Implementation of Multi Objective - Modulated Model ... · Generally, basic control in Alternate current (AC) MG can be divided into master-slave control [29], hierarchical control

Implementation of MultiObjective - Modulated ModelPredictive Control Into Virtual

Synchronous Machine

Edi MatijevicEnergy Technology, PED3-943, 2019-06

Master’s Project

ST

U

DE

NT R E P O R T

Copyright c© Aalborg University 2018

In this project, following tools and software were used for simulation and modelling:Simulink, Matlab. For design of figures and flowcharts, following tools and software wereused : Gimp, Visio 2013.

Energy DepartmentAalborg University

http://www.aau.dk

Title:Implementation of Multi Objective -Modulated Model Predictive ControlInto Virtual Synchronous Machine

Theme:Power Electronics Control

Project Period:01.09.2018 to 31.05.2019

Project Group:PED4-943

Participant(s):Edi Matijevic

Supervisor(s):Tomislav DragicevicFrede Blaabjerg

Copies: 1

Page Numbers: 81

Date of Completion:May 30, 2019

Abstract:

As more countries are focusing on im-lementation of clean and renewableenergy sources into electric distribu-tation systems, decentralized distribu-tion systems such as Microgrid arebecoming more popular and feasiblesolution. With rapid penetrations ofrenewables energy sources, inertia ofthe system will suffer due to inertia-less characteristics of renewable en-ergy sources. Solution to problem isprovided by modelling of power con-verter to emulate behaviour of syn-chronous machine. Standard vir-tual synchronous machine uses cas-cade control loops to regulate voltageand current, where voltage control isouter loop with frequency bandwidthan order of magnitude lower than in-ner current control loop. This project isfocused on developing Model Predic-tive Control (MPC) which uses multi-ple control targets integrated into sin-gle cost function, therefore, reducingcontrol loops which improves tran-sient response of virtual synchronousmachine. Furthermore, novel devel-oped MPC is compared with cascadecontrol. Experimental results of noveldeveloped MPC in islanded parallelVirtual Synchronous Machine (VSM)operation are obtained in laboratory.

The content of this report is freely available, but publication (with reference) may only be pursued due to

agreement with the author.

http://www.aau.dk

v

List of Abbreviations

RES Renewable Energy Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

MG Microgrids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

EES Electrical Energy Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

DG Distributed Generations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

THD Total Harmonic Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

VSM Virtual Synchronous Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

SM Synchronous Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

VSC Voltage Source Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

SRF Syncronous Reference Frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

PLL Phase Locked Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

HPF High pass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

VSI Voltage source inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

DC Direct current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

AC Alternate current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

PCC Point of common coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

MO-M2PC Multi Objective Modulated Model Predictive Control . . . . . . . . . . . 5

PWM Pulse Width Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

MPC Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

SVPWM Space Vector Pulse Width Modulation . . . . . . . . . . . . . . . . . . . . . . . . . 12

ESR Equivalent Series Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

ZOH Zero Order Hold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38

FFT Fast Fourier Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30

FCS-MPC Finite Control Set - Model Predictive Control . . . . . . . . . . . . . . . . . . 40

Contents

1 Project Formulation 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Low Carbon Future . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Microgrids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Problem analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Microgrid Challenges . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Control strategy in Microgrids . . . . . . . . . . . . . . . . . . . . . . 31.4 State of the art - Virtual Synchronous Machine . . . . . . . . . . . . . 41.5 Scope and project objective . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 Project Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Virtual Synchronous Machine 92.1 Voltage Source Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Space Vector Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Active and Reactive power control . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Reactive power control . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 Active power control and inertia emulation . . . . . . . . . . 16

2.4 Phase locked loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Virtual impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.6 Active damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7 Cascade loop Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7.1 Voltage control . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.7.2 Current control . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.7.3 Tuning of PI controllers . . . . . . . . . . . . . . . . . . . . . . 24

2.8 Simulation of VSM in islanded operation . . . . . . . . . . . . . . . . 262.8.1 Total harmonic distortion . . . . . . . . . . . . . . . . . . . . . 302.8.2 Power sharing of 2 parallel connected VSM . . . . . . . . . . 31

2.9 Small signal stability analysis . . . . . . . . . . . . . . . . . . . . . . . 322.9.1 Load Sweep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.9.2 Manual analysis of parametric sensitivity . . . . . . . . . . . . 34

vii

viii Contents

2.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Multi Objective - Modulated Model Predictive Control 373.1 State space model of the Inverter with LC filter . . . . . . . . . . . . 383.2 MPC Algorithm principle . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 Cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.1 Capacitor voltage . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3.2 Inductor current . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.3 PR controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3.4 Current overprotection . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 Simulation of MO-M2PC VSM in islanded operation . . . . . . . . . 423.4.1 Total harmonic distortion . . . . . . . . . . . . . . . . . . . . . 443.4.2 Power sharing of 2 parallel connected VSM . . . . . . . . . . 46

3.5 Parameter tuning and stability analysis . . . . . . . . . . . . . . . . . 463.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Comparison of simulation results 534.1 Total Harmonic Distortion . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Transient response of control strategies . . . . . . . . . . . . . . . . . 544.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Experimental Work 575.1 Benchmark strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Islanded microgrid operation of Parallel VSM . . . . . . . . . . . . . 59

5.2.1 FCS-MPC VSM . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2.2 MO-M2PC VSM . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3 Interpretation of experimental results . . . . . . . . . . . . . . . . . . 645.3.1 PCC current and voltage regulation . . . . . . . . . . . . . . . 645.3.2 Frequency response . . . . . . . . . . . . . . . . . . . . . . . . 645.3.3 Power sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6 Conclusion 676.1 Brief description of work done . . . . . . . . . . . . . . . . . . . . . . 676.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Bibliography 69

A Direct-quadrature-zero transformation 73

B Clarke transformation 75

C Linear VSM State space model 77

Contents ix

D Active and reactive power calculations 81

Preface

This master’s project was written during 3th and 4th semester of the Master pro-gram "Power electronics and drives" at department of Energy Technology, AalborgUniversity.

The project is motivated by research done by Prof. Tomislav Dragicevic on de-veloping novel model predictive control in order to improve overall performanceof microgrids. Main focus of project is control part of the system, as it can be testedon available hardware setup.

The Master’s thesis is structured in such way that both mathematical modelsof each subsystem and their implementation in Simulink are presented and ex-plained. This can give reader a deeper understanding of each model presentedand knowledge how to implement such a complex system into Simulink.

I would like to express my thankfulness to Prof. Tomislav Dragicevic and Prof.Frede Blaabjerg for valuable advice and mentorship during this project. Phd stu-dent Changming Zheng for helping me with experiments. Also, I would like tothank my family for support. Lastly, I would like to thank my girlfriend TajanaAndric for her love and support during last 2 years of studying in Denmark. Also,for motivation and encouragement in hard decision to leave home country to pro-ceed higher education.

