Design and Analysis of Proportional-Resonant Current Control for VSC

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DESIGN AND ANALYSIS OF PROPORTIONAL-RESONANT CURRENT CONTROL FOR VSC

Faculty of Information Technology and Communication Sciences Master of Science Thesis April 2020

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ABSTRACT

Mikko Suovirta: Design and Analysis of Proportional-Resonant Current Control for VSC Master of Science Thesis

Tampere University Electrical Engineering April 2020

The proportional-resonant (PR) controller offers certain advantages over the proportional- integral (PI) controller. The PI controller requires DC quantities which can be achieved when modeling is done in a synchronous reference frame. On the other hand, the PR controller can control sinusoidal quantities that appear in a stationary reference frame and can be tuned to compensate for selected harmonics without additional filters or detection methods.

The use of the PR controller is becoming more popular but information has to be sought from several sources. The main objective of this thesis is to increase the understanding of the PR current control and its tuning, and to serve as a general guidebook. Tuning the PR controller requires basic knowledge of control theory. For this reason, the Nyquist stability criterion and stability margins are reviewed in order to determine the stability and robustness of the control system. Moreover, the fundamentals, parameters and discrete transformation of the PR controller are reviewed in detail.

Every actual control system has a time delay and it lags the phase curve of the frequency response and reduces the stability margins. In this thesis, the adverse effect of the time delay is compensated for by the phase-lead compensator the operating principle of which is easy to understand, and tuning is straightforward.

To support the presented theoretical analysis a grid-connected three-phase voltage source converter (VSC) in delta configuration is used as a case study. The simulations are performed considering both a strong and weak grid conditions the voltage of which is distorted by low- frequency harmonics. The controller parameters are chosen based on the open-loop frequency responses which are measured by applying themaximum-length binary sequence(MLBS) excitation and frequency response methods. Furthermore, the stability margins are determined from the open-loop frequency responses, and the effects of thephase-lead compensator andgrid voltage feedforward(VFF) on the control performance are analyzed.

The total time delay of the control system, which consists of the measurement and control delay, proves to be significant and can thus deteriorate the control performance. Specifically, without any delay compensation, theharmonic compensators(HCs) can be utilized up to the 7^th harmonic. To address this, thephase-lead compensatoris used to increase the phase margin of the control system and the HCs can be utilized up to the 13^th harmonic. Halving the total time delay would have a significant effect on the control performance and the HCs could be utilized beyond the 13^thharmonic.

The VFF is commonly used in grid-connected converters but it has mixed effects on the PR control performance. In terms of control tuning, the VFF proves to be useful and the best stability margins are achieved. On the downside, the VFF amplifies the grid voltage harmonics in certain cases, especially with a weak grid condition.

Grid frequency variations affect the current control performance, as the PR controller is tuned to compensate for selected frequencies. The sensitivity to grid frequency variations can be slightly reduced by increasing the cutoff frequency of the PR controller. However, this also reduces the stability margins and the current control performance may be decreased. Thus, the cutoff frequencyof the PR controller should be set in terms of stability margins.

Keywords: PR controller, current control, discrete, frequency response, stability, time delay The originality of this thesis has been checked using the Turnitin OriginalityCheck service.

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TIIVISTELMÄ

Mikko Suovirta: VSC:n PR-virtasäädön suunnittelu ja analyysi Diplomityö

Tampereen yliopisto Sähkötekniikka Huhtikuu 2020

PR-säädin (engl.proportional-resonant controller) tarjoaa tiettyjä etuja verrattuna PI-säätimeen (engl. proportional-integral controller). PI-säädin vaatii toimiakseen DC-muotoiset suureet, jotka muodostetaan synkronisessa koordinaatistossa. Vastaavasti PR-säädin ohjaa stationäärisessä koordinaatistossa sinimuotoisia suureita, ja säädin voidaan virittää kompensoimaan valittuja harmonisia taajuuksia ilman erillisiä suotimia tai havaitsemismenetelmiä.

PR-säätimen käyttö on yleistymässä, mutta tietoa täytyy hakea useista lähteistä. Tämän työn päätavoite onkin lisätä ymmärrystä PR-virtasäädöstä ja sen virittämisestä sekä toimia yleisenä käyttöohjeena. PR-säätimen virittäminen vaatii ymmärrystä säätöteorian perusteista. Tästä syys- tä Nyquistin stabiiliuskriteeri ja yleiset stabiiliusmarginaalit käydään läpi, jotta säädettävän systeemin stabiilius ja käyttövarmuus voidaan määrittää. Lisäksi PR-säätimen perusteet, säätöpara- metrit ja diskreettimuunnos käsitellään yksityiskohtaisesti.

Jokaisessa käytännön säätösysteemissä on aikaviivettä, ja se laskee taajuusvasteen vaihe- käyrää pienentäen stabiiliusmarginaaleja. Tässä työssä aikaviiveen haitallista vaikusta kompen- soidaan vaiheenjohtopiirillä (engl.phase-lead compensator), jonka perusteet on helppo ymmär- tää ja virittäminen on suoraviivaista.

Teoreettisen analyysin tueksi työssä mallinnetaan kolmivaiheiseen sähköverkkoon kolmiokyt- kettyä VSC:tä (engl.voltage source converter). Simuloinnit suoritetaan vahvan ja heikon verkon olosuhteissa, missä verkkojännite on säröytynyt matalataajuisista harmonisista komponenteista.

Virtasäädön parametrit määritetään avoimen säätösilmukan taajuusvastekuvaajista, jotka mita- taan soveltamalla MLBS (engl.maximum-length binary sequence) viritys- ja taajuusvastemenetel- miä. Lisäksi taajuusvasteista määritetään systeemin stabiiliusmarginaalit sekä samalla tutkitaan vaiheenjohtopiirin ja verkkojännitteen myötäkytkennän (engl.grid voltage feedforward) vaikutuk- set virtasäädön suorituskykyyn.

