PROMOTioN – Progress on Meshed HVDC Offshore Transmission Networks Mail info@promotion-offshore.net Web www.promotion-offshore.net This result is part of a project that has received funding form the European Union’s Horizon 2020 research and innovation programme under grant agreement No 691714. Publicity reflects the author’s view and the EU is not liable of any use made of the information in this report.

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WP16 – MMC Test Bench Demonstrator

Documentation of analytical approach Implementation of an Analytical Method for Analysis of Harmonic Resonance Phenomena

PROJECT REPORT

i

DOCUMENT INFO SHEET

Document Name: Implementation of an Analytical Method for Analysis of Harmonic

Resonance Phenomena

Responsible partner: DNV GL

Work Package: WP 16

Work Package leader: Philipp Ruffing

Task: 16.5

Task lead: Yin Sun

DISTRIBUTION LIST

PROMOTioN partners, European Commission, PROMOTioN website

APPROVALS

Name Company

Validated by: Dragan Jovcic University of Aberdeen

Sertkan Kabul TenneT TSO B.V.

Task leader: Yin Sun/Yongtao Yang DNV GL

WP Leader: Philipp Ruffing RWTH Aachen University

DOCUMENT HISTORY

Version Date Main modification Author

1.0 01.03.2019 Final submission Yin Sun

2.0 17.06.2019 Revision of the

introduction

Yongtao Yang/

Philipp Ruffing

WP

Number WP Title Person months Start month

End

month

WP16 MMC Test Bench Demonstrator 106.8 M24 M48

Deliverable

Number Deliverable Title Type

Dissemination

level Due Date

D16.5 Implementation of an Analytical Method for Analysis of Harmonic Resonance Phenomena

Report Public 38

PROJECT REPORT

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LIST OF CONTRIBUTORS

Work Package and deliverable involve many partners and contributors. The names of the partners,

who contributed to the present deliverable, are presented in the following table.

PARTNER NAME

RWTH Aachen Matthias Quester, Philipp Ruffing

UPV Soledad Bernal-Perez, Salvador Añó-Villalba, Ramón

Blasco-Gimenez

DNV GL Yin Sun, Yongtao Yang

Orsted Mohammad Kazem Dowlatabadi

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LIST OF ABBREVIATIONS

ABBREVIATONS FULL NAMES

VSC Voltage Source Converter

WTG Wind Turbine Generator

DRU Diode Rectifier Unit

MMC Modular Multi-level Converter

WPP Wind Power Plant

OWF Offshore Wind Farm

PLL Phase Locked Loop

SRF Synchronous Reference Frame

LTI Linear Time Invariant

PI Proportional Integral

PR Proportional Resonance

ACC Alternating Current Control

SISO Single-Input-Single-Output

MIMO Multi-Input-Multi-Output

AD Active Damping

PM Phase Margin

FFT Fast Fourier Transformation

Table of Content

Document info sheet .............................................................................................................................................................. i

Distribution list ...................................................................................................................................................................... i

Approvals ............................................................................................................................................................................. i

Document history ................................................................................................................................................................. i

List of Contributors ............................................................................................................................................................... ii

List of Abbreviations ............................................................................................................................................................ iii

1 Introduction.................................................................................................................................................................... 1

1.1 Objective and Scope of Work .................................................................................................................................. 2

1.2 Reading guidance and Mathematical Symbol Conventions .................................................................................... 3

1.2.1 Two-level VSC General Conventions .............................................................................................................. 3

1.2.2 MMC-VSC General Conventions ..................................................................................................................... 3

2 Two-level VSC ................................................................................................................................................................ 6

2.1 AC current control loop ............................................................................................................................................ 6

2.2 Phase-locked loop effect ......................................................................................................................................... 9

2.2.1 SRF-PLL small-signal model ........................................................................................................................... 9

2.2.2 SRF-PLL impedance shaping effect .............................................................................................................. 10

2.3 αβ-frame input admittance derivation ................................................................................................................... 13

2.4 VSC with Proportional resonant Alternating Current controller ............................................................................. 16

2.5 Impedance-based Stability Analysis ...................................................................................................................... 19

2.6 Experimental Verifications ..................................................................................................................................... 20

2.6.1 High Frequency Oscillations .......................................................................................................................... 20

2.6.2 Low Frequency Oscillations ........................................................................................................................... 26

2.7 Summary ............................................................................................................................................................... 28

3 MMC VSC...................................................................................................................................................................... 29

3.1 Introduction to Modular Multilevel Converter ......................................................................................................... 30

3.2 Average Value Model ............................................................................................................................................ 32

3.3 Frequency Domain Model ..................................................................................................................................... 36

3.4 Multi-harmonic Linearized Model ........................................................................................................................... 38

3.5 Control Modeling ................................................................................................................................................... 41

3.5.1 Phase Current Control ................................................................................................................................... 41

3.5.2 Circulating Current Control ............................................................................................................................ 43

3.5.3 Phase Locked Loop ....................................................................................................................................... 44

3.5.4 Grid Forming Control ..................................................................................................................................... 47

3.6 Final Impedance Model ......................................................................................................................................... 49

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4 Grid-forming Inverter .................................................................................................................................................. 52

4.1 System description ................................................................................................................................................ 52

4.1.1 Wind Power Plant modelling .......................................................................................................................... 53

4.1.2 Wind Turbine Control ..................................................................................................................................... 55

4.1.3 DRU Modeling ............................................................................................................................................... 59

4.1.4 State Space Model Validation........................................................................................................................ 62

4.1.5 State-Space Stability analysis ....................................................................................................................... 63

4.2 Impedance-Based STABILITY Analysis ................................................................................................................ 65

4.2.1 Harmonic impedances from state space equations ....................................................................................... 65

4.2.2 Wind turbine impedances in dq-frame ........................................................................................................... 66

4.2.3 DRU Impedance in dq-frame ......................................................................................................................... 67

4.2.4 Small-signal stability analysis ........................................................................................................................ 69

4.3 Conclusions ........................................................................................................................................................... 73

4.4 System Parameters ............................................................................................................................................... 74

5 BIBLIOGRAPHY ........................................................................................................................................................... 76

6 Appendix ...................................................................................................................................................................... 78

6.1 MMC VSC: Multi-harmonic Linearized Model ........................................................................................................ 78

6.2 MMC VSC: Phase Current Control ........................................................................................................................ 79

6.3 MMC VSC: Circulating Current Control ................................................................................................................. 82

6.4 MMC VSC: Phase Locked Loop ............................................................................................................................ 84

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1 INTRODUCTION

The ongoing energy transition is transforming the fossil-fuel based energy system of today into a more

sustainable energy system of the future. This transition is further accelerated by the Paris agreement,

which aims to significantly cut down the global CO2 emission levels. The grid-connected Voltage-

Source-Converter (VSC) is the underpinning technology for the success of this energy transition. It

not only integrates the intermittent renewable generation into the electric power system, but also aids

in improving the end-use energy efficiency amidst the widespread electrification of the heating/cooling

and the transport sectors, for example. These are the first signs of a power electronics dominant grid

taking shape, confirmed by the rapid increase in penetration level of grid-connected VSCs in today’s

electric power system.

The power system stability, once dominated by the synchronous generation units, will be strongly

influenced by the grid-connected VSCs. The wide control bandwidth of VSCs therefore demands the

industry to investigate harmonic oscillations (i.e. up to 2.5 kHz following the power quality standard

EN-50160) in addition to the traditional electro-mechanical and transient stability. Consequently, the

wider harmonic stability is of great importance for the stable operation of a future power electronics

dominant grid, where controllers of various grid-connected VSCs should be inter-operable without

invoking resonances. The dynamic interaction among the power grid and VSCs tend to cause

oscillations in a wide frequency range and has recently been reported in both the wind [1] and PV [2]

power plants. Similar instability has also been reported in railway power systems [3], leading to the

standardization of the input admittance characteristics of concerned power electronics equipment.

For the study of small-signal stability of a power system, where multiple grid connected VSCs are

connected to the vicinity of each other, the impedance-based method has been identified as an

effective yet simple technique. The impedance-based method was first proposed in [4] for the study of

DC regulator stability with various concentrated output filter configurations (i.e. simple RLC circuit with

constant parameters). Since then this method has been adopted and extended for the study of low

and high frequency small-signal stability in the AC power system with grid-connected VSCs [5] [6] [7]

[8] [9]. The previous research work [5] [6] [7] [8] effectively bridges the original, DC system impedance-

based stability technique [4] to the AC system taking into account the AC Current Control (ACC) with

the PLL effect in the dq-frame [5] [6] [7] and later in the αβ-frame [8]. Several study cases are reported

[10] [11] [12] [13] [14] [15] showing the effectiveness of the impedance-based stability analysis.

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1.1 OBJECTIVE AND SCOPE OF WORK

The main objective of this document is to present a step-by-step theoretical derivation of the state-of-

the-art input admittance of the grid-connected power electronics applications (i.e. wind turbine

generators, HVDC, and Diode rectifier units) relevant for the offshore wind power integration. The

reader is suggested to treat this document as a summary of the state-of-the-art input admittance

modelling of grid-connected power electronics. It is by no means the purpose of this document to

publish or create a new methodology for the harmonic resonance analysis. The sequel document

D16.4 will elaborate on the model validation of the input admittance modelling presented in this

document and apply the methodology for the harmonic resonance analysis pertaining to offshore wind

farm grid integration. This deliverable will also include the frequency domain modelling of additional

components, such as cables.

Due to the different working principles, and the circuit topologies of the power electronics applications,

this document is naturally divided into three parts:

1. Two-level VSC Model, by DNV GL

2. Modular Multilevel Converter (MMC) VSC Model, by RWTH Aachen

3. Diode Rectifier Unit (DRU) Model, by UPV

D16.5 is served as the input for the forthcoming D16.4, where the harmonic resonance analysis

frequency domain model will be validated against the time-domain experimental results1 and used in

the study cases defined for the offshore wind farm grid integration. Overall, D16.4 will contain inputs

from several other WPs (e.g. WP2, WP3, WP4), and the theoretical background of the harmonic

resonance analysis will be addressed by D16.5.

The impedance admittance of the VSC, in this document, can be derived either in the 𝑑𝑞-frame, 𝛼𝛽-

frame or the sequence-frame. The conversion matrices from the 𝑑𝑞-frame to the 𝛼𝛽-frame is

elaborated in the Section 2.3. The conversion matrices from the 𝛼𝛽-frame to the positive and negative

sequence components are given in the equations below:

𝑣𝛼𝛽+ = [𝑇𝛼𝛽

+ ]𝑣𝛼𝛽; [𝑇𝛼𝛽+ ] =

1

2 [1 −𝑞𝑞 1

] (1.1)

𝑣𝛼𝛽− = [𝑇𝛼𝛽

− ]𝑣𝛼𝛽; [𝑇𝛼𝛽− ] =

1

2 [1 𝑞−𝑞 1

] (1.2)

where 𝑞 = 𝑒−𝑗𝜋

2 is the 90-degree lagging phase operator. The positive and negative sequence

impedance are not necessarily the same and this can be reflected in the derived input admittance of

the VSC. The conversion matrices allow for a holistic harmonic resonance analysis even though the

input admittance of VSC might be derived from different reference frames.

1 Within this report only first simulation-based verifications of the three topologies are carried out.

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1.2 READING GUIDANCE AND MATHEMATICAL SYMBOL CONVENTIONS

This document is by nature highly theory-oriented, where the authors tried to include the full details of

the theoretical foundations for the modelling work in T16.5 and T16.6, such that interested readers can

follow the work on his/her own. The contents might be a difficult for non-academic readers, for whom

we would like to advise to read the following sections and neglect the rest of the document:

• Section 2.7

• Section 3.5.4

• Section 4.3

Before entering the analytical equations that describe the input admittance of the VSC technologies,

the conventions of electrical symbols used in this document are introduced. Because of the

assumptions taken for the small-signal input-admittance modelling and the nature of operation for the

different power electronics topologies, the conventions used in this document are summarized for the

two-level VSC and the MMC-VSC exclusively.

1.2.1 TWO-LEVEL VSC GENERAL CONVENTIONS

In the case of the two-level VSC, typically applied for the type-4 wind turbine generator system, the

complex space-vector is originally used in the 𝑑𝑞-frame to describe a symmetrical system; due to the

unsymmetrical effect of the PLL, a more general 2x2real-space matrix is used to describe the system

considering both the ACC and PLL. The two-level VSC conventions of electrical symbols used in this

document is summarized.

Bold, non-italic variables with vector sign → on top of the symbol are used to denote complex space

vectors (e.g. 𝐄→

= 𝐸𝛼 + 𝑗𝐸𝛽 for PCC voltage) and complex transfer functions (e.g. 𝐆𝐭𝐜𝐥→

(𝑠) = 𝐺𝑡𝑐𝑙,𝛼(𝑠) +

𝑗𝐺𝑡𝑐𝑙,𝛽(𝑠) for the closed-loop ACC transfer function). Whenever referred to the grid 𝑑𝑞-frame, a

subscript "𝑑𝑞" is added for the complex space vectors and complex transfer functions (e.g. 𝐄𝐝𝐪→

= 𝐸𝑑 +

𝑗𝐸𝑞 and 𝐘𝐨𝐩,𝐝𝐪→

(𝑠) = 𝑌𝑜𝑝,𝑑(𝑠) + 𝑗𝑌𝑜𝑝,𝑞(s) for the PCC voltage and VSC output admittance in the rotating

𝑑𝑝-frame with reference to the grid, respectively). Whenever referred to the converter 𝑑𝑞-frame

synchronized by the PLL, a superscript "c" is added for the complex space vectors (e.g. 𝐄𝐝𝐪𝐜→ = 𝐸𝑑

𝑐 +

𝑗𝐸𝑞𝑐 and 𝐈𝐝𝐪

𝐜→ = 𝐼𝑑𝑐 + 𝑗𝐼𝑞

𝑐 refer to the PCC voltage and the output current in the converter PLL

synchronized 𝑑𝑞-frame).

The real-space matrices are expressed with italic letters and "m" in its superscript (e.g. 𝐸𝑚 =

[𝐸𝛼 𝐸𝛽]𝑇).

1.2.2 MMC-VSC GENERAL CONVENTIONS

In the case of the MMC-VSC - typically applied for the high voltage Flexible Alternating Current-

Transmission System (FACTS) devices and High Voltage Direct Current (HVDC) transmission - the

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model is built as a single arm, one phase representation for the positive and negative sequences. For

this reason, the MMC-VSC convention of electrical symbols used in this document is summarized.

In the frequency domain, different harmonics are considered by means of matrix and vector

representation of variables (e.g. see equations (1.6) and (1.7)). Lowercase letters indicate time or

frequency-dependent variables (e.g. 𝑣𝑎); whereas capital letters indicate the amplitude of time-or

frequency-dependent variables, or DC quantities (e.g. 𝑉1):

𝑣𝑎(𝑡) = 𝑉1 cos(2𝜋𝑓1𝑡 + 𝜑1) (1.3)

Variables in the dq-frame are indicated by the subscript d or q:

[𝑥𝑑𝑥𝑞] = 𝑻𝒅𝒒 [

𝑥𝑎𝑥𝑏𝑥𝑐

] (1.4)

The model differentiates between variables representing the steady state solution and operation point

and small signal variables resulting from the perturbation. There is no further indication for the steady

state variables. Small signal variables are indicated by a ”^” on top of the variable such as (𝒂𝒖.).

A specific element of a matrix is indicated in parenthesis in the subscript of the corresponding matrix.

In expression (1.5) Z(𝑓𝑝 ) is the (𝑛 + 1, 𝑛 + 1)th element of 𝐘.

Z(𝑓𝑝 ) =1

2𝐘(n+1,n+1) (1.5)

Time Domain

Time dependent variables (e.g. currents and voltages) are written in small, italic letters. Subscripts are

used for indicating the phase and arm. The first subscript indicates the phase, the second subscript

the arm:

𝑣𝑎𝑢 → instantaneous voltage of phase a, upper arm

𝑣𝑎𝑙 → instantaneous voltage of phase a, lower arm.

Frequency Domain

Steady state signals in the frequency domain are represented as vectors or matrices up to the nth

harmonic order with bold, non-italic, lowercase letters. For instance, in expression (1.6) 𝐢𝒂𝒖 is the

current of phase a, upper arm and is represented as vector.

𝐢𝒂𝒖 =

[ 𝐼𝑛𝑒

−𝑗𝛼𝑛

⋮

𝐼1𝑒−𝑗𝛼1

𝐼0𝐼1𝑒

𝑗𝛼1

⋮

𝐼𝑛𝑒𝑗𝛼𝑛 ]

(1.6)

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Subscripts are used for indicating the frequency within the Fourier coefficient representation. E.g. the

subscript 𝑝𝑘 indicates the frequency at 𝑓𝑝+ 𝑘𝑓

1. The first subscript indicates the perturbation

frequency 𝑓𝑝 and 𝑘 the multiple of the fundamental frequency 𝑓

1. In the expression (1.7), 𝒂𝒖is the

small-signal variable caused by perturbation the frequency 𝑓𝑝.

𝒂𝒖 =

[ 𝑝−𝑛𝑒

𝑗𝑝−𝑛

⋮

𝑝−1𝑒𝑗𝑝−1

𝑝𝑒𝑗𝑝

𝑝+1𝑒𝑗𝑝+1

⋮

𝑝+𝑛𝑒𝑗𝑝+𝑛]

(1.7)

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2 TWO-LEVEL VSC

For a two-level Voltage Sourced Converter (VSC), the coordinate transformations (i.e. Clarke and Park

transformations) as well as the Taylor series defines the small-signal linearization around an

equilibrium point [16]. By further assuming a constant DC link voltage, a small-signal linear time

invariant (LTI) model can be derived for a closed-loop controlled two-level VSC. For simplification, in

this chapter, VSC is referring to the two-level VSC.

The rest of this chapter is organized as follows: first, the VSC input admittance considering only the

alternating current control (ACC) loop is derived. Then, the phase-locked loop (PLL) effect is added

through small-signal linearization around a given operational point. In the end, the experimental results

are presented for the verification of the analytical prediction using the impedance-based stability

method.

2.1 AC CURRENT CONTROL LOOP

For three-phase VSC applications, synchronous-frame PI controllers are widely applied for the inner

ACC. For this reason, the input admittance of the VSC is firstly derived with the synchronous-frame PI

controller for the ACC. The small signal model of a grid-connected VSC interface with a synchronous-

frame PI controller for the ACC is shown in Figure 2-1.

Figure 2-1 Grid-connected VSC interface closed-loop transfer function with the 𝑑𝑞-frame PI controller

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In Figure 2-1, the variables are described as follows:

𝑣𝑐𝑓 capacitor voltage measured for the grid synchronization of the PLL

𝐻𝑃𝐿𝐿(𝑠) closed-loop transfer function of the PLL

𝑖𝑑𝑞∗ ,𝑖𝑑𝑞 reference current set-points and measured inverter side current in the 𝑑𝑞-frame

𝐺𝑣𝑐(𝑠) High-pass filter of the capacitor voltage feed-forward

𝐺𝑖𝑐(𝑠) Synchronous-frame PI controller for the ACC

𝐿1 inverter side inductor

𝐿2 grid side inductor2

𝐶𝑓 filter capacitance

𝐿𝑔 lumped grid impedance

𝑣𝑔 deal voltage source emulating the grid voltage

𝑣𝑑𝑐 constant DC link voltage upon which small-signal model is derived for the VSC

𝑣𝑜 VSC terminal output voltage

𝑣𝑐 digital controller output voltage reference signal

𝑒−𝑠𝑇𝑠 one sampling cycle delay considering sampling, computation, and update when

the synchronous PWM modulation is considered

𝑇𝑠 sampling frequency

𝑣𝑚 actual voltage input signal for the PWM modulation

𝜃 Phase angle output of the PLL

For the analysis in this chapter, current positive direction is defined from the VSC towards the grid

(black arrow shown in Figure 2-1 to the left of 𝐿1).

The input admittance3 of the VSC can be derived with regards to the filter capacitor terminal voltage

as shown in Figure 2-2, where the ACC loop and PLL control loop are explicitly considered.

2 In practice, this is typically the equivalent of the transformer leakage inductance. 3 Input admittance is defined as a small-signal property with the resulting small-signal current divided by the small-signal voltage perturbation imposed on a given terminal of the VSC. The direction is typically defined as from the grid towards the VSC.

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Figure 2-2 Grid-connected VSC interface input admittance transfer function with the 𝑑𝑞-frame PI controller

In Figure 2-2, 𝐺𝑖𝑐,𝑑𝑞(𝑠) is the 𝑑𝑞-frame PI controller:

𝐺𝑖𝑐,𝑑𝑞(𝑠) = 𝐾𝑝 +𝐾𝑖𝑠

(2.1)

𝐺𝑑(𝑠) indicates one and a half sampling cycle delay caused by the digital computation and the PWM

zero-order hold effect, respectively [17]:

𝐺𝑑(𝑠) = 𝑒−1.5𝑇𝑠𝑠 (2.2)

𝑌𝑜𝑝(𝑠) and 𝑌𝑖𝑝(𝑠) represents the passive L filter output and input admittance respectively in the 𝛼𝛽-

frame, their 𝑑𝑞-frame equivalent are complex transfer function obtained via the frequency translation [18]:

𝐘𝐨𝐩,𝐝𝐪→

(𝑠) = 𝑌𝑜𝑝(𝑠 + 𝑗𝜔1) =1

𝐿1(𝑠 + 𝑗𝜔1) + 𝑅1

𝐘𝐢𝐩,𝐝𝐪→

(𝑠) = 𝑌𝑖𝑝(𝑠 + 𝑗𝜔1) =1

𝐿1(𝑠 + 𝑗𝜔1) + 𝑅1

(2.3)

Consequently, the open-loop gain 𝑇𝑜 is obtained if we neglect the PLL and the active damping:

𝑇𝑜(𝑠) = 𝐺𝑖𝑐(𝑠)𝐺𝑑(𝑠)𝑌𝑜𝑝(𝑠 + 𝑗𝜔1) (2.4)

In the 𝑑𝑞-frame, the small-signal AC output current can be written below with Δ being omitted for

simplicity (see [5] for detailed derivation):

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𝐈𝐝𝐪→ = 𝐈𝐝𝐪

∗→ 𝑇𝑜(𝑠)

1 + 𝑇𝑜(𝑠)⏟ 𝐺𝑡𝑐𝑙(𝑠)

− 𝐄𝐝𝐪→ 𝑌𝑖𝑝(𝑠 + 𝑗𝜔1)

1 + 𝑇𝑜(𝑠)⏟ 𝑌𝑡𝑜(𝑠)

(2.5)

where 𝐺𝑡𝑐𝑙(𝑠)is the closed-loop ACC transfer function4 in the 𝑑𝑝-frame while 𝑌𝑡𝑜(𝑠) is defined as the

input admittance of the VSC in the 𝑑𝑞-frame considering the AC current controller only.

2.2 PHASE-LOCKED LOOP EFFECT

Despite the rapid development of innovative PLL concepts [19] [20], the core structure of PLL remains

unaltered but enhanced with the input filtering capability and the adaptive frequency tracking capability

(e.g. Dual Second Order Generalized Integrator PLL). For the analysis of PLL impedance shaping

effect, SRF-PLL is chosen for study in this chapter as it represents the most commonly applied PLL

structure found in VSC applications.

Firstly, the small-signal model of SRF-PLL is derived in the 𝑑𝑞-frame. Then the small-signal

perturbation theory is applied to describe the current feedback change and the output voltage change

caused by the Δ𝜃 because of the small-signal external voltage perturbation, 𝑒. Due to the asymmetrical

effect of the SRF-PLL on the impedance shaping of the VSC input admittance, the derived 𝑑𝑞-frame

input admittance is described as an asymmetric 2 × 2 real space matrix. In the end, this matrix can

be written as a complex-space matrix in the 𝑑𝑞-frame. Via the frequency translation, the input

admittance in the 𝛼𝛽-frame can be obtained for easier harmonic resonance analysis with the power

system components.

2.2.1 SRF-PLL SMALL-SIGNAL MODEL

With reference to [8], the small-signal mode of the SRF-PLL can be effectively represented as seen in

Figure 2-3, where Δ𝜃 via a gain constant 𝐸1𝑑 presents a negative feedback loop path.

4 Signal input is the AC current reference and the signal output is the actual measured AC current.

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Figure 2-3 PLL small-signal closed loop transfer function diagram (Note: ∆𝐸𝑞 is in the PLL controller frame after the

summation point)

In

Figure 2-3:

Δ𝐸𝑞 is the grid voltage perturbation projected on the q-axis

Δ𝜔 represents the PI controller output in rad/s

Δ𝜃 is the phase angle output

𝐸1𝑑 is the grid voltage space vector projected on the d-axis of the SRF when the perturbation Δ𝑞 is

small.