Aalborg University, May 30, 2019

Edi Matijevic<[email protected]>

xi

Chapter 1

Project Formulation

1.1 Background

1.1.1 Low Carbon Future

Rapid change in CO2 levels leading to climate changes [14] and constant increasein energy demand [15] due to population growth requires more intelligent use oflocalized energy systems and production. Problems mentioned above introducesa necessity for switching from traditional power production using fossil fuels toclean, Renewable Energy Source (RES). Europe is working intensively on a lowcarbon economy. One of the leading country in low carbon economy is Denmarkwhich has a plan to cover all energy supply with RES by 2050 [24]. In addition, pol-icymakers are also optimizing carbon tax in order to guarantee switching of largecompanies to low carbon technologies [32]. Analyzing all of the countermeasuresdone, RES seems like a good bet for the future of energy production[31].

1.1.2 Microgrids

While RES main advantages are low carbon emissions, RES energy generationis generally unpredictable and inconsistent due to its dependency on weather.Without additional systems, RES are unable to satisfy consumer energy demand.Adding Electrical Energy Storage (EES) into equation, surplus of energy generatedby RES can be stored and used when there is demand for it, directly decreasingRES unpredictability. While this solves RES unpredictability, it also introduceschallenges for control of complete system and energy management. RES like pho-tovoltaic cells or wind turbine needs additional power electronics, Voltage SourceConverters (VSC) for energy conversion to desired output. Connecting RES in par-allel with EES and local loads introduces concept of localized energy systems orMicrogrids (MG) which can be seen simplified in Figure 1.1.Main advantage of MG is ability to be work independently, in areas where energy

1

2 Chapter 1. Project Formulation

from power plants is not available . It also has ability to be controlled decentralizedwhen communication links are expensive and inapplicable to implement. In sys-tems where grid energy is available, MG has obvious advantage of cost efficiencydue to less energy transmission losses and better power quality. MG also provide apossibility to disconnect from grid during blackouts. This is crucial for specific in-stitutions like health care or educational institutions where power outage can costlives or research developments. Also, as already pointed out , in rural areas wherethere is not transmission lines or power plants, MG would be perfect solution forproviding power. Therefore, scientific community can agree that MG solves vul-nerability to power outages, improves power quality and reduces carbon footprint.

Figure 1.1: Microgrid

1.2 Problem analysis

1.2.1 Microgrid Challenges

While MG seems like good solution for the future of local energy grid, there arestill challenges which have to be handled accordingly. Due to AC load demand

1.3. Control strategy in Microgrids 3

and grid standard frequency, power converters are used for energy conversion toaccepted standard frequency and voltage. One of important disadvantages whenit comes to RES is lack of rotating moving parts. This corresponds in lack off or noinertia at all in renewables energy sources like solar panels or wind turbines. Thisintroduces new control challenges for power converters. Therefore, one of the mainchallenges for MG is low inertia characteristic [28]. Traditional power productionusing large synchronous machines provides high inertia(kinetic energy) to sys-tem which correlate to better transient response when introduced to disturbances,while for MG, due to low inertia characteristic during transients periods, systemcan suffer from high frequency deviation [30]. This can lead to stability issues andincrease in Total Harmonic Distortion (THD) when there change in non-linear loadapplied to MG or grid frequency. In order to solve this problem, researchers aredeveloping new control idea based on behavioural characteristics of SynchronousMachine (SM). Main idea is to replicate damping and inertial characteristics ofsynchronous machines used in traditional power systems into decentralized MG.

1.3 Control strategy in Microgrids

Generally, basic control in Alternate current (AC) MG can be divided into master-slave control [29], hierarchical control [22], and decentralized control [12], [33].Master-slave control uses one main Distributed Generations (DG) as a master unit,which provides voltage and frequency reference to other current controlled DGunits [29]. Since master-slave control uses communication link between masterand slaves DG, it provides good power sharing and synchronization characteris-tics [1]. Problem arises when main DG has to be shut down due to emergency,whole microgrid has to be shut down. It also adds additional economical cost forcommunication links which are required.

Hierarchical control consist of three control levels, primary, secondary and ter-tiary control [3]. Primary control is responsible for power sharing between DGunits, while secondary restores voltages and frequency to nominal values. Whenit comes to tertiary control, it deals with power flow between microgrid and maingrid. Hierarchical control can be centralized or decentralized. Centralized de-pends on central controller which collects information from communication linksand acts upon each DG unit, Decentralized does not use communication links, anduses local controller to provide necessary action.

When it comes to converter utilized in MG, they can be classified as grid feed-ing, grid forming and grid supporting converters [27]. Grid feeding convertersoperates as current source with high parallel impedance responsible for injectingactive and reactive power into utility grid. Voltage amplitude and frequency ofVoltage source inverter (VSI) are driven by the utility grid or grid forming/gridsupporting converters. Grid forming operates as AC voltage source with low se-


ries impedance when MG operates in islanded mode, responsible for providingvoltage amplitude and frequency reference in MG. Grid supporting can operate inboth, grid connected or islanded mode, including all features of grid feeding andgrid forming converters. Taking the above mentioned into account, proper MGcontroller has following responsibilities [35]:

• Voltage and frequency regulation under any circumstances

• Active and reactive power sharing between converters in grid connected/islandedmode

• Ideal transition from grid connected to islanded mode of operation

• Uninterrupted supply and power to critical loads( schools, hospitals andother essential services)

• Energy market optimized participation

1.4 State of the art - Virtual Synchronous Machine

The concept of VSM was introduced in the last decade. The research aim of VSMconverter control is to emulate advantageous characteristics of SM like reactivepower control, oscillation damping and rotating inertia. Concept of VSM is forthe first time proposed in [2], and since then different scheme of implementingSM characteristics was used for controlling VSC. There are several classification ofVSM [8] depending on a model complexity, but in this project virtual SM based on[6] was examined.In Figure 1.2, schematic of grid connected VSM is presented. VSM consist of eachblock with own function which will be detailed explained in chapter 2. VSM modelpresented in 1.2 uses cascade loop control for controlling voltage and current inrotating reference frame. In following section, model will be briefly explained.In depth explanation of each block and equation is provided in chapter 2. Firstly,measured grid voltage is used by "PLL" block in order to determine actual grid fre-quency. VSM uses traditional SM characteristics equation in order to control activepower and voltage amplitude control for reactive power. Actual grid frequencyprovided from Phase Locked Loop (PLL) is used by "Virtual inertia and powercontrol" block to calculate VSM phase angle for dq transformation explained inappendix A and VSM speed used in control loop. This means that three phasesignals (Capacitors voltage and Inductors current) will be transformed into VSM-oriented rotating reference frame. "Reactive power control" block provides voltageamplitude reference for inner loop virtual impedance, voltage and current con-trol. Reference voltage amplitude processed by virtual impedance block for betterpower sharing is then compared with measured capacitor voltage and converter

1.5. Scope and project objective 5

Figure 1.2: VSM.

currents, then regulated with Syncronous Reference Frame (SRF) PI controllers in-side of "Virtual impedance", "Voltage control" and "Current control" blocks andfinal reference voltage is provided into "PWM" block which is controlling gate sig-nal of VSC. "Active damping" is also included for suppressing LC oscillations. Asmain focus in standalone operation, full model is presented as grid forming modelbut it will only be simulated and tested in standalone operation.