Mittaus- ja säätöviiveestä muodostuva kokonaisaikaviive osoittautuu merkittäväksi, ja voi si- ten heikentää virtasäädön suorituskykyä. Ilman aikaviiveen kompensointia virtasäädön harmonisia kompensaattoreita voidaan hyödyntää korkeintaan 7. harmoniseen asti. Vaiheenjohtopiirin avulla kasvatetaan säädettävän systeemin vaihevaraa ja harmonisia kompensaattoreita voidaan hyödyntää 13. harmoniseen asti. Aikaviiveen puolittamisella olisi merkittävä vaikutus virtasäädön suorituskykyyn ja harmonisia kompensaattoreita voitaisiin hyödyntää yli 13. harmonisen.

Verkkojännitteen myötäkytkentää käytetään yleisesti sähköverkkoon kytketyissä konverttereis- sa, mutta sillä on ristiriitaisia vaikutuksia PR-virtasäädön suorituskykyyn. Säädön virittämisen kannalta myötäkytkentä osoittautuu hyödylliseksi ja parhaimmat stabiiliusmarginaalit saavutetaan.

Varjopuolena myötäkytkentä vahvistaa verkkojännitteen harmonisia komponentteja tietyissä tilan- teissa, erityisesti heikossa sähköverkossa.

Verkkotaajuuden vaihtelut vaikuttavat virtasäädön suorituskykyyn, sillä PR-säädin viritetään kompensoimaan tiettyjä taajuuksia. Herkkyyttä verkkotaajuuden vaihteluille voidaan hieman pie- nentää kasvattamalla PR-säätimen rajataajuutta (engl.cutoff frequency). Tämä kuitenkin pienen- tää myös stabiiliusmarginaaleja ja PR-virtasäädön suorituskyky voi heikentyä. Täten PR-säätimen rajataajuus tuleekin asetella stabiiliusmarginaalien ehdoilla.

Avainsanat: PR-säädin, virtasäätö, diskreetti, taajuusvaste, stabiilius, aikaviive Tämän julkaisun alkuperäisyys on tarkastettu Turnitin OriginalityCheck -ohjelmalla.

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PREFACE

This Master of Science thesis has been done forGE Grid Solutions.

I would like to thank Anssi Mäkinen and Tuomas Messo for the opportunity, interesting topic and guidance.

Special thanks to Petros Karamanakos, responsible supervisor at the university, for his developing, accurate and prompt feedback.

Tampere, 21st April 2020 Mikko Suovirta

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LIST OF FIGURES

2.1 Closed-loop system (taken from [12]) . . . 3

2.2 Nyquist contour and plot (taken from [12]) . . . 5

2.3 Phase and gain margins of stable and unstable systems (taken from [12]) . 5 2.4 Stability margins from Nyquist plot . . . 6

2.5 n-bit shift register with XOR feedback for MLBS generation . . . 7

2.6 8-bit-length MLBS generated at 1000 Hz in time-domain . . . 8

2.7 Power spectrum of 8-bit-length MLBS generated at 1000 Hz . . . 8

2.8 Typical MLBS measurement setup . . . 8

2.9 Bode diagram of ideal and non-ideal PR controller . . . 11

2.10 Bode diagram of PR controller, effect of proportional gainK_p . . . 12

2.11 Bode diagram of PR controller, effect of integral gainKi . . . 12

2.12 Bode diagram of PR controller, effect of cutoff frequencyω_c . . . 13

2.13 Bode diagram of PR controller, with and without HCs . . . 14

2.14 Bode diagram of error due to Tustin approximation . . . 16

2.15 Diagram of discrete-time harmonic compensatorG_HC,h(z) . . . 17

2.16 Diagram of discrete-time PR controllerG_PR(z) . . . 17

2.17 Bode diagram of PR controllerGPR(s|z) . . . 18

2.18 Bode diagram of pure time delayG_d(s) . . . 19

2.19 Diagram of discrete-time phase-lead compensatorG_kw(z) . . . 20

2.20 Bode diagram of phase-lead compensatorGkw(s|z) . . . 21

2.21 Diagram of grid voltage feedforward . . . 21

3.1 Diagram of the simulation model . . . 23

3.2 Diagram of discrete-time PR current control . . . 25

3.3 Open-loop frequency response in strong grid, case 1 . . . 28

3.6 Open-loop frequency response in weak grid, case 1 . . . 32

4.1 Grid current and voltage in strong grid, case 1 . . . 37

4.4 Grid current and voltage in weak grid, case 1 . . . 41

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A.1 Open-loop frequency response with increasedωcin strong grid, case 1 . . 53 A.2 Open-loop frequency response with increasedω_cin strong grid, case 2 . . 54 A.3 Open-loop frequency response with increasedω_cin strong grid, case 3 . . 55 A.4 Open-loop frequency response with increasedωcin weak grid, case 1 . . . 57 A.5 Open-loop frequency response with increasedω_cin weak grid, case 2 . . . 58 A.6 Open-loop frequency response with increasedω_cin weak grid, case 3 . . . 59 B.1 Open-loop frequency response with reducedTdin strong grid, case 1 . . . 61 B.2 Open-loop frequency response with reducedT_din strong grid, case 2 . . . 62 B.3 Open-loop frequency response with reducedT_din strong grid, case 3 . . . 63 B.4 Open-loop frequency response with reducedTdin weak grid, case 1 . . . . 64 B.5 Open-loop frequency response with reducedT_din weak grid, case 2 . . . . 65 B.6 Open-loop frequency response with reducedT_din weak grid, case 3 . . . . 66

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LIST OF TABLES

3.1 Simulation model parameters . . . 24

3.2 MLBS parameters . . . 25

3.3 Controller parameters and stability margins in strong grid . . . 26

3.4 Controller parameters and stability margins in weak grid . . . 31

4.1 Grid voltage harmonic levels . . . 35

4.2 Grid current and voltage in strong grid . . . 36

4.3 Grid current and voltage in weak grid . . . 40

4.4 Grid current and voltage TDDs in strong grid . . . 44

4.5 Grid current and voltage TDDs in weak grid . . . 45

4.6 Controller parameters and stability margins with reduced T_din strong grid . 46 4.7 Controller parameters and stability margins with reduced Tdin weak grid . . 47 A.1 Controller parameters and stability margins with increasedω_cin strong grid 52 A.2 Controller parameters and stability margins with increasedωcin weak grid . 56

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LIST OF SYMBOLS AND ABBREVIATIONS

DM Delay margin

f Frequency

f0dB Gain crossover frequency f-180^◦ Phase crossover frequency FFT Fast Fourier transform FT Fourier transform

