in Y-point Vprocess noisecapacitance Fphase-to-phase voltage in rms of the grid Vgain margininductance Hmodulation indexactive power Wphase margin degreesreactive power VArresistance £2apparent power VAshort-circuit apparent power of the grid VAtime constant for modified Pi-controller sconverter fundamental phase to phase voltage, rms V rated voltage low voltage oftrf Vrated voltage high voltage oftrf Vreactance £2impedance £2integral time constant in a Pi-controller snon-dimensional damping factorangular frequency rad/sresonance angular frequency rad/sphase displacement angle raderror signalphase displacement angle between current and voltage flux (i.e. flux linkage) Wbmatrices of state equation matrices of state equation feedforward gain vector filter gain matix feedback gain vector matrixweighting matrices

other important notations x scalar quantity or vector quantityx' x referred to the lower voltage side of a transformerx" x referred to the higher voltage side of a transformer

2

List of symbols

X vector in the complex planeX rms-value, Laplace-, or Fourier transform of a scalar quantityX matrixX peak values or estimated quantityX derivate of x with respect to timeAx small signal distortionE{x} expectation of x

Abbreviationsest estimaterms root mean squaretri triangleADC analog to digital converterDAC digital to analog converterFF Feed ForwardI Integral regulatorIGBT Insulated Gate Bipolar TransistorIQ Linear QuadraticP Proportional regulatorPC personal computerpi Proportional, Integral regulatorPIFF Proportional, Integral with Feed Forward regulatorPLL phase locked loopPWM Pulse Width ModulationVSC Voltage Source ConverterTHD Total Harmonic DistortionTHDi Total Harmonic Distortion, current weightTHDu Total Harmonic Distortion, voltage weightTRF Transformer

3

Introduction

Chapter 1

Introduction

Wind power plants, for alternative electrical energy production, with variable rotor speed, have two major advantages compared to constant rotor speed wind power plants. The use of variable rotor speed improves the dynamic behaviour of the turbine and hereby alleviates stresses and prolongs the lifetime of the wind turbine. By decreasing the stresses on the mechanical construction, the weight can be decreased and thereby the cost. Further, the noise from the wind power plant is reduced at low wind speeds [1]. It is also possible that the energy capture can be increased by about 5 % [2]. In addition, the power fluctuations occurring from constant rotor speed wind turbines can be eliminated. The major disadvantage is a more complex electrical system leading to a more expensive system. There is also a risk for excitation of structural resonances, for example in the tower.

Until now, the most used electrical system of a variable rotor speed wind power plant has consisted of a synchronous generator connected to the grid via a thyristor/diode rectifier and a thyristor inverter. Due to the fast development of rapid switching valves with high power ratings, the use of Voltage Source Converters (VSC) is increasing in wind power applications. For a specific current harmonic content a much smaller grid filter is needed if a VSC is used instead of a thyristor inverter. Moreover, a VSC can produce any desired power factor and the energy can flow in both directions. Another benefit is that the starting problem of a stall control wind turbine easily can be solved by reversing the energy flow.

The increased use of power electronic valves give rises to grid quality problems; i. e. increased harmonic content in the grid. Proposals for future standards deal with how much harmonics every single consumer/producer is allowed to contaminate the grid with. One possibility is that the local power company in the future will adjust its compatibility price paid for energy in relation to how much the producer is contaminating the grid with harmonics; by using VSC, future grid quality problems can be avoided.

By using a VSC connected to the grid it may be possible to install more wind power on an existing grid compared to the case where fixed speed wind turbines are installed. Today in Sweden, it is recommended that the short-circuit power of the grid should be more than 20 times the installed wind power. By controlling the reactive power flow, the voltage variations caused by the wind turbine can be reduced, and the ratio between grid short-circuit power and the installed wind power can thus be reduced.

There are different methods to control the currents of a VSC connected to the grid. A common method, used on strong grids, is to use inner current loops, see [3] for a survey.

4

Introduction

Another method, often used on weak grids is the current hysteresis control [4]. In this report, the possibility of using a cheap controller without grid current measurements is investigated. The control method is based on controlling the voltage angle of the inverter, where the active power is proportional to the phase displacement angle, [5]. This control method has been investigated for HVDC applications [6,7] and for a VAR compensator [8]. Further on, suitable models of the inverter system are examined and the system performance in a wind power plant application is determined. The influence of parameter values to the system performance is also analysed.

In Chapter 2 the voltage source converter is introduced and different modulation methods are investigated. In Chapter 3 the voltage harmonics on the grid, coming from the grid- connected inverter, are analysed. Different grid filters are introduced to minimize the voltage harmonics. In Chapter 4 the inverter system with voltage angle control is analysed. Further on, the system performance with different system parameters is determined. In Chapter 5 optimal control is adopted to increase the performance of the system. In Chapter 6 the results from Chapter 4 are verified by measurements on a laboratory set-up.

5

Properties of the voltage source converter

Chapter 2


2.1 Main circuit

The main circuit consists of 6 valves, two in each phase leg. A scheme of the main circuit of the converter is shown in Figure 2.1. The converter is connected to a symmetrical three-phase load, where each phase consists of an impedance and an emf.

L(f)

valve 3 valve 5valve 1

valve 2valve 4 valve 6

f,(0 +

Figure 2.1: Main circuit of a six-pulse converter.

In each phase leg only one of the two valves is conducting at a time and in order to prevent a short-circuit of the DC-link both valves have to be turned off before a valve starts to conduct. The delay caused by this switching is called blanking time and it depends on the type of valves used in the converter.

In this chapter it is assumed that the valves are ideal, i. e. the valves of the inverter have no losses, due to voltage drop or internal resistance. Further on the valves are operating as ideal switches, which means that the switching between the blocking and conducting state is instantaneous and without delay.

6


When the converter is analysed, it is easier to replace each valve by a switch sw. If the upper valve is conducting in phase leg 1 the switch sw, has the value 1 and if the lower valve is conducting the switch sw, has the value -1, see Figure 2.2.

dc O

SW, = -1

Figure 2.2: The modified main circuit of the six-pulse converter for mode analysing.

The three phase voltage potentials can have two potentials ±udc / 2, so for the whole converter there are eight combinations. The phase voltages can be written as the phase voltage potential minus the zero potential v0

«i(f) = v,(t)-v0(r)u2(t) = v2(t)-v0(t) (2.1)

«3(f) = v3(f)-vo(f)

Assume that the load of the converter is symmetric. This makes the zero potential

v0 (r) = | (v, (t) + v2 (r) + v3 (r)) (2.2)

7


22 Control of output voltage

2.2.1 Pulse width modulation (PWM) by means of the suboscillation method

By PWM method the fundamental output voltage of the converter can vary, and the harmonics are cancelled from the fundamental frequency to the vicinity of the switching frequency fm. For a converter using the PWM suboscillation method and which has asinusoidal reference curve, the rms-value of the fundamental phase voltage will be

u/m=^f~0.35Udc (2.3)

when the modulation index is 1. The maximum fundamental voltage vector n(1)(r) from a converter without overmodulation, i. e. 0 <M < 1, has the maximum amplitude

k')(z)L=^”0-61^ (2-4)

Definition of voltage and current vectors is shown in Appendix A.

2.2.2 Adding zero components to the sinusoidal reference curve

By adding a variable consisting of triplen harmonics to the reference curve, the fundamental PWM signal can be increased. By using this technique an increase in the converter output voltage by a factor of 2 / V3 can be achieved at most, see [9]. The rms- value of the fundamental voltage, when modulation index is 1, will be

Uni) = ^0AWic (2.5)

Using zero voltage components, the maximum, fundamental, voltage vector amplitude of um(t) will be

= P'S

8


2.2.3 On-line zero components

The principle of the on-line zero component determination is to centre the reference voltages around zero. By adding a variable consisting of half the mean values of the maximum and the minimum of the phase potentials to the reference curve, a centring is done and an on-line optimization will occur. In Figure 2.3 a block scheme shows how the on-line zero component can be implemented.

Figure 2.3: Block scheme of the PWM implementation with added on-line zero components.

23 The linearized switching state space vector

If the frequency ratio p is large enough, say 50, and the modulation index M< 1, the voltage harmonics can be neglected when controlling the converter. For the switching vector the linearization results in a linear modulation [10], i. e. the switching vector of the converter is proportional to the space vector of the pulse width modulator reference voltages. With this assumption, the switching function sw(t) rotates with a constant frequency and is a dc-quantity in the dq-plane in stationary state. The voltage space vector of the converter u(t) becomes the same as the reference value of the voltage space vector M*(r) of the converter

«(0 = «*(r) (2.7)

9


If the converter is controlled by a computer, the modulation method will be the space vector modulation, see [11], and the linearized voltage space vector of the converter u(t) becomes the same as the reference value of the voltage space vector of the converter u (r) except for a delay of, at least, the switching period Ts = 0.5//w.

u{kTs) = u{(k-l)Ts)

Only continuous time systems are considered in this thesis.

(2.8)

10

Harmonic distortion and filtering

Chapter 3


3.1 Background

In this chapter a voltage-stiff self-commutated converter is analysed. The inverter is ideal, which means that the valves of the inverter have no losses due to voltage drop nor internal resistance. Furthermore, the valves work as ideal switches. This means that the switching between off-state and on-state is instantaneous with no delay. When the valves are nearly ideal, the voltage flanks at the output voltage, which is pulse width modulated, will be steep resulting in high frequency components in the spectrum of the output voltage. Harmonics up to a frequency of 20 kHz are considered. Harmonics causing radio interference are not considered here.

3.2 Voltage harmonics of the converter

A self-commutated PWM inverter can be regarded as a voltage source. The lowest frequency harmonics in the output voltage appear in a band around the switching frequency. Higher frequency harmonics will be at frequencies that are a multiple of the switching frequency.

In Figure 3.1 below the spectrum of the phase voltage from the inverter has been calculated, using MATLAB / SIMULINK. The method is the sub-oscillating PWM, using added zero components, see Chapter 2. The rms-value of the fundamental phase voltage Ufm is 230 V at the frequency of 50 Hz and the modulation index M is one. The DC-link voltage udc is 654 V and the switching frequency fm is 5 kHz.

Usually, the total harmonic voltage distortion THDV is used to compare different system solutions. All harmonics are equally evaluated when the THDV is calculated. By using the total harmonic current distortion THD,, lower frequency harmonics become more important than high frequency harmonics because it is easier to remove high frequency harmonics by means of filters than harmonics of low frequencies. The magnitude of the generated n:th weighted harmonic voltage becomes reciprocally proportional to the frequency of the n:th harmonic voltage. The THD, is defined as

(3.1)

11


/fN

Figure 3.1: Phase voltage spectrum of the PWM-converter. <

where the fundamental voltage Um is the rms-value of the voltage at the fundamental frequency, /(I), usually 50 or 60 Hz. is the total amount of the harmonicvoltage, which has been weighted by the harmonic number. All voltage harmonics up to number are included. Uw is the rms-value of the harmonic voltage with number n.

If the converter is connected to the grid, which has a constant voltage amplitude, the fundamental voltage amplitude of the converter should also be constant, then the converter is running in steady state. When the DC-link voltage is fluctuating due to load variations, the reference modulation index must change so that the product M" -uic becomes constant and thereby, the fundamental voltage amplitude of the converter remains constant. In Figure 3.2 the fundamental voltage and the total harmonic distortion value have been plotted as a function of the reference modulation index M", by using MATLAB/SIMULINK. The ripple of the curves is caused due to finite time steps in the simulations.

When the reference modulation index M’ is lower or equal to one, the converter can compensate the DC-link voltage fluctuations. If the reference modulation index M* > 1, an over-modulation will occur and the amplitude of the fundamental voltage will decrease with increasing M\ because then the DC-link voltage is decreasing. If the DC-link voltage is decreasing, the control system of the converter increases the reference modulation index. If the DC-link voltage becomes too low, the reference modulation index becomes larger than one.

12


M*

0.02

0.015

0.01

I

0.005

0

Figure 3.2: The normalized fundamental voltage, the normalized harmonic voltage andthe THD as a function of the reference modulation index.

Figure 3.2 shows that for M‘< 1, the THD, and UWDl are decreasing with increased reference modulation index M'. For M’> 1, the THD, and UWD, are increasing with increased reference modulation index M\ because low frequency harmonics occur due to over-modulation. THD, is increasing faster than the UTHDI because the amplitude of the fundamental voltage is also decreasing. The fundamental voltage level is constant when the reference modulation index is below one, and above one the fundamental voltage level is decreasing linearly.

The total harmonic voltage distortion is constant when the switching frequency is altering. Because, the voltage harmonics will be transferred up in frequency when the switching frequency is increasing and when the switching frequency is decreasing the voltage harmonics are transferred down in frequency. The total harmonic current distortion decreases when the modulation frequency is increasing.

As a conclusion, the THD, from the PWM-converter will be minimized, if the modulation index is close to one. That means that the DC-link voltage should be as low as possible. The problem with a low DC-link voltage is that the controller bandwidth of the inverter system decreases, see Chapter 4.

In Figure 3.2 the THD is decreasing with increased reference modulation index M*. When the reference modulation index M* is greater than one, the normalized harmonic voltages is decreasing faster than the THD, because the amplitude of the fundamental

13


voltage is also decreasing. The fundamental voltage level is constant when the reference modulation index is below one, and above one the fundamental voltage level is decreasing linearly.

The normalized harmonic voltage is constant when the switching frequency is altering. Because, the voltage harmonics will be transferred up in frequency when the switching frequency is increasing and when the switching frequency is decreasing the voltage harmonics are transferred down in frequency.

As a conclusion, the THD from the PWM-converter will be minimized, if the modulation index is close to one. That means that the DC-link voltage should be as low as possible. The problem with a low DC-link voltage is that the controller bandwidth of the inverter system decreases, see Chapter 4. The THD does not decrease when the switching frequency increases but it is easier to remove high frequency harmonics by means of filters than harmonics of low frequencies.

33 Voltage harmonics on the grid induced by converter

To calculate the voltage harmonics on the grid all the parameters having influence on the harmonic content must be known. To simplify the calculations all resistances and capacitances of the grid are neglected. This section shows how the harmonic content on a grid is affected by a wind power plant connected to the grid via a self-commutated voltage-stiff inverter.