1.5 Scope and project objective

Main problem introduced with VSM presented earlier is inner cascade controlloops. Cascade control loop consist of inner current control loop and outer voltagecontrol loop. In order to regulate outer voltage control loop, firstly inner currentloop has to regulated, therefore, outer voltage loop bandwidth is at least order amagnitude lower, which when combined may provide slow response. This can benoticed during sudden changes as load step or external disturbances. Also cas-cade control loops introduces complexity of tuning SRF PI controllers and stabilitysensitivity. Problems introduced can be solved using Multi Objective ModulatedModel Predictive Control (MO-M2PC) which introduces one cost function wherevoltage and current can be simultaneously controlled resulting in better transientresponse. Besides better transient response, MO-M2PC can offer better THD dueto its nature of optimizing function, which also means that LC resonant frequencycan be damped without need of additional passive or active damping. Taking allof this in account, scope of project is following :

• Model: Develop a Cascade loop control - Virtual Synchronous machine


• Model: Develop and Implement MO-M2PC into Virtual Synchronous ma-chine model

• Benchmark: Compare a novel control strategy with Cascade loop control -Virtual Synchronous machine in simulation

• Experimental testing: Test a novel control strategy with benchmark strategyin islanded parallel vsm operation

1.5.1 Methodology

To obtain all the project objectives the following procedure will be followed:

1. Modelling of the Cascade loop control - Virtual Synchronous machine -For a control design, it is vital to develop accurate model of the system.VSM model is developed with cascade loop control which is be used as abenchmark strategy in simulations.

2. Modelling of novel VSM control strategy - Novel control strategy is devel-oped and implemented into VSM in order to improve performance of thesystem.

3. Validate novel and benchmark model through simulations - Cascade loopcontrol and novel VSM strategy are simulated in islanded MG operation un-der same conditions using MATLAB.

4. Compare simulations results - Simulations results of both control strategiesare compared and analyzed on important aspects and characteristics in orderto determine which strategy seems more beneficial.

5. Experiments of parallel connected converters in islanded operation withnovel control strategy and additional benchmark strategy - Novel VSM con-trol strategy is tested in islanded operation of 2 parallel connected inverterswith additional benchmark strategy. Both strategies are tested under thesame conditions on same setup.

6. Conclusion of final experimental results Novel VSM and benchmark controlstrategy are analyzed separately, then compared and conclusion has beendrawn.

1.6 Project Limitations

As every project has constraints, assumptions and limitations which could effectfinal representation of project results, they should be mentioned. The project hasfollowing limitations and assumptions :

1.6. Project Limitations 7

• Ideal input DC link voltage is assumed, neglecting all DC link dynamics

• Switching effects of converter are neglected

• Experimental tests are made on the existing setup

• Constant DC link source capable of delivering demanding power

Chapter 2

Virtual Synchronous Machine

This chapter will introduces full VSM model similarly developed in [6]. Each blockpresented in figure 1.2 will be represented as mathematical model. Model imple-mentation into simulink is presented with in depth explenation of block developed.

2.1 Voltage Source Inverter

In MG, power electronics have a big role in controlling the system. Using powerelectronics, it is possible to control DG active and reactive power in a way it isdesired or requested by central controller. Most common used power electronicdevice in link between DG and Point of common coupling (PCC) is 2 level VSI. VSIis the power conversion circuit responsible for conversion from Direct current (DC)to AC. In figure 2.1, a 2 level VSI with LC filter connected to PCC is shown. Basicthree phase inverter consists of 6 switches (upper and lower switches are com-plementary to avoid short circuit) and LC filter responsible for filtering switchingharmonics. Output voltage value is controlled by three gate signals (Sa, Sb, Sc),which are presented in table 2.1.

Based on table presented in 2.1, inverter can have 23 = 8 switch configurations.Inverter leg voltages with respect to neutral point can be expressed as :

vaN = Sa ∗Vdc

vbN = Sb ∗Vdc

vcN = Sc ∗Vdc

(2.2)

In order to present phase voltages, common mode voltage vnN 2.3 is subtractedfrom 2.2.

vnN =vaN + vbN + vcN

3(2.3)

9

10 Chapter 2. Virtual Synchronous Machine

Table 2.1: Switch configuration.

S = switching signal, Q = switch (2.1)

Sa = 1 ,if Q1 is ON and Q2 is OFFSa = 0 ,if Q1 is OFF and Q2 is ONSb = 1 ,if Q3 is ON and Q4 is OFFSb = 0 ,if Q3 is OFF and Q4 is ONSc = 1 ,if Q5 is ON and Q6 is OFFSc = 0 ,if Q5 is OFF and Q6 is ON

Resulting phase voltages are expressed by :

van = vaN − vnN

vbn = vbN − vnN

vcn = vcN − vnN

(2.4)

As modulation technique used in VSM will be space vector modulation, phasevoltages can be represented in stationary orthogonal α, β reference frame.

v = vα + jvβ (2.5)

After applying complex clark transformation explained in appendix B.1 to all pos-sible switch configurations, corresponding voltage output can be expressed by fol-lowing voltage input vector vi presented in table 2.2.

Table 2.2: Complex Voltage Vectors values in two level VSI

Switchstates

Sa Sb Sc Voltage vector vi

0 0 0 0 01 1 0 0 2

3 Vdc

2 1 1 0 13 Vdc+j

√3

3 Vdc

3 0 1 0 − 13 Vdc+j

√3

3 Vdc

4 0 1 1 - 23 Vdc

5 0 0 1 − 13 Vdc-j

√3

3 Vdc

6 1 0 1 13 Vdc-j

√3

3 Vdc

7 1 1 1 0

As an important note, complex voltage vector is later used for space vectormodulation, and has particularly important role in MO-M2PC strategy.

2.1. Voltage Source Inverter 11

After determining output inverter voltage, mathematical model of inverter with LCfilter presented in figure 2.1 connected to common coupling point is developed.

Figure 2.1: Inverter connected to point of common coupling.