Referring to the small-signal transfer function diagram in

Figure 2-3, the SRF-PLL closed-loop transfer function 𝐻𝑝𝑙𝑙(𝑠) can be written as:

𝐻𝑝𝑙𝑙(𝑠) =𝐺𝑃𝐼(𝑠)

𝑠 + 𝐺𝑃𝐼(𝑠)𝐸1𝑑 (2.6)

where 𝐺𝑃𝐼(𝑠) is the PI controller for the SRF-PLL defined as:

𝐺𝑃𝐼(𝑠) = 𝐾𝑝𝑝𝑙𝑙 +𝐾𝑖𝑝𝑙𝑙

𝑠 (2.7)

Eventually phase angle output in the controller frame because of the small-signal voltage

perturbation 𝚫𝐄𝐝𝐪→

can be written as:

Δ𝜃 = 𝐻𝑝𝑙𝑙(𝑠)ℑ 𝚫𝐄𝐝𝐪

→

= 𝐻𝑝𝑙𝑙(𝑠)Δ𝐸𝑞 (2.8)

Taking only the quadrature component, then:

⇒ ℑ𝚫𝐄𝐝𝐪𝐜→ = Δ𝐸𝑞 − 𝐸1𝑑Δ𝜃 (2.9)

2.2.2 SRF-PLL IMPEDANCE SHAPING EFFECT

The SRF-PLL impedance shaping effect can be expressed by the small-signal linearization of the

complex Parke transformation modulator 𝑒𝑗Δ𝜃. Applying the small-signal approximation for the VSC

current feedback 𝐈𝐝𝐪→

and the VSC output voltage 𝐕𝐜,𝐝𝐪→

around an equilibrium operation point in the

𝑑𝑞-frame gives:

PROJECT REPORT

11

𝐈𝐝𝐪→

= (𝐼𝛼 + 𝑗𝐼𝛽)𝑒−𝑗𝜃

= (𝐈𝟏,𝐝𝐪→

+ Δ 𝐈𝐝𝐪→ ) (1 − 𝑗Δ𝜃)

≈ 𝐈𝟏,𝐝𝐪→

+ Δ 𝐈𝐝𝐪→ − 𝐈𝟏,𝐝𝐪→

𝑗Δ𝜃⏟

Δ𝐈𝐏𝐋𝐋,𝐝𝐪→

(2.10)

Δ 𝐈𝐏𝐋𝐋,𝐝𝐪→

= 𝑗𝐼1𝑑Δ𝜃 − 𝐼1𝑞Δ𝜃 (2.11)

𝑉𝑐,𝛼 + 𝑗𝑉𝑐,𝛽 = 𝐕𝐜,𝐝𝐪→

𝑒𝑗𝜃

≈ (𝐕𝟏𝐜,𝐝𝐪→

+ Δ𝐕𝐜,𝐝𝐪→

) (1 + 𝑗Δ𝜃)𝑒𝑗𝜔1𝑡

⇒ 𝐕𝐜,𝐝𝐪→

= 𝐕𝟏𝐜,𝐝𝐪→

+ Δ𝐕𝐜,𝐝𝐪→

+ 𝐕𝟏𝐜,𝐝𝐪→

𝑗Δ𝜃⏟

Δ𝐕𝐏𝐋𝐋,𝐝𝐪→

(2.12)

Δ𝐕𝐏𝐋𝐋,𝐝𝐪→

= 𝑗𝑉1𝑐𝑑Δ𝜃 − 𝑉1𝑐𝑞Δ𝜃 (2.13)

where 𝐈𝟏,𝐝𝐪→

= 𝐼1𝑑 + 𝑗𝐼1𝑞 and 𝐕𝟏𝐜,𝐝𝐪→

= 𝑉1𝑐𝑑 + 𝑗𝑉1𝑐𝑞 indicate the steady-state VSC output current and

voltage respectively. Δ 𝐈𝐏𝐋𝐋,𝐝𝐪→

and Δ𝐕𝐏𝐋𝐋,𝐝𝐪→

denote the dynamic effect of PLL on the VSC output current

and voltage upon a given steady-state operating point (i.e. 𝐈𝟏,𝐝𝐪→

= 𝐼1𝑑 + 𝑗𝐼1𝑞 and 𝐕𝟏𝐜,𝐝𝐪→

= 𝑉1𝑐𝑑 + 𝑗𝑉1𝑐𝑞).

Substituting (2.8) into (2.11) and (2.13) gives:

Δ 𝐈𝐏𝐋𝐋,𝐝𝐪→

= 𝑗𝐼1𝑑Δ𝐻𝑝𝑙𝑙(𝑠)Δ𝐸𝑞 − 𝐼1𝑞Δ𝐻𝑝𝑙𝑙(𝑠)Δ𝐸𝑞 (2.14)

Δ𝐕𝐏𝐋𝐋,𝐝𝐪→

= 𝑗𝑉1𝑐𝑑Δ𝐻𝑝𝑙𝑙(𝑠)Δ𝐸𝑞 − 𝑉1𝑐𝑞Δ𝐻𝑝𝑙𝑙(𝑠)Δ𝐸𝑞 (2.15)

where only the 𝑞-axis voltage Δ𝐸𝑞 is considered other than the complex space voltage vector Δ𝐄𝐝𝐪→

.

Therefore, the resulting transfer function matrices are asymmetric and contain the cross-couplings

between the 𝑑-axis and the 𝑞-axis. To combine the ACC with the PLL effect on the VSC input

admittance, the single input single output (SISO) transfer function (2.5) shall be modified as a multi-

input-multi-output(MIMO) real space transfer function matrix. Rewriting (2.14) and (2.15) into real

space matrix form gives:

[𝐼𝑃𝐿𝐿,𝑑𝐼𝑃𝐿𝐿,𝑞

] = [0 −𝐻𝑝𝑙𝑙(𝑠)𝐼1𝑞0 𝐻𝑝𝑙𝑙(𝑠)𝐼1𝑑

]⏟

𝑌𝑃𝐿𝐿𝑚 (𝑠)

[𝐸𝑑𝐸𝑞]

(2.16)

[𝑉𝑃𝐿𝐿,𝑑𝑉𝑃𝐿𝐿,𝑞

] = [0 −𝐻𝑝𝑙𝑙(𝑠)𝑉1𝑐𝑞0 𝐻𝑝𝑙𝑙(𝑠)𝑉1𝑐𝑑

]⏟

𝐺𝑃𝐿𝐿𝑚 (𝑠)

[𝐸𝑑𝐸𝑞]

(2.17)

PROJECT REPORT

12

Incorporating the feedback current measurement change and the output voltage change as a result of

the PLL dynamics (i.e. (2.16) and (2.17) ), the input admittance transfer function diagram in Figure 2-1

and Figure 2-2 can be modified as shown in Figure 2-4, where 𝑌𝑃𝐿𝐿𝑚 (𝑠) and 𝐺𝑃𝐿𝐿

𝑚 (𝑠) are represent the

current reference modification and output voltage change as a result of the small-signal voltage

perturbation 𝐸𝑑𝑞.

Figure 2-4 Grid-connected VSC interface input admittance real space transfer matrix with the 𝑑𝑞-frame PI controller

In Figure 2-4, the space matrices of the ACC can be defined as the following:

𝐺𝑖𝑐,𝑑𝑞𝑚 (𝑠) ≜ [

𝐺𝑖𝑐,𝑑𝑞(𝑠) 0

0 𝐺𝑖𝑐,𝑑𝑞(𝑠)] (2.18)

𝐺𝑑𝑚(𝑠) ≜ [

𝐺𝑑(𝑠) 0

0 𝐺𝑑(𝑠)] (2.19)

𝐺𝑜𝑝,𝑑𝑞𝑚 (𝑠) ≜

1

(𝐿1𝑠 + 𝑅1)2 + (𝜔1𝐿1)

2[𝐿1𝑠 + 𝑅1 −𝜔1𝐿1𝜔1𝐿1 𝐿1𝑠 + 𝑅1

] (2.20)

𝐺𝑖𝑝,𝑑𝑞𝑚 (𝑠) ≜

1

(𝐿1𝑠 + 𝑅1)2 + (𝜔1𝐿1)

2[𝐿1𝑠 + 𝑅1 −𝜔1𝐿1𝜔1𝐿1 𝐿1𝑠 + 𝑅1

] (2.21)

Consequently, the ACC open loop gain transfer function 𝑇𝑜𝑚(𝑠) and closed-loop transfer function

𝐺𝑡𝑐𝑙𝑚 (𝑠) in real space matrices can be derived following the matrix left multiplication rule:

𝑇𝑜,𝑑𝑞𝑚 (𝑠) = 𝐺𝑜𝑝,𝑑𝑞

𝑚 (𝑠)𝐺𝑑𝑚(𝑠)𝐺𝑖𝑐,𝑑𝑞

𝑚 (𝑠) (2.22)

𝐺𝑡𝑐𝑙,𝑑𝑞𝑚 (𝑠) = (𝐼𝑚 + 𝑇𝑜,𝑑𝑞

𝑚 (𝑠))−1𝑇𝑜,𝑑𝑞𝑚 (𝑠) (2.23)

where 𝐼𝑚 is defined as the diagonal identity matrix:

𝐼𝑚 = [1 00 1

] (2.24)

PROJECT REPORT

13

Considering the PLL impedance shaping effect, the small-signal AC output current equation (2.5) in

the real space matrix is derived as:

[Δ𝐼𝑑Δ𝐼𝑞] = 𝐺𝑡𝑐𝑙,𝑑𝑞

𝑚 (𝑠) [Δ𝐼𝑑∗𝑐

Δ𝐼𝑞∗𝑐]

⏟

Δ𝐈𝐝𝐪∗𝐜→

− (−𝐺𝑡𝑐𝑙,𝑑𝑞𝑚 (𝑠)𝑌𝑃𝐿𝐿

𝑚 (𝑠)⏞ 𝑃𝐿𝐿−𝑑𝑦𝑛𝑎𝑚𝑖𝑐𝑠

+ 𝑌𝑡𝑜,𝑑𝑞𝑚

⏟ 𝑌𝑡𝑐𝑙𝑚 (𝑠)

) [Δ𝐸𝑑Δ𝐸𝑞

] (2.25)

where the 𝑌𝑡𝑐𝑙𝑚 (𝑠) is the input admittance considering both the ACC and PLL dynamics. 𝑌𝑡𝑜,𝑑𝑞

𝑚 is

defined as the output admittance reshaped by the PLL dynamics:

𝑌𝑡𝑜,𝑑𝑞𝑚 = (𝐼𝑚 + 𝑇𝑜,𝑑𝑞

𝑚 (𝑠))−1

(𝑌𝑖𝑝,𝑑𝑞𝑚 (𝑠) − 𝑌𝑜𝑝,𝑑𝑞

𝑚 (𝑠)𝐺𝑑𝑚(𝑠)𝐺𝑃𝐿𝐿

𝑚 (𝑠)⏟ 𝑃𝐿𝐿−𝑑𝑦𝑛𝑎𝑚𝑖𝑐𝑠

) (2.26)

If the controller bandwidth of the outer controller loops (i.e. DC link voltage control, AC voltage/reactive

power control) is kept low, then it is considered that the voltage perturbation only produces negligible

offset on the current reference set-point 𝐈𝐝𝐪∗𝐜→ in the converter 𝑑𝑞-frame. In such a case, (2.25) can be

simplified and written its general form with the Δ sign being omitted for simplicity:

[𝐼𝑑𝐼𝑞] = [

𝑌𝑑𝑑(𝑠) 𝑌𝑑𝑞(𝑠)

𝑌𝑞𝑑(𝑠) 𝑌𝑞𝑞(𝑠)]

⏟ 𝑌𝑡𝑐𝑙,𝑑𝑞𝑚

[𝐸𝑑𝐸𝑞]

(2.27)

2.3 𝛼𝛽-FRAME INPUT ADMITTANCE DERIVATION

In the previous section, the input admittance 𝑌𝑡𝑐𝑙𝑚(𝑠) considering the ACC and the PLL impedance

shaping effect is derived in the 𝑑𝑞-frame, where generalized Nyquist stability criterion can be applied

with the grid impedance being expressed in the same 𝑑𝑞-frame via a frequency translation. For the

stability assessment of multiple VSCs connected in the vicinity of each other, a reference 𝑑𝑞-frame

must be established so that the input admittance of all the VSCs can be evaluated under "one" 𝑑𝑞-

frame [21]. Such an approach will make the impedance-based stability assessment cumbersome as

the state-space approach, therefore this section demonstrates the frequency translation method

proposed in [8] to translate the real space, dq-frame input admittance matrix to a complex space vector

matrix in the 𝛼𝛽-frame.

With reference to [18], the real space matrix of a balanced system can be expressed by a complex

space matrix comprised of complex space vectors and their conjugates. For the case of VSC input

admittance expressed in its general form (2.27), its complex space matrix equivalent can be expressed

as:

𝐈𝐝𝐪→ = 𝐘𝐭𝐜𝐥+,𝐝𝐪 (𝑠)𝐄𝐝𝐪 + 𝐘𝐭𝐜𝐥−,𝐝𝐪 (𝑠)𝐄𝐝𝐪

∗ (2.28)

PROJECT REPORT

14

where 𝐄𝐝𝐪∗→ is the complex space vector conjugate of 𝐄𝐝𝐪

→ . 𝐘𝐭𝐜𝐥+,𝐝𝐪→

(𝑠) and 𝐘𝐭𝐜𝐥−,𝐝𝐪→

(𝑠) are complex

admittance pair defined as follows:

𝐘𝐭𝐜𝐥+,𝐝𝐪→

(𝑠) =𝑌𝑑𝑑(𝑠) + 𝑌𝑞𝑞(𝑠)

2+ 𝑗𝑌𝑞𝑑(𝑠) − 𝑌𝑑𝑞(𝑠)

2

𝐘𝐭𝐜𝐥−,𝐝𝐪→

(𝑠) =𝑌𝑑𝑑(𝑠) − 𝑌𝑞𝑞(𝑠)

2+ 𝑗𝑌𝑞𝑑(𝑠) + 𝑌𝑑𝑞(𝑠)

2

(2.29)

To capture the double-frequency coupling in the stationary frame (i.e. 𝛼𝛽-frame) as a result of the

positive and negative sequence component in the 𝑑𝑝-frame, [8] proposes a complex transfer function

matrix 𝐘𝐭𝐜𝐥±,𝐝𝐪𝐦→ given by:

[𝐈𝐝𝐪→

𝐈𝐝𝐪∗→ ] = [

𝐘𝐭𝐜𝐥+,𝐝𝐪→

(𝑠) 𝐘𝐭𝐜𝐥−,𝐝𝐪→

(𝑠)

𝐘𝐭𝐜𝐥−,𝐝𝐪∗→ (𝑠) 𝐘𝐭𝐜𝐥+,𝐝𝐪

∗→ (𝑠)]

⏟

𝐘𝐭𝐜𝐥±,𝐝𝐪𝐦→ (𝑠)

[𝐄𝐝𝐪→

𝐄𝐝𝐪∗→ ]

(2.30)

Replacing the real space vectors with the complex space vectors, 𝐸𝑞 can be written as:

𝐸𝑞 =𝐄𝐝𝐪→

− 𝐄𝐝𝐪∗→

2𝑗 (2.31)

Then (2.16) and (2.17) in the real space matrices can be re-written equivalently in their complex

space vectors as:

Δ 𝐈𝐏𝐋𝐋,𝐝𝐪→

=𝐈𝟏,𝐝𝐪∗𝐜→ 𝑗𝐻𝑝𝑙𝑙(𝑠) (Δ𝐄𝐝𝐪

→ − Δ𝐄𝐝𝐪

∗→ )

2𝑗

=𝐈𝟏,𝐝𝐪∗𝐜→ 𝐻𝑝𝑙𝑙(𝑠)

2⏟

𝐘𝐏𝐋𝐋→

(𝑠)

(Δ𝐄𝐝𝐪→

− Δ𝐄𝐝𝐪∗→ )

(2.32)

Δ𝐕𝐏𝐋𝐋,𝐝𝐪→

=𝐕𝟏𝐜,𝐝𝐪→

𝑗𝐻𝑝𝑙𝑙(𝑠) (Δ 𝐄𝐝𝐪→

− Δ𝐄𝐝𝐪∗→ )

2𝑗

=𝐕𝟏𝐜,𝐝𝐪→

𝐻𝑝𝑙𝑙(𝑠)

2⏟

𝐆𝐏𝐋𝐋→

(𝑠)

(Δ 𝐄𝐝𝐪→

− Δ𝐄𝐝𝐪∗→ )

(2.33)

where 𝐘𝐏𝐋𝐋→

(𝑠) and 𝐆𝐏𝐋𝐋 (𝑠) are the equivalent complex transfer function of 𝑌𝑃𝐿𝐿𝑚 (𝑠) and 𝐺𝑃𝐿𝐿

𝑚 (𝑠) given

in (2.16) and (2.17). Effectively, the small-signal input admittance transfer function shown in Figure

2-4 can be expressed by their symmetrical complex space transfer functions, see Figure 2-5.

PROJECT REPORT

15

Figure 2-5 Grid-connected VSC input admittance small-signal complex transfer function diagram with the synchronous-frame PI controller

In Figure 2-5, the modified output admittance of the VSC considering the PLL dynamics is expressed

as:

𝐈𝐝𝐪 |𝐕𝐜,𝐝𝐪 =0 = ( 𝐘𝐢𝐩,𝐝𝐪 (𝑠) − 𝐆𝐏𝐋𝐋 (𝑠)𝐺𝑑(𝑠)⏟

𝐘𝐭𝐨+,𝐝𝐪(𝑠)

)𝐄𝐝𝐪

+(𝐆𝐏𝐋𝐋 (𝑠)𝐺𝑑(𝑠)⏟

𝐘𝐭𝐨−,𝐝𝐪(𝑠)

)𝐄𝐝𝐪∗

(2.34)

where 𝐘𝐭𝐨+,𝐝𝐪→

(𝑠) and 𝐘𝐭𝐨−,𝐝𝐪→

(𝑠) are the complex transfer functions of the modified open-loop output

admittance. Then the real space matrices of the input admittance transfer function 𝑌𝑡𝑐𝑙𝑚(𝑠) given in

(2.25) can be expressed by two complex space transfer functions:

𝐈𝐝𝐪 |𝐈𝐝𝐪∗𝐜 =0

= (−𝐆𝐭𝐜𝐥,𝐝𝐪→

(𝑠) 𝐘𝐏𝐋𝐋→

(𝑠) +𝐘𝐭𝐨+,𝐝𝐪→

(𝑠)

1 + 𝐓𝐨,𝐝𝐪→

(𝑠)⏟

)

𝐘𝐭𝐜𝐥+,𝐝𝐪→

(𝑠)

𝐄𝐝𝐪→

+(𝐆𝐭𝐜𝐥,𝐝𝐪→

(𝑠) 𝐘𝐏𝐋𝐋→

(𝑠) +𝐘𝐭𝐨+,𝐝𝐪→

(𝑠)

1 + 𝐓𝐨,𝐝𝐪→

(𝑠))

⏟

𝐘𝐭𝐜𝐥−,𝐝𝐪→

(𝑠)

𝐄𝐝𝐪∗→

(2.35)

where 𝐓𝐨,𝐝𝐪→

(𝑠) and 𝐆𝐭𝐜𝐥,𝐝𝐪→

(𝑠) are the AC current control open loop gain and closed-loop complex

transfer functions respectively:

𝐓𝐨,𝐝𝐪→

(𝑠) = 𝐺𝑖𝑐,𝑑𝑞(𝑠)𝐺𝑑(𝑠)𝐘𝐨𝐩,𝐝𝐪 (𝑠) (2.36)

PROJECT REPORT

16

𝐆𝐭𝐜𝐥,𝐝𝐪→

(𝑠) =𝐓𝐨,𝐝𝐪→

(𝑠)

1 + 𝐓𝐨,𝐝𝐪→

(𝑠) (2.37)

Finally, the input-admittance matrix comprising complex transfer functions and their conjugates in the complex space matrix form can be written as:

[𝐈𝐝𝐪→

𝐈𝐝𝐪∗→ ] = [

𝐘𝐭𝐜𝐥+,𝐝𝐪→

(𝑠) 𝐘𝐭𝐜𝐥−,𝐝𝐪→

(𝑠)

𝐘𝐭𝐜𝐥−,𝐝𝐪∗→ (𝑠) 𝐘𝐭𝐜𝐥+,𝐝𝐪

∗→ (𝑠)]

⏟

𝐘𝐭𝐜𝐥±,𝐝𝐪𝐦→ (𝑠)

[𝐄𝐝𝐪→

𝐄𝐝𝐪∗→ ]

(2.38)

Applying the frequency translation, the 𝑑𝑞-frame complex transfer matrix (2.38) is transformed into

the 𝛼𝛽-frame as:

[ 𝐈→

𝑒𝑗2𝜃 𝐈∗→] = [

𝐘𝐭𝐜𝐥+,𝐝𝐪→

(𝑠 − 𝑗𝜔1) 𝐘𝐭𝐜𝐥−,𝐝𝐪→

(𝑠 − 𝑗𝜔1)

𝐘𝐭𝐜𝐥−,𝐝𝐪∗→ (𝑠 − 𝑗𝜔1) 𝐘𝐭𝐜𝐥+,𝐝𝐪

∗→ (𝑠 − 𝑗𝜔1)]

⏟

𝐘𝐭𝐜𝐥±𝐦→ (𝑠)

[ 𝐄→

𝑒𝑗2𝜃 𝐄∗→ ]

(2.39)

where 𝐘𝐭𝐜𝐥±𝐦→ (𝑠) denotes the 𝛼𝛽-frame VSC input admittance including the ACC and the PLL effect for

the VSC with a synchronous-frame PI controller.

In sum, the section 2.2 and the section 2.3 of this chapter provide the step-by-step derivation of the

input admittance of a grid-connected two-level VSC with the synchronous-frame PI controller

considering the PLL impedance shaping effect in the 𝑑𝑞-frame and the 𝛼𝛽-frame. The stationary-frame

PR controller, however, is another popular ACC. Utilizing the input admittance derivation procedure

presented in the section 2.2 and the section 2.3 for the synchronous-frame PI controller, the input

admittance of a grid-connected two-level VSC with the stationary-frame PR controller can be derived

in the following section.

2.4 VSC WITH PROPORTIONAL RESONANT ALTERNATING CURRENT CONTROLLER

Considering the PR-ACC of a grid-connected two-level VSC, Figure 2-6 illustrates the ACC transfer

function with the grid voltage synchronization using the PLL. Again, the SRF-PLL is considered for the

grid voltage synchronization. Since the PLL dynamics is defined in the 𝑑𝑞-frame, for the input

admittance derivation considering PLL dynamics, it is necessary to make frequency translation of the

stationary-frame PR controller to its equivalent 𝑑𝑞-frame PI controller.

PROJECT REPORT

17

In Figure 2-6 where 𝐆𝐢𝐜→ (𝑠) denotes the PR controller in the form of complex transfer function:

𝐆𝐢𝐜→ (𝑠) = 𝐾𝑝 +

𝐾𝑖𝑠 − 𝑗𝜔1

(2.40)

𝐆𝐝,𝐝𝐪→

(𝑠)is the 𝑑𝑞-frame equivalent of one and half sample time delay 𝐺𝑑(𝑠):

𝐆𝐝,𝐝𝐪→

(𝑠) = 𝐺𝑑(𝑠 + 𝑗𝜔1) = 𝑒−1.5𝑇𝑠(𝑠+𝑗𝜔1) (2.41)

In (2.40), 𝜔1 is the fundamental frequency angular velocity in rad/s at which the PR controller is

tuned. The 𝑑𝑞-frame equivalent of PR controller can be expressed via frequency translation:

𝐺𝑖𝑐,𝑑𝑞(𝑠) = 𝐆𝐢𝐜→ (𝑠 + 𝑗𝜔1) = 𝐾𝑝 +

𝐾𝑖(𝑠 + 𝑗𝜔1) − 𝑗𝜔1

= 𝐾𝑝 +𝐾𝑖𝑠

(2.42)

The SRF-PLL impedance shaping effect can be expressed by the small-signal linearization of the

complex Parke transformation modulator. Applying the small-signal approximation for a given VSC

current output setpoint 𝐈𝟏,𝐝𝐪∗𝐜→ gives:

Δ 𝐈𝐏𝐋𝐋,𝐝𝐪→

=𝐈𝟏,𝐝𝐪∗𝐜→ 𝑗𝐻𝑝𝑙𝑙(𝑠) (Δ𝐄𝐝𝐪

→ − Δ𝐄𝐝𝐪

∗→ )

2𝑗

=𝐈𝟏,𝐝𝐪∗𝐜→ 𝐻𝑝𝑙𝑙(𝑠)

2⏟

𝐘𝐏𝐋𝐋→

(𝑠)

(Δ𝐄𝐝𝐪→

− Δ𝐄𝐝𝐪∗→ )

(2.43)

PWM

+

+ +

- -

+

-

+

-

-

+

-

Figure 2-6 Grid-connected VSC interface closed-loop transfer function diagram with the 𝛼𝛽-frame PR controller

PROJECT REPORT

18

where 𝐘𝐏𝐋𝐋→

(𝑠) is the complex transfer function of the current reference set-point change because of

the small-signal perturbation 𝐄𝐝𝐪→

. Similarly to Figure 2-5, the input admittance of the VSC with a

stationary-frame PR controller is shown in Figure 2-7. The modified closed-loop control output

admittance can be shown as:

𝐈𝐝𝐪→ |𝐈𝐝𝐪∗𝐜→ =0

= −𝐆𝐭𝐜𝐥,𝐝𝐪→

(𝑠) 𝐘𝐏𝐋𝐋→

(𝑠) +𝐘𝐢𝐩,𝐝𝐪→

(𝑠)

1 + 𝐓𝐨,𝐝𝐪→

(𝑠)⏟

𝐘𝐭𝐜𝐥+,𝐝𝐪→

(𝑠)

𝐄𝐝𝐪→

+𝐆𝐭𝐜𝐥,𝐝𝐪→

(𝑠) 𝐘𝐏𝐋𝐋→

(𝑠)⏟

𝐘𝐭𝐜𝐥−,𝐝𝐪→

(𝑠)

𝐄𝐝𝐪∗→

(2.44)

Where𝐓𝐨,𝐝𝐪→

(𝑠) and 𝐆𝐭𝐜𝐥,𝐝𝐪→

(𝑠) are the open loop gain and closed-loop complex transfer functions of

the stationary-frame PR controller in the 𝑑𝑞-frame:

𝐓𝐨,𝐝𝐪→

(𝑠) = 𝐺𝑖𝑐,𝑑𝑞(𝑠) 𝐆𝐝,𝐝𝐪→

(𝑠) 𝐘𝐨𝐩,𝐝𝐪→

(𝑠) (2.45)

𝐆𝐭𝐜𝐥,𝐝𝐪→

(𝑠) =𝐓𝐨,𝐝𝐪→

(𝑠)

1 + 𝐓𝐨,𝐝𝐪→

(𝑠) (2.46)

As opposed to the input admittance derived for the synchronous-frame PI controller, the output

admittance of that of stationary-frame PR controller is not reshaped by the PLL dynamics since the

inverse Parke transformation is not present at the output voltage 𝐕𝐨,𝐝𝐪 (𝑠). In this regard, the impact of

the PLL in the reshaping of the input admittance with the stationary-frame PR controller is less in

comparison to that of the synchronous-frame PI controller.

Figure 2-7 Grid-connected VSC interface input admittance complex transfer function with the 𝛼𝛽-frame PR controller

PROJECT REPORT

19

Eventually, the input-admittance matrix can be written in the complex space:

[𝐈𝐝𝐪→

𝐈𝐝𝐪∗→ ] = [

𝐘𝐭𝐜𝐥+,𝐝𝐪→

(𝑠) 𝐘𝐭𝐜𝐥−,𝐝𝐪→

(𝑠)

𝐘𝐭𝐜𝐥−,𝐝𝐪∗→ (𝑠) 𝐘𝐭𝐜𝐥+,𝐝𝐪

∗→ (𝑠)]

⏟

𝐘𝐭𝐜𝐥±,𝐝𝐪𝐦 (𝑠)

[𝐄𝐝𝐪→

𝐄𝐝𝐪∗→ ]

(2.47)

Applying the frequency translation, (2.47) is transformed to the 𝛼𝛽-frame as:

[ 𝐈→

𝑒𝑗2𝜃 𝐈∗→] = [

𝐘𝐭𝐜𝐥+,𝐝𝐪→

(𝑠 − 𝑗𝜔1) 𝐘𝐭𝐜𝐥−,𝐝𝐪→

(𝑠 − 𝑗𝜔1)

𝐘𝐭𝐜𝐥−,𝐝𝐪∗→ (𝑠 − 𝑗𝜔1) 𝐘𝐭𝐜𝐥+,𝐝𝐪

∗→ (𝑠 − 𝑗𝜔1)]

⏟

𝐘𝐭𝐜𝐥±𝐦→ (𝑠)

[ 𝐄→

𝑒𝑗2𝜃 𝐄∗→ ]

(2.48)

where 𝐘𝐭𝐜𝐥±𝐦 (𝑠) denotes the 𝛼𝛽-frame input admittance including the AC current control and PLL

effect for a VSC with a 𝛼𝛽-frame PR controller.