GM Gain margin

G_kw Transfer function of phase-lead compensator GPR Transfer function of PR-controller

h Harmonic order

HC Harmonic compensator K_i Integral gain

K_ih Individual resonant gain Kp Proportional gain

Kw Gain of phase-lead compensator

L Inductance

MLBS Maximum-length binary sequence PI Proportional-integral

PLL Phase-locked loop

PM Phase margin

PR Proportional-resonant

PRBS Pseudo-random binary sequence

pu Per unit

R Resistance

RHP Right half-plane

s Continuous-time variable SM Stability margin

SNR Signal-to-noise ratio

STATCOM Static synchronous compensator

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T_d Time delay

TDD Total demand distortion TF Transfer function

THD Total harmonic distortion T_s Sampling period

VFF Grid voltage feedforward VSC Voltage source converter

ω Angular frequency

ω_c Cutoff frequency z Discrete-time variable

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1 INTRODUCTION

The proportional-resonant (PR) controller is becoming more popular in grid-connected converters and offers certain advantages over aproportional-integral (PI) controller. The PI controller requires DC quantities resulting from modeling in a synchronous reference frame¹(dq-frame), whereas the PR controller operates on sinusoidal quantities that exist in a stationary reference frame²(αβ-frame) [1]. Such a controller can be tuned to compensate for selected harmonic frequencies without additional filters or detection methods.

The PR controller does not require thedq-transformation or grid angleθinformation from thephase-locked loop(PLL), as aforementioned. This relieves the computational burden and allows for single-phase applications, as noted in [2–5]. In this thesis, the PR controller is used for the current control of avoltage source converter (VSC). The PR current control aims to mitigate the supply current harmonic distortions in a grid environment where the grid voltage is distorted by low-order harmonics. Only the PR current control is considered in this thesis.

The main objective of this thesis is to increase the understanding of the PR current control and its tuning, and to serve as a general guidebook. Tuning the PR controller requires a basic knowledge of control theory in order to determine the stability and robustness of the control system. Several tuning methods and strategies exist, as noted in [6–9]. However, these can be difficult to understand and may only be suitable for a specific application.

The tuning procedure presented in this thesis is based on the fundamentals of control theory and analysis of the measured frequency responses.

Chapter 2 presents the theoretical background. TheNyquist stability criterionandstability margins are discussed in detail. The required open-loop frequency responses are measured by applying themaximum-length binary sequence(MLBS) excitation and frequency response methods. The fundamentals, parameters and discrete transformation of thePR controller are reviewed in detail. The adverse effect of the time delay is compensated for by the phase-lead compensator. The need for thegrid voltage feedforward (VFF) is re-evaluated with the PR controller. Finally, thetuning procedureof the PR current control is outlined.

Chapter 3 presents the case study. The simulation model is a delta-connected VSC which is modeled as controlled voltage sources with internal resistances R and induc- tancesL. The discrete-time PR current control design procedure takes into account the MLBS, phase-lead compensator, VFF, and sampling intervals and time delays. Finally,

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the tuning results for the strong and weak grid conditions are presented with the controller parameters, stability margins and open-loop frequency responses.

Chapter 4 presents the simulation results for the strong and weak grid conditions from the perspective of how the phase-lead compensator and VFF affect the control performance.

The results are also presented at thevariable grid frequencywhile the effect of thecutoff frequency on the PR controller performance is investigated. Finally, the effect ofhalving the total time delay on the tuning and stability margins is analyzed.

1 Clarke transformation (abctoαβ0) [10]

⎡

⎢

⎣ xα

xβ

x0

⎤

⎥

⎦

=²₃

⎡

⎢

⎣

1 −¹₂ −¹₂ 0

√ 3

2 −

√ 3 2 1

2 1 2

1 2

⎤

⎥

⎦

⎡

⎢

⎣ xa

xb

xc

⎤

⎥

⎦

2 Park transformation (abctodq0) [11]

⎡

⎢

⎣ xd

xq

x0

⎤

⎥

⎦

= ²₃

⎡

⎢

⎣

cos (θ) cos(︁

θ−^2π₃ )︁

cos(︁

θ+^2π₃ )︁

−sin (θ) −sin(︁

θ−^2π₃ )︁

−sin(︁

θ+^2π₃)︁

1 2

⎤

⎥

⎦

⎡

⎢

⎣ xa

xb

xc

⎤

⎥

⎦

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2 THEORETICAL BACKGROUND

This chapter covers the necessary theories and methods for the study. The Nyquist stability criterion is needed for stability analysis which requires the open-loop frequency response of the control system. Thestability margins are required to understand when tuning the controllers and analysing the robustness of the control system. The required open-loop frequency response is measured by applying a broadband excitation method called themaximum-length binary sequence(MLBS).

ThePR controller is used for the current control, and thus its basic principles and discrete transformation are reviewed in detail. The effect of thetime delay is presented along with methods on how to compensate for its adverse effects with thephase-lead compensator.

Thegrid voltage feedforward (VFF) is commonly used in grid-connected converters, but its use should be re-evaluated with the PR controller. Finally, thetuning procedureof the PR current control is outlined.

2.1 Nyquist stability criterion

TheNyquist stability criterionis a method to study the stability of the closed-loop system from an open-loop frequency response [12–16]. This is convenient when the mathematical expression of the system is not known and only a measured frequency response is available. To understand the criterion the simple closed-loop system shown in Figure 2.1 is considered.

Figure 2.1. Closed-loop system (taken from [12])

The closed-loop system is represented as transfer functions which are the ratio of the output to the input in the Laplace transformeds-domain. The transfer function is a linear differential equation in which the roots of the numerator are zeros and the roots of the

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denominator arepoles. For example,G(s)can be represented as G(s) = L{output}

L{input} = C(s)

E(s). (2.1)

In the open-loop transfer function, B(s)

E(s) =G(s)H(s) =L(s), (2.2) poles are the roots of the denominator of L(s) that cause the function to go to infinity.

If there is a pole in the right half-plane (RHP), the time-domain response consists of the exponential growth function and the open-loop is unstable.