Mathematical model of inverter connected to PCC using kirkoff law can be repre-sented as :

Vabc −VabcO = L fdIabc

dt+ riabc (2.6)

Iabc − IabcO = C fdVabcO

dt(2.7)

Where Vabc represent inverter output voltage,VabcO voltage at filter capacitor orcommon coupling point,Iabc inverter output current, IabcO output current, L f is filterinductor, C f is filter capacitor and r is filter resistance. As in this section, controlwill be implemented in VSM reference frame, park transformation explained inappendix A.1 has to be performed, resulting in :

Vd −VdO = L fdid

dt+ r f id + ωVSMC f iq (2.8)

Vq −VqO = L fdiq

dt+ r f iq −ωVSMC f id (2.9)

id − idO = C fdVdO

dt+ ωVSML f iq (2.10)

iq − iqO = C fdVqO

dt−ωVSML f id (2.11)

This equation will be base for inner control of capacitor voltage and inductor cur-rent explained later in chapter. As it can be seen after park transformation, d andq axis are coupled which will be needed to take account in control model.


2.2 Space Vector Modulation

In order for inverter to generate AC voltage from DC, transistors has to be switchedin particular fashion. Switching is done using Pulse Width Modulation (PWM).There are few different modulation techniques used in VSI applications, but mainfocus is Space Vector Pulse Width Modulation (SVPWM) as it provides best perfor-mance in terms of improved voltage and current THD, and also 15 percent higherAC output voltage [18]. As stated in previous section 2.1, for two level VSC thereis 8 possible switching states, 6 of them(v1 − v6) correspond to active vectors withmagnitudes in complex form listed in table 2.2, and 2 of them (v0 and v7) zerostates with magnitudes of zero. Active vectors can be divided into 6 active re-gions by equally dividing space between them as showed in figure 2.2. In order

Figure 2.2: Switching sequence diagram in αβ frame.

to produce desired inverter output, reference voltage is synthesized by combiningand averaging number of vectors over sample time (Ts). As example, in figure2.2, reference voltage Vre f is between basic vectors (v1 and v2). By alternating be-tween these 2 vectors, rotating voltage vector with fixed magnitude correspondingto reference can be emulated. Two zero vectors (v0 and v7) are also used in orderto add dead time to switching pattern. Dead time will reduce voltage referencemagnitude if voltage reference magnitude is also below 100 %. (Ts) can be foundusing following expression 2.12.

Ts =1fs

(2.12)

2.2. Space Vector Modulation 13

Figure 2.3: Space vector PWM.

where fs is voltage reference sampling frequency. In order to implement SVPWM,and obtain higher output voltage compared to other PWM techniques, trianglewave is added as a common mode voltage, with triple the frequency and a peakto peak of half of the sinusoidal reference peak, resulting a reference signal as arump signal. Where triangle wave is calculated using 2.13

Vcm =max(Va, Vb, Vc) + min(Va, Vb, Vc)

2(2.13)

As can be noticed from figure 2.3, rump signal has a lower peak which results inpossibility of higher output voltage up to 15 %. In figure 2.4, input translationto switching state is presented. In upper figure, control rump signal is comparedwith triangle wave in middle figure. When control signal is higher than trianglewave, switching sequence is positive, and negative when it other way around.The triangle wave frequency, known as a carrier wave frequency will be switchingfrequency of the power converter. Switching sequence or gate signal for 1 transistorcan be seen in figure below.


Figure 2.4: Switching sequence.

2.3 Active and Reactive power control

Active and reactive power loop in VSM operation can be considered as outer loopresponsible for providing reference values to inner control cascade loop. Activeand reactive power control in VSM are similar to droop control [19] which is widelyused primary control technique in MG. They both use same principle controllingactive power with frequency regulation and reactive power with voltage amplituderegulation if line impedance is inductive. The only difference is that VSM is usingSM traditional swing equation to provide inertia and damping effect to voltage andcurrent control [8] but both techniques can be considered equivalent under specificcondition [7] . Active and reactive power control can be split into two blocks,where the reactive power control uses voltage droop control, while active poweruse traditional swing equation for inertia and damping emulation with frequencydroop control.

2.3.1 Reactive power control

In order to control active and reactive power, they need to be decoupled. If lineimpedance is inductive or emulated with virtual impedance, inverter output canbe consider as a highly inductive. Inverter connected to a common bus can be

2.3. Active and Reactive power control 15

mathematically modelled considering following figure 2.5. In figure 2.5, invertercoupled to DG unit is connected to common bus through impedance.

Figure 2.5: Equivalent model of inverter connected to common bus

In an inductive system, inverter active and reactive power drawn to the bus can beexpressed as

P =EVsin(α)

X. (2.14)

Q =EVcos(α)−V2

X. (2.15)

Where E is amplitude output of inverter voltage, V is amplitude output ofcommon bus voltage, alpha is power angle and X is output reactance of the inverter.Assuming α is very small, it can be determined that active power injected to thecommon bus is determined by power angle, while for reactive by difference inamplitude between common bus and inverter voltage magnitude. Reactive powerequation in 2.15 can be rewritten as

Vr∗ = V∗ + kq(q∗ − qm) (2.16)

Where Vr∗ represents voltage amplitude reference, V∗ external amplitude voltagereference, qm is real reactive power filtered through first order low pass filter whereωf represents cut-off frequency. kq is reactive power droop gain acting on thedifference between real and reference reactive power. kq is determined using

kq =∆E

qmax(2.17)

Where ∆E is maximum allowed deviation in voltage and qmax is nominal reactivepower supplied from inverter.In figure 2.6, mathematical model of reactive power control block implemented in


Figure 2.6: Reactive power block

simulink is presented. Real filtered reactive power qm is compared with referencereactive power q*. Error difference is then regulated with reactive power droopgain and added to external voltage amplitude reference. Given signal results involtage reference output is directed into virtual impedance block explained later inchapter. As reactive power demand changes, voltage reference will adjust throughthis equation.

2.3.2 Active power control and inertia emulation

As already mentioned, main difference between standard droop control and VSMis swing equation 2.18 in active power regulation which provides virtual inertiaand damping to the system [8]. Traditional SM in power production plants haveenergy stored into rotational inertia and rotate at same frequency. If there is afault in one of the SM or grid frequency changes, rest of SM will release kineticenergy from his rotating mass, or absorb energy in different case to keep the gridfrequency same. This can be implemented into inverter using following swingequation.

Jdω

dt= To − Te − D(ω−ωb) (2.18)

where J represents rotor inertia, ω is rotating speed of the SM, ωb is angular fre-quency of the grid, To is mechanical torque, Te is electromagnetic torque and Dis coefficient of real SM. Equation 2.18 can be represented in power by multiply-ing all equation terms with frequency ω and expressed in VSM reference frame.Now inertia momentum and change in VSM speed in time is expressed by powerbalance equation in 2.19.

JdωVSM

dtω = Pr∗ − P− Kd(ωvsm −ωPLL) (2.19)

2.3. Active and Reactive power control 17

Figure 2.7: Active power control and inertia emulation.