2.5 IMPEDANCE-BASED STABILITY ANALYSIS

The generalized Nyquist stability criterion can be applied for the stability analysis concerning an

asymmetric matrix:

𝐋𝐦→ (𝑠) = 𝐙𝐠

𝐦→ (𝑠) 𝐘𝐭𝐜𝐥±𝐦→ (𝑠) (2.49)

where 𝐘𝐭𝐜𝐥±𝐦→ (𝑠) is the input admittance matrix as described by (2.48), and 𝐙𝐠

𝐦→ (𝑠) is the grid impedance

matrix. The system stability can therefore be predicted by the frequency responses of the eigenvalues

of the impedance ratio, which can be calculated as:

det [𝜆𝐼𝑚 − 𝐙𝐠𝐦→ (𝑠) 𝐘𝐭𝐜𝐥±

𝐦→ (𝑠)] = 0 (2.50)

The equivalent grid impedance can be represented by the LC parallel circuit as an example:

𝑍𝑔 =𝐿2𝑠

1 + 𝐿2𝐶𝑓𝑠2 (2.51)

Considering a symmetrical grid impedance, the complex transfer impedance matrix 𝐙𝐠𝐦→

(𝑠) in the 𝛼𝛽-

frame is defined as:

𝐙𝐠𝐦→ (𝑠) = [

𝑍𝑔(𝑠) 0

0 𝑍𝑔(𝑠 − 2𝑗𝜔1)] (2.52)

PROJECT REPORT

20

2.6 EXPERIMENTAL VERIFICATIONS

To verify the impedance-based stability analysis method presented in above, this section gives the

experimental verification results. A single VSC setup is developed to demonstrate both high and low

frequency oscillations; the measured results are compared to the theoretical prediction. The

measurement results in this section is obtained by oscilloscope Tektronix MDO4104B-3 Firmware

version 3.14.

2.6.1 HIGH FREQUENCY OSCILLATIONS

The experimental setup is shown in Figure 2-8, where the power grid is simulated by the regenerative

grid simulator Chroma 61845. The VSC is implemented by a Danfoss FC103P11KT 11 inverter and

the control algorithms are programmed in dSPACE1007. The main circuit and controller parameters

of the VSC are summarized in Table 2-1 . A constant DC source is connected to the 𝑉𝑑𝑐 in Figure 2-8.

Table 2-1 Parameters of Single VSC Experiment Verification

Main electrical parameters Value Unit

Sn Rated Power 2.00 kVA

f1 Grid fundamental frequency 50 Hz

Vdc DC Link Voltage 730 V

Cdc DC Link Capacitor5 1.5 mF

Vrms Line to line AC Voltage 400 V

L1 Inverter side inductor 1.5 mH

R1 Equivalent series resistor of L1 0.2 Ohm

L2 Grid side inductor 1.5 mH

R2 Equivalent series resistor of L2 0.2 Ohm

C Filter capacitor 5 µF

fs Inverter control sampling frequency 10 kHz

5 DC link capacitance is inside the Danfoss FC103P11KT 11 VSC

Figure 2-8 Experiment setup for the impedance-based stability analysis

PROJECT REPORT

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ACC and PLL controller parameters

Kp proportional gain of AC current control 9.4 Ohm

Ki integral gain of AC current control 2961 Ohm/s

Kppll proportional gain of PLL 1 rad/s

Kipll integral gain of PLL 1000 rad/s2

Kd proportional gain of capacitor voltage feedback 1

Since typically the PLL has only a closed-loop bandwidth up to a few hundred Hz, for the stability

analysis of high frequency oscillations (i.e. a few hundred Hz and beyond), the MIMO input admittance

matrix can be considered as SISO owning to the fact that the PLL closed loop transfer function

bandwidth determines the significance of the non-diagonal elements in the MIMO input admittance

matrix.

Furthermore, due to the closed loop bandwidth, it is practical to consider only the ACC for the study of

high frequency phenomena.

The experimental setup is implemented with a stationary-frame PR controller for the ACC with the

capacitive voltage feedback loop to perform the active damping (AD) function. In Figure 2-9, the VSC

main circuit is illustrated with its ACC and PLL loops.

𝐆𝐢𝐜→ (𝑠) is the transfer function of the stationary-frame PR controller:

𝐆𝐢𝐜→ (𝑠) = 𝐾𝑝 +

𝐾𝑖𝑠 − 𝑗𝜔1

(2.53)

PWM

+- +

+ +

- -

+

-

+

-

-

Figure 2-9 Single VSC setup with a PR controller for the ACC and capacitive voltage feedback for the AD function

PROJECT REPORT

22

where 𝐾𝑝 and 𝐾𝑖 are the proportional and integral gain respectively (see Table 2-1). 𝜔1 is the

fundamental frequency angular velocity in rad/sec. 𝑒−𝑠𝑇𝑠 represents one sampling cycle digital delay.

𝐺𝑣𝑐(𝑠) is the capacitive voltage feedback transfer function implemented as a high-pass filter instead of

a pure differentiator to avoid the amplification of high frequency noise:

𝐺𝑣𝑐(𝑠) = −𝐾𝑑𝑠

𝑠 + 0.4𝜔𝑠 (2.54)

where 𝐾𝑑 is the gain constant for the high pass filter, 𝜔s is the angular velocity of sampling frequency

𝑓𝑠 in rad/s. Ignoring the impact of the PLL, the input admittance 𝑌𝑡𝑜(𝑠) of a single VSC setup with an

ACC and active damping can be derived following the illustration shown in Figure 2-10:

Figure 2-10 Input admittance illustration for the single VSC setup with the PR controller for the ACC and the capacitive voltage feedback for the AD function

𝑌𝑡𝑜(𝑠) =𝑌𝑖𝑝(𝑠)(1 − 𝐺𝑣𝑐(𝑠)𝐺𝑑(𝑠))

1 + 𝐺𝑖𝑐(𝑠)𝐺𝑑(𝑠)𝑌𝑜𝑝(𝑠)

(2.55)

The small-signal stability of the overall system can be evaluated by plotting the ratio of the VSC input

admittance to the equivalent grid admittance following the Nyquist stability criteria. Equivalently, the

small-signal stability of the overall system can be assessed by plotting the VSC input admittance )

+

-

+

-

+-+- +-

AD L filter

PROJECT REPORT

23

against the equivalent grid admittance in a Bode diagram and examining the phase difference at

intersection point of the admittance magnitudes. At the admittance intersection point, if the phase

difference between the equivalent grid admittance and the VSC input admittance is larger than 180

degree then the overall system will become small-signal unstable. Conversely, at the admittance

intersection point, if the phase difference between the equivalent grid admittance and the VSC input

admittance is within 180 degree, then the overall system maintains small-signal stability. In this section,

we apply the impedance-based stability method and assess the impedance intersection point with the

associated phase difference.

The input admittance 𝑌𝑡𝑜(𝑠) is plotted in a Bode diagram applying the controller parameters from Table

2-1. This plot is shown in Figure 2-11, with 𝐾𝑑 ∈ (0, 0.32, 1) indicating no active damping, critical active

damping, and full active damping, respectively; and the equivalent grid admittance 𝑌𝑔(𝑠) is described

as an LC parallel circuit:

𝑌𝑔(𝑠) =1

𝐿2𝑠 + 𝑅2+

𝐶𝑓𝑠

1 + 𝐶𝑓𝑅𝑓𝑠 (2.56)

Figure 2-11 Impedance based stability analysis w/wo active damping in Bode diagram - grid impedance 𝑌𝑔 in solid blue line,

VSC input admittance 𝑌𝑐𝑙 with 𝐾𝑑 = 0 in dash dotted organge line, VSC input admittance 𝑌𝑐𝑙 with 𝐾𝑑 = 0.32 in dotted yellow

line, VSC input admittance 𝑌𝑐𝑙 with 𝐾𝑑 = 1 in dashed purple line

In Figure 2-11, the fully damped plot intersects with the grid admittance near 3 kHz yet the phase

margin (PM) is above zero (i.e. less than 180-degree phase difference) indicating a stable system.

PROJECT REPORT

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Additionally, the critically damped plot and undamped plot both intersect with the grid admittance

around 2.72 kHz and the PM, for both cases, are below zero (i.e. more than 180-degree phase

difference) indicating unstable cases.

Figure 2-12 Experimental results of AC current control loop with and without active damping - (left) with active damping being switched off (right) normal operation with active damping

To verify the stability analysis using the impedance-based method, experimental verification is

performed. Figure 2-12 (left) demonstrates the three-phase VSC-side current output when the VSC is

operating with the AD active. Figure 2-12 (right) shows the case with the AD being switched off at 0.4

seconds. It is clearly visible that the current output grew out of control and went exponentially high,

which eventually tripped the device due to the internal over-current protection.

In Figure 2-13, the experiment is performed with 𝐾𝑑 = 0.32 showing high frequency oscillations with a

critical damping oscillation. Revisiting the stability analysis using the impedance-based method (Figure

2-11), the critically damped result falls just below -90 degrees indicating a phase difference between

𝑌𝑡𝑜(𝑠) and 𝑌𝑔(𝑠) of more than 180 degree. Due to the skin-effect, the actual resistive damping is more

than what is predicted in the analytical analysis. Figure 2-13 (upper right) gives the FFT analysis results

of the current with critical damping. There are visible harmonic components at both 2.66 kHz and 2.76

kHz. The FFT analysis results thus found a good match with the impedance intersection point (i.e. 2.72

kHz). Sampling time is 40 𝜇𝑠.

PROJECT REPORT

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Figure 2-13 Experimental results of AC current control loop with critical damping - (upper left) current output with 𝐾𝑑 = 0.32 (upper right) FFT analysis of current output with 𝐾𝑑 = 0.32 (lower left) 𝐾𝑑 changed from 1 → 0.32 (lower right) 𝐾𝑑 changed from 0.32 → 1

The lower left plot in Figure 2-13 presents the experimental results when 𝐾𝑑 is changed from 1 to 0.32

showing high frequency oscillation; whereas the lower right plot shows a regaining of stable operation

by modifying 𝐾𝑑 from 0.32 to 1. The experimental results confirm that the inverter-side current control

stable region can be extended above 𝑓𝑠/6 provided that the AD function is implemented. By careful

selection of the capacitor voltage feedback, high pass filter gain, 𝐾𝑑, the overall system stability can

be maintained without high frequency oscillations.

In summary, high frequency oscillations are investigated using the impedance-based stability analysis

considering only the inner ACC loop since the impedance shaping effect of the PLL dynamics is

negligible at frequencies well above 100 Hz. For this reason, the input admittance of the VSC is

considered as a SISO transfer function and the overall system stability is assessed by plotting the VSC

input admittance against the grid admittance in a Bode diagram. The experimental current waveforms

and their corresponding FFT analysis results affirm the accuracy and correctness of the analytical

prediction made in the Bode diagram following the impedance-based stability method.

PROJECT REPORT

26

2.6.2 LOW FREQUENCY OSCILLATIONS

For the stability analysis of low frequency oscillations (e.g. near 100 Hz), the generalized Nyquist

stability criterion should be applied to the 2 × 2 eigen value admittance matrix calculated from the

MIMO input admittance matrix ratio defined by (2.49). Note that the weak grid condition is realized by

adding a large inductor 𝐿𝑔 = 11 𝑚𝐻 behind the Chroma grid source shown in Figure 2-14. The main

circuit and controller parameters of the VSC used in the study case is summarized in Table 2-2.

Table 2-2 Parameters of Single VSC Experiment Verification

Main electrical parameters Value Unit

Sn Rated Power 3.05 kVA

f1 Grid fundamental frequency 50 Hz

Vdc DC Link Voltage 600 Volts

Cdc DC Link Capacitor 1500 µF

Vrms Line to line AC Voltage 134 V

L1 Inverter side inductor 1.5 mH

L2 Grid side inductor 1.5 mH

Lg Grid equivalent inductor 11.0 mH

Lt Equivalent grid-side inductor (L2 + Lg ) 12.5 mH

Cf Filter capacitor

fs Inverter control sampling frequency

fsw Inverter switching frequency

20

10

10

µF

kHz

kHz

ACC and PLL controller parameters

Kp proportional gain of AC current control 7.8 Ohm

Ki integral gain of AC current control 2056 Ohm/s

Kppll proportional gain of PLL (Tsettle = 20 ms) 3.43 rad/s

Kipll integral gain of PLL (Tsettl e = 20 ms) 789.5 rad/s2

Kppll proportional gain of PLL (Tsettle = 40 ms) 1.72 rad/s

Kipll integral gain of PLL (Tsettl e = 40 ms) 197.4 rad/s2

Kd proportional gain of capacitor voltage feedback 1 S

Figure 2-14 Experiment setup for the impedance-based stability analysis

PROJECT REPORT

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The VSC is operated with 𝐼𝑑0 = +11.2 A (delivering active power to the grid). Applying the generalized

Nyquist stability criteria, two eigenvalues (i.e. 𝜆1 and 𝜆2) of the overall system impedance ratio are

calculated and plotted in a Bode diagram, as seen in Figure 2-15. The phase angle of 𝜆2 is within ±180

degrees indicating that the small-signal stability of the system is maintained. The phase angle of 𝜆1

crossed -180 degrees at 86.2 Hz and the magnitude of 𝜆1 is just slightly below 0 dB implying an under-

damped system condition.

To verify the analytical results, an empirical analysis is performed with a VSC current output of 𝐼𝑑0 =

+11.2 A. Figure 2-16 provides the inverter side current output waveform and the corresponding FFT

results. The left-hand-side plot indicates an unstable current output when the VSC current output 𝐼𝑑0 =

+11.2 A. Additionally, FFT analysis is performed on the output current waveforms with 1 Hz resolution

and shown in the right-hand-side plot, where the dominant low order harmonics components at 86.5

Hz and 13.5 Hz are reported. Compared to the frequency domain analytical prediction (i.e. 86.2 Hz

and 13.7 Hz)6, a good match is found with the experimental verification results. Sampling time is 1 𝑚𝑠.

6 Note the frequency component 13.7 Hz is negative sequence component when the three-phase system is evaluated here.

Figure 2-15 Generalized Nyquist Stability Analysis of the overall system in Bode diagram (𝐼𝑑0 = +11.2A)

PROJECT REPORT

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Figure 2-16 Experimental results with 𝐼𝑑0 = +11.2 A considering AC current control loop and PLL control loop – (left) three phase current waveforms under harmonic instability. (right) FFT results of the three phase current waveforms under harmonic instability.

2.7 SUMMARY

This chapter elaborates on the principle of VSC input admittance derivation considering the inner ACC

including the PLL impedance shaping effect. First, the grid-connected VSC interface with a

synchronous-frame PI controller and the SRF-PLL is considered for the derivation of the input

admittance. Due to the use of only the q-axis for the PLL input, the PLL impedance shaping effect

presents asymmetry considering the PLL dynamics on the VSC output current and voltage. Therefore,

considering the ACC and the PLL dynamics on a given operating point, the input admittance of the

VSC can be derived as a real space transfer function. With the help of symmetrical components in the

𝑑𝑞-frame, the input admittance can be expressed by an equivalent symmetrical complex space

transfer function. In the end, a frequency translation is performed on the input complex space

admittance transfer function to obtain the input admittance in the 𝛼𝛽-frame. Using the same approach,

the input admittance of the grid-connected VSC interface with the 𝛼𝛽-frame PR controller is derived in

a complex space transfer function.

For the impedance-based stability analysis concerning multiple-devices connected in the vicinity of

each other, the 𝛼𝛽-frame input admittance derived in this chapter allows direct coupling to the network

impedance for the stability analysis using the generalized Nyquist stability criterion. The experimental

results demonstrate agreement with analytical prediction, and hence justifies the use of the modelling

approach presented here for the further experimental verification work in T16.6.

PROJECT REPORT

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3 MMC VSC

The frequency dependent impedance model of a modular multilevel converter (MMC) is needed to

assess the harmonic stability of a system including MMC by means of the impedance-based stability

criterion. Impedance models can be obtained either by measuring the frequency response of the MMC

for different operating points or by deriving analytically the frequency response if the electrical structure

as well as the control system detail is known. The objective of the chapter is amongst others to show

and derive an impedance model representing the frequency behaviour of an MMC on the AC side.

The model will be compared to the frequency response obtained from the MMC Test Bench laboratory

at RWTH Aachen in Task 16.6. For this, the impedance model has to include the main controllers as

used in the laboratory setup. Due to the presence of harmonics in arm currents and capacitor voltages

of the MMC the impedance model should include the effect of these harmonics on the frequency

response. This is why as modelling bases, the multi-harmonic linearization technique developed for

MMC is applied [22], [23]. The matrix formulation allows to include the impact of in theory infinite

number of harmonics on the frequency response [22], [23]. The model basis will be further adapted

according to the electrical system and control system of the MMC Test Bench. Different control

strategies can be included by defining coefficient matrices representing the influence of the control on

the phase current or perturbation voltage.

The modelling begins with the modelling of the electrical part of the MMC in steady-state and time

domain. The phase relationships between the six arms of the MMC are analysed, resulting in the

representing of the MMC by only one arm. Then, the time-domain averaged model is converted to

obtain a frequency-domain steady-state model of the electrical part of the MMC. After that, by

introducing a perturbation, the small-signal model in frequency domain can be derived from the steady-

state model. Later, the small-signal models of control loops of the MMC such as current control and

phase-locked-loop are built. Combining the model of the electrical part of the MMC and the model of

controls, the model of MMC for impedance analysis in frequency-domain is given at last.

The model is developed based on the following assumptions:

1) The three phase AC power grid which the MMC is connected to can be assumed as

symmetrical.

2) Both the structure and parameters of all six arms of MMC are identical.

3) The control signals of the six arms satisfy the following symmetry with 𝑓1 being the fundamental

frequency (i.e. 50 Hz for Europe)

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Table 3-1: Symmetries among control signals of an MMC

PHASE A PHASE B PHASE C

Upper

Arm

𝑚(𝑡) 𝑚(𝑡 −

1

3𝑓1) 𝑚(𝑡 −

2

3𝑓1)

Lower

Arm 𝑚(𝑡 −

1

2𝑓1) 𝑚(𝑡 +

1

6𝑓1) 𝑚(𝑡 −

1

6𝑓1)

4) Since the six arms are identical, the symmetry summarized in Table 3-1 also holds true among

the arm currents and capacitor voltages.

5) DC bus voltage is assumed to be constant.

6) AC terminal voltage is assumed to be harmonic free.

3.1 INTRODUCTION TO MODULAR MULTILEVEL CONVERTER

Figure 3-1 shows a diagram of the electrical part without the control loops, of a three-phase MMC

circuit.

Figure 3-1: MMC circuit diagram

PROJECT REPORT

31

An MMC has three phases and six arms. Every phase of the converter consists of an upper arm and

a lower arm, and every arm comprises a string of switching submodules (SM) and an arm inductor L.

The arm inductor L is used to limit fault currents. Each submodule consists of a bridge and a capacitor

to act as a controllable voltage source. The bridge of submodules employs two switches, 𝑆1 and 𝑆2,

each of them consists of a semiconductor device (usually an IGBT) with an anti-parallel connected

diode. In normal operation, only one of the two switches is switched on at a given instant of time.

The structure of a submodule is shown in Figure 3-2. It has two normal operation statuses, inserted

and bypassed. The blocking mode, in which all switches in a submodule are turned off, is not

considered as a normal operation status. Thus, it is not included in the model. As shown in Table 3-2,

when switch 𝑆2 is turned on and 𝑆1 is turned off, the capacitor of the submodule is inserted in the arm

circuit and this status is called inserted as shown in Figure 3-3 (a). In this status, the voltage crossing

the submodule equals the voltage of the capacitor of the submodule. On the contrary, as shown in

Figure 3-3 (b), when switch 𝑆1 is turned on and 𝑆2 is turned off, the submodule is in bypassed status

and the voltage drop over it is zero.

Figure 3-2: MMC submodule diagram

Figure 3-3: MMC submodule operation status

PROJECT REPORT

32

Table 3-2: Operation status of MMC submodule

STATUS 𝐒𝟏 𝐒𝟐 𝐕𝑨𝑩

Inserted off on 𝑣𝑐

Bypassed on off 0

3.2 AVERAGE VALUE MODEL

The modelling of the MMC starts from developing the steady-state impedance model of the electrical

part of the MMC. The control loops are neglected at first. The variables used for modelling have double

subscripts. The first subscript indicates the phase (𝑎,𝑏 and 𝑐) and the second subscript indicates the

arm (𝑢 for upper arm and 𝑙 for lower arm).

Since the six arms have the same submodule and arm inductor and the number of submodules is the

same for all arms, the six arms of an MMC are identical. Due to this the modelling is focused on the

upper arm of phase 𝑎. Assuming the number of submodules of an arm is 𝑁, the submodules of the

phase 𝑎 upper arm can be numbered from 1 to 𝑁. The submodules can be divided into two sets 𝑆0

and 𝑆1: The index 𝑘 of the 𝑘th submodule belongs to 𝑆1 if the submodule is inserted and to 𝑆0 if the

submodule is bypassed. The capacitance of each submodule is defined as 𝐶𝑠𝑚, the inductance and

resistance of the arm inductor are defined as L and 𝑟𝐿. The capacitor voltage of the kth submodule of

phase a of the upper arm is defined. as 𝑣𝑎𝑢(𝑘)

.

For phase 𝑎 upper arm, the voltage difference between the DC positive voltage and AC terminal

voltage equals the voltage drop over the arm inductor and capacitor of all inserted submodules. For

the submodule capacitors, the multiplication of the capacitance and the derivative of the capacitor

voltage equals the current flow through it, which is the arm current for the inserted capacitors, and is

zero for bypassed ones. Therefore, the dynamics of phase 𝑎 upper arm can be modeled as follows

[22]:

𝐿𝑑𝑖𝑎𝑢𝑑𝑡

+ 𝑟𝐿𝑖𝑎𝑢 = 𝑣𝑝 − 𝑣𝑎 − ∑ 𝑣𝑎𝑢(𝑘)

𝑘∈𝑆1

(3.1)

𝐶𝑠𝑚𝑑𝑣𝑎𝑢

(𝑘)

𝑑𝑡= 𝑖𝑎𝑢 𝑓𝑜𝑟 𝑘 ∈ 𝑆10 𝑓𝑜𝑟 𝑘 ∈ 𝑆0

(3.2)

If all submodule voltages are assumed to be balanced,

𝑣𝑎𝑢(1)= 𝑣𝑎𝑢

(2)= ⋯ = 𝑣𝑎𝑢

(𝑁)=𝑣𝑎𝑢

𝑁. (3.3)

The voltage 𝑣𝑎𝑢 is the sum of all submodule capacitor voltages inserted in the arm. To simplify the

equations above, the insertion index 𝑚𝑥𝑦 (𝑥 = 𝑎, 𝑏, 𝑐; 𝑦 = 𝑢, 𝑙) is introduced to the model. The insertion

index is a time varying signal which indicates the inserted portion of all submodules in an arm at any

instant. Supposed 𝑚𝑎𝑢 is the insertion index of phase 𝑎 of the upper arm, then 𝑚𝑎𝑢 ⋅ 𝑁 submodules

are inserted and 𝑆1 has that many elements. Consequently (1 − 𝑚𝑎𝑢) ⋅ 𝑁 submodules are bypassed,

PROJECT REPORT

33

defining the number of submodules of 𝑆0. The insertion indeces 𝑚𝑥𝑦 are generated by the control loops

of the MMC and should be rounded to the next integer of 𝑚𝑥𝑦𝑁. The number of submodules 𝑁 is

typically up to several hundreds, which is why the insertion index can be assumed to be a continuous

variable without losing accuracy. The sum of the voltages of all inserted capacitors in equation (3.1)

∑ 𝑣𝑎𝑢(𝑘)

𝑘∈𝑆1 can be substituted by the product of 𝑚𝑎𝑢𝑣𝑎𝑢, resulting into the following equation:

𝐿𝑑𝑖𝑎𝑢𝑑𝑡

+ 𝑟𝐿𝑖𝑎𝑢 = 𝑣𝑝 − 𝑣𝑎 −𝑚𝑎𝑢𝑣𝑎𝑢 (3.4)

For every submodule, the right side of equation (3.2) is 𝑖𝑎𝑢 or 0 depending on the operating status of

the submodule. Due to the fact that the number of inserted submodules of phase 𝑎 upper arm is 𝑚𝑎𝑢 ⋅

𝑁, equation (3.2) for all submodules 𝑘 = 1…𝑁 can be written as

𝐶𝑠𝑚𝑑(∑ 𝑣𝑎𝑢

(𝑘)𝑁𝑘=1 )

𝑑𝑡= 𝑚𝑎𝑢𝑁𝑖𝑎𝑢. (3.5)

Dividing both sides of (3.4) by 𝑁 gives

𝐶𝑠𝑚𝑁

𝑑(∑ 𝑣𝑎𝑢(𝑘)𝑁

𝑘=1 )

𝑑𝑡= 𝑚𝑎𝑢𝑖𝑎𝑢 (3.6)

It should be noted that the sum of all capacitor voltage in phase 𝑎 upper arm equals to

∑𝑣𝑎𝑢(𝑘)

𝑁

𝑘=1

= 𝑣𝑎𝑢 (3.7)

The equivalent capacitance of the 𝑁 submodule capacitors of capacitance 𝐶𝑠𝑚 in series equals to

𝐶 = 𝐶𝑠𝑚 𝑁⁄ (3.8)

Based on this equation (3.6) can be modified to

𝐶𝑑𝑣𝑎𝑢𝑑𝑡

= 𝑚𝑎𝑢𝑖𝑎𝑢 (3.9)

The model given by equation (3.4) and gives the equivalent circuit of phase 𝑎 upper arm shown in

Figure 3-4.