3.3.1 The electrical system of a wind power plant with variable speed

In the wind power plant the generator is connected to a converter, i.e. a rectifier and an inverter, and the other side of the converter is connected to the grid. The inverter is voltage-stiff and three inductors, one in each phase, must be connected between the inverter and the grid. To adapt the converter voltage to the grid voltage level a transformer is connected after the inductors. The three inductors, one in each phase are called the inductance grid filter. In Figure 3.3 a wind power plant is shown. Usually the converter is installed in the tower to reduce costs.

The electrical base parameters for the wind power plant and the grid being analysed are:

• The wind power plantThe rated power of the wind power plant is denoted Pn and the power factor is approximately one. The phase-to-phase voltage of the inverter is denoted U„. The electrical system has the same rated power as the wind power plant

14


rectifier and inverter are located in the tower

grid filtertransformer

Figure 3.3: The electrical system of a windpower plant with variable speed.

• The transformerThe rated power of the transformer is StTf n and the transformer has a voltage drop ut of 0.05 p.u. The primary side of the transformer has the rated voltage Uh„ n and the secondary side, towards the grid, has the rated voltage UUeh n.

• The gridThe phase-to-phase voltage is denoted E and the apparent short-circuit power is

3.3.2 The equivalent circuit of the plant

Below in Figure 3.4 the equivalent Y-phase circuit for the inverter of the wind power plant, the transformer, the grid filter and the grid are shown. All inductance values refer to the low-voltage side of the transformer. This equivalent circuit is adapted for low frequencies.

15


grid filter transformerinverter

inverter transformergrid filter

Figure 3.4: a) Equivalent circuit for inverter connected to the grid via grid filter and transformer, b) Equivalent circuit used for calculating harmonics.

The generator emf of the grid is equivalent to a short-circuit for the current harmonics. When the transformer is a part of the system, all variables refer to the low-voltage side of the transformer, i.e. to the voltage level of the inverter. The parameters in the equivalent circuit in Figure 3.4 are:

Up (r) instantaneous phase voltage of the grid.

My(r) instantaneous phase voltage of the inverter.

“/(n>i)Cr) ^e harmonic phase voltage content of the inverter, i. e. the output voltage waveform with the fundamental voltage subtracted.

e'f{t) the instantaneous phase voltage of the generator of the grid referring to the low- voltage side of the transformer. In harmonic calculations e'f{n>i)(t) = 0.

%/%(«>!) (0 the instantaneous harmonic voltage of the grid side of the transformer.

iN(t) the instantaneous phase current from the inverter to the transformer.

iN(ri>,j(r) the instantaneous phase current, without the fundamental current component

Ltlf the total leakage reactance of the transformer.

L'ff the inductance of the grid. It refers to the low-voltage side of the transformer.

16


3.3.3 The parameter of the equivalent circuit

In this section the equivalent circuit parameters are derived from the basic parameters of the wind power plant and the grid.

The grid inductance is derived from the short-circuit power of the grid Ssc and the phase- to-phase voltage of the grid£, see equation below. The phase reactance of the grid is

The grid inductance per phase is

=_2

(3.3)

In Equation (3.4) the grid inductance value refers to the low-voltage side of the transformer.

L'n = u,V. ^high,n )

(3.4)

The transformer inductance is derived from the voltage drop uk and refers to the low- voltage side of the transformer. The reactance of the transformer is

= (3.5)

and the leakage inductance of the transformer yields

L Ix.2»fN

(3.6)

3.3.4 Different types of grid filters

To connect two voltage-stiff systems a reactance must be installed between the systems to limit harmonic currents. If a simple control system is used, the inductance has to be well defined.

Here three types of grid filters are analysed. They are:

17


Filter# 1: Series inductance.

Filter #2: A series inductance and a parallel capacitor.

Filter #3: A series inductance, a parallel capacitor and a link, damping the resonances between inductances and capacitors in the circuit

In all further calculations in this chapter we assume that the grid-connected inverter has the following data:

The modulation method is a sub-oscillating PWM, using added zero components, see Section 2.3.2.

The switching frequency /„ is 5 kHz.

The modulation index M is 0.9.

The DC-link voltage uic is 629 V.

The fundamental phase voltage Uf of the inverter is 230 V.

No voltage harmonics above the frequency of 20 kHz.

3.3.5 The L-filter

The L-filter contains a series inductance. In Figure 3.5 the equivalent circuit of the inverter is connected to the grid via series inductances and a transformer.

inverter transformer

Figure 3.5: The equivalent circuit of the inverter connected, to the grid via seriesinductance and a transformer.

The harmonic voltage on the grid can be calculated considering a voltage division between the inductances Ls, Ltrf and L'N.

18


u7N(«> 1) (0 =l'n

L:+Lvf + LN 7(n>l) (0 (3.7)

By defining the harmonic reduction factor K the harmonic content of the grid can be calculated

K = - L'NLs+L,rf + LN

(3.8)

In Figure 3.6 the harmonic reduction factor has been calculated and the series inductance Ls has been set to 0.1,0.2 and 0.3 p.u. The voltage drop of the transformer uk has been chosen to 0.05 p.u., Sn is the rated apparent power of the wind power plant and Ssc is the short-circuit apparent power on the grid.

J I Lilli

Figur 3.6: The harmonic reduction factor as a function of the short-circuit apparentpower of the grid compared with rated apparent power of the wind power plant.

When the grid is weak, i. e. the short-circuit power of the grid is low, the grid inductance will be large. This results in a high harmonic level on the grid. The larger series inductance between the inverter and the transformer the smaller the voltage harmonics on the grid. A large transformer leakage inductance is good. It keeps the voltage harmonics on the grid down. But a transformer with a larger leakage inductance is more expensive than a standard transformer.

19


To calculate the total harmonic current distortion of the grid THD m the distortion of the inverter THD, is needed. The total harmonic current distortion of the grid becomes known by multiplying the distortion of the inverter THD, by the harmonic reduction factor k, see equation below

THDm = kTHD[ (3.9)

Suppose that the modulation index of the inverter is 0.9, the relation between the switching frequency and the fundamental frequency is 100 and the voltage drop of the transformer is 0.05 p.u. With Equation 3.9 the THD w can be calculated at different

short-circuit power on the grid and different series inductances, see Figure 3.7.

0.001

0.00010.250.05

Figure 3.7: The total harmonic current distortion on the grid, caused by the PWM-inverter. The series inductances and short-circuit power of the grid are varying.

As a conclusion, the short-circuit power of the grid can be seen as a damping factor. At higher short-circuit power the distortion level is lower.

3.3.6 The LC-filter

An LC-filter has larger damping at a specific frequency than an L-filter. In this application it results in less harmonics on the grid. The disadvantage with the LC-filter is that it has an undamped resonance at a specific frequency. Even if the inverter of the wind power plant does not affect the LC-filter at its resonance frequency, other harmonic sources on the grid can. Due to this disadvantage the LC-filter is not interesting in this application

20


and will not be further investigated. By using a complex control strategy, the currents and voltages in the filter can be controlled and the filter resonance becomes restrained, see [12].

3.3.7 The LC-filter with a damping link

The LC-filter with a damping link consists of three parts in each phase. These are series inductance, parallel capacitor and a damping link. The benefit with this grid filter is that the series inductance can be smaller without the grid distortion becoming larger. The disadvantage is that the filter needs more components, which makes the filter more expensive than the simpler filters. In Figure 3.8 the equivalent circuit is shown.

|( ~ V! /(”>i>w :

transformerinverter

Figure 3.8: The equivalent circuit of the wind power plant, the transformer, the gridand the grid filter No. 3.

Design of an LC-filter with a damping link

A proper basis design [13] of an LC-filter with a damping link, see Figure 3.9, is to choose the components of the filter as in Table 3.1. Then the resonance peak of the filter is suppressed.

Table 3.1: Parameters of the LC-filter with a damping link shown in Figure 3.14.

Filter parametersu 0.05 p.u.L, 0.20 p.u.Xa 10.0 p.u.Xci 10.0 p.u.2, 1.60 p.u.

Li

Figure 3.9: The equivalent circuit of the LC-filter with a damping link

If the LC-filter with the damping link is designed as shown in Table 3.1, the resonance frequency between L, and C, will be

21


0-10)

if the components are expressed in per unit. A grid frequency fN is 50 Hz and this results in a resonance frequency of 707 Hz. The Bode diagram of the filter is shown in Figure 3.10. When changing the filter impedance or resonance frequency, classical filter design methods are used. In Figure 3.10 a damped resonance frequency of 700 Hz can be seen. The filter has a damping of 40 dB per decade when the frequency is above the resonance frequency of the filter.

/ [Hz]

Figure 3.10: Bode diagram ofLC-filter with a damping link. The filter parameters are taken from Table 3.1.

Design of a grid filter with a damping link by method A

The design of the filter with method A is to adapt the basis filter parameters in Table 3.1 to a new impedance level and a new resonance frequency. The impedance level is adjusted to the chosen series inductance Ls, i. e. Lx —Ls.

The problem is that the leakage inductance of the transformer and the inductance of the grid are approximately equal to the series inductance Ls of the grid filter. Therefore thegrid filter design must take into consideration the leakage inductance of the transformer and the inductance of the grid.

22


The transfer functions and HM2(i) describe the frequency characteristics fromthe harmonic voltage of the inverter «/(ri>1)(f) to the harmonic voltage of the grid filter M/j>fe3(n>i)(?) to the harmonic voltage of the grid «Mn>1)W- When the voltagespectrum of the inverter is known, the transfer functions can show if the grid filter amplifies or damps the harmonic voltage of the inverter at the grid filter or at the grid at a specific frequency.

The grid filter should not have a resonance peak amplifying the grid voltage harmonics, if the grid voltage is distorted. The filter or the inverter can be damaged if the voltage over them becomes too high. The transfer function Hlhi3(s) describes the frequency response from the grid voltage u >n(t) to the voltage of the grid filter M/i/M(n>1)(?)- The transfer

functions are written as

W)(3.11)

(3.12)

(3.13)

Suppose that the plant and the grid have parameters as in Table 3.2. With Equation (3.10) and using Table 3.1 the grid filter parameters can be designed. The parameters of the grid filter are shown in Table 3.2. The transfer functions which are shown in Figure 3.11 are calculated with the grid filter parameters in Table 3.2.

23


20

-604

100 1 0 1 0f [Hz]

Figure 3.11: The transfer functions H[hJ}(s), HM1{s) and Htm{s) with parameters as in Table 3.2.

From Figure 3.16 a resonance top of the transfer function HM2(s) can be detected. The

peak is approximately at the frequency of

(3.14)

The Equation 3.14 is derived from Figure 3.8. It is the distortion level on the grid which is important. So, the resonance peak of the transfer function Hihd2(s) should be removed.

Figure 3.11 shows the transfer function H,m(s). The shape of the curve is approximately the same as for the other two transfer functions and no resonance frequency with a high magnitude peak can be detected. So harmonics on the grid can not damage the filter or the inverter.

Design of grid filter with method B

The problem with design method A is that taking the parameters directly from Table 3.1 and modify them to the right impedance level, i. e. L, = Ls, and resonance frequency, the shape of the transfer function in Figure 3.10 becomes different due to the other impedances of the system. The resonance frequency of the basis filter, Equation 3.10, differs from the resonance frequency of the transfer function HtkJ2(s), Equation 3.14.

24


By designing the grid filter with a new value of the inductance L,

--------!-j------- (3.15)

+ Ln

and then determine the other filter parameters from the new L, and the desired resonance frequency, Equation 3.10, the shape of the transfer function becomes equal to the shape of the curve in Figure 3.10. After the grid filter design procedure the series inductance will have the value of Ls. In Figure 3.12 the new transfer functions, which apply to thedesign method B, can be seen. The electrical system and the calculated filter parameters are shown in Table 3.3.

/ [Hz]

Figure 3.12: The transfer Junctions H:hdl(s) and HIhd2(s) with parameters from Table 3.3.

Table 3.2: Grid filter parametersderived using method A.

Plant and Grid Grid filterL,=0.1 p.u. &<-jT

Ltr= 0.05 p.u. L2=0.4 p.u.L'n=0.05 p.u. Xcl=40 p.u.4 =1000 Hz XC2=40 p.u.

J?2=4.53 p.u.

Table 3.3: Grid filter parametersderived using method B.

Plant and Grid Grid filterL$=0.1 p.u. Lj=0.05 p.u.Lrrf=0.05 p.u. L2=0.2 p.u.L'n=0.05 p.u. XC1=20 p.u./ =1000 Hz Xp.U.

R2=2.21 p.u.

.rti

25


A comparison of the transfer functions in Figures 3.11 and 3.12 shows that the transfer function Hthdl{s) with the method B design has a lower resonance peak than the transfer function HlM2 with design method A. Further on, the capacitor Cl of the grid filter design method B has twice the reactance value of capacitor C, of the grid filter design method A. To get a fair grading of the two different design methods, the two filter capacitors should have the same size. The grid filter with design method A has been designed so that the reactance of the parallel capacitor Cl is 20 p.u., see Table 3.4 for

electrical system and filter data.

Table 3.4: Grid filter parametersderived using method B.

Plant and Grid Grid filterL,=0.1 p.u. Lj=0.05 p.u.Lir=0.05 p.u. p.u.L^=0.05 p.u. XC1=20 p.u.f =707 Hz XCi=20 p.u.

i?2=3.2 p.u.

When the transfer functions HMl(s) and with data from Tables 3.3 and 3.4 arecompared, see Figure 3.13, the method B has the lowest gain at the resonance frequency.Above the resonance frequency the two transfer functions design method A and B have the same damping factor, i. e. 40 dB per decade. The two filters have the same low andhigh frequency gains. It depends on the fact that the filters have the same capacitor and <inductance values.


Figure 3.13: The transfer functions HthJ](s) and HM2(s) with parameters from Tables 3.3 and 3.4.

Since the resonance frequency of method B has a lower gain than method A, method B is to prefer.

The distortion level of the grid with varying parameters

Suppose that the inverter has the same data as those in Section 3.3.4, where the grid filter is a series inductance. The distortion level has been calculated both with method A and B. The grid filter inductance Ls and the capacitor C, have been altered and from thesevalues the other parameters of the grid filter have been calculated. The leakage inductance of the transformer is 0.05 p.u. and the ratio between the short-circuit power of the grid and the rated power of the wind power plant is 20. In Figures 3.14 and 3.15 the distortion level of the grid, due to the connected inverter, is plotted.