Where Pr∗ represents mechanical input power, P is measured electrical power fromVSM to grid and kd is damping constant in traditional SM reacting on differencebetween real VSM speed ωVSM and angular grid frequency ωPLL determined byPLL. Introducing mechanical time constant(Ta=2H) [8], mathematical model canbe expressed as 2.20.

dωVSM

dt=

Pr∗

Ta− P

Ta− Kd(ωVSM −ωPLL)

Ta(2.20)

This will provide emulation of rotating inertia and power balance to the system.Including external droop frequency control similar to droop control into model

will provide Pr∗ from 2.20 for power balance equation as a result of external powerreference p∗ and frequency droop effect. Frequency droop effect is a result ofdifference between VSM reference speed ω∗ and real speed ωVSM, and frequencydroop constant kω acting up on it. Thus, final mathematical model presented in2.16 is

dωVSM

dt=

Pr∗

Ta− P

Ta− Kd(ωVSM −ωPLL)

Ta− Kω(ωVSM −ω∗)

Ta(2.21)

In figure 2.7 implementation of active power block into simulink is presented.Output VSM frequency ωVSM will be provided into inner loops to control vir-tual impedance, current and voltage. In order to transform three phase measuredsignals into VSM rotating reference frame, resulting phase angle ΘVSM from 2.22will be used for transformation.

dΘVSM

dt= ωVSM ∗ωb (2.22)

where ωb is rated angular frequency.


Figure 2.8: Basic Phase locked loop.

2.4 Phase locked loop

Grid forming or grid supporting inverters requires proper technique in order tobe synchronized with the grid. This means that grid voltage frequency and phaseangle information has to be estimated in order to synchronize inverter with a grid.PLL is widely used technique for retrieving this necessary information. PLL has tobe precise and fast enough for adequate synchronization with a grid. It also has tobe immune to disturbances and distortions in utility voltage [17].Basic PLL is presented in figure 2.8 and it consists of three important blocks. Firsta phase angle error is produced by comparing an actual voltage signal from in-verter with a estimated one. Error is then run through loop filter which removesunwanted frequency and reduces disturbances in system. Loop filter output isestimated frequency of input signal. Estimated frequency is feed into voltage con-trolled oscillator block which is responsible for producing phase angle. Usingphase angle inverter can be synchronized with a grid. For the purpose of VSM,PLL will not be used for synchronization to the grid, but rather as an method toestimate grid frequency which is used in active power control 2.7 for implementingdamping effect of SM denoted by ωgrid. The reason why PLL synchronization isnot needed is VSM power balance swing equation which will ensure synchroniza-tion to the grid voltage. Whole system and control is in VSM reference frame. PLLused in [16] was implemented. Mathematical model for estimating grid frequencyωPLL can be expressed as

dVPLL

dt= −ωLP, PLL ∗VPLL + ωLP ∗Vo ∗ e− j(δΓPLL − δΓVSM) (2.23)

dδPLL

dt= tan−1(

VPLL, qVPLL, d

) (2.24)

δωPLL = kp ∗ tan−1 ∗ (VPLL, qVPLL, d

) + ki ∗ εPLL (2.25)

2.5. Virtual impedance 19

ωPLL = δωPLL + ωg (2.26)

where ωLP, PLL is first order low pass filter, kp proportional gain and ki integratorof PI controller.

Figure 2.9: PLL block.

PLL implementation in simulink is presented in figure 2.9. Measured voltage atfilter capacitors are transformed into rotational reference frame using cosine-parktransformation explained in appendix A.1. Signals d and q are filtered throughlow pass filters, using inverse tangent function (tan−1) and compared with 0 (errorreference), phase angle error of PLL is produced. Phase angle error is correctedusing PI controller. PI controller output is estimated grid frequency which is usedin modelling of active power and inertia. By integrating with ωb gain, a phaseangle is obtained used in feedback loop. Under steady state, ωPLL will be equal toωVSM which results in system being synchronized with a grid voltage. For islandedoperation, ωPLL can be set as a reference frequency.

2.5 Virtual impedance

Virtual impedance is control technique in microgrids commonly used with droopcontrol. Main purpose of using this control technique is to improve power sharingand voltage harmonic distortion. In MG applications, component tolerances anddifferent power line lengths will lead to mismatch in impedance and unbalancedpower sharing [23]. Changing output impedance of inverter or impedance match-ing using virtual impedance, current sharing will be improved [13]. Significantly,virtual impedance emulates behaviour of actual impedance without any real losses.


Figure 2.10: Virtual Impedance block.

Using following expression 2.27, virtual impedance can be implemented :

vo = v∗o − Io ∗ Zo (2.27)

where vo is output voltage provided to inner control loops, v∗o is voltage amplitudeprovided by reactive power control loop, Io is output current and Zo is virtualimpedance.

Equation 2.27 implemented in VSM can be represented as :

V∗o = Vr∗ − (rv + j ∗ωVSM ∗ lv) ∗ io (2.28)

Where V∗o represents output voltage used in voltage control loop, Vr∗ is inputamplitude voltage reference coming from reactive power control loop, io is inductorcurrent, rv is virtual resistance, lv is virtual inductance and ωVSM as a VSM speed.Equation 2.28 in VSM reference frame implemented in simulink can be seen infigure 2.10. Output dq voltage reference is later used in inner cascade controlloops.

2.6 Active damping

In order to fulfill power quality requirements, low pass filter has to be added toreduce inverter switching ripple. In this project, second order LC filter is used,which may introduce oscillations due to resonance of LC filter. Resonance canlead to stability issue in a system. Oscillations can be reduced by adding resistor,but due to undesired losses, efficiency of inverter decreases [20]. Another wayof reducing a oscillations without sacrificing inverter efficiency is adding virtual

2.7. Cascade loop Controller 21

resistor or active damping [5] [36]. Similar to virtual resistor method, simple activedamping can be implemented using method from [21]. Output V∗AD used in currentcontrol loop is expressed by following equations :

V∗AD = kAD(vo − φ) (2.29)

dφ

dt= ωAD ∗ vo −ωAD ∗ φ (2.30)

where ωAD is cut-off frequency, kAD is damping gain,vo is capacitor voltage and φ

are states of low pass filter. Principle of method is following: Filtering first orderharmonics using low pass filter on measured dq capacitor voltage and subtractingit from measured unfiltered dq capacitor voltage provides High pass filter (HPF)effect. Then scaling resulting HPF signal with gain kAD and providing it in acurrent control block in order to cancel oscillations in LC filter. Active dampingcontrol block in simulink is shown in 2.11.

Figure 2.11: Active Damping.

2.7 Cascade loop Controller

Inner control loop is based on cascade-loop control, where inner loop is for control-ling filter inductor current and outer loop for controlling filter capacitor voltage.