PROJECT REPORT

34

Figure 3-4: Equivalent circuit of phase a upper arm of MMC

It has to be noted that 𝑉+ and 𝑉− in Figure 3-1 are the voltages from the positive and negative terminal

of the DC bus to the AC neutral respectively. As next step, the variable 𝑉𝑚, is introduced as a reference

voltage for 𝑉+ and 𝑉−. The voltage 𝑉𝑚 represents the voltage from the floating middle point of the DC

bus to the neutral of the AC bus as shown in Figure 3-1. Thus, 𝑉+ and 𝑉− can be represented as follows:

𝑉+ = 𝑉𝑚 +𝑉𝑑𝑐

2⁄ (3.10)

𝑉− = 𝑉𝑚 −𝑉𝑑𝑐

2⁄ (3.11)

The voltage 𝑉𝑑𝑐 is the DC bus voltage and equals to:

𝑉𝑑𝑐 = 𝑉+ − 𝑉− (3.12)

As next step, an expression for 𝑣𝑚 is derived from the averaged model. For this, the voltage and

current relationships of the upper and lower arm are used. Similar to the derivation from equation

(3.1) to (3.4), the equations for phase 𝑎 lower arm can be developed as

𝐿𝑑𝑖𝑎𝑙𝑑𝑡

+ 𝑟𝐿𝑖𝑎𝑙 = 𝑣𝑎 − 𝑉− −𝑚𝑎𝑙𝑣𝑎𝑙 (3.13)

Subtracting equation (3.13) from equation (3.4) yields the voltage and current relationship of both

arms for phase 𝑎.

𝐿𝑑(𝑖𝑎𝑢 − 𝑖𝑎𝑙)

𝑑𝑡+ 𝑟𝐿(𝑖𝑎𝑢 − 𝑖𝑎𝑙) = (𝑉+ + 𝑉−) − 2𝑣𝑎 −𝑚𝑎𝑢𝑣𝑎𝑢 +𝑚𝑎𝑙𝑣𝑎𝑙 (3.14)

According to the defined reference direction of the current shown in Figure 3-1, the relationship

between the arm currents and phase current can be derived as follows:

PROJECT REPORT

35

𝑖𝑎𝑢 − 𝑖𝑎𝑙 = 𝑖𝑎 (3.15)

Adding up the expressions of 𝑣𝑝 (3.10) and 𝑣𝑛 (3.11) gives:

𝑉+ + 𝑉− = 2𝑉𝑚 (3.16)

By substituting (3.15) and (3.16) into (3.14), the equation can be modified as

𝐿𝑑𝑖𝑎𝑑𝑡+ 𝑟𝐿𝑖𝑎 = 2𝑉𝑚 − 2𝑣𝑎 −𝑚𝑎𝑢𝑣𝑎𝑢 +𝑚𝑎𝑙𝑣𝑎𝑙 (3.17)

For phase 𝑎 and phase 𝑏, the relationship can be developed in a similar way:

𝐿𝑑𝑖𝑏𝑑𝑡+ 𝑟𝐿𝑖𝑏 = 2𝑉𝑚 − 2𝑣𝑏 −𝑚𝑏𝑢𝑣𝑏𝑢 +𝑚𝑏𝑙𝑣𝑏𝑙 (3.18)

𝐿𝑑𝑖𝑐𝑑𝑡+ 𝑟𝐿𝑖𝑐 = 2𝑉𝑚 − 2𝑣𝑐 −𝑚𝑐𝑢𝑣𝑐𝑢 +𝑚𝑐𝑙𝑣𝑐𝑙 (3.19)

Adding up equations (3.17), (3.18) and (3.19), gives

𝐿𝑑(𝑖𝑎 + 𝑖𝑏 + 𝑖𝑐)

𝑑𝑡+ 𝑟𝐿(𝑖𝑎 + 𝑖𝑏 + 𝑖𝑐) =

6𝑉𝑚 − 2(𝑣𝑎 + 𝑣𝑏 + 𝑣𝑐) − ∑ (𝑚𝑥𝑢𝑣𝑥𝑢 −𝑚𝑥𝑙𝑣𝑥𝑙)

𝑥=𝑎,𝑏,𝑐

. (3.20)

It should be noted that according to the phase shift between voltages of three phases (2𝜋/3), the sum

of voltages of three phases is zero (𝑣𝑎 + 𝑣𝑏 + 𝑣𝑐 = 0). This relationship also holds true for the three-

phase currents. Thus, the expression of 𝑣𝑚 is obtained as follows:

𝑉𝑚 =1

6∑ (𝑚𝑥𝑢𝑣𝑥𝑢 −𝑚𝑥𝑙𝑣𝑥𝑙)

𝑥=𝑎,𝑏,𝑐

(3.21)

Substituting equation (3.10) into (3.4), the model given in equation (3.4) and (3.9) can be modified to

𝐿𝑑𝑖𝑎𝑢𝑑𝑡

+ 𝑟𝐿𝑖𝑎𝑢 =𝑉𝑑𝑐2− 𝑣𝑎 −𝑚𝑎𝑢𝑣𝑎𝑢 + 𝑉𝑚 (3.22)

𝐶𝑑𝑣𝑎𝑢𝑑𝑡

= 𝑚𝑎𝑢𝑖𝑎𝑢 (3.23)

The relationships between arm current and capacitor voltage of the other five arms can be described

by equations similar to (3.22) and (3.23). The equations of all six arms together form the three-phase

averaged circuit model of an MMC as seen in Figure 3-5.

PROJECT REPORT

36

Figure 3-5: Continuous approximate (averaged) model of a MMC

3.3 FREQUENCY DOMAIN MODEL

To develop the frequency dependent impedance model, it is necessary to convert the time-domain

model into the frequency domain. For this, a vectorised method of representation of variables in

frequency-domain is introduced. By means of this method, the average value model developed in the

previous part can be transformed into the frequency domain.

The right-hand-side of equation (3.23), the multiplication of the fundamental components of the arm

current and insertion index, would produce a second harmonic component in the capacitor voltage.

The multiplication of this second harmonic component of the capacitor voltage and the insertion index

in (3.22) would produce a third harmonic component in arm current. This sequence will continue,

producing an infinite number of harmonics in theory. Therefore, it’s essential to consider harmonics in

the modelling and analysis of MMC. The linearization of the MMC average value model represented

by equation (3.22) and (3.23) has to include the harmonics of variables such as arm current and

capacitor voltage as part of the steady state operation in order to ensure the accuracy of the model.

While only a few low-order harmonics need to be considered in practice, the method in general should

be able to include any number of harmonics.

For this purpose, vectors of 2n + 1 elements are defined representing the variables of the average

value model such as voltage, current and insertion index. By this approach complex Fourier

coefficients up to the 𝑛th harmonic can be represented as seen in equation (3.24) and (3.25). The

PROJECT REPORT

37

elements of the vector correspond to frequencies in the following order: −𝑛𝑓1, ⋯ , −𝑓1, 0, 𝑓1, ⋯ , 𝑛𝑓1,

where 𝑓1 is the fundamental frequency. The vector representation of the arm current 𝑖𝑎𝑢, the capacitor

voltage 𝑣𝑎𝑢 and the insertion index 𝑚𝑎𝑢 of phase 𝑎 upper arm are given in equation (3.24). The vectors

representing the variables for the other phases and arm can be defined in the same way.

𝐢𝒂𝒖 =

[ 𝐼𝑛𝑒

−𝑗𝛼𝑛

⋮𝐼1𝑒

−𝑗𝛼1

𝐼0𝐼1𝑒

𝑗𝛼1

⋮𝐼𝑛𝑒

𝑗𝛼𝑛 ]

, 𝐯𝒂𝒖 =

[ 𝑉𝑛𝑒

−𝑗𝛽𝑛

⋮𝑉1𝑒

−𝑗𝛽1

𝑉0𝑉1𝑒

𝑗𝛽1

⋮𝑉𝑛𝑒

𝑗𝛽𝑛 ]

, 𝐦𝒂𝒖 =

[ 𝑀𝑛𝑒

−𝑗𝛾𝑛

⋮𝑀1𝑒

−𝑗𝛾1

𝑀0𝑀1𝑒

𝑗𝛾1

⋮𝑀𝑛𝑒

𝑗𝛾𝑛 ]

(3.24)

The AC terminal voltage of phase 𝑎, 𝑣𝑎, is be assumed to be harmonic free and without any DC offset.

The DC voltage 𝑉𝑑𝑐 is assumed to be constant and without any harmonics.

𝑽𝒅𝒄 =

[ 0⋮0

𝑉𝑑𝑐2⁄

0⋮0 ]

, 𝒗𝒂 =1

2

[

0⋮

𝑉1𝑒−𝑗𝜑1

0𝑉1𝑒

𝑗𝜑1

⋮0 ]

(3.25)

The voltage 𝑉1 in the equation above is the amplitude of the fundamental component of the grid

voltage, 𝜑1 is the phase of the fundamental component of the grid voltage.

Based on the vectorized notation given in equation (3.24) and (3.25), the average value model given

in (3.22) and (3.23) can be transformed to the frequency domain model:

𝐙𝒍𝟎𝐢𝒂𝒖 = 𝑽𝒅𝒄 − 𝒗𝒂 −𝐦𝒂𝒖⨂𝐯𝒂𝒖 + 𝐯𝒎 (3.26)

𝐘𝒄𝟎𝐯𝒂𝒖 = 𝐦𝒂𝒖⨂𝐢𝒂𝒖 (3.27)

It has to be noted that the multiplication in time-domain equals a convolution in the frequency-domain.

The voltage 𝐯𝒎 is the complex Fourier coefficient vector of 𝑣𝑚 (3.28), 𝐙𝒍𝟎 is a diagonal matrix

representing the impedance of the arm inductor at different frequencies (3.29) and 𝐘𝒄𝟎 is the diagonal

matrix for the admittance of equivalent arm capacitor (3.30).

𝐕𝒎 =1

6∑ (𝐦𝑥𝑢⨂𝐯𝑥𝑢 −𝐦𝑥𝑙⨂𝐯𝑥𝑙)

𝑥=𝑎,𝑏,𝑐

(3.28)

𝐙𝒍𝟎 = 𝑗2𝜋𝐿 ∙ 𝑑𝑖𝑎𝑔[−𝑛𝑓1, −(𝑛 − 1)𝑓1, ⋯ , −𝑓1, 0, 𝑓1, ⋯ , (𝑛 − 1)𝑓1, 𝑛𝑓1]

+ 𝑑𝑖𝑎𝑔[𝑟𝐿 , 𝑟𝐿 , … , 𝑟𝐿] (3.29)

𝐘𝒄𝟎 = 𝑗2𝜋𝐶 ∙ 𝑑𝑖𝑎𝑔[−𝑛𝑓1, −(𝑛 − 1)𝑓1, ⋯ , −𝑓1, 0, 𝑓1, ⋯ , (𝑛 − 1)𝑓1, 𝑛𝑓1] (3.30)

PROJECT REPORT

38

The expressions (3.28), (3.29) and (3.30) are also valid for the five other arms. The approach to

develop the frequency domain model of phase 𝑎 upper arm can be repeated for the other five arms to

form a complete frequency domain model for the electrical part of the MMC.

The symmetry among the six arms allows the model to be significantly reduced. The expression of the

voltage of the floating middle point of DC bus Vm, which couples the six arms, can be expressed by

using the insertion index and capacitor voltage of a single arm and the symmetry above. Thus, the

model of an MMC can be represented by the upper arm of phase a instead of modeling all three phases

and six arms. The response of the other arms can be obtained based on the symmetry among the

arms.

3.4 MULTI-HARMONIC LINEARIZED MODEL

To obtain the frequency response of the MMC, the grid voltage is superimposed by a perturbation at

a specific frequency. A perturbation in the grid voltage leads to the perturbations in arm currents and

submodule capacitor voltages. By representing these variables as vectors, the elements of a vector

correspond to the Fourier coefficient of the variable at various frequencies. Therefore, the magnitude

and phase of the component at the perturbation frequency of AC terminal voltage and phase current

can be obtained from the vectors representing them. By dividing the voltage component by the current

component, the impedance of the MMC at a certain frequency can be determined.

For this, a perturbation at frequency 𝑓𝑝 with the initial phase 𝜑𝑝 is being superimposed on the voltage

of phase 𝑎. The perturbation can be of any arbitrary frequency.

𝑣𝑎(𝑡) = 𝑉1 cos(2𝜋𝑓1𝑡 + 𝜑1) + 𝑉𝑝cos (2𝜋𝑓𝑝𝑡 + 𝜑𝑝) (3.31)

This perturbation will cause harmonics of the same frequency in the arm current and also in the

insertion index due to the fact that the insertion index is produced by the current control functions.

When multiplied with the steady-state harmonic at frequency 𝑘 ⋅ 𝑓1, the perturbation at frequency 𝑓𝑝

would lead to new small-signal harmonics at frequency 𝑓𝑝 + 𝑘𝑓1 , 𝑘 = −𝑛,…− 1,0,1, … , 𝑛. For example,

when the frequency of the perturbation is 30 Hz, there will be small-signal harmonics produced by the

multiplication of the perturbation and steady-state harmonics at the frequencies

30𝐻𝑧, 80𝐻𝑧, 130𝐻𝑧, 180𝐻𝑧 …… (30 + 𝑘 ⋅ 50)𝐻𝑧. Analogous to the steady-state frequency-domain

modeling, vectors are also defined in the small-signal analysis to represent the arm current, the

equivalent capacitor voltage and the insertion index:

PROJECT REPORT

39

𝒂𝒖 =

[ 𝑝−𝑛𝑒

𝑗𝑝−𝑛

⋮

𝑝−1𝑒𝑗𝑝−1

𝑝𝑒𝑗𝑝

𝑝+1𝑒𝑗𝑝+1

⋮

𝑝+𝑛𝑒𝑗𝑝+𝑛]

, 𝒂𝒖 =

[ 𝑉𝑝−𝑛𝑒

𝑗𝑝−𝑛

⋮

𝑉𝑝−1𝑒𝑗𝑝−1

𝑉𝑝𝑒𝑗𝑝

𝑉𝑝+1𝑒𝑗𝑝+1

⋮

𝑉𝑝+𝑛𝑒𝑗𝑝+𝑛]

, 𝒂𝒖 =

[ 𝑝−𝑛𝑒

𝑗𝑝−𝑛

⋮

𝑝−1𝑒𝑗𝑝−1

𝑝𝑒𝑗𝑝

𝑝+1𝑒𝑗𝑝+1

⋮

𝑝+𝑛𝑒𝑗𝑝+𝑛]

(3.32)

The element in a vector with subscript 𝑝 + 1 represents the Fourier coefficient corresponding to

frequency 𝑓𝑝 + 𝑓1, the element with subscript 𝑝 + 𝑛 represents the Fourier coefficient corresponding to

frequency 𝑓𝑝 + 𝑛𝑓1, and the hat above indicates the small-signal nature of the variables in order to

distinguish them from the steady-state variables defined in (3.24) and (3.25).

With this notation, the frequency-domain small-signal model can be derived from the steady-state

frequency-domain model (3.26) and (3.27). The convolution of perturbations (m⨂au) is neglected

since second order small-signals are not considered. The convolution of the steady state operation

(𝐯au⨂𝐦au) is not considered here since it is included in the steady state part of the model.

𝐙𝐥 au = −p − 𝐯au⨂au −𝐦au⨂au + m (3.33)

𝐘𝐜𝐚𝐮 = 𝐢𝐚𝐮⨂𝐚𝐮 + 𝐦𝐚𝐮⨂𝐚𝐮 (3.34)

The voltage p is the frequency-domain representation of the second part on the right side of (3.31),

representing the perturbation in grid voltage.

𝑝 =1

2

[

⋮

0

𝑉𝑝𝑒𝑗𝜑𝑝

0

⋮ ]

(3.35)

𝐙𝒍 and 𝐘𝒄 represent the impedance of the arm inductor and the admittance of the equivalent capacitor

of an arm respectively. Similar as in the steady-state model, 𝐙𝒍 and 𝐘𝒄 are defined as

𝐙𝒍 = 𝑗2𝜋𝐿 ∙ 𝑑𝑖𝑎𝑔[𝑓𝑝 − 𝑛𝑓1, 𝑓𝑝 − (𝑛 − 1)𝑓1, ⋯ , 𝑓𝑝 − 𝑓1, 𝑓𝑝, 𝑓𝑝 + 𝑓1, ⋯ , 𝑓

𝑝

+ (𝑛 − 1)𝑓1, 𝑓𝑝+ 𝑛𝑓

1] + 𝑑𝑖𝑎𝑔[𝑟𝐿 , 𝑟𝐿 , … , 𝑟𝐿]

(3.36)

𝐘𝒄 = 𝑗2𝜋𝐶 ∙ 𝑑𝑖𝑎𝑔[𝑓𝑝 − 𝑛𝑓1, 𝑓𝑝 − (𝑛 − 1)𝑓1, ⋯ , 𝑓𝑝 − 𝑓1, 𝑓𝑝, 𝑓𝑝 + 𝑓1, ⋯ , 𝑓

𝑝

+ (𝑛 − 1)𝑓1, 𝑓𝑝+ 𝑛𝑓

1]

(3.37)

It should be noted that the small-signal model described by equations (3.33) and (3.34) only involves

the operation of small-signal variables. The voltage 𝑉𝑑𝑐 which is given in the steady-state model (3.26)

is absent in the small-signal model since the DC bus voltage is assumed to be constant.

PROJECT REPORT

40

Similar to the steady-state model, the small-signal dynamics of an MMC can be also represented with

only a single arm model. For this, the small-signal representation of the voltage of DC bus floating

middle point m in equation (3.33) which represents the coupling of the six arms needs to be eliminated.

Eliminating m (compare Appendix 6.1) gives:

= −𝐘𝒍(𝒑 + 𝐯⨂ + 𝐦⨂) (3.38)

𝐘𝐜 = 𝐢⨂ + 𝐦⨂𝐢 (3.39)

It should be noted that so far, the model involves convolution, which makes it difficult to manipulate

the model algebraically. To further simplify the model, the convolution operation can be eliminated by

converting the convolution into a matrix inner product form. For this, a matrix is defined replacing one

vector of the convolution operation, such that the product of this matrix with the other vector has the

same result as the convolution operation of the two vectors. Taking the insertion index 𝐦 as example,

the corresponding matrix 𝐌 is

𝑴 =

[ ⋱ ⋮ ⋮ ⋮ ⋮ ⋮⋯ 𝑀1𝑒

𝑗𝛾1 𝑀0 𝑀1𝑒−𝑗𝛾1 𝑀2𝑒

−𝑗𝛾2 𝑀3𝑒−𝑗𝛾3 ⋯

⋯ 𝑀2𝑒𝑗𝛾2 𝑀1𝑒

𝑗𝛾1 𝑀0 𝑀1𝑒−𝑗𝛾1 𝑀2𝑒

−𝑗𝛾2 ⋯

⋯ 𝑀3𝑒𝑗𝛾3 𝑀2𝑒

𝑗𝛾2 𝑀1𝑒𝑗𝛾1 𝑀0 𝑀1𝑒

−𝑗𝛾1 ⋯

⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ]

(3.40)

The (𝑛 + 1)th (center) row of the matrix 𝐌 is vector 𝐦 in inverse order. The 𝑘th row above the center

row is the center row shifted left by 𝑘 elements, with the last 𝑘 elements being zero.

Correspondingly, the 𝑘th row below the center row is the center row shifted right by 𝑘 elements, with

the first 𝑘 elements being zero. By this definition, the convolution operation 𝐦⨂ on the right side of

equation (3.39) gives the same result as 𝐌 ∙ . Thus, the convolution operation can be replaced.

Accordingly the matrix 𝐕 is defined for the equivalent capacitor voltage 𝐯 and the matrix 𝐈 for the arm

current 𝐢. As result the model given by equation (3.38) and (3.39) will be simplified to

= −𝐘𝒍(𝒑 + 𝐕 ∙ + 𝐌 ∙ ) (3.41)

𝐘𝐜 = 𝐈 ∙ + 𝐌 ∙ 𝐢 (3.42)

Equations (3.41) and (3.42) represent the small-signal model of the MMC. The relationship of the input

perturbation AC voltage 𝒑 and the resulting perturbation in phase current gives the impedance at

the perturbation frequency. Based on this, the frequency response can be determined for each

frequency if the matrices 𝐘𝐜 and 𝐘𝒍 as well as 𝐕, 𝐈 and 𝐌 and the vectors and 𝒑 as well as and

are known. Matrices 𝐘𝐜 and 𝐘𝒍 represent the physical characteristic of the MMC and can be calculated

based on the electrical parameters of the MMC. Matrices 𝐕, 𝐈 and 𝐌 are determined based on the

PROJECT REPORT

41

vectors 𝐯, 𝐢 and 𝐦. These vectors are steady-state values representing the operating status of the

MMC. The vector can be eliminated by substituting equation (3.42) into (3.41). However, the

perturbation in the insertion index ist not yet determined. This is why as next step, the small-signal

insertion index variable will be eliminated by modeling the respective controls and relating to

other variables of the model.

3.5 CONTROL MODELING

The control system of the MMC is used to relate the insertion index to the arm current and the AC

terminal voltage. In small-signal linearized form, this relationship can be expressed as

= 𝐐 ⋅ + 𝐏 ⋅ 𝒑 (3.43)

The coefficient matrices 𝐐 and 𝐏 depend on the design of the controls. The modelled controls of the

MMC comprise phase current control, circulating current control and phase-locked-loop.

The arm currents of an MMC can be separated into the portion of the current which is outputted to the

AC terminal of the three phases, being referred to as the phase current, and the portion of the current

circulating in the arms of the MMC, being referred to as circulating current. The control functions of the

MMC control the circulating current and the phase current independently. For all three phases, the

arm currents can be separated into the phase current 𝑖𝑠 and the circulating current 𝑖𝑐as follows:

𝑖𝑠 =𝑖𝑢 − 𝑖𝑙

2 (3.44)

𝑖𝑐 =𝑖𝑢 + 𝑖𝑙

2 (3.45)

It should be noted that the phase current 𝑖𝑠 as defined in (3.44) is the phase output component of one

arm and such only half of the corresponding phase current in Figure 3-1.

For the model coefficient matrices, Q and P are presented for phase current control, circulating current

control and phase locked loop. The matrices are shown for an example and generic control design

and serve as basis for the MMC Test Bench specific control design. Thus, the matrices presented will

be modified and parametrized accordingly for the test cases presented in D16.1 and the control design

used in Task 16.6.

3.5.1 PHASE CURRENT CONTROL

According to the reference directions defined in Figure 3-1, the phase current is the difference between

the arm currents of the upper and lower arms. Thus, the phase current control affects only the

differential-mode harmonics in the arm currents. The control of the phase current is implemented in a

rotating 𝑑𝑞 frame. Figure 3-6 shows the block diagram of the phase current control in 𝑑𝑞 frame with 𝑖𝑑

and 𝑖𝑞 being the three-phase phase currents (𝑖𝑎, 𝑖𝑏 and 𝑖𝑐) transformed into the rotating dq-frame by

dq-transformation. The currents 𝐼𝑑 and 𝐼𝑞 are the references for 𝑖𝑑 and 𝑖𝑞 and are given by outer control

PROJECT REPORT

42

functions such as voltage and power controls. The block 𝐻𝑖(𝑠) is the transfer function of the PI

controller of the phase current control, 𝐾𝑖𝑑 is the decoupling gain.

Figure 3-6: Phase current control in dq frame

The 𝑑𝑞 frame is synchronized with the phase voltage by means of a phase-locked-loop (PLL), which

acts as a feedback loop around the 𝑑𝑞-transformation of AC grid voltage. The output of the PLL is the

transformation angle 𝜃. The transformation from 𝑎𝑏𝑐 to 𝑑𝑞 frame is defined as

[𝑥𝑑𝑥𝑞] = 𝑻𝒅𝒒 [

𝑥𝑎𝑥𝑏𝑥𝑐

]. (3.46)

The transformation matrix 𝑻𝒅𝒒 is defined as

𝑻𝒅𝒒 = √2

3∙ [cos(𝜃) cos (𝜃 −

2𝜋

3) cos (𝜃 +

2𝜋

3)

−sin(𝜃) − sin (𝜃 −2𝜋

3) −sin (𝜃 +

2𝜋

3)]. (3.47)

The corresponding inverse transformation is given by the transposed transformation matrix 𝑻𝒅𝒒𝑻 .

[

𝑥𝑎𝑥𝑏𝑥𝑐

] = 𝑻𝒅𝒒𝑻 [𝑥𝑑𝑥𝑞] (3.48)

Because the dynamics of outer controls (voltage and power control) are considered slow and are not

included in the impedance modelling, the current reference for 𝑑 and 𝑞 axis 𝐼𝑑 and 𝐼𝑞 can be assumed

constant. When considering the influence of the dynamics of phase current control on the insertion

index, the transformation angle is assumed to track the grid voltage angle perfectly (𝜃 = 2𝜋𝑓1 + 𝜑1)

and is not affected by the grid voltage perturbation.

A positive-sequence perturbation in the grid voltage will lead to positive-sequence perturbation in the

arm current. A Fourier coefficient 𝑆𝑖𝑎𝑝𝑘

is assumed for the harmonic component in the arm current of

phase 𝑎 upper arm at frequency 𝑓𝑝 + 𝑘 ⋅ 𝑓1 , 𝑘 = −𝑛,…− 1,0,1, … , 𝑛:

𝑆𝑖𝑎𝑝𝑘= 𝐼𝑃𝑘𝑒

𝑗𝜑𝑝𝑘 (3.49)

The superscript 𝑝𝑘 indicates the frequency at 𝑓𝑝 + 𝑘𝑓1, the subscript 𝑖𝑎 of 𝑆𝑖𝑎𝑝𝑘

indicates that the Fourier

coefficient stands for the current of phase 𝑎. The magnitude of the Fourier coefficient 𝐼𝑃𝑘 is half of the

magnitude of the corresponding sinusoidal signal, and the phase 𝜑𝑝𝑘 represents the initial phase of

PROJECT REPORT

43

the corresponding sinusoidal signal. As explained in Appendix 6.2 the resulting relationship between

the Fourier coefficient of the perturbation in arm current and the Fourier coefficient of the perturbation

in the insertion index at frequency 𝑓𝑝 + 𝑘𝑓1.is:

𝑆𝑚𝑎𝑝𝑘 = 𝑆𝑖𝑎

𝑝𝑘𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 2𝜋𝑓1] − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3)

∙ 𝑗𝐾𝑖𝑑

(3.50)

As a result, the contribution of phase current control to the dynamics of the insertion index 1 can be

modeled in the frequency domain as:

1 = 𝐐i𝐢 (3.51)

The vector represents the small-signal harmonics in the arm current. The (2𝑛 + 1) × (2𝑛 + 1)

matrix 𝐐i is a diagonal matrix defined as:

𝐐i = 𝑑𝑖𝑎𝑔[𝑞𝑖𝑘|𝑘 = −𝑛, … ,0, … , 𝑛] (3.52)

For a positive-sequence perturbation, according to (6.22) and the coefficient defined above, 𝑞𝑖𝑘

equals to

𝑞𝑖𝑘(𝑝)

=1+(−1)𝑘

2 ∙ |𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3)| ∙ 𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙

2𝜋𝑓1] − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 𝑗𝐾

𝑖𝑑,

(3.53)

where the subscript (𝑝) indicates a positive sequence element 𝑞𝑖𝑘

of 𝐐i. Similar to equation (3.50), a

negative-sequence perturbation can be derived. The matrix element 𝑞𝑘 for a negative-sequence-

perturbation equals to:

𝑞𝑖𝑘(𝑛)

=1 + (−1)𝑘

2 ∙ |𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 − 1,3)|

∙ 𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 − 1,3) ∙ 2𝜋𝑓1]

− 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 − 1,3) ∙ 𝑗𝐾𝑖𝑑

(3.54)

3.5.2 CIRCULATING CURRENT CONTROL

The circulating current control reacts to the common-mode harmonics of the arm currents. Since the

dominant component of the circulating currents in steady-state operation is the second harmonic, the

circulating current control is implemented in the 𝑑𝑞 rotating frame at second-harmonic frequency. The

control structure is similar to the phase current control as seen in the block diagram in Figure 3-6.