27


0.0025

0.002 L, = 0.1 A

^ 0.0015L, = 0.3 A

0.001-

L = 0.1 B0.0005

k 0.2 BL, = 0.3 B

0 250 500 750 1000

Xa [p.u.]

Figure 3.14: The distortion level of the grid when the grid filter series inductance and the parallel capacitor have been altered. A stands for filter design method A and B stands for filter design method B.

L, = 0.1 BL, = 0.1 A

L, = 0.2 A

0.001--L, = 0.3 A

L, = 0.2 BL, = 0.3 B

0.00011000

Xa [p.u.]

Figure 3.15: The distortion level of the grid when the grid filter series inductance andthe parallel capacitor have been altered. A stands for filter design method A and B stands for filter design method B.

28


3.3.8 Comparison between the grid filters

When comparing the two filter design methods A and B, the inductance of the grid is 0.05 p.u. and the transformer leakage inductance is 0.05 p.u. Figure 3.14 shows that the two design methods A and B result in different harmonic levels of the grid for the same values of the series inductance and the parallel capacitor. For the methods A and B, the distortion level starts to differ from the reactance value of the parallel capacitor C1 of 250 p.u. Figure 3.15 is the same as Figure 3.14 but it has a logarithmic axis. The distortion level is almost constant up to the reactance value of the parallel capacitor C, of 100 p.u. To reach the same THDm, say 3-10-4, the reactance value of the parallel capacitor must approximately be 2 times larger for the filter using series inductance Ls 0.2 p.u. than for the filter using series inductance Ls 0.1 p.u. For the series inductance Ls 0.3 p.u. the reactance value of the parallel capacitor must approximately be 3 times larger than for the filter using series inductance Ls 0.1 p.u. The peak of the distortion level occurs when the filter resonance frequency is equal to the switching frequency of the converter.

When the L-filter is compared with LC-filter with a damping link and the THDm is lower than 3-10~4, the series inductance, the L-filter, must be at least 0.5 p.u. But using a series inductance of this size the bandwidth of the power angle control will decrease compared with a series inductance of 0.1 or 0.2 p.u.

29

Power angle control

Chapter 4

Power angle control

4.1 The system configuration

The electrical system of the wind power plant can be divided into two subsystems, the generator/rectifier system and the inverter system. The best alternative for the generator/rectifier system is to choose the synchronous generator and the diode rectifier, see [14]. It depends on that the cost is low and the efficiency is high compared with an induction generator connected to a forced-commutated rectifier. When the speed of the synchronous generator is altering, the voltage value at the DC-side of the diode rectifier will change. A voltage-stiff forced-commutated inverter utilizes DC-link voltage. A step- up chopper is used to adapt the rectifier voltage to the DC-link voltage of the inverter, see Figure 4.1. When an induction generator with a forced-commutated rectifier is used the chopper will not be necessary.

When the inverter system is analysed the generator/rectifier system can be modelled as an ideal current source, because the step-up chopper or the forced-commutated rectifier utilizes a high switching frequency. The bandwidth of the rectifier is much higher than the bandwidth of the generator, gear and turbine system. The load steps are also small. The current is denoted idc (?) and is independent of the value of the DC-link voltage.

The inverter system is a stiff-voltage forced-commutated inverter. This type of inverter must have a capacitor in the DC-link and a series inductance in each phase on the AC-side of the inverter, when the inverter is connected to a grid. The capacitor of the DC-link has the value Cie. The inductors at the AC-side of the inverter have the inductance Ls and the resistance Rs.

The three-phase voltage system of the grid has the phase voltages e,(t), e2(?) and e3(?) and the phase currents are denoted i,(?), z'2(?) and z'3(r). The DC-link voltage value is denoted udc(t) and the current in the DC-link to the inverter is called iv(t).

30

■’F

V

LO

\?<*&)

:-<>

r 7

vv ■'

• <

inverter

valve 5valve 3valve 1

valve 6valve 4 valve 2

diode-rectifiersynchronous

generator step-up converter

turbine

Power angle control

Power angle control

4.2 The basic idea of the power angle control

Assume that the inverter does not produce voltage harmonics, because the voltage harmonics have a much higher frequency than the fundamental voltage frequency. The basic idea of power angle control is to vary the phase displacement angle between the three-phase voltage of the inverter and the three-phase voltage of the grid. The voltage difference between the three-phase voltage of the inverter and the three-phase voltage of the grid will be placed over an inductance in each phase. The larger phase displacement angle the larger voltage difference which results in a higher current through the inductor.

The active power from the inverter to the grid is, if the resistance of the inductor is negligible

P = -^r~ sin(0) = -^-0

(OnL, cdnL.(4.1)

where E and U are the phase-to-phase rms-voltages of the inverter and the grid. The grid angular frequency is denoted (0N and the inductance in each phase is Ls. The phase displacement angle is 9. If the phase displacement angle is small the reactive power to the grid will be

Q = Ucos(0)) = —^—(E~ U) (4.2)conLs co1yLs

The phase displacement angle 9 can now be controlled so that the voltage over the capacitor in the DC-link will have a constant voltage independent of the power fluctuation in the generator of the wind power plant.

4.3 Non-linear model of the system

The three-phase voltage of the grid is written as

*i(f)

e2(?)

e3(t)

2Ecos((DNt)

•J|Ecos(nv-^)

■JjEcos(coNt—

(4.3)

If the voltage harmonics from the inverter are neglected, the three-phase voltage can be expressed as

32

Power angle control

«iW =

u2(t) =

u3(t) =

^Ucos(m)

2 2%-XJcos(m——)

2 . 4jt—TJcos(m------ )3 3

(4.4)

where Z7 is the phase-to-phase rms-voltage of the inverter and co is the angular frequency, expressed in radians per second, of the inverter.

The phase currents in space vector form can yielded from, see Figure 4.1,

u(t)-i(t)Rs-^Ls-e{t) = 0

where the space vector voltage of the grid is

e(t) = Eeia"'

(4.5)

(4.6)

and the space vector voltage of the inverter, derived from Equation 4.4, is

u(t) = Ueim (4.7)

In Figure 4.2 the space vector voltages and currents are shown. The resistances in the inductors have been neglected. a) b)

Figure 4.2: a) The space vectors of the voltages and current in an aft-coordinatesystem, b) The space vectors of the voltages and current in a dq-coordinate system.

33

Power angle control

When the voltages and current space vectors are transformed into the dq-coordinate system the quadrature axis will be parallel with the voltage vector e(t). This makes the dq-coordinate system rotating in the <xp-coordinate system. In Figure 4.2b the voltage and current space vectors in the dq-coordinate system are shown.

Equation 4.5 can be written in component form in the dq-coordinate system as follows

^~z"d(0 = (0Niq{t)

ig (0 = 7- (0 - 7- eq (t) - y-iq (0 - coNid{t)L, L, L.

d,dt

(4.8)

This equation is valid when the output voltages of the inverter are not sinusoidal. If the output voltages of the inverter are sinusoidal, the direct and quadrature voltage components become

“,/(*) =-sin(0(f))Z7

uq(t) = cos(8(t))U(4.9)

Here the phase displacement angle 0(f) is equal to

0(r) = j(<a(T)-<a„)dT 0

(4.10)

Equation 4.8 can be rewritten as

~ft V(0 = —j~ 17sin(0(f)) - -j- id(t) + CONiq(i)

jtiq(t) = j- C/cos(0(f)) -j-eq(t)-j±iq(t) - coNid(t)(4.11)

When the valves are ideal, the DC-power and electrical AC-power of the inverter are equal all the time, see Equation 4.12.

ud(t)id(t) + uq{t)iq{t) = iv(t)uic(t) (4.12)

and the DC-link current iv (f) can be written as

34

Power angle control

*V(0 ud(t)id(t) + uq{t)iq(t) “*(0

(4.13)

The voltage across the DC-link capacitor can be expressed as

udc{t) = ^-]{idc{f)-iv{x))dx (4.14)

By controlling the amplitude of the inverter three-phase voltage and the phase displacement angle 0(f) the DC-link voltage udc(f) can be controlled when the current from the generator/rectifier idc(t) is varying.

4.4 The reduced non-linear model

The generator/rectifier can be replaced by an ideal current source, because small current steps are not dependent on the voltage of the DC-link. The control of the inverter has two input signals. These are the phase displacement angle 0(f) and the relative voltage ux(t). In Figure 4.3 the reduced non-linear model is shown. The relative voltage is defined as

«iW =mE(t)

(4.15)

The grid is supposed to be constant. This means that the frequency and voltage amplitude of the grid are constant. Equation 4.15 can now be rewritten as

U{t) = ux{i)E (4.16)

positive power flow ----------------------- 5^"

0"(f) <(')

Figure 4.3: The reduced non-linear model of the system.

■> 1 \*{ s;

35

Power angle control

4.5 Linearization of the system

4.5.1 Operating points

Let the operating points be denoted index "0" and the variables perturbation around their variables operating points be denoted index " A The two control variables of the inverter can be written

6(t) = 90 + Ad(t) ux(t) = ux0 + Aux(t)

(4.17)

The operating points of the voltage of the inverter are

udo =-sin0o«,o equq0= cos G0ux0eq

and with Equation 4.11 the operating point of the currents of the inverter can be decided as

ho

.V

' Rs -1 r i .. i0)N ~~f~eqUx0Sm&0

Tei(uxo cos 0O-1)

and the operating point of the DC-link current iv (t) is

7 r i—

-M«0. V

(4.19)

(4.20)

4.5.2 Linearization of the inverter

When the operating points for the system are decided, the linearized voltages of the inverter will be, using Equations 4.9 and 4.16

'Aud(t)~ ~eqUxO COsQq -eqsin%~ 'AQ(t)'_-equx0sinQ0 eq cosQ0 _ _Aux(t)_

36

Power angle control

and the linearized currents of the inverter can be solved by means of the state equation, where A and B denote state equation matrices

d Af„(,)-— A

‘A iM , T> "A0(r)‘dt Ai,(f)

— A_Ai,(r)

T* Jt>_A“,(f).

where

and

B =

-y-ux0cos60 —^-sin0oLs Ls

—r-“*osin0o -f-cosBo Lt L,

(4.22)

(4.23)

(4.24)

The linearized DC-link current is written as

r -izUd0 TAi/f)l pd»TTAndW]

UqO Af,(r)+

A°. Auf(t)

UdcO(4.25)

The Equation 4.25 can be rewritten as

Aiv(r) = CAi/f)Ai,(t)

+D"A0(r)"_Aux(t\

(4.26)

where C and D denote state equation matrices

'Ud0 T

.V.

3

ui0

UdcO f 1 (4-27)

and

37

Power angle control

D = Vr p

B =>.

eqUx 0 (iM cos e0 + ig0 sin 0O) —[~id0 sin 60 + iq0 cos 0O)■ldcO

(4.28)

4.5.3 The state equation of the DC-Iink

The state equation of the DC-link is easy to find because there are no nonlinearities. The inputs are the DC-link current iv(t) and the generator/rectifier current idc{t). The output is the voltage across the DC-link capacitor. A resistor R& with a large value of the resistance is mounted parallel with the DC-link capacitor. The state equation becomes

^«*W = AMn<fc(t)+BM

uic{t) = C^u^it)

where

AM —

1

iS'ic Cdc CdclOil — 1

(4.29)

(4.30)

4.5.4 The state equation for the linearized system

To dimension a suitable regulator for the whole system, the state equation for the open loop system must be known. By putting the state equation for the inverter and the DC- link together a new state equation with three states and three inputs and one output will be achieved. The state equation becomes

'A i/t)" "AW" "A0(t)‘

Ai,(f) — Avm Ai,(f) + ®VM Mz(t)„An&(r)_ .Ak *(0. _Ai^(f).

and

Am*(0 — CyMAi„(f)Ai,(f)

.AK*(r)

(4.31)

(4.32)

38

Power angle control

The matrices Am, and D^, are written as

A-vm -

_&Ls

—COn

C0N

_& 0

?oUdcOCdc «dcO^dc Rifidc.

BVM “

---~MjOCOS(®o)

e?°^° (*<ro c°s(90) + :*o sin(0o))Udc0^dc

-“•sin(0o) 0

^COS (0O) 0

Ls£q*— (sin(0O) - z;0 cos(0o))

Udc0'-'dc t'A.

Cvm=[0 0 1]

39

Power angle control

4.5.5 The step response of the DC-Iink current

The performance of the DC-Iink current A iv{t) is essential in order to control the whole inverter system. Here the step responses and Bode-diagram of the DC-link current Aiv(t) will be analysed. Eight step responses have been simulated, numbered from a to h. It is the phase displacement angle A0(t) and the relative voltage Aux(t) which make steps. The parameters of the inverter system which alter are shown in Table 4.1. The necessary parameters, see Appendix B for parameters normalization, which are constant for the different steps are U=E = l p.u. and udcQ = 1.52?=1.0 p.u.

Table 4.1: Different parameters of the inverter systemNo a b c d e f S hLs [ p.u.] 0.044 0.044 0.044 0.044 0.305 0.305 0.305 0.305Rs [ p.u. ] 0.0028 0.0028 0.0083 0.0083 0.0028 0.0028 0.0083 0.00836o 5s 105 5> 10= 5s XT S’ 10PLs/Rs [ms] 50 50 169 169 347 347 117 117

In Figure 4.4 the different step responses are shown. Here the step of the DC-link current Aiv(t) is normalized by Aiv(°°) so the step size is one. Because, when the value of theseries inductance is changing, the size of the step response will also change. The step response of the DC-link value Aiv(t), when the relative voltage Aux(t) makes a step, is normalized by Azvf°°) which comes from the A6f t) step.

When comparing the step responses a and c in Figure 4.4 we see that the transient time becomes shorter when the resistances are increasing. From Figures 4.4a and 4.4e we see that the transient time increases when the series inductance becomes larger. When the working point of the phase displacement angle 80 increases, the transient time willdecrease, see Figure 4.4g and h.

The relative voltage Aujt) steps result in similar responses in the DC-link currents Aiv (t). The difference compared with AQ(t) step responses that the value of the DC-link current A iv(t) after the A ujt) step is near zero. Only the reactive power is changing when the relative voltage is altering in the steady state, see the Equations 4.1 and 4.2.