In the closed-loop transfer function, C(s)

R(s) = G(s)

1 +G(s)H(s) = G(s)

1 +L(s), (2.3)

the denominator of the function 1 +L(s) is known as the characteristic equation. For stability, all roots of the characteristic equation (1 +L(s) = 0) must lie in the left half- plane. However, it should be noted that the open-loop transfer function L(s) can be unstable and have RHP poles and zeros, but the closed-loop transfer function can still be made stable.

The stability is studied by the Nyquist contour which is based on a mapping theorem called Cauchy’s argument principle. The Nyquist contour encloses the entire RHP and all the zeros and poles of1 +L(s)that have positive real parts. The Nyquist contour in s-plane is shown in Figure 2.2a. The mapping shows the relative difference between the number of poles and zeros inside the contour based on how many times the plot circles the origin and in what direction.

Instead of plugging every value of the contour to1 +L(s), it is only necessary to replace swithjω, because1 +L(jω)and1 +L(−jω)are symmetrical with each other about the real axis. The semicircle arc with infinite radius maps to a single point because the gain of the transfer function goes to zero as the frequency increases to infinity. Furthermore, encirclement of the origin by1 +L(jω)is equivalent to encirclement of the−1 +j0point byL(jω). Thus, the stability of the closed-loop system can be studied by the open-loop plotL(jω). Nyquist plots inω-plane are shown in Figure 2.2b.

TheNyquist stability criterioncan be expressed as

Z =N+P, (2.4)

where Z=number of zeros of1 +L(s)in the RHP

N=number of clockwise encirclements of the−1 +j0point P=number of poles ofL(s)in the RHP.

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(a)Nyquist contour ins-plane (b)Nyquist plots inω-plane Figure 2.2.Nyquist contour and plot (taken from [12])

For a stable closed-loop control system,1 +L(s)must not have RHP-zeros (Z = 0).

• IfL(s) has RHP-poles (P ≥ 1), thenN =−P. Thus, for stability there must beP counterclockwise encirclements of the−1 +j0point byL(jω).

• IfL(s) has no RHP-poles (P = 0), thenN = Z = 0. Thus, for stability there must be no encirclements of the−1 +j0point byL(jω).

TheNyquist stability criteriontells whether the control system is stable or not; the robust control system must have sufficient stability margins. Some examples of stability and margin dependencies are shown in Figure 2.3. Stability margins are reviewed in more detail in Section 2.2.

Figure 2.3.Phase and gain margins of stable and unstable systems (taken from [12])

2.2 Stability margins

Thestability margin(SM) is a versatile indicator for analyzing the stability and robustness of the control system. The SM cannot easily be seen from the Bode diagram. Åströmet al. [14] have stated that the SM is the shortest distance of the open-loop plotL(jω)form

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the−1 +j0point and can be presented as SM = min

ω≥0|1 +L(jω)|= 1

M_s, (2.5)

whereMs is the maximum sensitivity. A sensitivity analysis is not covered in this thesis, butM_s is a measure of the largest amplification of the disturbances. Overall, the control system is more likely to be stable and robust ifSM≥0.5.

Other stability margins are the gain margin (GM) andphase margin (PM). The GM indicates how much the open-loop gain can be increased before the closed-loop system is marginally stable. Respectively, the PM indicates how much phase lag is required to reach the stability limit.

The stability margins viewed from the Nyquist plot are shown in Figure 2.4 which shows how the stability margins are linked to each other. As mentioned, ifSM ≥0.5 then most commonlyGM≥6 dBandPM≥45^◦. However, it is possible to have a sufficient GM and PM, but the SM may still be insufficient, e.g., a resonant peak at a higher frequency than the gain crossover frequency. For this reason, it is important to perform the an analysis with the Nyquist plot as well.

Im

-1 Re

-1/GM

PM SM

L(jω)

0

Figure 2.4.Stability margins from Nyquist plot

The last stability margin is thedelay margin (DM), which is less significant compared to the other aforementioned margins, indicates the smallest time delay required to make the system unstable. The DM is closely related to the PM and can be presented as

DM = PM·π/180^◦ 2πf0dB

= PM

360^◦·f0dB

, (2.6)

wheref_0dB is the gain crossover frequency.

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2.3 Maximum-length binary sequence

The broadband excitation is used to measure the frequency response of the system. The basic idea is simple; a certain type of perturbation signal is injected to the system and the excitation (input) and response (output) signals are measured. The perturbation injection can be an impulse, step, sinusoidal or binary signal. TheFourier transform(FT) is applied on the time-domain signals and thus the frequency response is determined.

This thesis focuses on the pseudo-random binary sequence (PRBS), and more specifically, themaximum-length binary sequence (MLBS). Roinilaet al. [17] have stated that the sequence is a deterministic and periodic signal, and has the lowest possible peak factor. Multiple injection periods can be applied through spectral averaging, and as a result, the amplitude of the excitation can be kept at a low level.

Based on [18–22], other PRBS-derived broadband excitation methods also exist. The inverse-repeat binary sequence (IRS) and ternary sequence are used in systems with non-linearities. Thediscrete-interval binary sequence (DIBS), on the other hand, is suitable for systems with a lot of noise. However, the MLBS is the most commonly used PRBS signal [23].

The reason for the popularity of the MLBS is that the design and generation of the sequences is straightforward and can be done by using feedback shift register circuits, as shown in Figure 2.5. Such MLBS sequences exist forN = 2ⁿ−1, wherenis a positive integer and denotes the degree of the shift register. Due to the deterministic nature of the sequence the signal can be repeated and injected precisely.

1 2 ... ... i ... n

c

_i

c

_m

XOR

SHIFT REGISTER

MLBS

Figure 2.5. n-bit shift register with XOR feedback for MLBS generation

Figure 2.6 shows one period of an 8-bit-length MLBS in the time-domain. The sequence is generated at 1000 Hz and has signal levels±1. In practice, the binary values 0 and 1 are mapped to±1 to produce a symmetrical sequence with an average close to zero.

The energy content of the MLBS, i.e., the power spectrum varies with frequency. The power spectrum has an envelope and drops to zero at the generation frequency and its multiples. The power spectrum is given by

Φ_MLBS(q) = a²(N+ 1) N²

sin²(πq/N)

(πq/N)² , q=±1,±2, . . . (2.7) whereq denotes the sequence number of the spectral line,ais the signal amplitude and N is the length of the sequence. An example of the power spectrum is shown in Figure

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0 0.05 0.1 0.15 0.2 0.25 Time (s)

-1 0 1

Amplitude

Figure 2.6.8-bit-length MLBS generated at 1000 Hz in time-domain

2.7 (frequency resolutionfresis reduced due to the visual clarity). The effective bandwidth of the power spectrum must be taken into account when designing the MLBS, especially in systems with a lot of noise.