For the second harmonic, the phase shift between three phases (for example, from 𝑏 to 𝑎) is (−2𝜋/3) ∙

2 = −4𝜋/3 ≜ 2𝜋/3, meaning the second harmonic is in negative-sequence. As result, the 𝑑𝑞

transformation used in the circulating current control should be a negative-sequence transformation.

PROJECT REPORT

44

𝑻𝒅𝒒−𝒄𝒄𝒄 = √2

3∙ [cos(2𝜃) cos (2𝜃 +

2𝜋

3) cos (2𝜃 −

2𝜋

3)

−sin(2𝜃) − sin (2𝜃 +2𝜋

3) −sin (2𝜃 −

2𝜋

3)] (3.55)

As in the phase current control, the transformation angle is assumed to track the grid voltage angle

perfectly (𝜃 = 2𝜋𝑓1 + 𝜑1) and is not affected by the grid voltage perturbation.

The contribution of the circulating current control to the dynamics of insertion index can be modelled

in the frequency domain as shown in Appendix 6.3:

2 = 𝐐c𝐢 (3.56)

The (2𝑛 + 1) × (2𝑛 + 1)matrix 𝐐c is defined as:

𝐐c = 𝑑𝑖𝑎𝑔[𝑞𝑐𝑘|𝑘 = −𝑛, … , −1,0,1, … , 𝑛] (3.57)

According to (6.34) and the coefficient defined above, 𝑞𝑐𝑘

equals for a positive-sequence

perturbation.

𝑞𝑐𝑘(𝑝)

=1−(−1)𝑘

2 ∙ |𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3)| ∙ 𝐻𝑐 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) + 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙

4𝜋𝑓1] + 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 𝑗𝐾

𝑖𝑑.

(3.58)

The subscript (𝑝) indicates that the element 𝑞𝑐𝑘 stands for a positive sequence. Similar to equations

(6.24)-(6.34) the negative-sequence perturbation can be derived. Therefore, 𝑞𝑘 is obtained for a

negative-sequence-perturbation as

𝑞𝑐𝑘(𝑛)

=1−(−1)𝑘

2 ∙ |𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 − 1,3)| ∙ 𝐻𝑐 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) + 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 − 1,3) ∙

4𝜋𝑓1] + 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 − 1,3) ∙ 𝑗𝐾

𝑖𝑑.

(3.59)

3.5.3 PHASE LOCKED LOOP

The Phase-Locked Loop (PLL) tracks the AC grid voltage angle. The output angle of the PLL is used

in the dq-transformation of the current control functions to synchronize the transformation angle with

the grid voltage angle. Figure 3-7 shows a PLL in 𝑑𝑞 frame that locks the grid voltage angle by driving

the q-axis voltage to zero using a PI regulator. The voltages 𝑣𝑑 and 𝑣𝑞 are the 𝑑𝑞-frame voltage

transformed from the abc-frame grid voltage. The transfer function of the PI regulator is denoted as

𝐻𝜃(𝑠). The output of the PI regulator, 𝜔 = 2𝜋∆𝑓, is the measured angular frequency deviation of the

grid voltage. The measured angular frequency is added to the rated angular frequency 𝜔0 = 2𝜋𝑓1, and

the result is integrated to contribute to the voltage angle 𝜃.

Figure 3-7: :Block diagram of PLL

PROJECT REPORT

45

Considering a positive perturbation at frequency 𝑓𝑝 is added to the steady-state balanced grid

voltage the Fourier coefficient of the perturbation is:

𝑆𝑣𝑎𝑝𝑘 =

1

2𝑉𝑝𝑒

𝑗𝜑𝑝 (3.60)

In this case the grid voltage would be:

𝑣𝑎 = 𝑉1 cos(2𝜋𝑓1𝑡 + 𝜑1) + 𝑉𝑝 cos (2𝜋𝑓𝑝𝑡 + 𝜑𝑝)

𝑣𝑏 = 𝑉1 cos (2𝜋𝑓1𝑡 + 𝜑1 −2𝜋

3) + 𝑉𝑝 cos (2𝜋𝑓𝑝𝑡 + 𝜑𝑝 −

2𝜋

3)

𝑣𝑐 = 𝑉1 cos (2𝜋𝑓1𝑡 + 𝜑1 +2𝜋

3) + 𝑉𝑝 cos (2𝜋𝑓𝑝𝑡 + 𝜑𝑝 +

2𝜋

3)

(3.61)

Assuming the PLL output consists of a steady-state grid angle 2𝜋𝑓1𝑡 + 𝜑1 and an angle (𝑡) caused

by the perturbation 𝜃 equals:

𝜃 = 2𝜋𝑓1𝑡 + 𝜑1 + (𝑡) (3.62)

In steady-state operation, the q-axis value 𝑣𝑞 should be zero. Under the influence of the perturbation,

the 𝑑𝑞 transformation would be

[𝑣𝑑𝑣𝑞] = 𝑻𝒅𝒒[2𝜋𝑓1𝑡 + 𝜑1 + (𝑡)] [

𝑣𝑎𝑣𝑏𝑣𝑐

]. (3.63)

As a result, under the influence of a voltage perturbation, the q-axis value 𝑣𝑞 would be

𝑣𝑞 = −𝑣𝑎 ∙ sin[2𝜋𝑓1𝑡 + 𝜑1 + (𝑡)] − 𝑣𝑏 ∙ sin [2𝜋𝑓1𝑡 + 𝜑1 + (𝑡) −2𝜋

3] − 𝑣𝑎

∙ sin [2𝜋𝑓1𝑡 + 𝜑

1+ (𝑡) +

2𝜋

3]

(3.64)

Substituting the expression of 𝑣𝑎, 𝑣𝑏 and 𝑣𝑐 (3.61) into (3.64) and simplifying the equation, gives:

𝑣𝑞 = √3

2𝑉𝑝 cos [2𝜋 (𝑓𝑝 − 𝑓1

) 𝑡 + 𝜑𝑝− 𝜑

1− (𝑡) −

𝜋

2] − 𝑉1 sin (𝑡)

≈ √3

2𝑉𝑝 cos [2𝜋 (𝑓𝑝 − 𝑓1

) 𝑡 + 𝜑𝑝− 𝜑

1−𝜋

2] − 𝑉1(𝑡)

(3.65)

The input 𝑣𝜃 of the PLL 𝑣𝜃 can be defined as

𝑣𝜃 = 𝑉𝑝 cos [2𝜋 (𝑓𝑝 − 𝑓1) 𝑡 + 𝜑

𝑝− 𝜑

1−𝜋

2]. (3.66)

It should be noted that the frequency of 𝑣𝜃 is shifted from the perturbation frequency by the

fundamental frequency. This indicates that the positive-sequence grid voltage perturbation at

frequency 𝑓𝑝 would produce a harmonic in the output angle of PLL at frequency 𝑓𝑝 − 𝑓1. The transfer

function of the PLL linearized model is

𝐺𝜃 =√3

2𝐻𝜃(𝑠)

𝑠+√3

2𝑉1𝐻𝜃(𝑠)

. (3.67)

The expression of (𝑡) is obtained:

(𝑡) = 𝑉𝑝 cos [2𝜋 (𝑓𝑝 − 𝑓1) 𝑡 + 𝜑

𝑝− 𝜑

1−𝜋

2+ 𝜑

𝐺] ∙ |𝐺𝜃 [𝑗2𝜋 (𝑓𝑝 − 𝑓1

)]| (3.68)

PROJECT REPORT

46

where 𝜑𝐺 is the phase of 𝐺𝜃 [𝑗2𝜋 (𝑓𝑝 − 𝑓1

)].

The perturbation in the transformation angle will cause additional terms within the 𝑑𝑞 transformation.

Defining 𝜃0 = 2𝜋𝑓1𝑡 + 𝜑1, including the perturbation in the transformation angle, the transformation

matrix 𝑻𝒅𝒒 would be

𝑻𝒅𝒒[𝜃0 + (𝑡)]

= √2

3∙ [cos(𝜃0 + (𝑡)) cos (𝜃0 + (𝑡) −

2𝜋

3) cos (𝜃0 + (𝑡) +

2𝜋

3)

−sin(𝜃0 + (𝑡)) − sin (𝜃0 + (𝑡) −2𝜋

3) −sin (𝜃0 + (𝑡) +

2𝜋

3)]

(3.69)

Applying small-signal estimates for sinus and cosinus, it should be noticed that

cos(𝜃0 + (𝑡)) = cos 𝜃0 ∙ cos (𝑡) − 𝑠𝑖𝑛 𝜃0 ∙ 𝑠𝑖𝑛 (𝑡) ≈ cos 𝜃0 − (𝑡) ∙ 𝑠𝑖𝑛 𝜃0 =

cos 𝜃0 + (𝑡) ∙ cos (𝜃0 +𝜋

2).

(3.70)

and

𝑠𝑖𝑛(𝜃0 + (𝑡)) = 𝑠𝑖𝑛 𝜃0 ∙ cos (𝑡) + cos 𝜃0 ∙ 𝑠𝑖𝑛 (𝑡) ≈ 𝑠𝑖𝑛 𝜃0 + (𝑡) ∙ cos 𝜃0 =

𝑠𝑖𝑛 𝜃0 + (𝑡) ∙ sin (𝜃0 +𝜋

2).

(3.71)

As a result, (3.69) can be rewritten as

𝑻𝒅𝒒[𝜃0 + (𝑡)] = 𝑻𝒅𝒒(𝜃0) + (𝑡) ∙ 𝑻𝒅𝒒 (𝜃0 +𝜋

2). (3.72)

The same can be applied to the inverse transformation:

𝑻𝒅𝒒𝑻 [𝜃0 + (𝑡)] = 𝑻𝒅𝒒

𝑻 (𝜃0) + (𝑡) ∙ 𝑻𝒅𝒒𝑻 (𝜃0 +

𝜋

2) (3.73)

When the second terms on the right side of (3.72) and (3.73) are multiplied with the steady-state

component of the phase currents or the 𝑑𝑞 frame control outputs, small-signal harmonics in the

insertion index would be generated. First, the fundamental component of the phase current is

converted to a DC quantity in the 𝑑𝑞 domain after the transformation 𝑻𝒅𝒒(𝜃0 + 𝜋/2) is applied. That dc

quantity would produce a component at frequency 𝑓𝑝 − 𝑓1 due to the multiplication with (𝑡). This

component would produce a component at frequency 𝑓𝑝 and another at frequency 𝑓𝑝 − 2𝑓1 in the

insertion index due to the inverse transformation 𝑻𝒅𝒒𝑻 (𝜃0). Appendix 6.4 shows the derivation of the

small-signal Fourier coefficient of the insertion index with respect to the steady state current 𝐈1 and

insertion index 𝐌1.

As a result, the contribution of the phase-locked loop to the dynamics of insertion index can be

modelled in the frequency domain as

3 = 𝐏𝐩 (3.74)

The (𝑛 + 1) × (𝑛 + 1)th element of the (2𝑛 + 1) × (2𝑛 + 1) matrix 𝐏 is:

PROJECT REPORT

47

𝐺𝜃 [𝑗2𝜋 (𝑓𝑝 − 𝑓1)] 𝐈1

∗ [𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 − 𝑓1 )] + 𝑗𝐾𝑖𝑑] − 𝐌1∗ 𝑒−𝑗𝜑1 (3.75)

Every other element of 𝐏 is zero.

Similar to the derivation for the positive perturbation, the (𝑛 + 1) × (𝑛 + 1)th element of 𝐏 of a negative

grid voltage perturbation is:

𝐺𝜃 [𝑗2𝜋 (𝑓𝑝 + 𝑓1)] −𝐈1

∗ [𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 + 𝑓1 )] + 𝑗𝐾𝑖𝑑] +𝐌1∗ 𝑒𝑗𝜑1 (3.76)

The (𝑛 + 3) × (𝑛 + 1)th element of 𝐏 is:

𝐺𝜃 [𝑗2𝜋 (𝑓𝑝 + 𝑓1)] 𝐈1 [𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 + 𝑓1 )] − 𝑗𝐾𝑖𝑑] −𝐌1 𝑒

𝑗𝜑1 (3.77)

3.5.4 GRID FORMING CONTROL

If the MMC is operated in grid forming mode, the grid angle 𝜃 is not tracked by the PLL to synchronize

the transformation angle with the grid voltage angle. Instead 𝜃 is generated by the control system.

Figure 1-9 shows the grid forming control where 𝑉1 is the AC voltage reference which is aligned with

the d-axis in the dq reference frame. The voltages 𝑉𝑑, 𝑉𝑞 and the currents 𝐼𝑑, 𝐼𝑞 result from applying

𝑑𝑞 transformation to the respective phase quantities. In this case, the angle θ is generated using the

constant frequency reference 𝜃 = 2𝜋𝑓1𝑡 + 𝜑1. The voltage control block generates the references for

the current control which further generates the 𝑑𝑞 frame modulating signals 𝑚 𝑑 and 𝑚𝑞.

Figure 3-8: Grid voltage and current control in dq frame

The small-signal linearized representation of the modulating signals is given by

= (𝑸𝒄 + 𝑸𝒊) ⋅ + 𝐏 ⋅ 𝒑, (3.86)

where 𝑸𝒄 and 𝑸𝒊 are diagonal matrices representing the effect of circulating current control and phase

current control respectively and given by equations (3.52) and (3.57). As the grid angle 𝜃 is generated

by the constant frequency reference, 𝜃 is not affected by the voltage perturbation. The diagonal matrix

𝐏 represents the effect of voltage control. Unlike arm currents and the capacitor voltage, the AC voltage

is assumed to have no steady state harmonics. Hence, only the fundamental and perturbation

frequency components exist in the grid voltage when superimposed with voltage perturbation. The grid

𝑉1

𝐾𝑣𝑑𝑣𝑑

𝐻𝑣(𝑠)

0𝐻𝑣(𝑠)

𝐾𝑣𝑞𝑣𝑞

𝐾𝑖𝑑𝑖𝑑

𝐻𝑖(𝑠)

𝐻𝑖(𝑠)

𝐾𝑖𝑞𝑖𝑞

𝑚𝑑

𝑚𝑞

PROJECT REPORT

48

voltage when superimposed with voltage perturbation at frequency 𝑓𝑝, is given by equation (3.61). The

product of the phase voltages and the 𝑑𝑞 transformation matrix results in 𝑉𝑑, 𝑉𝑞:

𝑉𝑑 = √3

2𝑝 cos[2𝜋(𝑓𝑝 − 𝑓1)𝑡 + 𝜑𝑝 − 𝜑1]

(3.78)

𝑉𝑞 = √3

2𝑝 sin[2𝜋(𝑓𝑝 − 𝑓1)𝑡 + 𝜑𝑝 − 𝜑1]

(3.79)

It can be observed that the 𝑑𝑞 frame voltage signals are a function of the perturbation voltage 𝑉𝑝.

Considering the reference currents generated by voltage control, the modulating signals from the

cascaded MMC control can be derived as follows:

𝑆𝑚𝑑𝑝𝑘 = (𝑆𝑖𝑑𝑟

𝑝𝑘− 𝑆𝑖𝑑

𝑝𝑘). 𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 2𝜋𝑓1] − 𝐾𝑖𝑑 ∙ 𝑆𝑖𝑞

𝑝𝑘

(3.80)

𝑆𝑚𝑞𝑝𝑘 = (𝑆𝑖𝑞𝑟

𝑝𝑘− 𝑆𝑖𝑞

𝑝𝑘). 𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 2𝜋𝑓1] + 𝐾𝑖𝑑 ∙ 𝑆𝑖𝑑

𝑝𝑘

(3.81)

Consider

𝑆1 = 𝑆𝑖𝑑𝑟𝑝𝑘 . 𝐻𝑖[𝑗2𝜋(𝑓𝑝 + 𝑘𝑓1 ) − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 2𝜋𝑓1] (3.82)

𝑆2 = 𝑆𝑖𝑞𝑟𝑝𝑘 . 𝐻𝑖[𝑗2𝜋(𝑓𝑝 + 𝑘𝑓1 ) − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 2𝜋𝑓1] (3.83)

where 𝑆𝑖𝑑𝑟𝑝𝑘

and 𝑆𝑖𝑞𝑟𝑝𝑘

generated by the voltage control and are given by:

𝑆𝑖𝑑𝑟𝑝𝑘= 𝑆𝑣𝑑

𝑝𝑘. 𝐻𝑣 [𝑗2𝜋 (𝑓𝑝− 𝑓1)]−𝐾𝑣𝑑.𝑆𝑣𝑞𝑝𝑘

(3.84)

𝑆𝑖𝑞𝑟𝑝𝑘= 𝑆𝑣𝑞

𝑝𝑘. 𝐻𝑣 [𝑗2𝜋 (𝑓𝑝− 𝑓1)]+𝐾𝑣𝑑. 𝑆𝑣𝑑𝑝𝑘

(3.85)

The terms 𝑆1 𝑎𝑛𝑑 𝑆2 are in dq frame and are converted to phase values by applying inverse dq

transformation:

𝑆1𝑎 = √2

3∙ [𝑆1 ∙ cos(2𝜋𝑓1𝑡 + 𝜑1) − 𝑆2 ∙ sin(2𝜋𝑓1𝑡 + 𝜑1)]

(3.86)

𝑆1𝑎𝑝𝑘= 𝑉𝑝𝑒

𝑗𝜑𝑝𝐻𝑣[𝑗2𝜋 (𝑓𝑝 − 𝑓1)] − 𝑗𝐾𝑣𝑑 .

𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 2𝜋𝑓1] (3.87)

The reference voltage is not considered as it is assumed to be constant. After expanding the equations

(3.80) and (3.81), the terms 𝑆1 and 𝑆2 are functions of the perturbation voltage. The reference currents

𝑆𝑖𝑑𝑟𝑝𝑘

and 𝑆𝑖𝑞𝑟𝑝𝑘

have the frequency component 𝑓𝑝 − 𝑓1 only. The product of the current compensator

matrix with the reference currents results in zero values of other steady state harmonic components.

PROJECT REPORT

49

Therefore, for the grid forming control, the matrix 𝐏𝐆𝐅𝐩𝐨𝐬

, containing the coefficients of the perturbation

voltage 𝒑 for positive sequence perturbation from the equation (1.96) is given by:

𝐏𝐆𝐅𝐩𝐨𝐬 = [𝐻𝑣 [𝑗2𝜋 (𝑓𝑝− 𝑓1)] − 𝑗 𝐾𝑣𝑑 ][𝐻𝑖 [𝑗2𝜋 (𝑓𝑝− 𝑓1)]] (3.88)

The negative sequence perturbation matrix 𝐏𝐧𝐞𝐠 is given by accordingly:

𝐏𝐆𝐅𝐧𝐞𝐠 = [𝐻𝑣 [𝑗2𝜋 (𝑓𝑝+ 𝑓1)] + 𝑗 𝐾𝑣𝑑 ][𝐻𝑖 [𝑗2𝜋 (𝑓𝑝+ 𝑓1)]] (3.89)

3.6 FINAL IMPEDANCE MODEL

In summary, the small-signal model of the controls, including phase current control, circulating

current control and PLL is built as follows:

= 1 + 2 + 3 = (𝐐i + 𝐐c) + 𝐏𝒑 (3.90)

The matrix 𝐐i represents the effect of the phase current control, defined by (3.51)-(3.54); 𝐐c represents

the effect of circulating current control, defined by (3.56)-(3.59); 𝐏 represents the effect of the PLL,

defined by (3.74)-(3.77). If the MMC is operated in grid forming mode, 𝐏 is defined by 𝐏𝑮𝑭 defined in

equation (3.88) and (3.89).

The control model can be substituted into (3.41) and (3.42) to eliminate in order to obtain the final

impedance model. To simplify (3.42), the following is defined:

𝐙𝐜 = 𝐘𝐜−𝟏 (3.91)

Equation (3.41) and (3.42) can be rewritten as:

= −𝐘𝒍(𝒑 + 𝐕 ∙ + 𝐌 ∙ ) (3.92)

= 𝐙𝐜(𝐈 ∙ + 𝐌 ∙ 𝐢 ) (3.93)

Substituting (3.93) into (3.92) and using 𝐔 to represent a (2𝑛 + 1) × (2𝑛 + 1) unity matrix gives:

(𝐔 + 𝐘𝐥𝐌𝐙𝐜𝐌)𝐢 + 𝐘𝐥𝐩 + 𝐘𝐥(𝐕 + 𝐌𝐙𝐜𝐈) = 𝟎 (3.94)

Substitute the control model (3.90) into (3.94) gives:

[𝐔 + 𝐘𝐥𝐌𝐙𝐜𝐌 + 𝐘𝐥(𝐕 + 𝐌𝐙𝐜𝐈)(𝐐𝐢 + 𝐐𝐜)]𝐢 + 𝐘𝐥[𝐔 + (𝐕 + 𝐌𝐙𝐜𝐈)𝐏]𝐩 = 𝟎 (3.95)

Based on this, the small-signal frequency responses to a perturbation in grid voltage of the current of

the upper arm can be modelled as

𝐘 = [𝐔 + 𝐘𝐥𝐌𝐙𝐜𝐌+ 𝐘𝐥(𝐕 +𝐌𝐙𝐜𝐈)(𝐐𝐢 + 𝐐𝐜)]−𝟏 ∙ 𝐘𝐥[𝐔 + (𝐕 +𝐌𝐙𝐜𝐈)𝐏]. (3.96)

PROJECT REPORT

50

In (3.96), matrices 𝐌, 𝐕 and 𝐈 represent the dependence of impedance on the converter operation and

matrices 𝐐𝐢, 𝐐𝐜 and 𝐏 represent the effects of the controls. According to (3.44), the harmonic at 𝑓𝑝 in

phase current is twice the corresponding harmonic in arm current. Thus, the input impedance of

converter at 𝑓𝑝 is:

Z(𝑓𝑝 ) =1

2 ⋅ 𝐘(n+1,n+1) (3.97)

The admittance Y at perturbation frequency 𝑓𝑝 is the (𝑛 + 1, 𝑛 + 1)th element of 𝐘. It should be noted

that the model represents the frequency behaviour on the AC side and is valid only for investigating

interactions on the AC side of converter. Table 3-3 shows the electrical parameters used for obtaining

the frequency response by the impedance model. Table 3-4 gives the parameters of the control system

and Table 3-5 shows the index used as input for the steady state solution of the model.

Table 3-3: MMC electrical parameters [24]

PARAMETER VARIABLE NAME VALUE

DC Bus Voltage 𝑉𝑑𝑐 640 kV

AC Bus Voltage 𝑉𝑎𝑏𝑐 350 kV (RMSL-L)

Number Submodules per Arm 𝑁 400

Capacitance Submodule 𝐶𝑠𝑚 8.8 mF

Arm Inductance 𝐿 0.15 H

Arm Inductor Resistance 𝑟𝐿 0.08 Ω

Table 3-4: Control system parameters

CONTROL SYSTEM 𝑲𝒑 𝑲𝒊 𝑲𝒊𝒅

Phase Current Control 1 50 0

Circulating Current Control 1 50 0

Phase Locked Loop 18 320 0

Table 3-5: Fourier coefficients insertion index of phase a, upper arm

𝒎𝒂𝒖 PHASE A

DC 0.4297

50Hz -0.355+j0.07134

100Hz -0.004079+j0.004421

150Hz 0

PROJECT REPORT

51

Figure 3-9 shows the open loop and closed loop response given by the impedance model for positive

and negative sequence. The closed loop response includes phase current control, circulating control

and phase locked loop.

Figure 3-9: Open Loop and closed loop response given by impedance model

The model will be tested, and the accuracy will be assessed in T16.6 by comparison with the MMC

test bench laboratory at RWTH and corresponding digital models [24].

PROJECT REPORT

52

4 GRID-FORMING INVERTER

This section presents a procedure for the harmonic impedance-based analysis of the stability of

electrical systems including grid-forming wind turbine generators (WTG) and/or Diode Rectifier Units

(DRU).

Harmonic impedance analysis is one of the techniques that allow stability analysis without knowing

the internal details of the different converters in the system (WTG control, filters and other elements).

In order to ensure the validity of the proposed impedance-based stability method, this section includes

the small-signal analytical model of grid-forming converters, offshore ac-grid and DRU station. The

state-space small-signal analysis [24] is then compared to a detailed time-domain simulation in

PSCAD, to ascertain the validity of the model. Then, the harmonic impedance is obtained from both

the analytical and the PSCAD models [25], in the latter case by means of disturbance injection. Finally,

it is shown that the conclusions reached regarding system stability are the same using: a) the state

space model, b) the impedance-based model, c) the PSCAD detailed simulation.

In this way, the impedance-based methodology will be clearly validated for grid-forming converters

and DRU stations. This methodology will be used in subsequent tasks of WP16, e.g. T16.6, where

further validation will be carried out against real-time HIL systems and, if time allows, against

experimental measurements.

Thus, the aim of this section is to develop a harmonic model for the aforementioned converters and

validate a procedure for harmonic stability studies of grid forming wind turbines connected through

DRU stations. The system under study has been taken from WP3 considering an adequate level of

detail. The wind power plant model has been aggregated as per [26], which ensures the main

resonance characteristics of the wind farm array are kept during the aggregation process.

4.1 SYSTEM DESCRIPTION

For the studies carried out in this chapter, the system proposed is shown Figure 4-1. The wind power

plant (WPP) is modelled considering an aggregated equivalent of 50 8MW full converter WTGs.

PROJECT REPORT

53

Figure 4-1 Off-shore wind farm connected to the on-shore MMC through a diode based HVDC rectifier station.