40

A M

l) A

fyfO

AM

O

Aiv(

i)A

lyM

A

iyf~

J ,

Aiy M

, Afy (~J

Power angle control

2 - !

* A9 step response

- 1.5-

A..

A9 step response

r

A Au, step response

t\5. Jvv---------

Au, step response

.f--------------------------------- ---------------- r0 0.1 0.2 0.3 0.4 0

time l s]

e

.5 0

f

0.2 0.3 0.4 0.5

time [ s]

A9 step response

relative voltage Aux(t) make steps. The parameters of the transfer function can be seen in Table 4.1.

41

Power angle control

I (s)The Bode diagram of the Equation 4.26 is shown in Figure 4.5. The parameters

used come from Table 4.1. The Bode diagram is normalized so that the transfer functions have the same gain at low frequencies. A large resonance peak can be found at the frequency of 50 Hz. Depending on the value of the parameters of the inverter system the resonance peak will have different amplitudes. The gain decreases by 40 dB per decade for frequencies over 50 Hz.

f [Hz]

Figure 4.5: The Bode diagram of Equation 4.26. The parameters come from Table 4.1.

The different Bode diagram in Figure 4.5 above has large resonance peaks. The resonance frequency is equal to the grid frequency. From the state Equation 4.26 the transfer function of the Hv(s) = Iv(s) / 6(s) can be made. The denumerator den of thetransfer function obtained from Equation 4.22 is

( D Vden[Hv ($)]=$+—*■ + a>l

V L>)(4.33)

and the resonance frequency cora of the transfer function yields

1- R. (4.34)

42

Power angle control

The transfer function has the damping factor of —. Here —— «1 so the resonanceLs

frequency becomes approximately

cores = coN (4.35)

The larger inductor resistance Rs and smaller inductor inductance Ls are the smaller theresonance peak for the Bode-diagram in Figure 4.5 will be and the faster the system transients will be damped out. To use a large resistance Rs the losses of the system willincrease and of cource the efficiency will decrease.

The bandwidth of the system will be limited due to the large resonance at the resonance frequency of 50 Hz, see the Figure 4.5. The bandwidth of the system will always become less than 50 Hz.

43

Power angle control

4.6 Regulators

The regulator should control the phase angle Q( t). The relative voltage ux(t) will always have a value so that the amplitude of the three-phase voltage system of the inverter will be equal to the amplitude of the three-phase grid voltage, see Equation 4.15. Here are two regulators analysed. The first one is a Pi-regulator that feedbacks the DC-link voltage udc(t). The second regulator is the same Pi-regulator with a feedforward of the generator/rectifier current idc(t). In Figure 4.6 the reduced non-linear model with the regulator is shown. The only parameter that can be changed by the user is the reference voltage value of the DC-link udc(t)

inverter

generator/rectifier

udcV) regulator

Figure 4.6: The regulated non-linear system.

4.6.1 The Pi-regulator

The Pi-regulator reduces steady state errors. Define e(f) as the error signal, i.e. the difference between reference value of the DC-link voltage udc(t) and the value of the DC- link voltage udc(t). The Pi-regulator is defined as

Ms(t) 1+i (4.36)

where kP is the proportional gain and T, is the integral time constant. To make it easy to implement the Pi-regulator into an analogue control system, the Pi-regulator will be modified to

(4.37)

44

Power angle control

where k, and T„ are constants to the integral time constant in the modified Pi-regulator. Here T„ is »1 and

The state equation becomes

(4.39)Q{f) — CPI0/(r)+Dpi(udc udc)

where

4.6.2 The PIFF-regulator

When using feedforward the system becomes faster than when only Pi-regulator is used. The PIFF-regulator is the same as the one for the Pi-regulator except for the feedforward of the known process disturbances. Here the generator/rectifier current is used for a feedforward. The PIFF-regulator expressed in the state equation is

z4(f)-“&(*). WO .

(4.40)

G(t) — Cp[FF5;(t) + DPIFFudc{t)-udc{i)

where

here kFF is the feedforward constant.

45

Power angle control

4.7 The total linearized system with a regulator

By using the open loop of the system, where the linearized inverter and the regulator are included, the parameters of the regulator can be decided for a specific stability margin. When the regulator parameters are determined, the system can be analysed by the state equation for the closed loop.

4.7.1 The open loop of the linearized system with a Pi-regulator

The open loop can be described by means of a state equation. The input to the state equation is the error signal Ae(r), i. e. the difference between the reference value of the DC-link voltage Audc(t) and that of the DC-link voltage Audc(t). The state equation hasfour states. The matrices are built up with matrix elements from the Equations 4.31 and 4.32. The first row and the first column element in the matrix AW1 are denoted amU.

'A iditY 'Aij(f)-

ddt

Ai,(f)Audc(t)

— A0LPjAi,(f) Ak *(f)

_A0,(r). .A0/CO.

(4.41)

where

Aqlpi —

^wnll Q'vmll ^vml3 bvmll -bvmllkPaim21 tivm22 °m23 ~bvm2l ~bvm21 kPavm3l avm32 avm33 ~^vm3l ®OLPI~ ~bvm3l^P

1 k,0 0 0 -----L T"\ . .

4.7.2 The closed loop of the linearized system with a Pi-regulator

When the parameters of the Pi-regulator are known we can see how the total linearized system reacts (with the Pi-regulator). The system is described using the state equation 4.41. The state equation has three inputs. These are the relative voltage Aux(t), the generator/rectifier current Aidc(t) and the reference value of the DC-link voltage Audc(t). The matrices ACLPI and B^p, have been built up with matrix elements from the state equations 4.31 and 4.32.

46

Power angle control

'Af/r)d_ Ai,(r)dt Audc(t)

„A6,to.

= Ar

Ai/t)Ai,(f)

AM*(f)A6/r)

"*"®CLPI

An^f)Az^(f)Am*(0.

(4.42)

where

avmll tivml2 avml3 + ^P^vmll ~^vmll ^vral2 ^vral3 —^vmll^P

Sm21 °vm22 ^vm23 ^p^vm 21 ^vm22 ^wu23 ~^vm21^P

avm31 avm32 avm33+^P^vm3l ~^vm3\ ®CLPI — bvm32 b\-m33 ~^vm31^P

k, 1 k,0 0 —^ -------—

L Tu \

4.7.3 The closed loop of the linearized system with a PIFF-reguIator

The state equation 4.43 represents the total linearized system with a PIFF-reguIator. The state equation has the same inputs and outputs as the state equation 4.42. Some matrix elements of the equation come from the state equations 4.31 and 4.32.

ddt

Ai„(f)Ai,(f)AKj/f)M(r)J

= Ar

a idity Ai,(f) A«dc(r) .A d,(t)

+BrAux(t)Af^(0

|A%L(f)

where

^ CLP IFF -

avm\l avml2 °vml3 kpbymii

avm21 avm22 avm23 + kp^wnll -bvm2\

avm3l avm32 avm33 +kpbym3l ~^vm31

0 0_A. 1

T„

B,CLPEFF"

^vml2 ^vth13 + ^vmU^FF ^vmil^P bvmtt ^vm23 + ^vm21^FF ~^vm2l^P ^vm32 ^vm33 + ^vmil^FF ~^vm31^P

k,

(4.43)

47

Power angle control

4.8 The step responses of the linearized system with different regulators

Based on the open loop state equation of the linearized system with the regulator the PI- regulator was tuned using gain margin criterion Gm >3 and the phase margin criterion Pm > 40°,[10, 15]. The feedforward constant is determined so that the DC-link current Aiv(t) has the same value as the generator/rectifier current Aidc(t) in steady state, see Figure 4.7.

Aza(') Equations 4.22K FF and 4.26

Figure 4.7: The feedforward block scheme.

Suppose that the inverter system has the parameters that are shown in Table 4.2. The parameter values have been determined by an iterative approach. Using the closed loop state Equations 4.42 and 4.43 the step responses and Bode diagrams can be made.

Table 4.2: Parameters of the inverter system.U = E 1.0 p.u. Rec 18.9 104 p.u.u,c0=l-5E 1.0 p.u. kP 1.39 IQ"4

So 5s k, 8.16 103

L, 0.175 p.u. Tlt 13.9R, 5.6 10-3 p.u. kFF 2.4 IQ'3

X Cdc 0.10 p.u.

Three transfer functions will be analysed here. These are the DC-link voltage as a function of the relative voltage Anx(r), reference value of the DC-link voltage Audc(t) and generator/rectifier current Aidc(t). The regulators are PI and PIFF controllers. The Bode-diagrams and the step responses of the relative voltage and the reference value of the DC-link voltage are the same for PI- and PIFF-regulators, see Figures 4.8,4.9,4.10 and 4.11.

48

Power angle control

' mini i i i Mini t mrml t i mini

mag(10°,PIFF) -120

-180

phase(5°,PIFF)phase(5°,PI)

phase(10°,PIFF)yphase(10°,PI)

-300

-360TTTTTTT1]—I i IWUlj—I I Mllllj I]—I I llllllj

/ [Hz]

Figure 4.8: The Bode diagram of the DC-link voltage Audc{t) as a function of therelative voltage Aux{t). The operating points are 60=5°and 60=10° and the controllers are of the PI and PIFF types.

' 11 uml i < » uml i » mini i » i i mini

-120

phase(5°,PI)—- phase(5°,PIFF)- phase(10°,PIFF) phase(10°,PI)

-180

-24 0

-300

mag(10°,PI)"mag(10°,PIFF)-150 -360

f [Hz]

Figure 4.9: The Bode diagram of the DC-link voltage Audc(f) as a function of thereference value of the DC-link voltage Auic{t). The operating points are 60 =5° and 60 =10°. The Bode diagrams are the same for the PI- as for the PIFF-regulators.

49

Power angle control

200-

t> -200-1

-40 0—

-600-

-8 0 00 0.5 1 1.5 2 2.5 3

t [s]

Figure 4.10: The step responses of the DC-linfc voltage Auic(t) when the relative voltage Aux(t) makes a unity step. The operating points are 80 =5° and 0Q=10°. The step responses are the same for the PI- and the P1FF- regulators.

Figure 4.11: The step response of the DC-link voltage Au^t) when the reference value of the DC-link voltage Audc{t) makes a unity step. The step responses are the same for the PI- and PIFF-regulators and for the operating points 60 =5°and 60 =10°.

50

Power angle control

The step responses of the DC-link voltage Audc(t), when the reference value of the Delink voltage Audc(t) makes a unity step, show that the steady state value of Audc(f) is the same as the reference value of the DC-link voltage Audc(t). The DC-link voltage Audc(t) responds with an overshoot of 38%. When the relative voltage makes a unity step, the linearized DC-link voltage step response returns to 0 V. Because the active power does not change in the system, the DC-link voltage remain constant.

When the generator/rectifier current Aidc(t) is varying, the DC-link voltage Audc(t)behaves differently when the regulator is a PI- or PEFF-controller. Figures 4.12 and 4.13 show the Bode diagram and the step response for the DC-link voltage Audc(t), respectively, as a function of Aidc(t), when the PI- and PEFF-controllers are used.

i mutt , , mini t i mini , , mini

phase(5°,PIFF).

f mag(10'

-1 50 -1 20' phase(10°,PI)

phase(5°,PI) -180-200T I llllllj—TTTTTTTIj—TTTTTTTTJ—I I IllllJf—TTTTTTnj- ■muj—rrrTTiTTj—i-1 mni]'

/ [Hz]

Figure 4.12: The Bode diagram of the DC-link voltage Audc(t) as a function of the generator/rectifier current Aidc(t). The controllers are PI and PIFF. The operating points are 90 =5°and 60 =10°.

51

Power angle control

Figure 4.13: The step responses of the DC-link voltage Audc(t) when the generator/rectifier current Aidc(t) makes a unity step. The regulators are PI- and PIFF-controllers and the operating points are 90 =5° and 60 =10°.

When comparing the step responses An&(r), when Ai&(r) makes a unity step, the overshoot is reduced approximately by a factor of 20 and the settling time is reduced approximately by a factor of 2 compared with if the Pi-regulator should be used instead of a PIFF-regulator.

The use of feedforward of the generator/rectifier current results in lower step responses of the DC-link voltage when the generator/rectifier current makes steps. The disadvantage is an increased cost of the controller and the system becomes more sensitively to noise.

52

Power angle control

4.9 Simulation of the non-linear system

When the regulator has been determined for the linearized system at an operating point, the total non-linear system can be simulated. By only simulating the fundamental voltage of the inverter, the simulating time is heavily decreased, approximately by a factor of 300. Here simulations both with and without harmonics are carried out. The input to the system is the generator/rectifier current idc(t) and one of the most important output signals is the DC-link voltage udc(t). The system is simulated in Simulink which is based on Matlab [16].

Figure 4.14 shows the Simulink model of the total inverter system. The regulators, PI and PIFF, are based on the state equations 4.39 and 4.40. The inverter system block named "InvSys" in Figure 4.14 has two configurations depending on if the simulations are with or without harmonics. If the system is simulated without harmonics the block "InvSys" consists of Equations 4.11, 4.13, 4.14 and 4.16. This model is non-linear and is from now on called the "non-linear" model.

timevectorClock!

theta ScopeRadToDegudcref

udcref inputid scopet

uxConstant udc '9 scope

udc scope

iv scopelGeneratorRectifier InvSys

Figure 4.14: The Simulink model of the non-linear inverter system. Reg=PI- or PIFF-regulator; InvSys ^Inverter system; GeneratorRectifier=The input of the Generator/rectifier current to the system.

When the system is simulated with harmonics an ideal PWM-converter is used. Figure 4.15 shows the structure of the inverter system in Simulink.

53

Power angle control

In/Sys

grkLvolages

heta in

phaseToZero gddjnterface voltages

Unit Delayudcln u do outDC link

-H udcudcws

Figure 4.15: The ideal PWM inverter system block structure in Simulink.

4.9.1 Step responses of the generator/rectifier current

The total inverter system step response is here simulated when the generator/rectifier current makes steps. At the beginning of the simulation the generator/rectifier current makes a step from 0 to 0.8 p.u. and at the time of 2.0 s the generator/rectifier current makes a step from 0.8 to 1.0 p.u. and at the time of 3.5 s the generator/rectifier current makes a step from 1.0 to 0.8 p.u., see Figure 4.16.

1.2-

1-

d1 0.8-cL

0.6-

3js 0.4-

0.2-

0-0 1 2 3 4 5

t [s]

Figure 4.16: The steps of the generator/rectifier current in the Simulink simulations.