2.7.1 Voltage control

Voltage control loop acts as a outer regulation loop. Voltage control is based oncontrolling a voltage in filter capacitors in VSM reference frame orientation usingPI controllers. Using equation 2.8,2.9, and adding feedforward term of measuredoutput current, voltage controller can be represented in following equation:

Icvd = K f f iio + Kpv(v∗od −Vod) + Kiv(v∗od −Vod)− C f ωVSMVq (2.31)


Icvq = K f f iio + Kpv(v∗oq −Voq) + Kiv(v∗oq −Voq) + C f ωVSMVd (2.32)

Where K f f i is feed-forward current gain(1 for enable and 0 for disable), Kpv isproportional gain and Kiv is integral gain of PI controller. C f is capacitor filter,ωVSM is VSM speed, Icvd and Icvq are output current references used in currentcontrol loop. Figure 2.12 represents voltage control loop implemented in simulink.Voltage control input signals, or voltage reference in dq frame coming from virtual

Figure 2.12: Voltage Control block.

impedance block are compared with measured filter capacitor voltages. Resultingerror is then corrected using PI controller. After inverter mathematical model parktransformation (appendix A.1), d and q axis are coupled. In order to decouple dand q axis, decoupling term is added, (−C f ωVSMVq) for d axes and (+C f ωVSMVd)for q axis. Output is current reference used in current control loop which is a resultof addition of output of PI controller, decoupling term and feed-forward term ifenabled. Outer control has to be at least an order of magnitude slower then inner


loop which can cause slower performance during sudden changes in system.

2.7.2 Current control

Current control is inner loop of cascade controller and main goal is to controlinductor current. Using equations 2.10,2.11, adding feed-forward term and activedamping term, current controller can be represented in following equation :

Vcvd = K f f vvo + Kpv(i∗cvd − icvd) + Kiv(i∗cvd − icvd)− L f ωVSMiq −V∗ADd (2.33)

Vcvq = K f f vvo + Kpc(i∗cvq − icvq) + Kic(i∗cvq − icvq) + L f ωVSMid −V∗ADq (2.34)

Where K f f v is feed-forward voltage gain(1 for enable and 0 for disable), Kpc is pro-portional gain and Kic is integral gain of PI controller. L f is inductor filter, ωVSM isVSM speed, V∗ADq represent active damping term, Vcvd and Vcvq are output voltagereferences used for controlling inverter. Figure 2.13 represents current controllerimplemented in simulink.

Figure 2.13: Current Control block.

Similarly to voltage control, current control output is result of addition of PIcontroller output(in this case, difference of reference current and real measuredfilter inductor current), decoupling term and feed-forward term [25]. Also, inorder to damp LC filter resonance, active damping term is subtracted from current


control output. In order to control inverter accordingly, modulation index has tobe implemented into output reference using following equation :

m =V∗cvVDC

(2.35)

where m represents modulation index, V∗cv is voltage output reference from 2.33,2.34 and VDC is dc link voltage. Final voltage reference is in dq reference frameand depending on switching technique can be transformed into desired frame toproduce gate signals to inverter switches.

2.7.3 Tuning of PI controllers

Tuning of PI controllers of cascade control loops can be of significant importancefor system stability and transients response. In [9], automatic tuning is presentedbut offers rather complicated and tiresome method of tuning, so conventionalmethod of tuning cascade controllers used in 3.5 will be implemented. PI con-trollers tuned have to take account of bandwith difference between inner(current)control loop and outer control loop which is in range of decades. Therefore, outercontrol loop is limited by inner loop control bandwith.

Tuning of inner current control

Assuming ideal decoupling between d and q axis, feedforward of measured volt-age and approximation of PWM effects, open loop transfer function can be repre-sent as :

hc, dq = (Kpc +Kic

s) ∗ ( 1

1 + Tv ∗ s) ∗ ( 1

r(1 + T1 ∗ s) (2.36)

where (Kpc +Kics ) is PI controller, ( 1

1+Tv∗s ) is first order PWM approximation and( 1

r(1+T1∗s ) is filter inductor. Time constant associated with filter inductor can beexpressed as :

T1 =l1

r1 ∗ωb(2.37)

where l1 is filter inductor, r1 is Equivalent Series Resistance (ESR) of inductor andωb is angular frequency. Tv as delay PWM delay approximation can be representedas :

Tv ≈1

2 ∗ fsw(2.38)

where fsw is switching frequency of converter. Using modulus optimum criterion[9], open-loop pole cancelation technique and selecting a gain to achieve critical


damping, resulting controller parametars can be expressed as :

Kpc =l1

2ωbTv

Kic =r1

2Tv

(2.39)

From equation 2.39, it can be seen that current controller bandwidth is affected byswitching frequency of the converter.

Tuning of outer voltage control

Similarly to current control, assuming ideal decoupling of d and q axis, open looptransfer function can be expressed as :

hvc, dq = (kpv +kiv

s) ∗ ( 1

Teq,ccs) ∗ ( 1

Tc1s) (2.40)

Where (kpv +kivs ) are PI controllers, ( 1

Teq,ccs ) is closed loop curent controller and

( 1Tc1s ) is filter capacitor. Time constant of Teq,cc of current closed loop controller can

be approximated by :Teq,cc ≈ 2Tv (2.41)

While integral time Tc1 of filter capacitor can be represented by :

Tc1 =c1

ωb(2.42)

Where c1 is filter capacitor and ωb is fundamental frequency. Using criterion ofsymmetrical optimum which can provide maximum phase margin at crossoverfrequency, tuning PI controllers gain can be expressed in following equations :

kpv =Tc1

α∗eq,cc

kiv =Tc1

α3 ∗ T2eq,cc

α = 2ζ + 1

(2.43)

where ζ is desired damping factor and α is design parameter correlated with damp-ing factor. As it can be seen from conventional tuning method, inner controlledloop is constrained by switching frequency, while outer is constrained by innerbandwidth (difference between outer and inner bandwidth is in decades) which atend results is sensitive stability and slow response of the system.


2.8 Simulation of VSM in islanded operation

As main focus of thesis is VSM islanded operation, VSM performance will be testedthrough simulations when VSM is disconnected from grid. Simulations will becarried out in timeframe of 1 second. In order to assess VSM transient behaviour,additional load will be connected at 0.3 seconds. After simulations, THD will becalculated to determine power quality of the system. Therefore, simulations willbe executed using following steps.

1. 0 s - Start of simulation (resistive load)

2. 0.3 s - Inductive load connected

3. 1 s - Simulation stop

System parameters can be seen in table(2.3), while load is presented in 2.4. Chang-ing certain parameters can change behaviour of VSM which will briefly be investi-gated later.