4.1.1 WIND POWER PLANT MODELLING

As mentioned in the introduction, as a first step, a state space model of the complete system is

developed as a benchmark to validate the results of the impedance-based stability analysis. The WTG

state space model will include both the controls and the system physical elements.

The WTG grid side converters are modelled as averaged models considering that the machine side

converter can effectively control the dc-link voltage [27]. Therefore, the dc-link, generator and wind

turbine mechanical dynamics are not included in this analysis.

Both state space model and harmonic impedance stability analysis are carried out in d-q frame, instead

of stationary frame of reference, as this is a more suitable choice for DRU modelling.

The considered WPP consists of 50 WTGs arranged in 6 strings, and it is aggregated using system

order reduction techniques which ensure a faithful representation of the array cables main resonant

peak for each considering operating condition [26].

Figure 4-2 shows the aggregated 400MW WPP, with the aggregated ac cable up to the WPP PCCF.

Figure 4-2 Aggregated WT which comprises the grid-side converter, ac-filter and transformer.

Based on Figure 4-2, the following state space equations for the WTG are developed, referred to a

local dq-frame rotating at a constant frequency 𝜔𝑇0.

PROJECT REPORT

54

WTG grid-side differential equations are developed as follows, considering the variables shown in

Figure 4-2. The VSC terminal dynamics are:

𝑉𝑊𝑑 = 𝑅𝑊𝐼𝑊𝑑 + 𝐿𝑊𝑑𝐼𝑤𝑑𝑑𝑡

− 𝜔𝑇0𝐿𝑊𝐼𝑊𝑞 + 𝑉𝑇𝑑

(4-1)

𝑉𝑊𝑞 = 𝑅𝑊𝑞𝐼𝑊𝑞 + 𝐿𝑊𝑑𝐼𝑤𝑞

𝑑𝑡+ 𝜔𝑇0𝐿𝑊𝐼𝑊𝑑 + 𝑉𝑇𝑞

( 4-2)

The capacitor dynamics are as follows:

𝐼𝑊𝑑 − 𝐼𝑇𝑑 = 𝐶𝑊𝑑𝑉𝑇𝑑𝑑𝑡

− 𝜔𝑇0𝐶𝑊𝑉𝑇𝑞

( 4-3)

𝐼𝑊𝑞 − 𝐼𝑇𝑞 = 𝐶𝑊𝑑𝑉𝑇𝑑𝑑𝑡

+ 𝜔𝑇0𝐶𝑊𝑉𝑇𝑑

( 4-4)

The WTG transformer is modelled considering only its copper losses and leakage reactance:

𝑉𝑇𝑑 = 𝑅𝑇𝐼𝑇𝑑 + 𝐿𝑇𝑑𝐼𝑇𝑑𝑑𝑡

− 𝜔𝑇0𝐿𝑇𝐼𝑇𝑞 + 𝑉𝑎𝑐,𝑑

( 4-5)

𝑉𝑇𝑞 = 𝑅𝑇𝐼𝑇𝑞 + 𝐿𝑇𝑑𝐼𝑇𝑞

𝑑𝑡+ 𝜔𝑇0𝐿𝑇𝐼𝑇𝑑 + 𝑉𝑎𝑐,𝑞

( 4-6)

For this particular study, the ac-cable is modelled considering aggregation technique [26], which

ensures that both losses and the frequency of the main resonance peak of the off-shore ac-grid is kept

in the model equivalent. This method is acceptable for the considered system, as the relatively large

DRU capacitor filters imply that off-shore grid resonance frequencies are relatively low. However, skin

effect and higher frequency resonance modes have not been taken into account. If skin effect and

multiple resonance modes need to be considered, more accurate cable modelling techniques can be

used [28] [29]. The parameters of the cable model included shown in Figure 4-3 are listed in Table

4-1, at the end of the chapter.

Figure 4-3: WT filter, transformer and ac-cable.

The state space dynamics of the considered -equivalent are shown as follows. The left-side shunt

capacitor dynamics are:

PROJECT REPORT

55

𝐼𝑇𝑑 − 𝐼𝑎𝑐,𝑑 = 𝐶𝑎𝑐𝑑𝑉𝑎𝑐,𝑑𝑑𝑡

− 𝜔𝑡0𝐶𝑎𝑐𝑉𝑎𝑐,𝑞

( 4-7)

𝐼𝑇𝑞 − 𝐼𝑎𝑐,𝑞 = 𝐶𝑎𝑐𝑑𝑉𝑎𝑐,𝑞

𝑑𝑡+ 𝜔𝑡0𝐶𝑎𝑐𝑉𝑎𝑐,𝑑

The series RL branch dynamics are:

( 4-8)

𝑉𝑎𝑐,𝑑 = 𝑅𝑎𝑐𝐼𝑎𝑐,𝑑 + 𝐿𝑇𝑑𝐼𝑎𝑐,𝑑𝑑𝑡

− 𝜔𝑇0𝐿𝑎𝑐𝐼𝑎𝑐,𝑞 + 𝑉𝐹𝑑

( 4-9)

𝑉𝑎𝑐,𝑞 = 𝑅𝑎𝑐𝐼𝑎𝑐,𝑞 + 𝐿𝑇𝑑𝐼𝑎𝑐,𝑞

𝑑𝑡+ 𝜔𝑇0𝐿𝑎𝑐𝐼𝑎𝑐,𝑑 + 𝑉𝐹𝑞

( 4-10)

The right-side capacitor is effectively in parallel with the DRU capacitor and filter bank and, hence, its

dynamics are considered together with those of the DRU capacitor bank 𝐶𝐹 in (4-38) and (4-39).

State-space variables for WPP and array cable:

[𝑥]𝑊𝑇 = [𝐼𝑊𝑑 , 𝐼𝑊𝑞 , 𝑉𝑇𝑑, 𝑉𝑇𝑞 , 𝐼𝑇𝑑 , 𝐼𝑇𝑞 , 𝑉𝑎𝑐,𝑑 , 𝑉𝑎𝑐,𝑞 , 𝐼𝑎𝑐,𝑑, 𝐼𝑎𝑐,𝑞 ]𝑡

( 4-11)

4.1.2 WIND TURBINE CONTROL

Figure 4-4 shows a typical control strategy for DRU-connected WTGs. The inner current and voltage

loops are designed in a stationary (−) frame using proportional-resonant (PR) controllers. As the

system is DRU-connected and it is capacitive- dominated, the outer loop includes P-V and Q- control

loops.

Figure 4-4 Grid-side converter control.

PROJECT REPORT

56

The internal current control loops are based on stationary frame proportional-resonant (PR) controllers,

whose transfer function 𝐺𝑃𝑅,𝐼(𝑠) is:

𝐺𝑃𝑅,𝐼 = 𝐾𝑃𝐼 + 𝐾𝑅𝐼𝑠

𝑠2 + 𝜔𝑃𝑅2

( 4-12)

Where KPI and KRI are the proportional and resonant gains, respectively and PR is the resonant

frequency of the PR controller.

For analysis purposes, the two − current PR controllers need to be expressed as d-q oriented state

space variables. Therefore, the resulting d-current (IWd) control law is:

𝑉𝑊𝑑∗ = 𝑥𝐼𝑊,1𝑑 + 𝐾𝑃𝐼(𝐼𝑊𝑑

∗ − 𝐼𝑊𝑑) + 𝑉𝑇𝑑 ( 4-13)

where the PR controller state space equations for the auxiliary state variables xIW,1d and xIW,2d are:

𝑑𝑥𝐼𝑊,1𝑑𝑑𝑡

− 𝜔𝑇0𝑥𝐼𝑊,1𝑞 = 𝑥𝐼𝑊,2𝑑 + 𝐾𝑅𝐼(𝐼𝑊𝑑∗ − 𝐼𝑊𝑑)

( 4-14)

𝑑𝑥𝐼𝑊,2𝑑𝑑𝑡

− 𝜔𝑇0𝑥𝐼𝑊,2𝑞 = −𝜔𝑃𝑅2 𝑥𝐼𝑊,1𝑑

( 4-15)

The control law for the q-axis current (IWq) is similar to that of the d-axis:

𝑉𝑊𝑞∗ = 𝑥𝐼𝑊,1𝑞 + 𝐾𝑃𝐼(𝐼𝑊𝑞

∗ − 𝐼𝑊𝑞) + 𝑉𝑇𝑞 ( 4-16)

where the PR controller state space equations for the auxiliary state variables xIW,1q and xIW,2q are:

𝑑𝑥𝐼𝑊,1𝑞

𝑑𝑡+ 𝜔𝑇0𝑥𝐼𝑊,1𝑑 = 𝑥𝐼𝑊,2𝑞 + 𝐾𝑅𝐼(𝐼𝑊𝑞

∗ − 𝐼𝑊𝑞) ( 4-17)

𝑑𝑥𝐼𝑊,2𝑞

𝑑𝑡+ 𝜔𝑇0𝑥𝐼𝑊,2𝑑 = −𝜔𝑃𝑅

2 𝑥𝐼𝑊,1𝑞 ( 4-18)

Similarly, the outer voltage loops are also based on stationary frame PR controllers, which, for analysis

purposes, will also be expressed in a synchronous d-q frame of reference. The transfer function of the

voltage loop PR controllers is:

𝐺𝑃𝑅,𝑉 = 𝐾𝑃𝑉 + 𝐾𝑅𝑉𝑠

𝑠2 + 𝜔𝑃𝑅2

( 4-19)

therefore, the d-axis voltage (VTd) control law is:

𝐼𝑊𝑑∗ = 𝑥𝑉𝑇,1𝑑 + 𝐾𝑃𝑉(𝑉𝑇𝑑

∗ − 𝑉𝑇𝑑) + 𝐼𝑇𝑑

( 4-20)

where the corresponding synchronous frame PR controller equations for the auxiliary state variables xVT,1d and xVT,2d are:

𝑑𝑥𝑉𝑇,1𝑑𝑑𝑡

− 𝜔𝑇0𝑥𝑉𝑇,1𝑞 = 𝑥𝑉𝑇,2𝑑 + 𝐾𝑅𝑉(𝑉𝑇𝑑∗ − 𝑉𝑇𝑑)

( 4-21)

PROJECT REPORT

57

𝑑𝑥𝑉𝑇,2𝑑𝑑𝑡

− 𝜔𝑇0𝑥𝑉𝑇,2𝑞 = −𝜔𝑃𝑅2 𝑥𝑉𝑇,1𝑑

( 4-22)

Finally, the synchronous q-axis voltage (VTq) control law is:

𝐼𝑊𝑞∗ = 𝑥𝑉𝑇,1𝑞 + 𝐾𝑃𝑉(𝑉𝑇𝑞

∗ − 𝑉𝑇𝑞) + 𝐼𝑇𝑑

( 4-23)

where the corresponding synchronous frame PR controller equations for the auxiliary state variables xVT,1q and xVT,2q are:

𝑑𝑥𝑉𝑇,1𝑞

𝑑𝑡− 𝜔𝑇0𝑥𝑉𝑇,1𝑑 = 𝑥𝑉𝑇,2𝑞 + 𝐾𝑅𝑉(𝑉𝑇𝑞

∗ − 𝑉𝑇𝑞)

( 4-24)

𝑑𝑥𝑉𝑇,2𝑞

𝑑𝑡− 𝜔𝑇0𝑥𝑉𝑇,2𝑑 = −𝜔𝑃𝑅

2 𝑥𝑉𝑇,1𝑞

( 4-25)

where KPV and KRV are the proportional and resonant gains, respectively and PR is the resonant

frequency of the PR controller. The meaning of the different variables is shown in Figure 4-4.

Therefore, the space state variables related with the current and voltage control loops are:

[𝑥]𝑊𝑇𝑣𝑖 = [𝑥𝐼𝑊,1𝑑, 𝑥𝐼𝑊,1𝑞 , 𝑥𝐼𝑊,2𝑑 , 𝑥𝐼𝑊,2𝑞,𝑥𝑉𝑇,1𝑑, 𝑥𝑉𝑇,1𝑞 , 𝑥𝑉𝑇,2𝑑, 𝑥𝑉𝑇,2𝑞 ]𝑡

( 4-26)

The model for the grid-forming converter is completed by including the P and Q controls. As the

considered system is capacitive and, moreover, active power delivered through the HVDC link

depends on the ac-grid voltage level, a P-V, Q- control has been chosen.

First of all, the measured active and reactive powers are low pass filtered with a cut of frequency

(PQ). The low pass filter transfer functions are:

𝑃 =𝜔𝑃𝑄

𝑠 + 𝜔𝑃𝑄𝑃𝑇

( 4-27)

𝑄 =𝜔𝑃𝑄

𝑠 + 𝜔𝑃𝑄𝑄𝑇

( 4-28)

where PT, QT are the measured active and reactive powers measured at the low voltage side of the

WTG transformer, and 𝑃, 𝑄 their filtered counterparts.

The corresponding state space equations are:

𝜔𝑃𝑄𝑃 + 𝑑𝑃𝑑𝑡

= 3𝜔𝑃𝑄𝑉𝑇𝑑𝐼𝑇𝑑 + 3𝜔𝑃𝑄𝑉𝑇𝑞𝐼𝑇𝑞 ( 4-29)

𝜔𝑃𝑄𝑄 + 𝑑𝑄𝑑𝑡

= 3𝜔𝑃𝑄𝑉𝑇𝑞𝐼𝑇𝑑 − 3𝜔𝑃𝑄𝑉𝑇𝑑𝐼𝑇𝑞 ( 4-30)

PROJECT REPORT

58

where active and reactive powers are calculated from the low voltage side WTG transformer d-q

voltage and current components (VTd, VTq, ITd, ITq).

As previously mentioned, the reference for the voltage magnitude |𝑉𝑇|∗ is obtained from the active

power control loop. The active power is therefore controlled by means of a PI controller (GPI,Pm in Figure

4-4), which includes a droop term (proportional gain) and also an integral term, which ensures zero

error in steady state (similar to a secondary control). Therefore, active power controller dynamics are:

|𝑉𝑇|∗ = 𝐾𝑃,𝑃𝑚(𝑃𝑇

∗ − 𝑃) + 𝐾𝐼,𝑃𝑚𝑥𝑃𝑚 ( 4-31)

Where the auxiliary integral variable is defined as:

𝑑𝑥𝑃𝑚𝑑𝑡

= (𝑃𝑇∗ − 𝑃)

( 4-32)

The reactive power control law (GPI,Pm in Figure 4-4), is:

𝜃𝑇∗ = 𝐾𝑃,𝑄𝑚(𝑄𝑇

∗ − 𝑄) + 𝜔𝑇0 ⋅ 𝑡 ( 4-33)

where T0 is the off-shore ac-grid constant frequency reference.

For analysis purposes, the voltage reference vector, obtained from the active and reactive power controllers, should be

expressed in the 𝑑𝑞 synchronous frame-T rotating at 𝜔𝑇0 =𝑑𝜃𝑇0

𝑑𝑡 as shown in

Figure 4-5. Therefore:

𝑉𝑇𝑑∗ = |𝑉𝑇|

∗ cos 𝛽𝑇 ( 4-34)

𝑉𝑇𝑞∗ = |𝑉𝑇|

∗ sin 𝛽𝑇 ( 4-35)

where 𝛽𝑇 is defined as:

Figure 4-5. Wind turbine grid side converter frames of reference

PROJECT REPORT

59

𝛽𝑇 = 𝜃𝑇∗ − 𝜔𝑇0 ⋅ 𝑡 = 𝐾𝑃,𝑄𝑚(𝑄𝑇

∗ − 𝑄𝑇) ( 4-36)

and hence, the space state variables for active and reactive power droops are:

[𝑥]𝑊𝑇𝑝𝑞 = [𝑃 , 𝑄 , 𝑥𝑃𝑚 ]𝑡

( 4-37)

4.1.3 DRU MODELING

The DRU model is shown in Figure 4-6, where ZFR represents the ac-filter and capacitor banks of the

uncontrolled rectifier.

Figure 4-6 Diode-based HVDC rectifier unit

The DRU dynamic equations are expressed in a global 𝑑𝑞-frame-F, which rotates at a frequency ωF =

𝑑𝜃𝐹 𝑑𝑡⁄ . Note DRU dynamic equations are expressed in a synchronous frame of reference which is

generally not the same as the one used for the derivation of WTG dynamics. The reason for this is that

naturally, the equations for the diode rectifier are simpler if expressed in a frame of reference which is

oriented to the diode rectifier ac-grid voltage (VF in Figure 4-6).

PROJECT REPORT

60

Figure 4-7 shows the relationship between WTG (T) and DRU (F) synchronous reference frames. As

mentioned, the DRU reference frame is defined as:

ωF = 𝑑𝜃𝐹

𝑑𝑡

( 4-38)

Therefore:

βF = 𝜃𝑇0 − 𝜃𝐹 ( 4-39)

Using the definition 𝜔𝑇0 =𝑑𝜃𝑇0

𝑑𝑡 , we have:

𝑑

𝑑𝑡βF = 𝜔𝑇0 − 𝜔𝐹

( 4-40)

where, VFq = 0 as 𝑑𝑞-frame-F is aligned with the capacitor voltage VF.

Figure 4-8 DRU ac-filter ZFR

Figure 4-7. WTG and DRU reference frames

PROJECT REPORT

61

The structure of the DRU ac capacitor and filter bank shown in Figure 4-8, is based on the Cigre

Benchmark [30]. The dynamic equations for the capacitor and filter bank, lead to 12 new state

equations.

The capacitor Cac due to the cable model Figure 4-3) and the capacitor bank CF (Figure 4-5are

connected in parallel. Therefore, their corresponding dynamic equations are:

𝐼𝑎𝑐,𝑑 − 𝐼𝐹𝑑 = (𝐶𝑎𝑐 + 𝐶𝐹)𝑑𝑉𝐹𝑑𝑑𝑡

− 𝜔𝑇0(𝐶𝑎𝑐 + 𝐶𝐹)𝑉𝐹𝑞 ( 4-41)

𝐼𝑎𝑐,𝑞 − 𝐼𝐹𝑞 = (𝐶𝑎𝑐 + 𝐶𝐹)𝑑𝑉𝐹𝑞

𝑑𝑡 − 𝜔𝑇0(𝐶𝑎𝑐 + 𝐶𝐹)𝑉𝐹𝑑

( 4-42)

The variables in the following equations are defined in Figure 4-8. The dynamic equations

corresponding to the ac-filter banks are:

𝑉𝐹𝑑 = 𝑉𝐶𝑎1𝑑 + 𝑉𝐶𝑎2𝑑 + 𝐿𝑎𝑑𝐼𝐿𝑎𝑑𝑑𝑡

− 𝜔𝐹 𝐿𝑎𝐼𝐿𝑎𝑞 + 𝑅𝑎1𝐼𝐿𝑎𝑑 ( 4-43)

𝑉𝐹𝑞 = 𝑉𝐶𝑎1𝑞 + 𝑉𝐶𝑎2𝑞 + 𝐿𝑎𝑑𝐼𝐿𝑎𝑞

𝑑𝑡− 𝜔𝐹 𝐿𝑎𝐼𝐿𝑎𝑑 + 𝑅𝑎1𝐼𝐿𝑎𝑞

( 4-44)

𝑉𝐹𝑑 = 𝑉𝐶𝑏𝑑 + 𝐿𝑏𝑑𝐼𝐿𝑏𝑑𝑑𝑡

− 𝜔𝐹𝐿𝑏𝐼𝐿𝑏𝑞

( 4-45)

𝑉𝐹𝑞 = 𝑉𝐶𝑏𝑑 + 𝐿𝑏𝑑𝐼𝐿𝑏𝑑𝑑𝑡

+ 𝜔𝐹𝐿𝑏𝐼𝐿𝑏𝑞

( 4-46)

𝑉𝐹𝑑 − 𝑉𝐶𝑎1𝑑 = 𝑅𝑎2𝐶𝑎1𝑑𝑉𝐶𝑎1𝑑𝑑𝑡

− 𝜔𝐹𝑅𝑎2𝐶𝑎1𝑉𝐶𝑎1𝑞 − 𝑅𝑎2𝐼𝐿𝑎𝑑 ( 4-47)

−𝑉𝐶𝑎1𝑞 = 𝑅𝑎2𝐶𝑎1𝑑𝑉𝐶𝑎1𝑞

𝑑𝑡− 𝜔𝐹𝑅𝑎2𝐶𝑎1𝑉𝐶𝑎1𝑑 − 𝑅𝑎2𝐼𝐿𝑎𝑞

( 4-48)

𝑉𝐹𝑑 − 𝑉𝐶𝑏𝑑 = 𝑅𝑏𝐶𝑏𝑑𝑉𝐶𝑏𝑑𝑑𝑡

− 𝜔𝐹𝑅𝑏𝐶𝑏𝑉𝐶𝑏𝑞 − 𝑅𝑏𝐼𝐿𝑏𝑑

( 4-49)

−𝑉𝐶𝑏𝑑 = 𝑅𝑏𝐶𝑏𝑑𝑉𝐶𝑏𝑞

𝑑𝑡+ 𝜔𝐹𝑅𝑏𝐶𝑏𝑉𝐶𝑏𝑑 − 𝑅𝑏𝐼𝐿𝑏𝑞

( 4-50)

𝐼𝐿𝑎𝑑 = 𝐶𝑎2𝑑𝑉𝐶𝑎2𝑑𝑑𝑡

− 𝜔𝐹𝐶𝑎2𝑉𝐶𝑎2𝑞

( 4-51)

𝐼𝐿𝑎𝑞 = 𝐶𝑎2𝑑𝑉𝐶𝑎2𝑞

𝑑𝑡+ 𝜔𝐹𝐶𝑎2𝑉𝐶𝑎2𝑑

( 4-52)

PROJECT REPORT

62

The corresponding state space variables are:

[𝑥]𝐷𝑅𝑈𝑎𝑐 = [𝛽𝐹 , 𝑉𝐹𝑑 , 𝑉𝑐𝑎1,𝑑 , 𝑉𝑐𝑎1,𝑞 𝑉𝑐𝑎2,𝑑, 𝑉𝑐𝑎2,𝑞 , 𝐼𝐿𝑎,𝑑, 𝐼𝐿𝑎𝑞 , 𝑉𝑐𝑏,𝑑, 𝑉𝑐𝑏,𝑞 , 𝐼𝐿𝑏,𝑑 , 𝐼𝐿𝑑,𝑞 ]𝑡

( 4-53)

Note VFq is not included as a state variable as the DRU frame is aligned with the voltage vector VF and

hence VFq=0. The additional dynamic equation is (4-40).

The dynamic model for the diode rectifier transformer and diodes is:

𝑉𝑅𝑑𝑐 =3 √6

𝜋 𝐵𝑁 𝑉𝐹𝑑 −

3

𝜋 𝐵 𝜔𝐹𝐿𝑇𝑅𝐼𝑅𝑑𝑐 − 2𝐵𝐿𝑇𝑅

𝑑𝐼𝑅𝑑𝑐𝑑𝑡

( 4-54)

𝐼𝑅𝑎𝑐𝑑 = 𝐵√6

𝜋𝑁 𝐼𝑅𝑑𝑐 −

𝐵𝜋𝜔𝐹𝐿𝑇𝑅𝐼𝑅𝑑𝑐

2

𝑉𝐹𝑑

( 4-55)

𝐼𝑅𝑎𝑐𝑞 = −𝑁

√2

𝐵√3

𝜋𝐼𝑅𝑑𝑐𝑠𝑖𝑛𝜇 + 3𝐵𝑁

2𝑉𝐹𝑑

2𝜋𝜔𝐹𝐿𝑇𝑅(𝑠𝑖𝑛 𝜇 − 𝜇)

( 4-56)

𝜇 = cos−1 (1 −2𝜔𝐹𝐿𝑇𝑅𝐼𝑅𝑑𝑐

√6𝑁𝑉𝐹𝑑)

( 4-57)

Where B is number of bridges in series, N is the transformer ratio, LTR the leakage impedance of the

diode rectifier transformer and µ the overlap angle of the diode rectifier. The variable frequency 𝜔𝐹

used in equation (4-40) is obtained by solving equations (4-51) to (4-54).

Only the DRU dc-filter smoothing reactor (LR in Figure 4-6) remains to be included:

𝑉𝑅𝑑𝑐 = 𝐿𝑅𝑑𝐼𝑅𝑑𝑐𝑑𝑡

+ 𝐸𝑅

( 4-58)

These equations do not add any new state-space variable as the term 2𝐵𝐿𝑇𝑅 𝑑𝐼𝑅𝑑𝑐

𝑑𝑡 is equivalent to an

inductor in series with LR. The only one new state space variable is that of smoothing reactor:

[𝑥]𝐷𝑅𝑈𝑑𝑐 = [𝐼𝑅𝑑𝑐]

( 4-59)

4.1.4 STATE SPACE MODEL VALIDATION

The model of the aggregated WTG and DRU connected to the HVDC link has been validated

comparing the analytical results with that of a detailed PSCAD simulation. The system parameters are

provided in Table 4-1. Figure 4-7 shows the system response to a step change of P*T. The analytical

results clearly agree with PSCAD, hence validating the proposed system model.

PROJECT REPORT

63

Figure 4-9 System response to a P*T step for model validation.

4.1.5 STATE-SPACE STABILITY ANALYSIS

The considered state space model includes the off-shore wind farm dynamics and their corresponding

controllers. Therefore, the system stability analysis will be carried out by means of eigenvalue analysis.

Figure 4-10 shows the root locus of the system as a function of the current PR controller proportional

gain (KPI) when PF* = 0.5 pu and QF

* = 0 pu. The values of KPI go from 1500 (dark) to 6000 (light).

Figure 4-11 shows a detailed of the root locus. The system becomes unstable when KPI decreases

from 2100 to 1900.

PROJECT REPORT

64

Figure 4-10 Root locus of the system (WT and DRU) as a function of .

Figure 4-11 Root locus of the system (WT and DRU) as a function of (zoom).

PROJECT REPORT

65

4.2 IMPEDANCE-BASED STABILITY ANALYSIS

4.2.1 HARMONIC IMPEDANCES FROM STATE SPACE EQUATIONS

In the impedance-based method, the system is split between the source and the load subsystems, as

in [22]. The stability of the system is assessed based on the source-load impedance ratio. Figure 4-12

shows the location of the interfacing point between the source (WT) and the load (DRU). The source

consists of WT ac-grid converter, LC-filter and transformer, while the load consists of ac-Cable, and

DRU ac-filter, transformers, diode bridges and dc-smoothing reactor.

Figure 4-12 Injection of a small disturbance IZ to compute WT and DRU impedances.

To validate the accuracy of the analytical study, a PSCAD frequency scan is numerically performed.

This frequency scan is obtained by applying a small current disturbance at a certain frequency to the

WPP current. The PCC voltage is measured, and the harmonic components of the disturbance

frequency are obtained.