The regulators are both of PI and PIFF types. Due to the different bandwidths of the two systems the parameters of the systems must be different so the performances match each

J____ I____ I____ L

T

54

Power angle control

other, i. e. the DC-link voltage fluctuation. Table 4.3 shows the parameters of the two inverter systems, using PI- and PIFF regulators.

Table 4.3: Parameters of the inverter system.System parameters when using the Pi-controller

System parameters when using the PIFF-controller

U = E 1 p.u.

k)lito 1 p.u.U'dc 1 p.u. 1 p.u.4 0.175 p.u. 4 0.175 p.u.a, 1.39 10-2 p.u. a, 1.67 10-2 p.u.

3.33 10-2 p.u. Xcdc 0.10 p.u.Rdc 13.9 103 p.u. a* 13.9 103 p.u.kP 1.0 10-3 kp 4.1 10-4k, 10 k, 4.1T„ 3.39 102 T„ 2.88 102

kFF 0 kFF 2.4 10-3

The PWM-inverter system has the same parameters as in Table 4.3 with the exception of the switching frequency which is 5 kHz. The DC-link capacitor of the inverter system with a Pi-regulator is three times larger than the capacitor of the inverter system with a PIFF-controller. The other parameters are the same except for the values of the regulator parameter.

Step responses of the generator/rectifier current, with a Pi-controller

The system has been simulated with and without harmonics. The input to the system is the generator/rectifier current, and how the current varies with the time can be seen in Figure 4.16. Figures 4.17 and 4.18 show the step response of the DC-link voltage with and without harmonics. The first step response of the DC-link voltage has a high peak, but for the step responses at the times of 2.0 and 3.5 s the DC-link voltage fluctuates approximately by 2.5%. Figures 4.19 and 4.20 show the step responses of the phase displacement angle. There is no difference between the phase displacement angles in the simulations with and without harmonics.

55

[ n d] (j)opn

Power angle control

1.05-

0.95-

Figure 4.17: Step response of the Delink voltage, simulated with a PI- regulator and no harmonics.

Figure 4.18: Step response of the Delink voltage, simulated with a PI- regulator and PWM.

Figure 4.19: Step response of the phase displacement angle, simulated with a PI- regulator and no harmonics.

Figure 4.20: Step response of the phase displacement angle, simulated with a PI- regulator and PWM.

56

[•tvd] (tf’n

Power angle control

Step responses of the generator/rectifier current, with a PIFF-controller

Here the system is using a PIFF-regulator. The inverter system is simulated with and without harmonics and the variation of the input current to the system can be seen in Figure 4.16. Figures 4.21 and 4.22 show the DC-link voltage both for non-harmonics inverter and the PWM-inverter. The step responses are approximately the same. When the generator/rectifier current is 1 p.u. the phase displacement angle is 1.0 p.u. and this results in a higher fluctuation of the DC-link voltage because the regulator parameters have been designed at the phase displacement angle of 0.5 p.u.

0.95

Figure 4.21: Step response of the DC- link voltage, simulated with a PIFF- regulator and no harmonics.

0.75

Figure 4.23: Step response of the phase displacement angle, simulated with a PIFF-regulator and no harmonics.

Figure 4.22: Step response of the DC- link voltage, simulated with a PIFF- regulator and PWM.

Figure 4.24: Step response of the phase displacement angle, simulated with a PIFF-regulator and PWM.

57

Power angle control

The DC-link capacitor of the inverter system with a Pi-regulator is three times larger than the capacitor of the inverter system with a PIFF-controller. The other parameters are the same for the two regulators except for the value of the regulator parameters.

This section clearly shows that the PIFF-controller has better performance than the PI- controller. There is nearly no difference in the result of the simulations. The two models are the complete inverter model, which use PWM and harmonics and the second model is the reduced inverter model, which only use the fundamental voltage of the three-phase voltage system of the inverter. The simulation time is heavily decreased if the model without harmonics is used instead of the complete harmonic model, because the simulation of the PWM-pattem in the complete model requires a small time step in the simulation program to maintain accuracy. At the same time the other parts of the simulated system have much longer time constants and therefore do not require so small time steps as the PWM does. However, it is the subsystem that has the highest bandwidth which determined which time step should be used. Observe that the non-harmonic model is used in an application which has a low bandwidth compared by the switching frequency of the inverter.

58

Power angle control

4.10 Parameter variation

Due to the high inertia of the wind turbine rotor, the time constant of the wind power plant is about 1 second. The highest frequency, where essential power effects influence on the wind power plant, is the drive line resonance, which is about 5 Hz, see [17]. This makes it possible to control the wind power plant with the power angle control. Of course the system should be as cheap as possible. By minimizing the DC-link capacitor and the series inductance the cost can be low. The value of the series inductance can not be chosen freely, because the inductance value determines partially the total harmonic distortion on the grid.

The generator/rectifier current amplitude and frequency are important as to how the parameters should be used. Assume that the DC-link voltage can fluctuate ±5% before it affects the total performance of the total wind power plant. The generator/rectifier current idc{t) is here supposed to vary sinusoidally from zero to 1 p.u., i.e. with an amplitude of 0.5 p.u. and a main value of 0.5 p.u, at the frequency f(idc).

By simulating the inverter system in Simulink, without harmonics, the "non-linear" model and by letting the transients be damped out before the DC-link voltage fluctuation is calculated as a function of the frequency of the generator/rectifier current idc(t) and theparameters of the inverter system, the fluctuation of the DC-link voltage can be determined. Because the fluctuations of the DC-link voltage are small, the input DC-link voltage will be set to its reference value in the inverter model.

In Sections 4.10.1 and 4.10.2 the fluctuation has been determined for the PI- and the PIFF-regulator, respectively. Two methods have been used, called "non-linear" and "linear", see Section 4.9. The "non-linear" method is the Simulink simulation described above and the "linear" method is to calculate the fluctuation by the linear state equations

4.42 and 4.43 and set the operating point — to 0.5. By stipulating the maximum DC-0.

link voltage fluctuation to 5 %, the necessary value of the DC-link capacitor can be expressed as a function of the series inductance and the frequency of the generator/rectifier current.

4.10.1 DC-link voltage fluctuation when using a PI- controller

The DC-link voltage fluctuation using a Pi-regulator is investigated. When comparing the simulated fluctuations "non-linear" with calculated fluctuations "linear" from the state equation 4.42 it shows that the fluctuations are the same for the two methods, see Figures

59

udc f

luct

uatio

n [%

] ud

c flu

ctua

tion

[%]

Power angle control

4.25, 4.26 and 4.27. The parameters of the system are listed in Table 4.4 and it is the value of the series inductance which is varying, and of course the regulator parameters.

non-linear

frequency of idc{t) [Hz]

Figure 4.25: The DC-link voltage fluctuation in per cent as a function of the frequency of the generator/rectifier current. The series inductance Ls is0.087 p.u.

non-linear

linear

frequency of i^tf) [Hz]


■non-linear

linear

frequency of iic(t) [Hz]

Table 4.4: Parameters of theinverter system.

parameters for the Pi-controllerU=E 1 p.u.

1 p.u.Ls (Fig. 4.25) 0.087 p.u.L, (Fig. 4.26) 0.175 p.u.Ls (Fig. 4.27) 0.262 p.u.Rs 1.39 IQ"2 p.u.X-Cdc 3.33 10-2 p.u.R* 13.9 103 p.u.


60

u,lc

fluc

tuat

ion [

%]

Power angle control

4.10.2 DC-link voltage fluctuation when using a PIFF- controller

The DC-link voltage fluctuation using a PIEF-regulator is investigated. The "non-linear" simulated fluctuations are compared with the calculated fluctuations from the state equation 4.43, called "linear". Figures 4.28, 4.29 and 4.30 show the fluctuations for three different value series inductance. The parameters of the system are listed in Table 4.5 and it is the value of the series inductance which is varying, and of course the regulator parameters.

non-linear

lineari i i muj


Figure 4.28: The DC-link voltage fluctuation in percent as a function of the frequency of the generator/rectifier current. The series inductance Ls is0.087 p.u.

non-lindar

linear

1—I I I llilj



61

udc f

luct

uatio

n [%

]

Power angle control

non-linear

0.001-

frequency of i&(Z) [Hz]


Table 4.5: Parameters of theinverter system.

parameters for the PIFF-controllerU = E 1 p.u.

1 p.u.L, (Fig. 4.28) 0.087 p.u.Ls (Fig. 4.29) 0.175 p.u.L, (Fig. 4.30) 0.262 p.u.2, 1.39 10-2 p.u.

0.10 p.u.a* 13.9 103 p.u.

When the value of the series inductance is small the phase displacement angle is kept small and the non-linearities become small. As a result the linear and non-linear models better fit small values of series inductances. When the frequency of the generator/rectifier current is around the resonance frequency of 50 Hz, the difference between the linear and non-linear models is small. Round 1 Hz the "non-linear" model has a resonance. This resonance frequency is the same as for the system with a Pi-regulator (Section 4.9.1).

The main reason for using the "linear" calculated fluctuations is that it does not require a powerful computer to keep down the calculation time of the calculation of the "non-linear" fluctuations. The disadvantage is that the "linear" method is not accurate when the series inductance becomes large, see Figures 4.28,4.29 and 4.30. When normal values of the series inductance are used, from 0.05 to 0.15 p.u., the "linear" model can be used.

4.10.3 Allowed parameter value variations to maintain the fluctuation below 5%

In Figure 4.31 it is shown what value of the DC-link capacitor that is needed for a specific value of series inductance and for a specific frequency of the generator/rectifier current. Both Pl-and PIFF-controllers are used. The fluctuation has been calculated by using the "linear" model. A finite number of reactance values of the DC-link capacitor, from 1 p.u. to 0.0154 p.u., has been used to calculate the fluctuation. Due to this fact the

62

Power angle control

curves in Figure 4.31 are discontinuous. For the same value of the DC-link capacitor and series inductance the allowed maximum frequency, until the fluctuation increases by 5 %, is increased significantly when the regulator is of a PIFF-type instead of a Pi-type. The curves using PEFF-regulators have approximately the same form as those using a PI- regulator. The maximum frequency for the PEFF-regulated system is about 20 to 30 Hz independent of the size of the series inductance and of the size of the DC-link capacitor. For the system that uses a Pi-regulator the maximum frequency of the generator/rectifier current depends on the series inductance used and what value of the DC-link capacitor that is used. The lower value of the series inductance the higher frequency of the generator/rectifier current can be allowed. For example, the inverter system, using PI- regulator, has a maximum DC-link voltage fluctuation of 5 % when the generator/rectifier current has a maximum frequency of 1 Hz, the reactance value of the DC-link capacitor is less than 0.04 p.u. and the series inductance is 0.087 p.u.

0.1-

frequency of idc(t) [Hz]

Figure 4.31: The relation between the values of the DC-link capacitor, series inductance and the frequency of the generator/rectifier current to get the fluctuation of the DC-link voltage below 5 %.

63

Power angle control

4.11 Conclusion of the power angle control

In this chapter an analysis method of the inverter system has been introduced. The inverter system has been linearized at different operating points and a PI- and a PIFF- regulator using proper parameters have been designed. By only using the fundamental three-phase voltage of the inverter and neglect the voltage harmonics caused by the PWM, the simulation time had been decreased significantly. To design the right values of the regulator parameters, series inductance and the DC-link capacitor the inverter system has been simulated, without harmonics, both using a "linear" model and a "non-linear" model. The result shows that the maximum frequency of the variations of the generator/rectifier current is approximately 0.1 to 1 Hz when using the Pi-regulator. The PEFF-regulator has a higher maximum frequency of the variations of the generator/rectifier current, approximately 20 to 30 Hz.

64

Power angle control using linear quadratic method

Chapter 5


The power angle control of the inverter system, when both the phase displacement angle Q(t) and the relative voltage ux(t) are controlling the system, is called the multiple input multiple output system (MIMO). To analyse and design a MIMO system the classical control methods are difficult to use or can not be used at all, because every output is coupled to the inputs. In this chapter the linear quadratic method will be used to control the inverter system. When both the phase displacement angle and the relative voltage are controlled, the system should approximately have the same bandwidth as the systems using fast current loop.

One of the best advantages with linear quadratic control is that the system will always be stable. Unfortunately this is only correct when the system is linear. If the system which should be controlled is non-linear, the system must be linearized in an operating point before the control parameters can be decided. If the system is located in one operating point but the control parameters have been designed for an other operating point, the system can become unstable. If the system becomes unstable or not depends on how robust the system is. Every control parameter in this chapter has been designed with linearized state equations where the harmonics, which come from the pulse width modulation of the inverter, have been neglected.

5.1 Linear quadratic control

Suppose that the system, which is expressed as the state equation

y = Cr+Dzt

will be controlled by feedback of the states with

u = —Lx (5.2)

In linear quadratic control, the control is sought which minimizes the value of a performance index J, which is often of the standard form

o(5.3)

65


The matrices Q and R are diagonal weighting matrices and xTQx and nrRn are scalar quadratic forms which measure the performance and the cost of the control, respectively. The optimal control is a constant-gain state feedback

-1* (5.4)

andL=R-'BrP (5.5)

where P is a symmetric matrix obtained by the solution of the algebraic matrix Riccati equation

PA+ArP+Q-PBR-'BrP = 0 (5.6)

For further information about linear quadratic control see [17].

5.2 LQ-control of the linearized inverter system

5.2.1 State space equation of the linear inverter system

The inverter can be formulated as the State equations 4.31 and 4.32, which have been taken from Chapter 4. The inputs to the system are the phase displacement angle A0(r), the relative voltage Aux(t) and the generator/rectifier current Aidc(l). Observe that the values of these variables are perturbations from an operating point.

" Aij(f)" "Ai,(f)- "Am"

Ai,(f) -^VM Ai,(f) + ®VM Aux(t).A udc(t)_ .A«*(0.

and the output yields

a“*(0 = Cvm

A idit)A(,(f)

.Audc(t)

(5.7)

(5.8)

66


5.2.2 Derivation of the LQ-control with integral error feedback and feedforward

The method used here can be found in [18]. The state equation of the inverter system is rewritten so the generator/rectifier current Aidc(t) can be seen as a known process disturbance, denoted £(r), which is separated from the other inputs to the system. The matrices of the new state equation are denoted A, B, C, D, E and F. In Figure 5.1 below the system in open loop is shown.

f lAL(')

A fl(f) Aux(t)

x = Ax+Bk+E| }> = Gc+Dh+F|

JA«*(*)

A ij,i) A i?(0 A M*(f)

Figure 5.1: The linearized state equation.