Table 2.3: Parameters of Islanded VSM

Parameter Value Parameter ValueVoltage reference, V∗ 300 V DC link 700 V

VSM inertia constant 2 sVSM damping coeffi-cient,kd

30000

Current gain, kpc, kic 6.3091, 100000 Voltage gain, kpv, kiv 6.1244, 9050Power reference, p∗ 4000 W Reactive reference, q∗ 0 VARω∗VSM 50 Filter resistance, R f 1 ΩFilter inductor, L f 20 mH Filter capacitor C f 5 uF

Power droop gain, kω 3 ∗ 10−10 Reactive power droopgain, kq

5 ∗ 10−4

Table 2.4: Investigated load

Load ValueResistive Load 90 ΩInductive Load 90 Ω, 40 mH

Simulations are carried out in Simulink. In figure 2.14, simulink model usingSimscape addon is presented. Simspace enables rapid creation of physical electricalmodel and power electronic devices like inverters. Simspace block universal bridgeis modelled as 2 level three phase inverter connected to dc source. Cascade controlloops described in previous section with SVPWM are responsible for providing

2.8. Simulation of VSM in islanded operation 27

input gate signals to inverter. LC filter model connected to inverter output can beseen in figure 2.15. 2 kind of load are connected to LC filter output. Resistive whichis connected through three phase line and inductive connected through three phasebreaker. Initial breaker switch state is open, and step signal is sent to three phasebreaker at 0.3, and at that point, switch closes and inductive load is connected to athree phase line. Same simulation setup is used in simulation in chapter 3.

Figure 2.14: Simulink simulation setup.

Figure 2.15: LC filter block.

Figure 2.16 represents filter capacitor voltage and filter inductor current. Asit can be seen from figure 2.16, at beginning of simulation there is some over-shoot where system needs time to reach to steady state. At 0.3 seconds, inductiveload is connected and system respond acceptable to transients. It should be notedthat high load can degrade system performance which will be detail discussed insection 2.9. Same results can been seen in measured active and reactive powerpresented in figure 2.17, where reactive power is approximately zero until induc-tive load is connected, and gradually increases to load requirement. By changing


Figure 2.16: Capacitor voltage and Inductor current.

active and reactive droop gains, better power sharing could be achieved but canalso degrade voltage regulation [4]. Therefore, tuning active and reactive droopgains is trade off between power sharing and voltage regulation. Figure 2.18 rep-resent speed of the VSM where it can been seen that swing equations emulateSM behaviour and VSM frequency adjust itself during transients (Load change),increasing or decreasing frequency, depending on load and power reference, butkeeps frequency approximately around 50 Hz. On figure 2.19, it can been seen howVSM speed decreases, and align with reference speed, when total power suppliedto load is similar to power reference. Therefore, it can be concluded that converterresponse emulates SM characteristics.


Figure 2.17: Active and Reactive power.

Figure 2.18: VSM Speed.

Figure 2.19: VSM Speed during transients.


2.8.1 Total harmonic distortion

THD has important role in power distribution. High THD can affect power qual-ity and reliability of the system which can lead to failures due to heating directlycaused by high THD. Also, THD of general system has to meet voltage and cur-rent THD requirements described in IEEE 519 standard. Therefore, THD has tocalculated. THD is calculated using Fast Fourier Transform (FFT) algorithm inMATLAB.

THD =IH

I[F](2.44)

Where, IH is rms value of total harmonics of signal

IH =√

I22 +

√I23 + .... +

√I2n (2.45)

IF is rms value of fundamental signal. Figure 2.20 and 2.21 represent load currentsTHD. From a figures, it can be seen that load current THD of resistive load is1.17%, while THD for inductive load is 0.53%.

Figure 2.20: THD of resistive load current.

Robustness to LC filter parameters variation

In the practical implementation of VSM and real world, LC filter parameters willvary and not be same value as ones implemented into control. Therefore, it isimportant to asses how parameters uncertainty affects load current THD. Table 2.5presents different THD values for LC filter parameter change of 50 %. It can beseen how in some cases, LC filter uncertainty can affect system THD badly.


Table 2.5: LC parameters value variations

Parameters varia-tion

THD(Resistive load)THD(Inductiveload)

C f + 50% 1.10% 0.55%L f + 50% 0.93% 0.49%C f - 50% 1.11% 0.46%L f - 50% 1.52% 0.52%C f + 50% L f - 50% 1.44% 0.52%C f + 50% L f + 50% 1% 0.55%C f - 50% L f + 50% 0.93% 0.45%C f - 50% L f - 50% 1.76% 0.48%

Figure 2.21: THD of inductive load current.

It should be noted that VSM requires active damping in order to THD be ac-ceptable, where simulations without active damping provide THD result over 8 %due to LC filter resonance. Therefore, it can be concluded that VSM power qualitydepends on active damping block, where parameter active damping gain has to beadjusted for better results.

2.8.2 Power sharing of 2 parallel connected VSM

In order to determine transient response of cascade control loop, islanded sim-ulation of 2 parallel connected VSM with load is presented in following section.Simulation is executed in following steps.


Figure 2.22: Active Power Sharing between 2 VSM

1. 0 s - Start of simulation (resistive load), 1 VSM is connected

2. Second VSM is connected



At beginning of simulation, 1 VSM is supplying a load, and after 0.1 s, anotherVSM is connected. At 0.3 inductive load is connected. Since both VSM have smalldifference in line impedance and virtual impedance implemented, power sharingshould between VSM should be accurate. Figure 2.22 presents active power sharingand figure 2.23 reactive power sharing between 2 parallel VSM. Good power shar-ing can be seen from figures where inner control responds adequate to commandsfrom VSM.

2.9 Small signal stability analysis

After developing and simulating models, full system should be analyzed in orderto assess its stability and dynamical behaviour. Therefore, using state space model,system can be determined how load variations affect stability of the system. Inorder to analyze system stability, a non linear state space model of VSM has tobe developed. By reducing equations described in previous subsections, fully pre-sented in appendix C.6, state space model of 18-th order will be obtained. Statespace model have 18 states x as it can be seen in 2.46 and 4 inputs u in 2.47.

x =

[vo,d vo,q Icv,d Icvq γd γq Iod Ioq ϕd ϕq

vPLL,d vPLL,q εPLL ζd ζq qm ωVSM δθPLL

]T

(2.46)

2.9. Small signal stability analysis 33

Figure 2.23: Reactive Power Sharing between 2 VSM.

u =[p∗ q∗ v∗ ω∗

](2.47)

Linearized small signal state space model is given by following form :

∆x = A ∗ ∆x + B ∗ ∆u (2.48)

where states and inputs are denoted by ∆ which represent small signal deviationsaround linearization point. Full developed state space model is presented in ap-pendix C.

For given parameters in table 2.3 used in simulations, system eigenvalues canbe plotted. In order to system satisfy stability requirements, all poles must lie onleft side of plane. As it can be seen in 2.24, all poles lies on left side plane andtherefore system is stable. While system is stable for given system state, this canchange and therefore, further investigation is desired.