The source or WT impedance ZWT can be calculated from the input-output relation as:

𝑉𝑎𝑐 = 𝑍𝑊𝑇𝐼𝑊𝑇

( 4-60)

Similarly, the load or DRU impedance ZDRU can be calculated as:

𝑉𝑎𝑐 = 𝑍𝐷𝑅𝑈𝐼𝐷𝑅𝑈

( 4-61)

The impedance based analysis is carried out in a rotating d-q. The WTG dq-impedance will be:

[𝑉𝑎𝑐,𝑑𝑉𝑎𝑐,𝑞

] = [𝑍𝑊𝑇,𝑑𝑑 𝑍𝑊𝑇,𝑑𝑞𝑍𝑊𝑇,𝑞𝑑 𝑍𝑊𝑇,𝑞𝑞

] [𝐼𝑊𝑇,𝑑𝐼𝑊𝑇,𝑞

]

( 4-62)

and the DRU dq-impedance is:

[𝑉𝑎𝑐,𝑑𝑉𝑎𝑐,𝑞

] = [𝑍𝐷𝑅𝑈,𝑑𝑑 𝑍𝐷𝑅𝑈,𝑑𝑞𝑍𝐷𝑅𝑈,𝑞𝑑 𝑍𝐷𝑅𝑈,𝑞𝑞

] [𝐼𝐷𝑅𝑈,𝑑𝐼𝐷𝑅𝑈,𝑞

]

( 4-63)

Therefore, both 𝑍𝑊𝑇 and 𝑍𝐷𝑅𝑈 can be obtained from both the state-space equations and by disturbance

injection to the detailed PSCAD model.

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4.2.2 WIND TURBINE IMPEDANCES IN DQ-FRAME

This section compares the d-q frame WTG impedances obtained from the state space equations and

those obtained from PSCAD by means of signal injection (Figure 4-13). The PSCAD model is based

on an averaged model of the grid-side converter, including the discrete time equivalent of the control

strategy programmed in C and including all relevant time delays. Transformers are modelled

considering saturation and the off-shore ac-grid is modelled by means of their aggregated equivalent,

as previously explained.

The considered operating point is defined by PT* = 0.5 pu and QT* = 0 pu. Figure 4-13 and Figure 4-14

show the resulting WTG dq-impedances (ZWT,dq).

Figure 4-13 ZWT_dd and ZWT_dq impedance of the wind turbine as a function of the frequency (Hz)

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Figure 4-14 ZWT_qd and ZWT_qq impedance of the wind turbine as a function of the frequency (Hz)

Clearly, both PSCAD and state space WTG impedance calculations closely match each other,

therefore validating the procedure carried out to obtain the WTG impedance.

To verify the theoretical analysis, a simulation has been carried out in PSCAD. The resulting dq-

impedances are shown as dots in Figure 4-13 and Figure 4-14 and they closely match the analytical

results.

4.2.3 DRU IMPEDANCE IN DQ-FRAME

The same procedure used for the calculation and validation of WTG d-q impedances is now applied

to the DRU station (which includes capacitor and filter banks, diode rectifiers, transformers and dc

smoothing reactor). The resulting DRU 𝑑𝑞-impedances are shown in Figure 4-15 and Figure 4-16.

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Figure 4-15 impedances of the DRU as a function of the frequency (Hz).

Figure 4-16 impedances of the DRU as a function of the frequency (Hz).

As in the previous case, the impedances obtained from PSCAD simulations show a very close match

to the analytical calculation. The few existing discrepancies are at frequencies that correspond to the

DRU switching harmonics.

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4.2.4 SMALL-SIGNAL STABILITY ANALYSIS

The small-signal stability of the system can be assessed by the impedance-based approach through

the frequency domain Generalized Nyquist Criterion (GNC). For this approach the system is split

between the impedance of the source (ZWT,dq) and the load (ZDRU,dq) as shown in Figure 4-17.

At the interfacing point, voltages and currents can be represented by the control system shown in

Figure 4-17.

The closed-loop transfer function of the negative feedback control system is:

𝑉𝑎𝑐,𝑑𝑞 = (𝐼 + 𝑍𝑊𝑇,𝑑𝑞𝑌𝐷𝑅𝑈,𝑑𝑞)−1𝑍𝑊𝑇,𝑑𝑞𝐼𝑍,𝑑𝑞

( 4-64)

and the stability of the system can be found by checking the generalised Nyquist criterion on the loop

impedance Ldq=ZWT,dqYDRU,dq.

As previously shown, the dq impedance matrices obtained from the analytical studies and those

obtained from PSCAD identification of the detail system agree to a very large extent. Therefore, those

impedances can be used in order to analyse the harmonic stability of the connected system.

The loop impedance Ldq=ZWT,dqYDRU,dq is a 2x2 complex matrix that varies with frequency. For each

individual frequency, the two eigenvalues of the loop impedance are calculated.

Figure 4-18 shows the two loop impedance eigenvalues, both from the analytical model and from the

PSCAD identified impedances for the baseline case. Both results have an excellent degree of

agreement up to 600 Hz, as previously mentioned. At 600 Hz the phase of one of the eigenvalues

calculated from PSCAD is very different from that obtained from the analytical model. Reasons for this

discrepancy are that the analytical DRU model does not consider the commutation harmonics which

are present in the detailed PSCAD simulation.

Figure 4-17 and relationship.

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Figure 4-18. Loop impedance matrix eigenvalues. Analytical study (1 in blue and 2 in red); circles show eigenvalues

obtained from impedance identification by PSCAD signal injection

Therefore, it can be concluded that the loop impedance (Ldq=ZWT,dqYDRU,dq) identified by PSCAD signal

injection and that obtained from the analytical model lead to very close eigenvalues and hence, it

validates the use of the identified d-q impedances for stability studies. From these values, a

generalised Nyquist diagram is obtained by plotting the open loop impedance eigenvalues for different

frequencies in a complex plane (Figure 4-19).

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Figure 4-19. Generalised Nyquist plot. Analytical study (1 in blue and 2 in red); circles show eigenvalues obtained from

impedance identification by PSCAD signal injection

Figure 4-19 shows the complex eigenvalues of the loop impedances for different frequency

(generalised Nyquist plot) corresponding to the baseline analytical model and also including those

obtained from the PSCAD impedance identification. Clearly, the system is stable and the values

obtained from PSCAD show a very good agreement with those from the analytical model.

Figure 4-20 Generalised Nyquist plots. Analytical study (1 in blue and 2 in red)

Figure 4-20 shows the eigenvalues of the loop transfer function ZWT,dqYDRU,dq for different frequencies

and for two different values of the WTG current loop resonant controller proportional gain (KRI). From

the root locus analysis, the system is stable for KRI=2100 and instable for KRI=1900.

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Figure 4-21 Zoom of the generalised Nyquist diagram. Analytical study (1 in blue and 2 in red)

Figure 4-21 shows the zoom of the previous generalised Nyquist diagrams around -1. Clearly, the

generalised Nyquist diagram does not encircle the -1 point for Kri=2100, whereas, when the Kri gain

is reduced to Kri=1900, the -1 point is now encircled by the first eigenvalue (1 shown in blue). As both

WTG and DRU are open loop stable, the fact that one of the eigenvalues crosses the point -1, clearly

shows that the system is instable for Kri=1900.

Figure 4-22 Eigenvalues of loop transfer function ZWT,dqYDRU,dq as a function of frequency. (1 in blue and 2 in red)

Cross-over at 230Hz

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Figure 4-22 shows clearly that the cross-over frequency of the eigenvalue causing instability is around

230 Hz (shown by the arrow), since at this frequency is where its value is 0dB and its phase 180º.

This result is in clear agreement with results of the root-locus analysis carried out in section 4.1.5 and

with the detailed PSCAD EMT simulation results shown in Figure 4-23.

Figure 4-23 shows the WTG d-axis voltage, DRU dc-side voltage and angle obtained both from the

PSCAD detailed simulation and the results obtained by using the analytical model.

Clearly, when Kri gain is 1900, the system shows an unstable 230 Hz oscillation. As previously

mentioned, this result agrees with both root-locus and generalised Nyquist analysis.

Figure 4-23 PSCAD detailed simulation of the complete system showing unstable oscillations at 232Hz (time shown in milliseconds).

4.3 CONCLUSIONS

This section included the use of the impedance based method for studying the harmonic stability of

grid forming converters connected to DRU-HVDC stations, both using an analytical approach and by

identifying the DRU and WTG impedances by means of signal injection. As mentioned in the

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introduction, small-signal state space analysis has been used in order to validate both the harmonic

impedance derived from PSCAD and the stability results obtained by means of the generalised Nyquist

criterion.

Obviously, the state space representation allows for a more detailed model of the considered system

and for its small-signal simulation. However, in many cases a state space representation is not

practical, as a full model of the system and all the controllers would be required.

Therefore, the harmonic impedance of both grid-forming converter and DRU station have been

obtained directly from the PSCAD model by means of perturbation injection.

As expected, both analytical and PSCAD-derived impedances show an excellent degree of agreement.

Based on the previous results, the generalised Nyquist diagrams have been obtained for two different

values of the WTG current resonant controller proportional gain (Kri), showing the transition between

stability and instability. The results obtained by means of the generalised Nyquist criterion agree with

those obtained from root-locus analysis, with the advantage that the impedance matrices used for the

generalised Nyquist criterion can be obtained experimentally without needing to know the details of

the WTG controllers.

The presented study has some limitations, including the limited usability of the generalised Nyquist

criterion for WTG control design, as physical meaning of the closed loop eigenvalues is not, generally,

straightforward.

Moreover, the presented study has been carried out for a particular operating point and for a relatively

simplified WTG aggregated model.

Therefore, although the results are sufficient to validate the application of impedance-based stability

to grid-forming DRU-connected WTGs, a comprehensive study should consider a more representative

number of operating points, include a thorough sensitivity study, a more detailed model of the WTG

converter and WPP, as well as a detailed frequency dependant model of the HVDC export cable.

These studies are planned for task 16.6.

4.4 SYSTEM PARAMETERS

The parameters used in the test system are listed in Table 4-1.

Table 4-1 Main parameters of the system under study

Aggregated wind turbine Value Unit

Grid side converter 0.69 kVac

50 Hz

400 MW

Transformer: 0.69/66 kV

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RT 0.004 pu

LT 0.1 pu

LC-Filter:

RW 0.008 pu

XW 0.1 pu

CW 20 pu

HVDC Rectifier

Capacitor Bank: CF 15.11 µF

DRU ac-side filter

Ca1 48.24 µF

Ca2 536 µF

La 18.9 mH

Ra1 4.12 Ω

Ra2 36.29 Ω

Cb 48.24 µF

Lb 1.88 mH

Rb 11.55 Ω

DC-smoothing reactor LR 0.2 H

PR controller parameters

KPI 0.0024400

KRI 1.5401835

KPV 394.87503

KRV 604914.93

Power controller parameters

KP,PM 0.5478x10-6

KI,PM 15.44x10-6

KP,QM 0.7x10-9

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5 BIBLIOGRAPHY

[1] J. Sun, C. Buchhagen and M. Greve, “Impedance Modeling and Simulation of Wind Turbines for Power System Harmonic Analysis,” in Wind Integration Workshop (WIW) Session 3A-1, Berlin, 2017.

[2] C. Li, “Unstable Operation of Photovoltaic Inverter from Field Experiences,” IEEE Transactions on Power Delivery, vol. 8977, no. c, pp. 1-8, 2017.

[3] E. Mollerstedt and B. Bernhardsson, “Out of control because of harmonics - an analysis of the harmonic response of an inverter locomotive,” IEEE Control Systems Magaine, vol. 20, no. 4, pp. 70-81, 2000.

[4] R. D. Middlebrook, “Input filter considerations in design and application,” in IEEE Industry Applications Society Annual Meet, 1976.

[5] L. Harnefors, M. Bongiorno and S. Lundberg, “Input-admittance calculation and shaping for controlled voltage-source converters,” IEEE Transactions on Industrial Electronics, vol. 54, no. 6, pp. 3323-3334, 2007.

[6] X. Wang, F. Blaabjerg and P. C. Loh, “An Impedance-Based Stability Analysis Method for Paralleled Voltage Source Converters,” in The 2014 International Power Electronics Conference, 2014.

[7] M. Cespedes and J. Sun, “Impedance modeling and analysis of grid-connected voltage-source converters,” IEEE Transactions on Power Electronics , vol. 29, no. 3, pp. 1254-1261, 2014.

[8] X. Wang, L. Harnefors and F. Blaabjerg, “A Unified Impedance Model of Grid-Connected Voltage-Source Converters,” IEEE Transactions on Power Electronics, vol. 33, no. 2, pp. 1775 - 1787, 2017.

[9] J. Sun, “Impedance-based stability criterion for grid-connected inverters,” IEEE Trans. Power Electron, vol. 26, no. 11, pp. 3075-3078, 2011.

[10] G. Wu, J. Liang, X. Zhou, Y. Li, A. Egea-Alvarez, G. Li, H. Peng and X. Zhang, “Analysis and design of vector control for VSC-HVDC connected to weak grids,” CSEE Journal of Power and Energy Systems, vol. 3, no. 2, pp. 115-124, 2017.

[11] L. Fan and Z. Miao, “An Explanation of Oscillations Due to Wind Power Plants Weak Grid Interconnection,” IEEE Transactions on Sustainable Energy, vol. PP, no. 99, 2017.

[12] X. Wang, F. Blaabjerg and M. Liserre, “An active damper to suppress multiple resonances with unknown frequencies,” in IEEE Applied Power Electronics Conference and Exposition - APEC 2014, 2014.

[13] S. Zhang, S. Jiang, X. Lu, B. Ge and F. Z. Peng, “Resonance Issues and Damping Techniques for Grid-Connected Inverters With Long Transmission Cable,” IEEE Transactions on Power Electronics, vol. 29, no. 1, pp. 110-120, 2014.

[14] C. Li, R. Burgos, Y. Tang and D. Boroyevich, “Impedance-based stability analysis of multiple STATCOMs in proximity,” in 2016 IEEE 17th Workshop on Control and Modeling for Power Electronics (COMPEL), 2016.

[15] C. Li, R. Burgos, Y. Tang and D. Boroyevich, “Application of D-Q frame impedance-based stability criterion in power systems with multiple STATCOMs in proximity,” in IECON 2017 - 43rd Annual Conference of the IEEE Industrial Electronics Society, 2017.

[16] Y. Sun, The impact of voltage-source-converters' control on the power system: the stability analysis of a power electronics dominant grid, Eindhoven: Technische Universiteit Eindhoven, 2018.

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[17] D. Pan, X. Ruan, C. Bao, W. Li and X. Wang, “Capacitor-current-feedback active damping with reduced computation delay for improving robustness of LCL-type grid-connected inverter,” IEEE Transactions on Power Electronics, vol. 29, no. 7, pp. 3414 - 3427, 2014.

[18] L. Harnefors, “Modeling of Three-Phase Dynamic Systems Using Complex Transfer Functions and Transfer Matrices,” IEEE Transactions on Industrial Electronics, vol. 54, no. 4, pp. 2239-2248, 2007.

[19] M. Josep, J. C. Vasquez, S. Golestan, S. Member and J. M. Guerrero, “Three-Phase PLLs : A Review of Recent Advances,” IEEE Transactions on Power Electronics , vol. 32, no. 3, pp. 1894-1907, 2016.

[20] F. Blaabjerg, R. Teodorescu, M. Liserre and A. V. Timbus, “Overview of control and grid synchronization for distributed power generation systems,” IEEE Transactions on Industrial Electronics, vol. 53, no. 5, pp. 1398 - 1409, 2006.

[21] A. Rygg, M. Molina, E. Unamuno, C. Zhang and X. Cai, “A simple method for shifting local dq impedance models to a global reference frame for stability analysis,” in Cornell University Computer Science Systems and Control, USA, 2017.

[22] J. Sun and H. Liu, “Sequence Impedance Modeling of Modular Multilevel Converters,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 5, no. 4, pp. 1427-1443, December 2017.

[23] H. Liu, HVDC Converters Impedance Modeling and System Stability Analysis, Troy, New York, 2017.

[24] A. Mohammad and M. Molinas, “Small-Signal Stability Assessment of Power Electronics Based Power Systems: A discussion of Impedance and Eigenvalue-Based Methods,” IEEE Transactions on Industry Applications, vol. 53, no. 5, pp. 5014-5030, 2017.

[25] S. Sudhoff, S. Glover, P. Lamm, D. Schmucker and D. Delisle, “Admittance Space Stability Analysis of Power Electronic Systems,” IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, vol. 36, no. 3, pp. 965-973, 2000.

[26] J. Martínez-Turégano, S. Añó-Villalba, G. Chaques-Herraiz, S. Bernal-Perez and R. Blasco-Gimenez, “Model aggregation of large wind farms for dynamic studies,” IECON 2017 - 43rd Annual Conference of the IEEE Industrial Electronics Society, Beijing, pp. pp. 316-321., 2017.

[27] R. Blasco-Gimenez, S. Añó-Villalba, J. Rodriguez-D'Derlée, S. Bernal-Perez and F. Morant, “Diode-Based HVdc Link for the Connection of Large Offshore Wind Farms,” Energy Conversion, IEEE Transactions on, vol. 26, pp. 615-626, 2011.

[28] A. Semlyen and A. Deri, “Time Domain Modeling of Frequency Dependent Three-Phase Transmission Line Impedance,” IEEE Power Engineering Review, Vols. PER-5, pp. 64-65, 6 1985.

[29] J. A. R. Macias, A. G. Exposito and A. B. Soler, “A comparison of techniques for state-space transient analysis of transmission lines,” IEEE Transactions on Power Delivery, vol. 20, pp. 894-903, 4 2005.

[30] M. Szechtman, T. Wess and C. V. Thio, “A benchmark model for HVDC system studies,” in International Conference on AC and DC Power Transmission, 1991, 1991.

[31] PROMOTioN - WP2, “Deliverable 2.1 - Grid topology and model specification,” Brussels, 2016.

[32] PROMOTioN WP16, “Deliverable D16.7 - Lab Documentation,” Brussels, 2019.

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6 APPENDIX

6.1 MMC VSC: MULTI-HARMONIC LINEARIZED MODEL

Elimination of m

𝐙𝐥 au = −p − 𝐯au⨂au −𝐦au⨂au + m (6.1)

Since there are no zero-sequence harmonics in the phase current, the differential-mode zero-

sequence harmonics in arm currents which would form the zero-sequence harmonics in phase current

should not exist. Through the implementation of the model, it’s observed that the effect of m in

equation (6.1) respectively equation (3.33) of chapter 3.4 is to cancel out the differential-mode zero-

sequence component that should not occur in the arm current. If m is dropped from equation (6.1)

without any other modification to this equation, (6.1) will wrongly predict zero-sequence differential-

mode harmonics in arm current au, which means the model would wrongly predict zero-sequence

harmonics in phase current. To eliminate m while preventing the zero-sequence component from

appearing in phase current, the phase relationship of small-signal harmonics should be reviewed.

For a positive -sequence perturbation, the phase shift among phases is 2𝜋/3. When combined with

the phase shift of steady-state harmonics at frequency 𝑘 ⋅ 𝑓1, the phase shift of a small-signal harmonic

at frequency 𝑓𝑝 + 𝑘𝑓1 between adjacent phases will be

2𝜋

3+2𝑘𝜋

3=2(𝑘 + 1)𝜋

3 (6.2)

Hence the sequence of this harmonic at frequency 𝑓𝑝+ 𝑘𝑓

1 can be determined by a remainder-

calculating function 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3), which calculates the remainder of 𝑘 + 1 divided by 3. The result

of this function can be 1,0 and -1. The reminder-calculating function is defined as

𝑚𝑜𝑑𝑢𝑙𝑜(𝑘, 3) =

1 𝑖𝑓 𝑘 = 3𝑛 + 10 𝑖𝑓 𝑘 = 3𝑛

−1 𝑖𝑓 𝑘 = 3𝑛 − 1 (6.3)

Similarly, for a negative sequence perturbation, the sequence of the harmonic corresponding to

frequency 𝑓𝑝+ 𝑘𝑓

1 can be determined by 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 − 1,3).

The phase difference between upper and lower arms can be obtained from Table 3-1. The phase shift

of the 𝑘th steady-state harmonic between the two arms of the same phase can be determined by

𝑚𝑜𝑑𝑢𝑙𝑜(𝑘, 2) ∙ 𝜋. When combined with the phase shift of the perturbation signal, the phase shift of a

small-signal harmonic at frequency 𝑓𝑝 + 𝑘𝑓1 between upper and lower arms in a single phase can be

calculated by 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,2) ∙ 𝜋.

If 𝑘 is an even integer, the harmonics in upper arm and lower arm have the same magnitude but a

phase difference of 𝜋. Harmonics with this characteristic will be referred to as differential-mode

harmonics. According to the defined direction of the arm current, as seen in Figure 3-5, the differential-

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mode harmonics in the arm currents are outputed to the AC terminal to form the phase current. On the

contrary, when 𝑘 is odd, the harmonics in upper and lower arms have the same magnitude and the

same phase. Those harmonics will be referred to as common-mode harmonics. These harmonics in

arm currents are not outputed to the AC terminal, insteadt hey form the circulating current.

To eliminate m in the model, equation (3.33) can be modified to

𝐚𝐮 = −𝐘𝐥(𝐩 + 𝐯𝐚𝐮⨂𝐚𝐮 + 𝐦𝐚𝐮⨂𝐚𝐮 − 𝐦) (6.4)

In which 𝐘𝐥 is the inverse of 𝐙𝒍.

𝒀𝒍 = 𝑑𝑖𝑎𝑔 [1

𝑗2𝜋 (𝑓𝑝− 𝑛𝑓

1) 𝐿 + 𝑟𝐿

, ⋯ ,1

𝑗2𝜋𝑓𝑝𝐿 + 𝑟𝐿

, ⋯ ,1

𝑗2𝜋 (𝑓𝑝+ 𝑛𝑓

1) 𝐿 + 𝑟𝐿

] (6.5)

The zero-sequence harmonics in the phase current correspond to the zero-sequence differential-mode

harmonic in the arm current. In order to force the zero-sequence differential-mode harmonics to zero

without relying on m, 𝐘𝐥 can be modified such that the elements to be multiplied with differential-mode

zero-sequence component are set to zero.

𝐘𝒍 = 𝑑𝑖𝑎𝑔 [⋯ ,0,1

𝑗2𝜋(𝑓𝑝−3𝑓1)𝐿+𝑟𝐿, ⋯ ,

1

𝑗2𝜋𝑓𝑝𝐿+𝑟𝐿,

1

𝑗2𝜋(𝑓𝑝+𝑓1)𝐿+𝑟𝐿, 0,

1

𝑗2𝜋(𝑓𝑝+3𝑓1)𝐿+𝑟𝐿, ⋯ ] (6.6)

Correspondingly, when the perturbation is of negative-sequence, the small-signal harmonics at

frequency 𝑓𝑝+ 𝑘𝑓

1 are zero-sequence differential-mode when 𝑘 satisfies 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 − 1,3) = 0 and

𝑚𝑜𝑑𝑢𝑙𝑜(𝑘, 2) = 0, which includes 𝑘 = ⋯ ,−8,−2,4,10,⋯. The modified 𝐘𝒍 is

𝐘𝒍 = 𝑑𝑖𝑎𝑔 [⋯ ,0,1

𝑗2𝜋(𝑓𝑝−𝑓1)𝐿+𝑟𝐿,

1

𝑗2𝜋𝑓𝑝𝐿+𝑟𝐿, ⋯ ,

1

𝑗2𝜋(𝑓𝑝+3𝑓1)𝐿+𝑟𝐿, 0,

1

𝑗2𝜋(𝑓𝑝+5𝑓1)𝐿+𝑟𝐿, ⋯ ] (6.7)

Due to the modification made in 𝐘𝒍, 𝑚 can be dropped from (6.4). As result the model involves only

variables of a single arm. Subscripts are dropped in the following to simplify the expression for the

resulting equations of the small-signal frequency-domain model:

= −𝐘𝒍(𝒑 + 𝐯⨂ + 𝐦⨂) (6.8)

𝐘𝐜 = 𝐢⨂ + 𝐦⨂𝐢 (6.9)

6.2 MMC VSC: PHASE CURRENT CONTROL

Relationship between perturbation in arm current and perturbation in insertion index

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A positive-sequence perturbation in the grid voltage will lead to positive-sequence perturbation in the

arm current. A Fourier coefficient 𝑆𝑖𝑎𝑝𝑘

is assumed for the harmonic component in the arm current of

phase 𝑎 upper arm at frequency 𝑓𝑝 + 𝑘 ⋅ 𝑓1:

𝑆𝑖𝑎𝑝𝑘= 𝐼𝑃𝑘𝑒

𝑗𝜑𝑝𝑘 (6.10)

The superscript 𝑝𝑘 indicates the frequency at 𝑓𝑝 + 𝑘𝑓1, the subscript 𝑖𝑎 of 𝑆𝑖𝑎𝑝𝑘

indicates that the Fourier

coefficient stands for the current of phase 𝑎. The magnitude of the Fourier coefficient 𝐼𝑃𝑘 is half of the

magnitude of the corresponding sinusoidal signal, and the phase 𝜑𝑝𝑘 represents the initial phase of

the corresponding sinusoidal signal. Thus, the three-phase arm current perturbations in time-domain

are

𝑎

𝑝𝑘= 2𝐼𝑃𝑘 cos (2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) 𝑡 + 𝜑𝑝𝑘)

𝑏𝑝𝑘= 2𝐼𝑃𝑘 cos (2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) 𝑡 + 𝜑𝑝𝑘 −

2(𝑘 + 1)𝜋

3)

𝑐𝑝𝑘= 2𝐼𝑃𝑘 cos (2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) 𝑡 + 𝜑𝑝𝑘 +

2(𝑘 + 1)𝜋

3)

. (6.11)

The corresponding perturbation in 𝑑 and 𝑞 axis 𝑑𝑝𝑘

and 𝑞𝑝𝑘

are:

𝑑𝑝𝑘= √

2

3∙ 2𝐼𝑃𝑘 ∙ cos [2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) 𝑡 + 𝜑𝑝𝑘] ∙ cos(2𝜋𝑓1𝑡 + 𝜑1)

+ cos [2𝜋 (𝑓𝑝+ 𝑘𝑓

1 ) 𝑡 + 𝜑

𝑝𝑘−2(𝑘 + 1)𝜋

3]

∙ cos (2𝜋𝑓1𝑡 + 𝜑

1−2𝜋

3)

+ cos(2𝜋 (𝑓𝑝+ 𝑘𝑓

1 ) 𝑡 + 𝜑

𝑝𝑘+2(𝑘 + 1)𝜋

3)