By using the integral control, a stable design with no steady state errors can be provided. With integral control a new state in the state equation of the system must be created

*4i(0=P(0 = JCy(T)-y’(T)>iT (5.9)o

The reference value y* (r) of the output y(t) is here defined as the reference value of the DC-link voltage Audc. The state equation can with the new state xM (t) be written as

Ai,(r)- Ai,(f)'

d_ Ai,(f)= Af

Ai,(f)dt A M*(0 Am*(0

„ pit) . . Pit) .

+Bra e(ty

Aux(t),+E,

.Audc{t)(5.10)

where

E O' F -/Ai =

o

BBj = Ej =c 0 D

Observe that / is an eye-matrix. The weighting matrix Q, who weights the states of the state equation will be a four times four diagonal finite matrix, so the states can be weighted independent of each other.

67


qu 0 0 0

0 qn 0 0

0 0 433 0

.0 0 0 qM

(5.11)

The weighting matrix R to weight the inputs of the state equation is a two times two symmetric finite matrix, so the two inputs can be weighted independently.

R=0

0(5.12)

When the Riccati equation is solved using the matrices A,, B,, Q and R, the feedback gain matrix becomes L which can be split into two pieces

L=[L[ L2]

where

Li =hi hi

^23.

(5.13)

Figure 5.2 below shows the linearized system using integrate LQ-control.

Figure 5.2: The linearized state equation with IQ- and integrate-control.

The state equation for the closed loop system can now be written as

'Ai/f)- "AWAi,(f)A«*(0

=[a,-b,l]Ar,(f) A«*(f)

+E,

II1____1

(5.14)

. Pi*) . „ Ptt) .

68


When the generator/rectifier current Aij.it) is known, a feedforward can be used. The same is valid for the reference value of the DC-link voltage Au'j.it). With feedforward the input vector u can be written as

u(t) = -Lx{t) +Hf(f)

(5.15)

where

H=[H, H2] = -[L1 JjG-'E,

and

G="A B C D

(5.16)

(5.17)

A picture of the linearized system with LQ-, integrate- and feedforward control can be seen in Figure 5.3.

Ai*(f)

x- Ax+Bk+E§ y = Cx+Dk+F£

A id(.t) Aig(t)

Lj [■■■■ AUj.it)

AM&(f)

Figure 5.3: The linearized system with LQ-, integrate- andfeedforward control.

The linearized state equation for the closed loop system can be written as

'AijW 'AijWAi,(r)Audc(t)

~ ALQPIFtAi,(f)Au^t)

+ ®LQPEFIAi^(f)'

_Auj.it) _

. XO . . pV) .

(5.18)

and the output vector becomes

69


~&UjM ' Ai'/f)"A0(r)

— ClQPJEIAi/f)

Aux{t) Audc(t)

.A «,(0. . Ap(f) .

+DLQPIHE (5.19)

where

^•LQPEF-[^1 ®LQPIFF— [®I + H]

-'LQPIEF"

0 0 10 -L

^'dc([tivm31 °vm32 ® ®] —[^vm31 ^vmSz]*-1)

(5.20 a:e)

DLQPIFF-

0 0

H~Cjc{[bvm31 ^vm3z]^)

The feedback gain L is given by the linear quadratic method.

5.2.3 LQ-controller for the linearized inverter system

To simulate the inverter system a regulator is needed. The outputs of the regulator are the phase displacement angle 0(f) and the relative voltage ux(t). The inputs of the regulator are the active and reactive current components of the phase current i(f), the DC-link voltage udc{t), the generator/rectifier current idc{t) and the reference value of the DC-link voltage K*(f). The parameter values of the regulator are decided for the phase displacement operating point angle 0O. Figure 5.4 shows a block scheme of the regulator.

(„(f)(,(')«*(0

L(0»*(0

X —Alqreg^- "h BlQREG u

y = Clqregx+DLqeeg u

0(f)«,(0

Figure 5.4: The block scheme of the LQ-regulator.

The state equation of the regulator is written as

70


x(f) — ALQREGi(f)+BLqREG

iq(t)«*to

“*(0.

(5.21)

and the output vector becomes

"6(f)"

A to.

',to: Q.QREGtto + DLQREG Udc(t)

“*(*).

(5.22)

where

ALQREG—® ®LQREG**[® 0 10 ~l]

O.QREG = —^2 Dlorsc^-L, H]

71


5.2.4 Analysis of the linearized inverter system with LQ- PI- and LQ-PIFF controllers

In Chapter 4 the Pi-controller contrails the phase displacement angle and the relative voltage is chosen so that the amplitude of the three-phase voltage system of the inverter is equal to the amplitude of the three-phase voltage system of the grid. In this chapter both the phase displacement angle and the relative voltage will change when the inputs of the inverter system change. To get a feeling of the bandwidth of the system, Bode diagrams and step responses have been created for an operating point, both for LQ-PI and LQ- PIFF controllers. Observe that the analyses are only valid when the variables make small perturbations and that the variables are not saturated. Assume that the parameters of the inverter system are the same as in Section 4.8. They are shown in Table 5.1. From here on the LQ-PI and LQ-PIFF controllers will be denoted without the "LQ."

Table 5.1: Parameters of the inverter system.E 1 p.u. 4,i 1

Udc0 1 p.u. 4e 1

% 5> 433 100

4 0.175 p.u. 444 3 107

R. 5.6 10"3 p.u. 4, 4000.1 p.u. r-n 50

18.9 104 p.u.

Figures 5.5 and 5.6 show the Bode-diagrams of the DC-link voltage AuJc(t) as a function of the reference value of the DC-link voltage Au’dc(t) and the generator/rectifier current Aidc(t). The regulators are both of PI and PIFF types. Step responses of thelinearized system are displayed in Figures 5.7 to 5.10. Figures 5.7 and 5.8 show the step responses of the DC-link voltage An&(r) when the reference value of the DC-link voltage Audc(f) makes a unity step. The controllers are both of PI and PIFF types. Further on Figures 5.9 and 5.10 show the step responses of the DC-link voltage Audc (t) when the generator/rectifier current Ai&(f) makes a unity step, both with PI and PIFF controllers. When the PIFF-controller is used, the system becomes a mixed phase system. Comparing the PI to the PIFF controller the main difference is the settling time. Feedforward makes the system faster compared with using Pi-controller only, approximately 6 times, because the integration term L2 slows down the system. For both PI- and PBFF-controlled linearized systems the step responses have no overshoots and no ripple.The resonance peak at 50 Hz has disappeared here when comparing with the controlled system in Chapter 4. The reason is that the amplitude of the three-phase voltage inverter now is controlled. When feedforward is used the DC-link voltage fluctuations are decreased up to 700 Hz, and the DC-link voltage fluctuation from the system with a PI- controller becomes larger.

72


phase(5°,PIFF)

-120

-160

-200

-240TTTTTTj-I I I llllj

/ [Hz]

Figure 5.5: The Bode diagram of the DC-link voltage Audc(t) as a function of thereference value of the DC-link voltage Audc(f).

.1..1.1.1 ll.l

-120phase(5°,PIFF)

-180

-150-240

I I I I 111 j

f [Hz]

Figure 5.6: The Bode diagram of the DC-link voltage Audc(t) as a function of thegenerator/rectifier current Aidc(t).

73


0.03

Figure 5.7: The step response of the DC-link voltage Audc(t) when the reference value of the DC-link voltage Audc(t) makes a unity step (1.0 V), using a PI- regulator.

0.75_

0.25_

Figure 5.8: The step response of the DC-link voltage Audc(t) when the reference valueof the DC-link voltage &udc(t) makes a unity step (1.0 V), using a PIFF- regulator.

74 I


0.06-

0.04-

0.02-

0.04

Figure 5.9: The step response of the DC-link voltage Auic(t), with a Pi-regulator, when the generator/rectifier current Ai&(r) makes a unity step (1.0 A).

0.005

„ 0.005-

-0.01-

0.015-

-0.02

Figure 5.10: The step response of the DC-link voltage Audc (t), with a PIFF regulator, when the generator/rectifier current &idc(t) makes a unity step (1.0 A).

75


53 The Kalman estimator

It is not common to use continuous time Kalman estimators in controllers for grid- connected converters. But discrete time Kalman estimators are often used in drive systems to determine the states of the electrical machine model. One of the disadvantages with the LQ-control is that process noises with high frequencies disturb the regulator. For the inverter system the grid current contains harmonics due to the PWM technique. These current harmonics affect the LQ-controller. One way to eliminate the current harmonics influence is to use lowpass filter at the inputs of the regulator but then the input signals become phase-shifted and the stability of the system becomes weaker. By using an estimator the current harmonics level is reduced without any phase shifts. Suppose that the inverter system is linearized around an operating point 60(t). A state equation with the matrices A, B, C and D describes the dynamics of the inverter system. When process and measuring noise are added to the inverter system the state equation becomes

x = Ax+Bk + Gw„ _ (5.23)

y = Cx+Du + v

where the process and measurement noise means and covariances are

£[w] = £[v] = 0

2s[wh’t] = Rj,

4wr] = R2

£[wvr] = 0

(5.24)

The estimated state vector x can be solved using the state equation

5=Air + BH + k(y-Cr-D«) (5.25)

The filter gain matrix k comes from

k = PC%' (5.26)

and by solving the static Riccati equation 5.27 the matrix P will be found.

P = AP+PAt-PCtR^1CP+R1 = 0 (5.27)

For further information about the theory of the Kalman estimator see [18]. For the linearized inverter system the process and measurement noise covariance matrices become

76


>m 0 0 ' 0 0 '0 rm 0 " 0 rm 0

0 0 ^33. 0 0

Figure 5.11 shows the state equation of the linearized inverter and the Kalman estimator.

process

estimator

Figure 5.11: The linearized inverter system with a block scheme of a Kalman estimator.

When the operating point of the inverter Q0(t) is changing, the gain matrix of the estimator k(0o(f)) must also change.

5.3.1 The non-linear Kalman estimator

When the inverter system is non-linear the estimator should also be non-linear. From Chapter 4 Equations 4.9 to 4.14 are used.

The fundamental voltage of the inverter, in dq-components, can be written as

«,=-sin(0(r)K(O*,(O (52g)ug=cos(6(t))ux(t)eq(t)

and the non-linear estimator can now be written as

77


Rs0

r i. 1

id L,coN

idl = -coN

Ls0 + +k(e0)

Ud K 1 ids.. Cdfidc Cdfidc R-dfidc. i—

and the output vector of the non-linear estimator can be written as

*dc

h

(5.29)

i o o' o i o o o 1

(5.30)

To get a picture of the inputs and outputs of the estimator, Figure 5.12 has been outlined. To determine the operating point of the system the DC-link current iv(t) can be used. In steady state the phase displacement angle 6(t) is proportional to the DC-link current

6(t)ux(t)L(')|

i,(f)»*(0

nonlinearKalmanestimator

Figure 5.12: The inputs and. outputs of the estimator block

5.3.2 Stability of the system when using LQ-controller and Kalman estimator

When the Kalman estimator estimates the states of the system for the LQ-controller, the system can become unstable. By adding fictitious noise to the plant input model, representing in a perhaps loose way plant variations, uncertainty, or unmodelled dynamics, there is an adjustment to the estimator design. In coping with the fictitious noise, the LQ-Kalman controller becomes more robust to gain and phase changes at the plant input. The analysed system is a MIMO-system and it is difficult to determine the stability by using analytic methods. Here two practical methods are used. By changing the parameters of the process and looking at the eigenvalues of the transfer function, the robustness of parameter variations becomes known. The second method is to simulate the total non-linear system and choose the parameters, weighting matrices and process and measurement noise covariances so the system becomes stable, by an iterative approach. The weighting matrices and process-, measurement-noises have been determined by the

78

imag

inar

y axi

s


last method. To verify the stability of the system the poles of the closed system have been analysed, when the parameters have been altered.

Stability due to parameter variations

Here the parameters of the system; series resistance, series inductance and DC-link capacitance are varying. The eigenvalues of the system transfer function from the reference value of the DC-link voltage to the DC-link voltage are determined. The parameters vary from 50% to 150% and the data of the system are presented in Table 5.2. The interesting poles are those near the imagine axis. Figure 5.13 shows the poles of the transfer function, when rated parameters are used. Figure 5.14 shows the eigenvalues when the resistance, the inductance and the capacitance are changed independent of each other.

Table 5.2: Parameters of the inverter system.F. 1 p.u. #44 3 107*4 1 p.u. rn 400L, 0.175 p.u. *22 50R. 5.6 10"3 p.u. r,u 10

Xcdc 0.1 p.u. rm 30R* 13.9 103 p.u. ri33 100

#ii 1 rm SIC"6

#22 1 *233 1 lo-io#33 100

------Q—

-200-

-400--400--1000-2000-5 10-1 10

real axis real axis

Figure 5.13: Left: The eigenvalues of the system. Right: The eigenvalues near the imagine axis. The parameters have rated values.

79

imag

inar

y axi

s


200-

•O—OGG>—0E>0-6-—

-200-

-400--5 10-1 10

real axis

■(^i...................... —

E -200

-400--2000 -1000

real axis

Figure 5.14: Left: The eigenvalues of the system. Right: The eigenvalues near the imagine axis.The parameters vary from 50% to 150% of the rated values independent of each other.

The result shows that no poles are near the imagine axis. Thus, the system is robust due to small parameter variations.

80


5.4 Simulation of the inverter system

5.4.1 The simulation model

The inverter model consists of a PWM inverter, which has no losses and which switches ideally. The same model is used in Section 4.9.1. Figure 5.15 shows the Simulink model of the total inverter system with the estimator, the controller and the inverter system. To locate the operating point of the system, which is used both in the controller and in the estimator, the DC-link current is used.

theta ScopeRadToD 3g

id scope

udc estiq scope

udc scopeudcrel

udcref input

GeneratorRectifier

timevectorClock!

Figure 5.15: The Simulink model of the LQ-controlled inverter system using anestimator. Est =Kalman estimator; Reg=LQ-regulator; InvSys=inverter system using ideal PWM.