0 1 2 3 4

Energy

10^-3

0 1000 2000 3000

Frequency (Hz)

Figure 2.7.Power spectrum of 8-bit-length MLBS generated at 1000 Hz

A typical measurement setup is shown in Figure 2.8, where an impulse response function g(t) presents the system under test and is to be identified. The system is perturbed by the excitation x(t)which yields the corresponding output responsey(t). The measured excitation and response signals are corrupted with noise, as presented bye(t)andr(t), respectively. The measured, noise-affected excitation and output response can now be denoted byxe(t)andyr(t).

g(t) y(t)

r(t) x(t)

e(t)

x_e(t) y_r(t)

EXCITATION RESPONSE

SIGNAL GENERATOR

Figure 2.8. Typical MLBS measurement setup

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The ideal, noise-free frequency response functionG(jω)can be presented as G(jω) = Y (jω)

X(jω), (2.8)

where X(jω) =F{x(t)}=FT of excitation signal Y(jω) =F{y(t)} =FT of response signal

The noise-affected frequency response functionGn(jω)can be presented as G_n(jω) = Y_r(jω)

Xe(jω) =G(jω) 1 + [R(jω)/Y (jω)]

1 + [E(jω)/X(jω)], (2.9) where X_e(jω) =F{x_e(t)}=FT of measured excitation signal

Yr(jω) =F{y_r(t)} =FT of measured response signal E(jω) =F{e(t)} =FT of excitation noise signal R(jω) =F{r(t)} =FT of response noise signal

The four main parameters and the design procedure of the MLBS injection are outlined below. The selection of MLBS parameters is reviewed in more detail in Section 3.2.

1. Generation frequencyf_gen

• Limited by the bandwidth of the control, and switching or sampling frequency

• The maximumf_genis half the limiting frequency 2. Length of sequenceN = 2ⁿ−1

• Higher than the settling timeT of the systemN ≥fgen·T

• Defines the frequency resolutionfres=fgen/N 3. Amplitude of injectiona

• High enough to provide adequatesignal-to-noise ratio(SNR)

• Low enough to avoid non-linearities or meet other restrictions 4. Number of excitation periodsP

• Defines the accuracy (variance) of the measurement

• The effect of noise is reduced by1/√ P

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2.4 Proportional-resonant controller

The PR control offers certain advantages over a proportional–integral (PI) control. The PR controller operates in astationary reference frame (αβ-frame) and can be tuned to compensate for selected harmonics without additional filters or detection methods. Sev- eral current control methods exists, but this thesis focuses only on the PR control. This section provides the basic relevant background, the effect of parameters on the controller performance, harmonic mitigation, and the discretization method used in this thesis.

2.4.1 Basic principles

Teodorescu et al. [24] have stated that the PR controller can be obtained from the PI controller through a reference frame transformation. The PI controller implemented in a synchronous reference frame(dq-frame) is given as

GPI(s) =Kp+Ki

s , (2.10)

whereK_pis the proportional gain andK_ithe integral gain. It can be transformed into the αβ-frame through a frequency modulated process, mathematically expressed as

G_PR(s) =G_PI(s−jω) +G_PI(s+jω)

=K_p+K_i (︃ 1

s−jω + 1 s+jω

)︃

. (2.11)

The transfer function of the ideal PR controller with the ideal integratorKi/s, given as GPR(s) =Kp+Ki

2s

s²+ω², (2.12)

has an infinite gain at the selected resonant frequencyω and it is sensitive to frequency variation. This can be an issue for the stability of the control system, and to overcome this, the controller must be damped. The transfer function of the non-ideal PR controller with the non-ideal integratorK_i/[1 + (s/ω_c)]can be given as

GPR(s) =Kp+Ki

2ω_cs

s²+ 2ωcs+ω², (2.13) whereωcis the cutoff frequency. It has finite gain at the resonant frequency and it is less sensitive to frequency variation. The non-ideal controller is used in practical applications, because the characteristics of the ideal controller can cause instability and the infinite gain is impossible to achieve in practical applications.

The differences between the ideal and non-ideal controller can be seen from the Bode diagrams in Figure 2.9 (K_p = 1,K_i = 20,ω_c = 2π,ω = 2π50). As mentioned, the non- ideal controller has a finite gain and the peak of the resonant frequencyωis broadened.

In rest of the thesis, only the non-ideal controller is considered.

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10¹ 10² 0

20 40 60 80

Magnitude (dB)

Ideal Non-ideal

10¹ 10²

Frequency (Hz) -90

-45 0 45 90

Phase (deg)

Figure 2.9. Bode diagram of ideal and non-ideal PR controller

2.4.2 Effect of parameters

The proportional gainK_p of the PR controller is tuned in the same way as for a PI controller, and it determines the dynamics of the system in terms of bandwidth, phase and gain margins. In particular, the gain crossover frequency f_0dB and the corresponding phase margin (PM) is determined by Kp. The Bode diagram of the PR controller with differentKp values is shown in Figure 2.10 (K_i= 20,ωc= 2π,ω= 2π50).

The integral gain K_i of the PR controller must have a relatively big value in order to minimize the steady-state error, but small enough to maintain the stability of the control system. Ki is used to set the desired gain for the selected resonant frequency ω. The Bode diagram of the PR controller with differentK_ivalues is shown in Figure 2.11 (K_p= 1,ω_c= 2π,ω= 2π50).

Thecutoff frequency ωc is used to broaden the peak of the resonant frequencyω which helps to reduce the sensitivity in slight frequency variations. Teodorescuet al. [24] have stated that, in practice, ω_c values between 5–15 rad s⁻¹ have been found to provide a good compromise. The Bode diagram of the PR controller with different ωc values is shown in Figure 2.12 (K_p= 1,K_i= 20,ω= 2π50).

It is important to understand how the parameters affect the phase. In a way, increasing theKp leads the phase, and increasing theKiorωclags the phase. In systems affected by the time delay, it is essential to minimize phase lag caused by the control parameters.