∙ cos (2𝜋𝑓1𝑡 + 𝜑

1+2𝜋

3)

(6.12)

𝑞𝑝𝑘= −√

2

3∙ 2𝐼𝑃𝑘 [cos [2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) 𝑡 + 𝜑𝑝𝑘] ∙ sin(2𝜋𝑓1𝑡 + 𝜑1)

+ cos [2𝜋 (𝑓𝑝+ 𝑘𝑓

1 ) 𝑡 + 𝜑

𝑝𝑘−2(𝑘 + 1)𝜋

3]

∙ sin (2𝜋𝑓1𝑡 + 𝜑

1−2𝜋

3)

+ cos [2𝜋 (𝑓𝑝+ 𝑘𝑓

1 ) 𝑡 + 𝜑

𝑝𝑘+2(𝑘 + 1)𝜋

3]

∙ sin (2𝜋𝑓1𝑡 + 𝜑

1+2𝜋

3)]

(6.13)

Simplification equations (6.12) and (6.13) gives:

𝑑𝑝𝑘= √

3

2∙ 2𝐼𝑃𝑘 ∙ cos [2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) 𝑡 + 𝜑𝑝𝑘 − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ (2𝜋𝑓1𝑡 + 𝜑1)]

(6.14)

PROJECT REPORT

81

𝑞𝑝𝑘= √

3

2∙ 2𝐼𝑃𝑘 ∙ 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3)

∙ sin [2𝜋 (𝑓𝑝+ 𝑘𝑓

1 ) 𝑡 + 𝜑

𝑝𝑘− 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ (2𝜋𝑓

1𝑡 + 𝜑

1)]

(6.15)

A perturbation in 𝑖𝑑 and 𝑖𝑞 causes perturbations in 𝑚𝑑 and 𝑚𝑞. Denoting the perturbations in 𝑚𝑑 and

𝑚𝑞 as 𝑑𝑝𝑘

and 𝑞𝑝𝑘

, and representing the Fourier coefficients of 𝐼𝑑𝑝𝑘

, 𝐼𝑞𝑝𝑘, 𝑑

𝑝𝑘 and 𝑞

𝑝𝑘 as 𝑆𝑖𝑑

𝑝𝑘, 𝑆𝑖𝑞

𝑝𝑘,

𝑆𝑚𝑑𝑝𝑘

and 𝑆𝑚𝑞𝑝𝑘 , the perturbation in the insertion index is derived according to the block diagram in Figure

3-6 as follows:

𝑆𝑚𝑑𝑝𝑘 = 𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 2𝜋𝑓1] ∙ 𝑆𝑖𝑑

𝑝𝑘+ 𝐾𝑖𝑑 ∙ 𝑆𝑖𝑞

𝑝𝑘 (6.16)

𝑆𝑚𝑞𝑝𝑘 = 𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 2𝜋𝑓1] ∙ 𝑆𝑖𝑞

𝑝𝑘− 𝐾𝑖𝑑 ∙ 𝑆𝑖𝑑

𝑝𝑘 (6.17)

The perturbations in 𝑚𝑎, 𝑚𝑏 and 𝑚𝑐 are

[

𝑎𝑝𝑘

𝑏𝑝𝑘

𝑐𝑝𝑘

] = 𝑻𝒅𝒒𝑻 [𝑑𝑝𝑘

𝑞𝑝𝑘]. (6.18)

Thus, the perturbation in the insertion index of phase 𝑎 is:

𝑎𝑝𝑘= √

2

3∙ [𝑑

𝑝𝑘∙ cos(2𝜋𝑓

1𝑡 + 𝜑

1) − 𝑞

𝑝𝑘∙ sin(2𝜋𝑓

1𝑡 + 𝜑

1)] (6.19)

Substituting equations (6.14)-(6.17) into (6.19) and simplification gives:

𝑎𝑝𝑘= 2𝐼𝑃𝑘 |𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 2𝜋𝑓1]|

∙ cos [2𝜋 (𝑓𝑝+ 𝑘𝑓

1 ) 𝑡 + 𝜑

𝑝𝑘+ 𝜑

𝐻𝑖] + 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 𝐾𝑖𝑑

∙ cos [2𝜋 (𝑓𝑝+ 𝑘𝑓

1 ) 𝑡 + 𝜑

𝑝𝑘−𝜋

2]

(6.20)

Where 𝜑𝐻𝑖

is the phase of 𝐻𝑖[𝑗2𝜋(𝑓𝑝 + 𝑘𝑓1 ) − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 2𝜋𝑓1].

The Fourier coefficient of 𝑎𝑝𝑘

(at frequency 𝑓𝑝+ 𝑘𝑓

1) is

𝑆𝑚𝑎𝑝𝑘 = 𝐼𝑃𝑘𝑒

𝑗𝜑𝑝𝑘 𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 2𝜋𝑓1] − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3)

∙ 𝑗𝐾𝑖𝑑

(6.21)

The substitution of (6.10) into (6.21) gives the relationship between the Fourier coefficient of the

perturbation in arm current and the Fourier coefficient of the perturbation in the insertion index at

frequency 𝑓𝑝 + 𝑘𝑓1.

𝑆𝑚𝑎𝑝𝑘 = 𝑆𝑖𝑎

𝑝𝑘𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 2𝜋𝑓1] − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3)

∙ 𝑗𝐾𝑖𝑑

(6.22)

PROJECT REPORT

82

It has to be noted that this relationship stands for the phase current control only and affects the

differential-mode non-zero-sequence harmonics in the arm currents which form the phase current

harmonics. As pointed out in section 3.4 when discussing the phase relationship of harmonics , the

Fourier coefficients can be used to distinguish the differential-mode non-zero-sequence harmonics in

arm currents. A coefficient 1 + (−1)𝑘/2 is used to distinguish the differential-mode harmonics. When

𝑘 is an even integer, the harmonic at frequency 𝑓𝑝+ 𝑘𝑓

1 is a differential-mode harmonic, the coefficient

1 + (−1)𝑘/2 equals 1. When 𝑘 is an odd integer, the harmonic at frequency 𝑓𝑝+ 𝑘𝑓

1 is a common-

mode harmonic and the coefficient 1 + (−1)𝑘/2 equals 0. The coefficient |𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3)| as defined

in (6.3) is used to distinguish the sequence of the harmonics. The coefficient |𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3)| equals

1 for positive and negative sequence harmonics and equals 0 for zero-sequence harmonics. As a

result, the contribution of phase current control to the dynamics of the insertion index 1 can be

modeled in the frequency domain as:

1 = 𝐐i𝐢 (6.23)

6.3 MMC VSC: CIRCULATING CURRENT CONTROL

Relationship between perturbation in arm current and perturbation in insertion index

Assuming a positive-sequence perturbation, the Fourier coefficient for a harmonic component in the

arm current of phase 𝑎 upper arm at frequency 𝑓𝑝 + 𝑘𝑓1, 𝑘 = −𝑛,…− 1,0,1, … , 𝑛, the corresponding

perturbation in 𝑑 and 𝑞 axis 𝑑𝑝𝑘

and 𝑞𝑝𝑘

are as follows when using using the same time-domain

expression of the perturbation current as defined in (6.10) and (6.11):

𝑑𝑝𝑘= √

2

3∙ 2𝐼𝑃𝑘 ∙ cos [2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) 𝑡 + 𝜑𝑝𝑘] ∙ cos(4𝜋𝑓1𝑡 + 2𝜑1)

+ cos [2𝜋 (𝑓𝑝+ 𝑘𝑓

1 ) 𝑡 + 𝜑

𝑝𝑘−2(𝑘 + 1)𝜋

3]

∙ cos (4𝜋𝑓1𝑡 + 2𝜑

1+2𝜋

3)

+ cos(2𝜋 (𝑓𝑝+ 𝑘𝑓

1 ) 𝑡 + 𝜑

𝑝𝑘+2(𝑘 + 1)𝜋

3)

∙ cos (4𝜋𝑓1𝑡 + 2𝜑

1−2𝜋

3)

(6.24)

PROJECT REPORT

83

𝑞𝑝𝑘= −√

2

3∙ 2𝐼𝑃𝑘 [𝑐𝑜𝑠 [2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) 𝑡 + 𝜑𝑝𝑘] ∙ 𝑠𝑖𝑛(4𝜋𝑓1𝑡 + 2𝜑1)

+ 𝑐𝑜𝑠 [2𝜋 (𝑓𝑝+ 𝑘𝑓

1 ) 𝑡 + 𝜑

𝑝𝑘−2(𝑘 + 1)𝜋

3]

∙ 𝑠𝑖𝑛 (4𝜋𝑓1𝑡 + 2𝜑

1+2𝜋

3)

+ 𝑐𝑜𝑠 [2𝜋 (𝑓𝑝+ 𝑘𝑓

1 ) 𝑡 + 𝜑

𝑝𝑘+2(𝑘 + 1)𝜋

3]

∙ 𝑠𝑖𝑛 (4𝜋𝑓1𝑡 + 2𝜑

1−2𝜋

3)]

(6.25)

Simplifying equations (6.24) and (6.25) gives:

𝑑𝑝𝑘= √

3

2∙ 2𝐼𝑃𝑘 ∙ cos [2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) 𝑡 + 𝜑𝑝𝑘 + 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ (4𝜋𝑓1 + 2𝜑1)]

(6.26)

𝑞𝑝𝑘= −√

3

2∙ 2𝐼𝑃𝑘 ∙ 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3)

∙ sin [2𝜋 (𝑓𝑝+ 𝑘𝑓

1 ) 𝑡 + 𝜑

𝑝𝑘+ 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ (4𝜋𝑓

1+ 2𝜑

1)]

(6.27)

The perturbation in 𝑖𝑑 and 𝑖𝑞 will cause perturbation in 𝑚𝑑 and 𝑚𝑞. Perturbations in 𝑚𝑑 and 𝑚𝑞 are

denoted as 𝑑𝑝𝑘

and 𝑞𝑝𝑘

, the Fourier coefficients of 𝐼𝑑𝑝𝑘

, 𝐼𝑞𝑝𝑘, 𝑑

𝑝𝑘 and 𝑞

𝑝𝑘 are denoted by 𝑆𝑖𝑑

𝑝𝑘, 𝑆𝑖𝑞

𝑝𝑘,

𝑆𝑚𝑑𝑝𝑘

and 𝑆𝑚𝑞𝑝𝑘

. According to the block diagram seen in Figure 3-6 perturbations of the insertion index is

as follows:

𝑆𝑚𝑑𝑝𝑘 = 𝐻𝑐 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) + 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 4𝜋𝑓1] ∙ 𝑆𝑖𝑑

𝑝𝑘+ 𝐾𝑖𝑑 ∙ 𝑆𝑖𝑞

𝑝𝑘 (6.28)

𝑆𝑚𝑞𝑝𝑘 = 𝐻𝑐 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) + 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 4𝜋𝑓1] ∙ 𝑆𝑖𝑞

𝑝𝑘− 𝐾𝑖𝑑 ∙ 𝑆𝑖𝑑

𝑝𝑘 (6.29)

The perturbations in a,b,c frame are:

[

𝑎𝑝𝑘

𝑏𝑝𝑘

𝑐𝑝𝑘

] = 𝑻𝒅𝒒−𝒄𝒄𝒄𝑻 [

𝑑𝑝𝑘

𝑞𝑝𝑘] , (6.30)

with 𝑻𝒅𝒒−𝒄𝒄𝒄𝑻 being the transpose of matrix 𝑻𝒅𝒒−𝒄𝒄𝒄 (3.55). Thus, the perturbation in the insertion index

of phase 𝑎 is:

𝑎𝑝𝑘= √

2

3∙ [𝑑

𝑝𝑘∙ cos(4𝜋𝑓

1𝑡 + 2𝜑

1) − 𝑞

𝑝𝑘∙ sin(4𝜋𝑓

1𝑡 + 2𝜑

1)] (6.31)

Substituting (6.26)-(6.29) into (6.31) and simplifying the equation, gives:

𝑎𝑝𝑘= 2𝐼𝑃𝑘 |𝐻𝑐 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) + 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 4𝜋𝑓1]|

∙ cos [2𝜋 (𝑓𝑝+ 𝑘𝑓

1 ) 𝑡 + 𝜑

𝑝𝑘+ 𝜑

𝐻𝑐] − 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 𝐾𝑖𝑑

∙ cos [2𝜋 (𝑓𝑝+ 𝑘𝑓

1 ) 𝑡 + 𝜑

𝑝𝑘−𝜋

2]

(6.32)

Within (6.32) 𝜑𝐻𝑐

is the phase of 𝐻𝑐 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) + 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 4𝜋𝑓1].

PROJECT REPORT

84

The Fourier coefficient of 𝑎𝑝𝑘

(at frequency 𝑓𝑝 + 𝑘𝑓1) is:

𝑆𝑚𝑎𝑝𝑘 = 𝐼𝑃𝑘𝑒

𝑗𝜑𝑝𝑘 𝐻𝑐 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) + 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 4𝜋𝑓1] + 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3)

∙ 𝑗𝐾𝑖𝑑

(6.33)

Substituting (6.10) into (6.33) gives the relationship between the Fourier coefficient of the perturbation

in the arm current and the Fourier coefficient of the perturbation in the insertion index at frequency

𝑓𝑝 + 𝑘𝑓1:

𝑆𝑚𝑎𝑝𝑘 = 𝑆𝑖𝑎

𝑝𝑘𝐻𝑐 [𝑗2𝜋 (𝑓𝑝 + 𝑘𝑓1 ) + 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3) ∙ 4𝜋𝑓1] + 𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3)

∙ 𝑗𝐾𝑖𝑑

(6.34)

Within equation (6.34) 𝐻𝑐(𝑠) is the transfer function of the PI controller of the circulating current control.

Due to the fact that the circulating current control acts only on the common-mode harmonics of the

arm currents, the coefficient 1 − (−1)𝑘/2 is used to distinguish between the differential-mode and

common-mode harmonics. Because the circulating current control is implemented in 𝑑𝑞 frame, the

controller has no effect on zero-sequence harmonics. The coefficient |𝑚𝑜𝑑𝑢𝑙𝑜(𝑘 + 1,3)| is used to

distinguish between the sequence of harmonics similar to the phase current control.

6.4 MMC VSC: PHASE LOCKED LOOP

Fourier coefficient small-signal insertion index

Considering a positive perturbation at frequency 𝑓𝑝 is added to the steady-state balanced grid voltage

the Fourier coefficient of the perturbation is:

𝑆𝑣𝑎𝑝𝑘 =

1

2𝑉𝑝𝑒

𝑗𝜑𝑝 (6.35)

Assuming the Fourier coefficient for the fundamental component of phase 𝑎 steady-state current is

𝐈1 = 𝐼1𝑒𝑗𝜑𝑖1, (6.36)

And the time-domain expression of the fundamental component of the steady-state current equals to

𝑖𝑎 = 2𝐼1 cos(2𝜋𝑓1𝑡 + 𝜑𝑖1)

𝑖𝑏 = 2𝐼1 cos (2𝜋𝑓1𝑡 + 𝜑𝑖1 −2𝜋

3)

𝑖𝑐 = 2𝐼1 cos (2𝜋𝑓1𝑡 + 𝜑𝑖1 +2𝜋

3)

. (6.37)

Applying the transformation 𝑻𝒅𝒒 (𝜃0 +𝜋

2) gives:

[𝑖𝑑𝑖𝑞] = 𝑻𝒅𝒒 (𝜃0 +

𝜋

2) [

𝑖𝑎𝑖𝑏𝑖𝑐

] (6.38)

The resulting 𝑑 and 𝑞 axis values for the current are:

PROJECT REPORT

85

𝑖𝑑 = √2

3∙ 2𝐼1 ∙ [cos (2𝜋𝑓1𝑡 + 𝜑1 +

𝜋

2) ∙ cos(2𝜋𝑓

1𝑡 + 𝜑

𝑖1)

+ cos (2𝜋𝑓1𝑡 + 𝜑

1+𝜋

2−2𝜋

3) ∙ cos (2𝜋𝑓

1𝑡 + 𝜑

𝑖1−2𝜋

3)

+ cos (2𝜋𝑓1𝑡 + 𝜑

1+𝜋

2+2𝜋

3) ∙ cos (2𝜋𝑓

1𝑡 + 𝜑

𝑖1+2𝜋

3)]

= √3

2∙ 2𝐼1 ∙ cos (𝜑𝑖1 − 𝜑1 −

𝜋

2)

(6.39)

𝑖𝑞 = −√2

3∙ 2𝐼1 ∙ [sin (2𝜋𝑓1𝑡 + 𝜑1 +

𝜋

2) ∙ cos(2𝜋𝑓

1𝑡 + 𝜑

𝑖1)

+ sin (2𝜋𝑓1𝑡 + 𝜑

1+𝜋

2−2𝜋

3) ∙ cos (2𝜋𝑓

1𝑡 + 𝜑

𝑖1−2𝜋

3)

+ sin (2𝜋𝑓1𝑡 + 𝜑

1+𝜋

2+2𝜋

3) ∙ cos (2𝜋𝑓

1𝑡 + 𝜑

𝑖1+2𝜋

3)]

= √3

2∙ 2𝐼1 ∙ sin (𝜑𝑖1 − 𝜑1 −

𝜋

2)

(6.40)

The multiplications of (𝑡) with 𝑖𝑑 and 𝑖𝑞 gives:

𝑖𝑑 ∙ (𝑡) = √3

2∙ 2𝐼1 𝑐𝑜𝑠 (𝜑𝑖1 − 𝜑1 −

𝜋

2)

∙ 𝑉𝑝 cos [2𝜋 (𝑓𝑝 − 𝑓1) 𝑡 + 𝜑

𝑝− 𝜑

1−𝜋

2+ 𝜑

𝐺]

∙ |𝐺𝜃 [𝑗2𝜋 (𝑓𝑝 − 𝑓1)]|

(6.41)

𝑖𝑞 ∙ (𝑡) = √3

2∙ 2𝐼1 sin (𝜑𝑖1 − 𝜑1 −

𝜋

2)

∙ 𝑉𝑝 cos [2𝜋 (𝑓𝑝 − 𝑓1) 𝑡 + 𝜑

𝑝− 𝜑

1−𝜋

2+ 𝜑

𝐺]

∙ |𝐺𝜃 [𝑗2𝜋 (𝑓𝑝 − 𝑓1)]|

(6.42)

Supposing that the small-signal harmonic in the output of the PI regulator are 𝑑 and 𝑞, representing

their Fourier coefficients by 𝑆𝑚𝑑 and 𝑆𝑚𝑞 and representing the Fourier coefficients of 𝑖𝑑 ∙ (𝑡) and 𝑖𝑞 ∙

(𝑡) by 𝑆𝑖𝑑𝜃 and 𝑆𝑖𝑞𝜃 their relationship would be:

𝑆𝑚𝑑 = 𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 − 𝑓1 )] ∙ 𝑆𝑖𝑑𝜃 + 𝐾𝑖𝑑 ∙ 𝑆𝑖𝑞𝜃 (6.43)

𝑆𝑚𝑞 = 𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 − 𝑓1 )] ∙ 𝑆𝑖𝑞𝜃 − 𝐾𝑖𝑑 ∙ 𝑆𝑖𝑑𝜃 (6.44)

The small-signal harmonic in 𝑎𝑏𝑐 frame can be obtained through inverse 𝑑𝑞 transformation:

PROJECT REPORT

86

[

𝑎′

𝑏′

𝑐′

] = 𝑻𝒅𝒒𝑻 (𝜃0) [

𝑑𝑞] (6.45)

The small-signal harmonic in the insertion index of phase 𝑎 is

𝑎′= √

2

3∙ [𝑑 ∙ cos(2𝜋𝑓1𝑡 + 𝜑1) − 𝑞 ∙ sin(2𝜋𝑓1𝑡 + 𝜑1)]. (6.46)

Substituting (6.41)-(6.44) into (6.46) gives:

𝑎′= 𝑉𝑝 ∙ 𝐼1 ∙ |𝐺𝜃 [𝑗2𝜋 (𝑓𝑝 − 𝑓1

)]|

∙ |𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 − 𝑓1 )]|

∙ [−cos (2𝜋𝑓𝑝𝑡 + 𝜑

𝑝+ 𝜑

𝑖1− 𝜑

1+ 𝜑

𝐺+ 𝜑

𝐻𝑖)

+ cos (2𝜋 (𝑓𝑝− 2𝑓

1) 𝑡 + 𝜑

𝑝− 𝜑

𝑖1− 𝜑

1+ 𝜑

𝐺+ 𝜑

𝐻𝑖)] + 𝐾𝑖𝑑

∙ [cos (2𝜋𝑓𝑝𝑡 + 𝜑

𝑝+ 𝜑

𝑖1− 𝜑

1+𝜋

2+ 𝜑

𝐺)

+ cos (2𝜋 (𝑓𝑝− 2𝑓

1) 𝑡 + 𝜑

𝑝− 𝜑

𝑖1− 𝜑

1+𝜋

2+ 𝜑

𝐺)]

(6.47)

where the phase 𝜑𝐻𝑖

represents the phase of 𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 − 𝑓1 )].

Secondly, the dc component of the phase current control output, 𝑚𝑑 and 𝑚𝑞, would produce

fundamental component due to the inverse transformation 𝑻𝒅𝒒𝑻 (𝜃0 + 𝜋/2). In addition a component at

frequency 𝑓𝑝 and another at frequency 𝑓𝑝 − 2𝑓1 would be generated due to the multiplication with (𝑡).

Assuming that the Fourier coefficient for the fundamental component of phase 𝑎 of the steady-state

insertion index is

𝐌1 = 𝑀1𝑒𝑗𝜑𝑚, (6.48)

in 𝑑𝑞 frame, 𝑚𝑑 and 𝑚𝑞 can be calculated by applying the 𝑑𝑞 transformation to:

[𝑚𝑑𝑚𝑞] = 𝑻𝒅𝒒(𝜃0) [

𝑚𝑎𝑚𝑏𝑚𝑐

] (6.49)

𝑚𝑑 = √2

3∙ 2𝑚1 [cos(2𝜋𝑓1𝑡 + 𝜑1) ∙ cos(2𝜋𝑓1𝑡 + 𝜑𝑚)

+ cos (2𝜋𝑓1𝑡 + 𝜑

1−2𝜋

3) ∙ cos (2𝜋𝑓

1𝑡 + 𝜑

𝑚−2𝜋

3)

+ cos (2𝜋𝑓1𝑡 + 𝜑

1+2𝜋

3) ∙ cos (2𝜋𝑓

1𝑡 + 𝜑

𝑚+2𝜋

3)]

= √2

3∙ 2𝑚1 ∙ cos(𝜑𝑚 − 𝜑1)

(6.50)

PROJECT REPORT

87

𝑚𝑞 = −√2

3∙ 2𝑚1 [sin(2𝜋𝑓1𝑡 + 𝜑1) ∙ cos(2𝜋𝑓1𝑡 + 𝜑𝑚) + sin (2𝜋𝑓1𝑡 + 𝜑1 −

2𝜋

3)

∙ cos (2𝜋𝑓1𝑡 + 𝜑

𝑚−2𝜋

3) + sin (2𝜋𝑓

1𝑡 + 𝜑

1+2𝜋

3)

∙ cos (2𝜋𝑓1𝑡 + 𝜑

𝑚+2𝜋

3)] = √

2

3∙ 2𝑚1 ∙ sin(𝜑𝑚 − 𝜑1)

(6.51)

Transforming 𝑚𝑑 and 𝑚𝑞 by the inverse transformation matrix 𝑻𝒅𝒒𝑻 (𝜃0 + 𝜋/2) into 𝑎𝑏𝑐 frame gives:

[

𝑚𝑎"

𝑚𝑏"

𝑚𝑐"

] = 𝑻𝒅𝒒𝑻 (𝜃0 +

𝜋

2) [𝑚𝑑𝑚𝑞] (6.52)

𝑚𝑎" = 2𝑚1 [cos (2𝜋𝑓1𝑡 + 𝜑1 +

𝜋

2) ∙ cos(𝜑

𝑚− 𝜑

1)

− sin (2𝜋𝑓1𝑡 + 𝜑

1+𝜋

2) ∙ sin(𝜑

𝑚− 𝜑

1)]

= 2𝑚1 ∙ cos (2𝜋𝑓1𝑡 + 𝜑𝑚 +𝜋

2)

(6.53)

The small-signal harmonic in the insertion index of phase 𝑎 is

𝑎"= 𝑚𝑎

" ∙ (𝑡) = 𝑉𝑝 ∙ 𝑚1

∙ |𝐺𝜃 [𝑗2𝜋 (𝑓𝑝 − 𝑓1)]| cos (2𝜋𝑓

𝑝𝑡 + 𝜑

𝑝+ 𝜑

𝑚− 𝜑

1+ 𝜑

𝐺)

− cos (2𝜋 (𝑓𝑝− 2𝑓

1) 𝑡 + 𝜑

𝑝− 𝜑

𝑚− 𝜑

1+ 𝜑

𝐺)

(6.54)

Therefore, the small-signal harmonics in the insertion index caused by the dynamics of PLL is:

𝑎 = 𝑎′+ 𝑎

" (6.55)

The Fourier coefficient of 𝑎 at frequency 𝑓𝑝 is:

𝑆𝑚𝑎𝑓𝑝−𝑓1 =

1

2𝑉𝑝𝑒

𝑗𝜑𝑝𝐺𝜃 [𝑗2𝜋 (𝑓𝑝 − 𝑓1)] 𝐼1𝑒

−𝑗𝜑𝑖1 [𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 − 𝑓1 )] + 𝑗𝐾𝑖𝑑]

− 𝑚1𝑒−𝑗𝜑𝑚 𝑒−𝑗𝜑1

(6.56)

Substituting (6.35), (6.36) and (6.48) into (6.55) and (6.56) and using 𝐈1∗ and 𝐌1

∗ to represent the

conjugate of 𝐈1 and 𝐌1 the Fourier coefficient is:

𝑆𝑚𝑎𝑓𝑝= 𝑆𝑣𝑎

𝑝𝑘𝐺𝜃 [𝑗2𝜋 (𝑓𝑝 − 𝑓1)] −𝐈1 [𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 − 𝑓1 )] − 𝑗𝐾𝑖𝑑] +𝐌1 𝑒

−𝑗𝜑1 (6.57)

𝑆𝑚𝑎𝑓𝑝−𝑓1 = 𝑆𝑣𝑎

𝑝𝑘𝐺𝜃 [𝑗2𝜋 (𝑓𝑝 − 𝑓1)] 𝐈1

∗ [𝐻𝑖 [𝑗2𝜋 (𝑓𝑝 − 𝑓1 )] + 𝑗𝐾𝑖𝑑] − 𝐌1∗ 𝑒−𝑗𝜑1 (6.58)