The parameters of the total system used in the simulations are listed in Table 5.3 below. The weighting matrices and the process and measuring covariance matrices have been determined using an iterative method.

81

[n-d] 0)v

?


Table 5.3: Parameters of the inverter system.E 1 p.u. iu 3107

»* 1 p.u. 'll 4004 0.175 p.u. r71 50a, 5.6 10-3 p.u. rn\ 10

0.1 p.u. r\n 30R*c 13.9 103 p.u. rm 100

frw 5000 Hz r7A\ 510-6

in 1 rm 510-6

in 1 r233 i 10-10

in 100

5.4.2 Step response of load steps

Here the generator/rectifier current steps are simulated. The parameters of the inverter system are shown in Table 5.3. When the inverter system is in steady state and the generator/rectifier current is 0.8 p.u. a positive current step of the generator/rectifier current idc(t) of 0.2 p.u. is made at the simulation time of 0.01 s. At the simulation timeof 0.025 s the generator/rectifier current makes a negative step of 0.02 p.u., see Figure 5.16. The step response of the DC-link voltage is small, see Figure 5.17, compared with a control only of the phase displacement angle 6(t), Chapter 4 Figure 4.23.

1.1

0.9-----

0.8

0.70.01 0.02 0.03

t [s]

1.001

a,

0.9990.01 0.02 0.03

t [s]

Figure 5.16: The generator/rectifier current. Figure 5.17: The DC-link voltage.

The direct current, Figure 5.18, is approximately zero except at the generator/rectifier current flanks. The current has a ripple of approximately 0.05 p.u. The estimated direct current, Figure 5.19, has less ripple and when the generator/rectifier current makes steps, the direct current will increase momentarily, both for positive and negative generator/rectifier current steps.

82


0.1

0 0.01 0.02 0.03t [s]

Figure 5.18: The direct current.

-0.05

0.02

Figure 5.19: The estimated direct current.

The output signals of the controller, 6(t) and ux(t), make momentary spikes at the flanks of the load steps, see Figures 5.20 and 5.21. At steady state the phase displacement angle is proportional to the generator/rectifier current idc{t) and the relativevoltage is one.

Figure 5.20: The phase displacement angle. Figure 5.21: The relative voltage.

The quadrature current iq(t) is the active current and when the generator/rectifier current makes steps the quadrature current iq(t) follows, see Figures 5.22 and 5.23.

83

[•tvd] (,yp! •(;)''?


Figure 5.23: The estimated quadrature current.Figure 5.22: The quadrature current.

The DC-link current zv(r) and generator/rectifier current are plotted at the positive step and negative step of the generator/rectifier current, see Figures 5.24 and 5.25. Except for the ripple caused by the PWM switching frequency of the inverter, the DC-link current has a ripple of six times the grid frequency, 50 Hz. Because the reactance value of the DC-link capacitance is small, the ripple voltage over the DC-link capacitance is rippling. The LQ-reguIator feels the voltage ripple and tries to compensate it. That is why the variables of the system have a small ripple.

1.5-K(t)

1 4

0.5

00.005 0.01 0.015

t [s]

Figure 5.24: The DC-link and generator/rectifier current. Positive current step.

Figure 5.25: The estimated quadrature current. Negative current step.

i

84


The phase currents of the inverter, expressed in ap-coordinates, are shown in Figures 5.26 and 5.27. The current ripple caused by the PWM is small and can be neglected.

-0.5- step-up-0.55

-1.1 -0.55 0 0.55 1.1z'a0) [p-u.]

Figure 5.26: The phase currentsin aP-coordinates.

Figure 5.27: The phase currents in afi-coordinates.

85


5.5 Conclusions of the inverter system with an LQ-controller

By using the LQ-control it is easy to control the inputs to the system (MEMO system). The inputs are the phase displacement angle and the relative voltage. If the harmonics of the inverter are neglected and the system is linearized, the comparison between the LQ- control method and the Pi-controlled phase displacement angle used in Chapter 4 shows that the LQ-controlled method is much faster and smoother and has no overshoots and no ripple.

The disadvantages of the LQ-controlled power angle-controlled inverter system are that the inverter system is non-linear and that the LQ-regulator needs new parameters when the operating point of the system is changing. When the LQ-control is used directly to the inverter system using PWM-technique, the current harmonics disturb the LQ-controller. This depends on a low roll-off rate, 20 dB per decade, of the amplitude characteristic of the system transfer function.

To avoid the harmonics disturbing the LQ-controller an estimator can be used. To estimate the non-linear inverter system the estimator has to be non-linear too. To find the filter gain constant k the system has to be linear. When the operating point is changing the filter gain constant k must also change. This fact complicates the design of the estimator.

To get a high bandwidth of the total inverter system using the LQ-PIFF control the harmonics, due the PWM, of the inverter system affect the control. To increase the bandwidth of the total inverter system, the harmonics must be included in the model, because the PWM-technique is a discrete system and should be sampled at specific times. This makes it natural to control the inverter system by means of a discrete time control instead of a continuous time control.

86

Laboratory set-up and measurements

Chapter 6


In this chapter the implementation of the power angle control, presented in Chapter 4, in a laboratory set-up is described. Measurements have been carried out to verify the control strategy.

6.1 Laboratory set-up

The laboratory set-up consists of a PWM-converter connected to the grid. To simulate the generator/rectifier in a wind power plant a thyristor-converter is connected to the grid, via a transformer. The transformer separates, in a galvanic way, the three-phase voltage of the generator/rectifier from the three-phase voltage of the PWM-converter. Figure 6.1 shows the laboratory set up. The PWM-converter consists of a main-circuit, DC-link, grid filter and a control unit.

Transformer

PWM-converterThyristor-converter

___ Pl-controllerimplemented in a PC

Figure 6.1: The laboratory set-up for simulating the electrical system in a windpower plant with variable speed.

6.1.1 Main circuit

The valves of the main circuit are IGBTs, called MG200Q1US1, rated at 1200 V and 200 A. To decrease voltage spikes and resonances on the main circuit an RCD-filter at each phase-leg has been implemented, see Figure 6.2. The capacitance value is 30 pF, the resistance value 4.70 and the diode name is BYT230PIV for the RCD-filter. The main circuit is designed for 600 V.

87


positive DC-link bus

phase-leg

RCD-filter

negative DC-link bus

Figure 6.2: Main circuit configurationfor one phase-leg.

6.1.2 DC-link

The DC-link consists of electrolytic capacitors, each rated Ur=400 V, C=3300 pF, Iac(100 Hz)=12.7 A. The capacitors are placed in two groups, which are connected in series. Each group has four parallel-connected capacitors. To maintain the voltage balance over the two groups of capacitors, two resistances of 10 k£2 have been parallel-connected to the groups. The total capacitance of the DC-link is 6.6 mF.

6.1.3 Grid filter

The grid filter that is used here is a series inductor. One series inductor in each phase between the converter and the grid. Different values of the series inductance are tested.

6.1.4 Current and voltage measurements

The currents are measured by current LEM-modules, LT300-S. The voltages are measured using resistors as voltage dividers and an isolating operation amplifier, AD210J.

6.1.5 Control unit

Two variables must be controlled when using the power angle control. The two variables are the phase displacement angle and the modulation index.

The modulation index control

Here the amplitude of the grid voltage and the converter voltage will be the same. By using the quadrature grid voltage eq(t) and the DC-link voltage uic(t) the reference

88


modulation index can be determined, see Figure 6.3. When using the on-line zero component the gain should be 1.41.

uq(t) >j—£

M*)—-Figure 6.3: The control of the reference modulation index.

gainirj. vv —>

Phase displacement implementation

By using a phase-locked loop, which is locked to the grid, and a 12-bit counter, the converter is synchronized to the grid voltage eft). The 12-bit number from the counter is called the grid phase bus. To make a phase displacement an 8-bit number is added to the grid phase bus. To realize both negative and positive phase displacements, an 8-bit sum, which represents 350° is added to the grid phase bus. The phase displacement span is 22.5° and the resolution is 0.09°.

To get the three-phase voltage reference values to the PWM-circuit, three EPROM are used together with three 8-bit DAC. The EPROMs contain sinusoidal tables. The sinusoidal tables are displaced 120° to each other. The EPROMs and the DACs represent the three-phase system of the converter. The voltage reference value of the DACs is the reference modulation index. Before the voltage reference values are compared to the triangular curve, the zero components are added to the voltage reference values. In Figure 6.4 the phase displacement implementation scheme is shown.

grid voltage phase 1

phase I displacement

reference modulation index triangular]wave

zerocomponents

EPROM

EPROM

EPROM

EPROM

EPROM

+1 —

+1 —

Figure 6.4: The phase displacement implementation scheme.

89


The phase displacement control

The phase displacement control is implemented in a personal computer. The computer measures the DC-link voltage. The output is the phase displacement angle.

6.2 Measurements on laboratory set-up

Here measurements of the system are presented. The regulator is of Pi-type. The measurements are divided into two groups. First, measurements using the series inductance 0.122 p.u. are performed; Second, measurements using the series inductance 0.175 p.u. are carried out

6.2.1 Measurement 1

The parameter value of the inverter is displayed in Table 6.1. The regulator is of the PI- type and has been tuned so the gain margin is 3.3 times and the phase margin is 60°.

Table 6.1: Parameters of the inverter system.M&o—2.02? 1.0 p.u.Bo 3.1°4 0.122 p.u.Rs 23 10-3 p.u.

0.188 p.u.kp 15.7 10-3T, 60-10-3 [s]

% O

Constant generator/rectifier current at the operating point

In the following Figures, 6.5 to 6.12 where the variables are shown in steady state, the generator/rectifier current is 0.28 p.u. A low-pass filter working as an anti-alias filter is tuned to 720 Hz, the data have been digitally filtered to 600 Hz.

90

[•n-d] (7)c

? XlflVYl [

*n d] (ifpn


1.05

1.025

0.975

0.95

1

0 0.05t [s]

Figure 6.7: The DC-link voltage.

0.1

Figure 6.6: The grid voltages.

Figure 6.8: The generator/rectifier current.

0.6 -I-------------------- 1----------------------

3 U/-^K;-/yv6 0.4----------------------------- 1----------------------------

S' 0.3--------------------- i--------------------

0.2 -|--------------- 1-----------------0 0.05 0.1

t [s]

Figure 6.10: The phase displacement angle.

l

UA-VVVvyv/U

T

91

[•trd] (i)pi

[-n-d] (;)3p

z


0.1-

■0.1—'

t [s]

Figure 6.11: The direct current.

0 0.05 0.1t [s]

Figure 6.12: The quadrature current.

Sinusoidal generator/rectifier current variation around the operating point

The inverter system is exposed to sinusoidal generator/rectifier current ripples around an operating point, see Figure 6.13. In Figures 6.14 to 6.16 the responses of the DC-link voltage, the DC-link current and the phase displacement angle are shown. The generator/rectifier current ripples with 15 Hz.

Figure 6.13: The generator/rectifier current. The current oscillates with 15 Hz.

Figure 6.14: The DC-link current, due to the oscillating generator/rectifier current.

92

0(0 [p-

u.]


Figure 6.15: The phase displacement, due to the oscillating generator/rectifier current.

1.025

s 0.975

Figure 6.16: The DC-lirik voltage, due to the oscillating generator/rectifier current.

The system frequency response from the generator/rectifier current

Here, measurements at different frequencies of the generator/rectifier current have been carried out at the operating point. The amplitude of the generator/rectifier current ripple is kept small so the frequency response is referred to the vicinity of the generator/rectifier current operating point. The Bode diagram starts at the frequency 0.25 Hz and stops at 40 Hz. Three measurements have been made at each frequency. Figure 6.17 shows the DC-link voltage response of the generator/rectifier current The frequency response from the generator/rectifier current to the DC-link current is shown in Figure 6.18.

10 10/ [Hz]

Figure 6.17: Bode diagram from the generator/rectifier current to the DC-link voltage.

93


L-l I I I

-120

-180

/ [Hz]

Figure 6.18: Bode diagramfrom the generator/rectifier current to the DC-lirik current.

6.2.2 Measurement 2

The parameter values are shown in Table 6.2. The regulator is of the Pi-type and has been tuned so the magnitude margin and phase margin are 3.3 times and 60°, respectively. The major difference is a larger series inductance here than in the previous measurements.

Table 6.2: Parameters of the inverter system.% 7.4°4 0.175 p.u.Rs 16 10-3 p.u.

0.134 p.u.kP 3.13 10-3Tr 60 10-3 [s]id0=-0.84iq0

Step responsesHere, a 10 Hz square disturbance in the generator/rectifier current is implemented. Figure 6.19 shows the generator/rectifier current. The disturbance responses in DC-link current, DC-link voltage and phase currents are shown in Figures 6.20 to 6.22.

94

[•n-d] oypi


Figure 6.19: The generator/rectifier current. The frequency of the square wave disturbance is 10 Hz.

1.005

s 0.995

0 0.1 0.2t [s]

Figure 6.20: The DC-link voltage response to square wave disturbance in the generator/rectifier current.

Figure 6.21: The DC-link current response to square wave disturbance in the generator/rectifier current.

Figure 6.22: The phase currents response to square wave disturbance in the generator/rectifier current.

95

[•n-d] 0YP!

[-n-d] to*

1?


The Figures 6.24 to 6.26 show the DC-link voltage, DC-link current and phase currents response to the generator/rectifier current disturbances of 15 Hz which are displayed in Figure 6.23.

1.005

0.995

0.6-

vigure 6.23: The generator/rectifier :urrent. The frequency of the square vave disturbance is 15 Hz.


t [s]

Figure 6.25: The DC-link current response to square wave disturbance in the generator/rectifier current.

t [s]


96

[•tvd] Q

)*t [*nd]

oyt


The 20 Hz responses for the DC-link voltage, DC-link current and for the grid currents are shown in Figures 6.28 to 6.30. The generator/rectifier current is displayed in Figure 6.27.

0.6-

1.005

0.995

figure 6.27: The generator/rectifier :urrent. The frequency of the square vave disturbance is 20 Hz.


Figure 6.29: The DC-link current response to square wave disturbance in the generator/rectifier current


97

[•n-d] (;)l9‘(z)1?