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10¹ 10² 0

5 10 15 20 25 30

Magnitude (dB)

K_p = 1.00 K_p = 1.78 K_p = 3.16

10¹ 10²

Frequency (Hz) -90

-45 0 45 90

Phase (deg)

Figure 2.10.Bode diagram of PR controller, effect of proportional gainK_p

10¹ 10²

0 5 10 15 20 25 30

Magnitude (dB)

K_i = 10 K_i = 20 K_i = 30

10¹ 10²

Frequency (Hz) -90

-45 0 45 90

Phase (deg)

Figure 2.11. Bode diagram of PR controller, effect of integral gainKi

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10¹ 10² 0

5 10 15 20 25 30

Magnitude (dB)

c = 2

c = 3

c = 4

10¹ 10²

Frequency (Hz) -90

-45 0 45 90

Phase (deg)

Figure 2.12. Bode diagram of PR controller, effect of cutoff frequencyω_c

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2.4.3 Harmonic compensators

Selective harmonic compensation can be achieved by cascading several integrators, known as harmonic compensators (HCs), tuned to resonate at the desired low-order harmonic frequencies [25]. The generalized transfer function of the HCs can be given as

G_HC(s) =

m

∑︂

h=1

K_ih 2ω_cs

s²+ 2ωcs+ (hω)², (2.14) wherehis the harmonic order to be compensated for,mis the maximum harmonic order andK_ihis the individual resonant gain. The PR controller and HC transfer functions can be generalized and presented as

G_PR+HC(s) =K_p+

m

∑︂

h=1

K_ih 2ω_cs

s²+ 2ωcs+ (hω)². (2.15) The Bode diagram of the PR controller, with and without HCs, is shown in Figure 2.13 (Kp = 1.41, K_ih = 20, ωc = 2π, ω = 2π50, h = 1,5,7,11,13). The HCs do not affect the dynamics of the fundamental PR controller (h = 1), as they compensate only for frequencies that are very close to the selected resonant frequencies. Admittedly, the disadvantage is that additional HCs increase the system complexity and complicate the controller design procedure [26].

10¹ 10² 10³

0 5 10 15 20 25 30

Magnitude (dB)

GPR

GPR+HC

10¹ 10² 10³

Frequency (Hz) -90

-45 0 45 90

Phase (deg)

Figure 2.13. Bode diagram of PR controller, with and without HCs

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2.4.4 Discrete transformation

Nowadays, almost all control systems implemented in practice are discrete. The conver- sion from continuous to discrete-domain and its effects are often overlooked, although the transformation has a clear effect on the system performance. Various discretization methods exist, but this thesis is focused on the bilinear transformation, also known as the Tustin approximation. It is stated in [27] that this method produces the best frequency- domain match between the continuous-time and discretized systems.

The discretization is based on the approximation of the continuous-time variable s. The approximation is placed in the continuous-time transfer function in place of thesvariable.

TheTustin approximationof the PR controller can be expressed as s= 2

Ts

z−1

z+ 1 −→GPR+HC(s), (2.16)

where Ts is the sampling interval of the discrete-time model and z is the discrete-time variable. After mathematical simplifications, the discrete-time PR controller and its coefficients can be presented as

G_PR(z) =K_p+

m

∑︂

h=1

K_ih k_e0,h+k_e2,h·z⁻²

k_c0,h+k_c1,h·z⁻¹+k_c2,h·z⁻², where k_e0,h= 4ω_cT_s

k_e2,h=−4ω_cT_s

k_c0,h= (hωT_s)²+ 4ω_cT_s+ 4 kc1,h= 2(hωTs)²−8

kc2,h= (hωTs)²−4ωcTs+ 4.

(2.17)

Exact match at the resonant frequencieshωis essential for the PR controller. Although theTustin approximation gives a good match, the error between to the continuous-time system increases with the frequency. The error can be seen from Figure 2.14 (Kp= 1.41, K_ih = 20, ω_c = 2π,ω = 2π50,h = 1,5,7,11,13, T_s = 100 µs). The error at 650 Hz is already around 9 Hz.

Exact match at the resonant frequencieshωcan be achieved by theTustin approximation with frequency prewarping. It is stated in [27] that this method ensures a match between the continuous and discrete-time responses at the prewarp frequency. However, this method can easily lead to really complex equations. TheTustin approximation with frequency prewarping of the PR controller can be expressed as

s= hω tan (hωT_s/2)

z−1

z+ 1 −→G_PR+HC(s), (2.18)

where the prewarp frequency is the same as the resonant frequencyhω. After mathematical simplification, the discrete-time PR controller and its coefficients can be presented as

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200 300 400 500 600 700 800 900 0

5 10 15 20 25 30

Magnitude (dB)

GPR(s) GPR(z)

200 300 400 500 600 700 800 900

Frequency (Hz) -90

-45 0 45 90

Phase (deg)

Figure 2.14. Bode diagram of error due to Tustin approximation

G_PR(z) =K_p+

m

∑︂

h=1

K_ih ke0,h+ke2,h·z⁻² k_c0,h+k_c1,h·z⁻¹+k_c2,h·z⁻²

=K_p+

m

∑︂

h=1

K_ihG_HC,h,

where k_e0,h= 2ωctan (hωTs/2) k_e2,h=−2ω_ctan (hωT_s/2)

k_c0,h=hωtan²(hωT_s/2) + 2ω_ctan (hωT_s/2) +hω k_c1,h= 2hωtan²(hωT_s/2)−2hω

k_c2,h=hωtan²(hωT_s/2)−2ω_ctan (hωT_s/2) +hω

(2.19)

andz⁻¹ corresponds to a one discrete time-step delay. The discrete-time HC is denoted as G_HC,h(z). Before implementing the discrete-time PR controller on a simulation level, the output signal ofG_HC,hneeds to be solved, and can be presented as

GHC,h(z) = C_h(k)

E(k) = k_e0,h+k_e2,h·z⁻² kc0,h+kc1,h·z⁻¹+kc2,h·z⁻²

⇐⇒C_h(k) = k_e0,h kc0,h

E(k) +k_e2,h kc0,h

E(k−2)−k_c1,h kc0,h

C_h(k−1)−k_c2,h kc0,h

C_h(k−2)

=k_1,h·E(k) +k_2,h·E(k−2)−k_3,h·C_h(k−1)−k_4,h·C_h(k−2),

(2.20)

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where E(k) is the controller input and Ch(k) is the HC output signal. The diagram of the discrete-time HC based on (2.20) is shown in Figure 2.15. Each selected resonant frequency hω has its own HC blockG_HC,h and coefficients k_1-4,h. The coefficients may seem complex, but after calculations, they are just real numbers.