[ n d] (r)'9


Grid currents wave-form

When using the power angle control, the grid current waveform is dependent of the grid voltage waveform and of course the shape of the inverter voltages. In Figures 6.31 to 6.33 the grid currents are shown for three operating points 60=6.3°, 60=7.4° and #o=8.4°. The figures show that the waveform of the grid currents is not pure sinusoidal.

phase voltage e

-0.75

phase current ij

Figure 6.31: The phase current and the grid voltage. The operating point 80=6.3°.

phase [voltage e

7 0.75

-0.75

phase currejnt i

0.040.02t [s]

Figure 6.32: The phase current and the grid voltage. The operating point 60=7.4°

phase -foltage e

0.75-

-0.75

phase current i

Figure 6.33: The phase current and the grid voltage. The operating point 80=8.4°.

98


63 Conclusion of the measurements

The measurements show that the inverter system using power angle control works well for the two series inductors. The system handles variations of the generator/rectifier current up to 40 Hz without problems. The system responses well to load steps and the DC-link voltage fluctuation is modest.

If the inverter voltage or the grid voltage contains harmonics, the grid current will also contain harmonics. As the measurements show, the currents contain harmonics. The measured Bode-diagram from the generator/rectifier current to the DC-link voltage has a higher attenuation than the theoretical calculations from Chapter 4 can explain. By in other words the dynamic resistance is higher than the static resistance, which is the resistance in the cables and in the series inductors. The harmonics or the non sinusoidal grid current depend on the blanking time and on the voltage drop over the valves. See the references [21] and [22] for further information about non-linearities in SVCs. When the series inductance is relative small and the grid voltage is pure sinusoidal, a small phase displacement angle is needed to create a large grid current. The voltage over the series inductor is small and the voltage drop is due to blanking time, the transistor voltage drop and the anti parallel diode voltage drop makes a large impact of the current harmonics. For the same fundamental current, the 5th current harmonic is 40% larger and the 7th current harmonic is 30 % larger than half the series inductance value (from 2 mH to 1 mH).

99

Conclusions and future work

Chapter 7


7.1 Conclusions

In this work, the power angle control of a Voltage Source Converter (VSC) connected to the grid in a wind power application has been investigated. The wind turbine time constant is around one second and small power fluctuations can occur at 5 Hz due to the drive train resonance. The inverter system using power angle control fulfils the required bandwidth to operate on a wind turbine.

The investigation has been carried out based on steady state analysis, small signal analysis, digital time-domain simulations and tests on a laboratory model. The control systems are examined using linearized space vector models in the dq-ffame. Once the system has been designed using the previous approximate methods, the complete system and parameters can be tested using a more exact non-linear model. All the methods are based on modelling the converter output voltage as a switching vector.

In Chapters 2 and 3 the grid voltage harmonic distortion caused by the VSC is analysed. The main advantage using a VSC instead of a grid-commutated inverter is the power quality. To improve the power quality an LC-filter with a damping link is suggested. The filter design considers the short-circuit inductance of the grid.

In Chapter 4, the power angle control is analysed. The performance limit is a large resonance peak at the grid frequency. The characteristics of the system depend on the Delink capacitor value and on the inductance and resistance of the filter inductor between the inverter and the grid. Large values of the resistance damp the system well but increase the energy losses. A high inductance leads to a reduced harmonic level on the grid but makes the system slower. By using feedforward of the generator/rectifier current signal, the performance is increased.

The limit of the DC-link voltage fluctuation bounds the parameters of the system. At the normal size of the series inductance, due to current harmonics on the grid, the linearized model of the system can be used to determine the fluctuation. The controller with feedforward is insensitive to series-inductance variations and the voltage fluctuations depend on the size of the DC-link capacitor and the frequency and amplitude of the generator/rectifier current.

In Chapter 5 the Linear Quadratic control method is used to control the two control variables. By controlling both the phase displacement angle and the modulation index, the bandwidth of the system is increased and the typical power angle resonance at the grid frequency disappears. An extended Kalman filter is used to filter the input signals, because the LQ-control method does not suppress the harmonics which occur around the

100


switching frequency. By using the LQ-controller, the performance of the system increases compared with the Pi-controller using feedforward of the generator/rectifier current signal.

In Chapter 6 laboratory tests have been performed. The power angle control, using a PI- controller operates well. Step responses up and down at an operating point are well- behaved. The frequency response from the generator/rectifier current to the DC-link voltage is more attenuated than the theoretical calculation shows. This is due to the dynamic resistance.

7.2 Future work

The disadvantages of the VSC compared with the grid-commutated inverter are the efficiency and the price. Compared with the inner-current loop system, the power angle control does not have a controller which actively tries to deliver sinusoidal currents to the grid. If the grid contains voltage harmonics, the current from the inverter also contains harmonics. This is important when the inverter operates on weak grids.

By using inner current control loops (vector control) the bandwidth of the VSC can be considerably increased. This makes it possible to implement extra options. For instance in a wind park using different electrical systems, both constant and variable speed systems, the VSC can control the voltage level on the grid and actively filter the grid current harmonics. This results in higher electricity quality for the whole wind park and the wind park can operate with a lower short-circuit ratio to the grid.

101

References

References

[1] W. E. Leithead, “Variable Speed Operation - Does it Have Any Advantages?,” Wind Engineering, vol. 13, pp. 302-314,1989.

[2] A. Gravers, “Higher electrical effiency with variable speed,” presented at ECWEC, Travemunde, 1993.

[3] L. Malesani and P. Tomasin, “PWM Current Control Techniques of Voltage Source Converters - A Survey,” presented at IECON'93, Hawaii, 1993.

[4] J. T. G. Pierik, A. T. Veltman, S. W. H. d. Haan, G. A. Smith, D. G. Infield, and A. D. Simmons, “A new class of converters for variable speed wind turbines,” presented at EWEC '94, Thessaloniki, 1994.

[5] P. Verdelho and G. D. Marques, “Digital simulation and applications of the PWM voltage converter connected to the AC mains,” presented at EPE, Firenze, 1991.

[6] A. Ekstrom, “Calculation of Transfer Functions for a Forced-Commutated Voltage-Source Converter,” presented at PESC'91, Boston, MA, 1991.

[7] B. T. Ooi and X. Wang, “Voltage Angle Lock Loop Control of the Boost Type PWM Converter for HVDC Application,” IEEE Trans, on Power Electronics, vol. 5, pp. 229-235, 1990.

[8] G. Joos, L. Moran, and P. Ziogas, “Performance Analysis of a PWM Inverter VAR Compensator,” IEEE Trans, on Power Electronics, vol. 6, pp. 380-391, 1991.

[9] J. Holtz, “Pulsewidth Modulation for Electronic Power Conversion,” IEEE, vol. 82, pp. 1194-1214, 1994.

[10] J. Ollila, “Analysis of PWM-Converters using Space Vector Theory - Application to a Voltage Source Rectifier,” Tampere University of Technology, Finland, Tampere 111, 1993.

[11] J. Holtz, “On the performance of optimal pulsewidth modulation techniques,” EPE Journal, vol. 3, pp. 17-24,1993.

[12] M. Lindgren, “Filtering and Control of a Grid-connected Voltage Source Converter,” Electric Power Engineering, Chalmers University of Technology, Sweden, Technical Report 208L, 1995.

[13] K. Thorborg, Power Electronics - in Theory and Practice. Lund, Sweden: Studentlitteratur, 1993.

102

References

[14] O. Carlson, A. Grauers, J. Svensson, and A. Larsson, “A comparision between electrical systems for variable speed operation of wind turbines,” presented at EWEC '94, Thessaloniki, 1994.

[15] B. Schmidtbauer, Analog och digital REGLERTEKNIK. Lund, Sweden: Studentlitteratur, 1988.

[16] “MATLAB,” , 4.2 ed. Natick, Mass: The MathWorks, Inc, 1992.

[17] P. Novak, I. Jovik, and B. Schmidtbauer, “Modeling and Identification of Drive- System Dynamics in a Variable-Speed Wind Turbine,” presented at Proc. IEEE 3rd Conf. on Control Applications, Glasgow, 1994.

[18] B. D. O. Anderson and J. B. Moore, Optimal Control - linear quadratic methods. Englewood Cliffs, NJ 07632: Prentice-Hall, Inc, 1989.

[19] J. V. D. Vegte, FEEDBACK CONTROL SYSTEMS. Englewood Cliffs, NJ 07632: Prentice-Hall, 1986.

[20] B. D. O. Anderson and J. B. Moore, Optimal Filtering. Englewood Cliffs, NJ 07632: Prentice-Hall, Inc, 1979.

[21] J. K. Pedersen, F. Blaabjerg, J. W. Jensen, and P. Thogersen, “An Ideal PWM- S VI Inverter with Feedforward and Feedback Compensation,” presented at EPE'93, Brighton, 1993.

[22] R. B. Sepe and J. H. Lang, “Inverter Nonlinearities And Discrete-Time Vector Current Control,” IEEE Trans. On Industry Application, vol. 30, pp. 62-70, 1994.

103

Appendix A

Appendix A

Transformations

A.l The transformation between a three-phase system and an a (3-system

The three-phase properties s2(t) and s3(t) can be transformed into a vector in a complex plane, called aP-plane

s(t) = sa(t)+jsp(t) = ^[x^CO +x2s2(t)+x3s3(t)] (A.1)

where the sum of the three-phase quantities will be zero when no conductor is connected to the mid-point of the three-phase system, Equation A.2.

(t)+s2 (r)+s3(r) = 0 (A.2)

Each phase in the three-phase system has a specific direction

(A3)

The Equation A.1 can now be expressed as a matrix equation.

X(;)' rjf o o iV2n 1 1

s2(t). VI VlJ

and the inverse becomes

\M o 1

-w) V 31 1

V6 V21 1

V6 ^

(A.4)

(A.5)

104

Appendix A

A.2 VSC current and voltage space vectors

The output voltages of the converter can be represented in state space vectors by the phase voltages u,(t), u2(t) and u3(t), see Appendix A. The voltage vector in thecomplex ap-plane is

«(0 = Ka(0 + jK/)(0 (A.6)

From now on the valves are replaced by switches. The switches xw,, sw2 and ^w3 of the converter can totally have eight different combinations. For each state of the total number of combinations of switches a state space vector sw can be written as

—W = 3Jj * j2r Ate 'X

jw,(r)+e 3sw2(t)+e 3 sw3(t) = sw£r(r)+jsw/,(r) (A.7)

In Table A.1 the eight different mechanical switch combinations and the state space vectors of the switch values are shown.

Table A.1: The mechanical switch combinationsJW, sw2 sw3 sw

1 -1 -11

[8 jf —e 33

1 1 -1i

8 if3e

-1 1 -1

-1 1 1

-1 -1 1

18 jf3S

1 -1 1 8 jf36

1 1 1 0-1 -1 -1 0

105

Appendix A

In Figure A.1 the different switch states are presented in the form of state space vectors.

Figure A.1: State space vector representation of the voltage vectors forthe six-pulse converters.

The output voltages of the converter can now be written as a state space vector

u(t) = ^-sw(t) (A.8)

Of course, the three phase currents z, (r), i2(t) and i3(t) can also be expressed as a state space vector

i(t) = +e^hit) + e%(t) j = ia(t)+}ip(t) (A.9)

The sum of the three phase currents is always zero. The DC-link current" iv(f) from the capacitor of the DC-link to the converter can be written as

z"v(r) = 5w, (Z)z, (t)+sw2(f)i2(t) +sw3(t)i3(t) = Re[z(r)-conj(jw(r))] (A.10)

106

Appendix A

A.3 The connection between an a|3-system and a dq-system

Let the vectors v(t) and w(z) rotate with the angular frequency coN and to in the emplane, respectively. The vector v(t) becomes a fix vector in the dq-plane if the vector w(r) forms the direct axis in the dq-plane and the angular frequencies to and tow are equal. See Figure A.2 below.

Figure A.2: The relation between the afi-plane and the dq-plane.

The angular 0(f) in Figure A.2 is denoted

0(Z) = J[toN(T)-to(T)]c?T (A.ll)

o

By means of Figure A.2 the components in the dq-plane can be determined. The transform equation from ap-plane to dq-plane becomes in matrix form

'vd(i) " cos(0) sin(0)‘yq(f\ -sin(0) cos(0)_ 7/0

and the inverse yields

~cos(0) -sin(0)" 'Vrfto".V). sin(0) cos(0) _

(A.13)

107

Appendix A

A.4 Voltage and current vectors in the a (3-system and the dq- system, respectively

Suppose that a symmetrical sinusoidal three-phase voltage is transformed into a vector u(t) = ua(t)+jup(t) in the a(3-plane. Define the q-axis in the dq-plane as parallell to the voltage vector u(t) in the a^-plane. This definition comes from a flux vector which is parallel to the d-axis in the dq-plane and the voltage vector is proportional to the time derivative of the flux vector. As a consequence of the chosen reference vector, the voltage vector u(t) will only contain a q-component in the dq-plane. The transform equation for a current-vector in the ap-plane to the dq-plane becomes in matrix form

"W cos (cot-—-) sin "Ley-sin(fi)f-y) cos

and the inverse yields

"UO" cos(fi)f--j) -sin(o)t - -^)

sin(fflf--^) cos(cof--^)

(A.14)

(A.15)

Of course the voltage vector transformations from the ap-plane to the dq-plane will be the same as those for the current vectors.

Appendix B

Appendix B Normalization

Assume that the rated voltages and the rated apparent power are known in a system; then the other parameters and variables can be normalized.

Let the base voltages and Udc base and the base apparent power be designed as

(B.l)

^dc.base ^dc.n (B.2)

Sbase ~ (B.3)

where U„, „ and Sn are the rated AC-, DC-voltages and the apparent power of thesystem.

The base currents Ibase and IiCibase and base impedance Ztoe of the system yield:

T __ ^base(B.4)

J _ Rbase

J debase jjU dc,base

(B.5)

TT TJ2'V __ u base __ u baset'base “ fX7 c

v 51 base ^base(B.6)

The capacitor, inductor and resistance values in per unit will be:

L’-=fkt'base

(B.7)

(B.8)

Rp.u. - z^base

(B.9)

where coN is the angular frequency of the grid.

109

Appendix B

B.l Normalization of power angle controlled inverter

The base apparent power for the power angle controlled inverter is defined as

Sbase~JLg

(B.10)

there the maximum phase displacement angle is denoted 0^ and is the base angle.

110

power angle control of grid-connected voltage source

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