Figure 2.15. Diagram of discrete-time harmonic compensatorG_HC,h(z)

The final form of the discrete-time PR controller based on (2.19) can now be assembled and is shown in Figure 2.16. Kpand HCs are in parallel and summed together to obtain the controller outputC(k).K_ihis excluded fromG_HC,hand presented as a separate gain because to facilitate tuning of the controller.

Figure 2.16. Diagram of discrete-time PR controllerGPR(z)

The comparison between continuous and discrete-time PR controllers can be seen in Figure 2.17 (K_p = 1.41,K_ih = 20,ω_c = 2π, ω = 2π50, h = 1,5,7,11,13,T_s = 100 µs).

TheTustin approximation with frequency prewarpinggives an exact match at the resonant frequencieshωcompared to the continuous-time system. Only a slight difference in the phase can be observed at higher than the resonant frequencies.

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10¹ 10² 10³ 0

5 10 15 20 25 30

Magnitude (dB)

GPR(s) GPR(z)

10¹ 10² 10³

Frequency (Hz) -90

-45 0 45 90

Phase (deg)

Figure 2.17. Bode diagram of PR controllerG_PR(s|z)

2.5 Effect of time delay

The time delay has a significant effect on the control system, and it lags the phase curve of the transfer function drastically. From stability point of view, margins decrease. A delayed system needs to have sufficient phase margin, especially when the time delay is uncertain. As stated in [12–16], the transfer function of the pure time delay is given as

G_d(s) =e^−T^d^s, (2.21)

whereT_dis the time delay. The gain of the pure time delay is 0 dB at all frequencies, but the phase lags linearly as a function of frequency.

In this thesis, only the measurement delay of 40 µs and the control delay of 100 µs are considered. Thus, the total time delay used in the simulations is 140 µs. At first this doesn’t seem like much, but it has a significant impact on the simulation results. Even changes of few tens of microseconds have a clear effect, as can be seen from Figure 2.18.

The effect of the time delay needs to be compensated for before harmonic compensators until 650 Hz (h= 13) can be used.

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10¹ 10² 10³ -1

-0.5 0 0.5 1

Magnitude (dB)

T_d = 70 s T_d = 120 s T_d = 140 s

10¹ 10² 10³

Frequency (Hz) -180

-135 -90 -45 0

Phase (deg)

Figure 2.18.Bode diagram of pure time delayG_d(s)

2.6 Phase-lead compensator

A phase-lead compensator can be used to increase the phase margin in the control system and compensate for the effect of the time delay. The phase-lead compensator has a stabilizing effect, but it also increases the gain in the system. In a sense, the phase-lead is a trade-off between the phase and gain margin. It must be ensured that the increased gain does not cause instability in the feedback system.

Lead-lag compensators, where the maximum phase-lead and design frequency can be determined separately, are discussed in [12–16]. The phase-lead compensatorG_kw(s) can be given as

G_kw(s) =K_w1 +ατ s

1 +τ s , (2.22)

where the proportional gain Kw is used to set the gain to 0 dB at the desired gain crossover frequency f_0dB. Parameters α and τ are calculated based on the maximum phase-leadϕmand the design angular frequencyωm, given as

ϕm= arcsin

(︃α−1 α+ 1

)︃

⇐⇒α=−sin (ϕm) + 1 sin (ϕm)−1 ωm= 1

τ√

α ⇐⇒τ = 1

ω_m√ α.

(2.23)

The continuous-time phase-lead compensatorG_kw(s)is discretized in the same way as

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the PR controller to achieve an exact match at the design frequency compared to the continuous-time transfer function. The Tustin approximation with frequency prewarping of the phase-lead compensator can be expressed as

s= ωm

tan (ωmTs/2) z−1

z+ 1−→Gkw(s), (2.24)

where the prewarp frequency is the same as the design angular frequency ω_m. After mathematical simplification, the discrete-time phase-lead compensator and its coefficients can be presented as

G_kw(z) =K_wk_e0+k_e1·z⁻¹

k_c0+k_c1·z⁻¹ =K_wG_w(z) where k_e0= tan (ω_mT_s/2) +ατ ω_m

k_e1= tan (ω_mT_s/2)−ατ ω_m k_c0= tan (ω_mT_s/2) +τ ω_m kc1= tan (ωmTs/2)−τ ωm,

(2.25)

where the discrete-time transfer function is denoted asGw(z). The output signal ofGw(z) needs to be solved before implementing the discrete-time phase-lead compensator on a simulation level, and can be presented as

Gw(z) = Cw(k)

E(k) = ke0+ke1·z⁻¹ kc0+kc1·z⁻¹

⇐⇒C_w(k) = k_e0 kc0

E(k) +k_e1 kc0

E(k−1)−k_c1 kc0

C_w(k−1)

=k₁·E(k) +k₂·E(k−1)−k₃·C_w(k−1),

(2.26)

where E(k) is the input and C_w(k) the output signal. The final form of the discrete- time phase-lead compensator, based on (2.25) and (2.26), can now be assembled and is shown in Figure 2.19. The gainKwis added back to the diagram to obtain the phase-lead compensator outputC(k).

Figure 2.19. Diagram of discrete-time phase-lead compensatorG_kw(z)

Differences between the continuous and discrete-time phase-lead compensator can be seen from Figure 2.20 (ϕ_m = 25^◦, ωm = 2π750). The proportional gain Kw is adjusted

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so the gain is 0 dB at 750 Hz (K_w = 0.637). The most obvious difference is that the discrete-time phase-lead compensator increases the gain and lags the phase compared to the continuous-time at frequencies higher than the design angular frequencyω_m.

10¹ 10² 10³

-6 -4 -2 0 2 4 6

Magnitude (dB)

Gkw(s) Gkw(z)

10¹ 10² 10³

Frequency (Hz) 0

5 10 15 20 25 30

Phase (deg)

Figure 2.20. Bode diagram of phase-lead compensatorG_kw(s|z)

2.7 Grid voltage feedforward

The grid voltage feedforward (VFF) is easy to understand and implement because the measured grid voltage is added directly to the controller output. The diagram of the VFF is shown in Figure 2.21. In a way, the VFF gives the controller an initial value which speeds up the controller operation, e.g., during converter commissioning and fault conditions. On the other hand, this may lead to amplification of grid voltage disturbances.

Figure 2.21.Diagram of grid voltage feedforward

Suntioet al. [28] have stated that the VFF is often utilized in grid-connected inverters to improve transient performance and impedance behavior. Originally, the method has been

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developed to improve dynamic response of electrical drives but due to its benefits and ease of implementation it has also become a popular control method in grid-connected three-phase converters.

Teodorescu et al. [24] have stated that there is no more need for the VFF and cross- coupling terms in case of the PR control. The cross-coupling terms are not needed because the control is done by sinusoidal quantities and nodq-transformation is required.

Although the VFF is not necessary with the PR control, there may be other requirements that support the use of the VFF.

The case study simulations are done with and without the VFF. The effects on stability margins, frequency responses and simulation results is analyzed in following sections.

2.8 Tuning procedure

The tuning procedure can be started by measuring the frequency response of the simulation model itself by disabling all controls. This provides a good guideline where the desired gain crossover frequency f_0dB can be set and how many HCs can be enabled;

f_0dB must be set higher than the highest HC to avoid multiple gain crossovers.

The tuning ofGPR(z)begins by adjustingKp, e.g., so thatf_0dB= 450 Hzand the HCs are enabled until the 7^th harmonic. It can be deducted from the Bode diagram how much the gain must be increased to achieve the desiredf_0dB. In the Bode diagram the magnitude is given in decibels which can be converted to absolute value as

K_p = 10^K^dB^/20, (2.27)

where K_dB is the desired gain in decibels. It is advisable to change K_p gradually and enable the HCs one by one, because increasingKpreduces the GM.

It is difficult to give unambiguous guidance on how to set upK_ihfor each HCs; it requires a few iterations and the Bode diagram analysis of the phase curve. K_i1is set equal to 15 for the fundamental HC, while the values ofKih, if needed, are set to values decreasing with the order of higher-order HCs. LowerK_ihvalues are set for the higher-order HCs to maintain as much PM as possible. The HC can be disabled by settingK_ih= 0.

The tuning of G_kw(z) is quite straightforward and it can be used to increase the PM and to utilize more HCs. The design frequency fm is set the same as the desiredf0dB. The maximum phase-leadϕ_mis set based on how much PM is desired to increase. The phase cannot be led indefinitely, as it increases gain and reduces the GM. The magnitude ofGkw(z)is set to 0 dB at the desiredf0dB by adjustingKw. Thus,f0dB remains almost unchanged, as previously set byK_p.

The final tuning results may require several iterations. After each iteration, the stability margins must be analyzed and necessary changes to the parameters be made.

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3 CASE STUDY

This chapter introduces thesimulation model and its parameters used in the study. The current-controlled voltage source converter (VSC) is delta-connected and modeled as controlled voltage sources. Thediscrete-time PR current control for the VSC includes all the components mentioned in Chapter 2. The control tuning is performed by analyzing the open-loop frequency responses measured by applying the MLBS. The selection of MLBS parameters is also discussed in more detail. Finally, the tuning results for the strong and weak grid conditions are presented with the controller parameters, stability margins and open-loop frequency responses.

3.1 Simulation model

The simulation model, shown in Figure 3.1, presents astatic synchronous compensator (STATCOM) in a simplified form. Shahnia et al. [29] have stated that the STATCOM is a reactive power compensation device that is shunt-connected to the grid, and the basic topology is based on the VSCs.

i_sa

i_sc

Figure 3.1. Diagram of the simulation model

The VSC is delta-connected and modeled as controlled voltage sources, so harmonics produced by the VSC itself are neglected. The advantage of the delta connection is that the rated current of the VSC can be dimensioned smaller than the actual current demand

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(I_g = 3Is). On the other hand, the disadvantage is that the nominal voltage of the VSC must be greater than the line voltage of the grid (V_s=√

3V_g).

The parameters of the simulation model are shown in Table 3.1. The purpose of the VSC is to supply reactive power to the grid, so there is an inductanceLs with an internal resis- tanceRs at each phase of the VSC. The grid is star-connected and the grid impedance is modeled as the inductanceL_g and resistanceR_g. The simulations are performed with strong and weak grid conditions; for both grid conditions the other parameters are kept the same.

Table 3.1. Simulation model parameters

Parameter Symbol Value Unit

VSC voltage (delta) V_s 32.0 kV

VSC rated current Is 1000 A

VSC inductance L_s 11 mH

VSC resistance Rs 17.3 mΩ

Base voltage V_B =√

2V_s 45.3 kV

Base current IB =√

2Is 1414 A

Grid frequency f 50 Hz

Grid voltage (star) Vg 18.5 kV

Grid voltage harmonics h_g 5 7 11 13 6.0 5.0 3.5 3.0

and levels %

Strong Weak

Grid short-circuit level SCL 3000 500 MVA

Grid impedance ratio X/R 30 10

Grid inductance L_g 1.09 6.49 mH

Grid resistance Rg 11.4 204 mΩ

The aim is to study the PR control performance in the grid environment where the grid voltage is already distorted by low-order harmonics from the positive (h = 7,13) and negative sequence (h= 5,11) [30]. The amplitude of the grid voltage harmonics used in the simulations are chosen based on the compatibility limits in low and medium voltage networks according to the IEC 61000-3-6 standard [31].

3.2 Discrete-time PR current control

Each phase of the VSC has its own discrete-time PR current control shown in Figure 3.2.

Usually the PR controller is used in theαβ-frame, but it can also be used in theabc-frame.

The controller design is performed in theper unit (pu) system [32]. Thebase values are determined in order to normalize the nominal or rated peak values to 1.

The control inputs, the measured instantaneous valuesisandvg, are divided by the base

Design and Analysis of Proportional-Resonant Current Control for VSC