2.9.1 Load Sweep

For a given system and parameters, It is desired to assess how can load affectssystem performance and stability. Load sweep will be performed by graduallyincreasing load resistance, therefore, decreasing the load. State space model stateswill be calculated 20 times by increasing load resistance by values of 3 ohms eachstep. This will give us information about system eigenvalues trajectories. On figure2.25 it can be seen how load affect eigenvalue trajectories . As Load resistance isincreased, load decreases and complex eigenvalue which lies almost on imaginaryaxis is moving towards right plane but it never move into positive real axis andtherefore it does not affect system stability. Similarly, complex eigenvalue closeto the origin associated with LC resonance, are affected with load changes. If


Figure 2.24: Pole and zero map

the load is too low, eigenvalues are can be found right next to the origin whichwill affects system performance. As load increases eigenvalue trajectory is movingtowards left until some specific point, after that, eigenvalue moves towards origin,where system performance will be changed. Therefore, we can conclude that loaddoes not affect system stability, but it affect performance which can be changeddepending on load change.

2.9.2 Manual analysis of parametric sensitivity

By doing manual analysis and changing certain parametric values in matlab, it canbe seen that only few parameters can change system performance and possiblymake system unstable. Parameters which mainly affect system performance areTa, Lv, kω, rv. Manual tuning of these parameters could lead into faster systemresponse but if not done properly could also possibly make system unstable. Sincethis is not in the scope of the project, parametric sensitivity will not be investigatedany further.

2.10 Conclusion

In this chapter, VSM with cascade control loop was developed. Control strategywas simulated in inslanded operation where performance in terms of THD andTHD when parameters uncertain was introduced was assesed. Also, transientsresponse at load changes was investigated and frequency response of VSM duringthose changes. Power sharing in 2 parallel VSM connected was briefly investigated.At the end, small signal stability analysis was investigated.

2.10. Conclusion 35

Figure 2.25: Load affect on system eigenvalues

Chapter 3

Multi Objective - Modulated ModelPredictive Control

In this chapter, an improved MPC strategy similarly demonstrated in [10] will bepresented. MPC is advanced optimization strategy used in feedback loop systems.This method is becoming more and more popular due to many advantages overclassical control. In MG applications, MPC can be implemented by using inverterdiscrete state space model to predict future behaviour of controlled state variables,and by minimizing formulated cost function, and selecting approximately closestcontrol input vector vi , MPC can follow reference signal fast without producinghigh THD.

Particularly, in MG and VSM point of view, MPC can provide better transientresponse and better THD by removing cascade control loop. Drawback could bemore intense number of calculations which would require more powerfull con-troller, but as computational power of microcontrollers is constantly increasing,this does not provide excessive problem for implementation of MPC.

Proposed VSM with MO-M2PC is presented in figure 3.1. Similarly to VSMpresented in earlier chapter, VSM reference frequency and voltage is controlledthrough swing equation inside of virtual inertia block and reactive power con-trol. Main difference is removing cascade control loops with function block ofMO-M2PC algorithm which correspond to one control loop, therefore control re-sponse speed should increase. In order to develop proper working MO-M2PC,state space model of system is developed in order to predict future behaviour ofstate variables. Then, cost function is formed, describing each subfunction insideof cost function and advantage over MPC which uses only one control objective.Lastly, parameters inside of algorithm are tuned and simulation of MO-M2PC im-plemented in VSM is presented and analyzed.

37

38 Chapter 3. Multi Objective - Modulated Model Predictive Control

Figure 3.1: VSM with MO-M2PC Implemented.

3.1 State space model of the Inverter with LC filter

In order to accurately predict future state of controlled variable, proper state spacemodel of system has to be developed. Mathematical equations 2.6, 2.7 of LC filterpresented in section 2.1 can be expressed in state space model form as 3.1. Foreasier notation, (Vabc = vi, VabcO = Vf , Iabc = i f , IabcO = io)

ddt

[i fVf

]= A

[i fVf

]+ B

[viio

](3.1)

where A is :

A =

[− r

L f− 1

L f1

C f0

](3.2)

and B

B =

[ 1L f

0

0 − 1C f

](3.3)

State space model 3.1 represent continuous state space model of LC filter connectedto inverter, where inputs are voltage vector vi from table 2.2 and output current io.For practical applications, digital control is employed, therefore, state space modelin 3.1 has to be discretized. As discussed in project limitation, dc link voltage isassumed to be constant. Voltage vector vi from table 2.2 has a staircase input tothe controlled inverter. For staircase inputs, Zero Order Hold (ZOH) match well totime domain staircase input. Therefore, state space model discretized using ZOH

3.2. MPC Algorithm principle 39

method is following :

ddt

[i f (k + 1)Vf (k + 1)

]= A

[i f (k)Vf (k)

]+ B

[vi(k)io(k)

](3.4)

where Ad is :Ad = eATs (3.5)

and Bd as : ∫ Ts

0eAτBdτ (3.6)

Ts is controller sampling time.

3.2 MPC Algorithm principle

MO-M2PC can be explained through following flowchart 3.2. i f , v f , io are mea-sured at beginning of every sampling instant. Using this measurements, and dis-crete state space model 3.4, future trajectories of i f and v f are predicted for every of23 voltage vectors vi. It should be mentioned that in real system application, io isoccasionally not measured [26], and there should be io estimator using measuredi f , v f implemented into control. After that, multi objective cost function is mini-mized for every of predicted values, and voltage vectors vi that correspond closestto minimized function is chosen using SVPWM. Same principle is used for everysampling instant.

3.3 Cost function

As it can be seen from section 3.2, cost function is foundation of MPC algorithm.Cost function is expressed in equation 3.7. In order to easier explain cost func-tion, it can be define by four important sub functions which will be explained infollowing sections. All notation will be expressed in terms of VSM.

CF = Gc + GL + Gerr + hlim (3.7)

3.3.1 Capacitor voltage

As capacitor voltage has to follow reference voltage, firstly reference voltage has tobe determined. Voltage reference can be formed by :

V∗f (t) = Vre f sin(ωVSMt) + jVre f cos(ωVSMt) (3.8)


Figure 3.2: Flowchart of MPC sampling instant.

Where Vre f is output reference voltage from virtual impedance block and ωVSMoutput speed of VSM. In order to algorithm follow the capacitor voltage reference,function can be formed using euclidean distance as 3.9.

Gc = (V∗f α −Vf α)2 + (V∗f β −Vf β)

2 (3.9)

where v f is predicted state using 3.4. This will produce voltage error at everysampling instant which main goal of algorithm is to minimize. This function canbe regarded as conventional single objective Finite Control Set - Model PredictiveControl (FCS-MPC). Even though this will provide satisfactory results for first or-

3.3. Cost function 41

der systems, problem arises when trying to control second order system, LC filterin this project. Coupling between state variables, introduces problem of control-ling capacitor voltage, and can only be done through indirectly controlling inductorcurrent. As current cannot change value instantaneously, therefore, capacitor volt-age cannot be regulated instantanous. This problem essentialy introduces urgencyto include capacitor derivative into cost function. If not included, there will besignifical increase in voltage THD detail explained in [10].

3.3.2 Inductor current

Knowing that current reference is produced using following equation 3.10.

i∗L = C fdv∗fdt

+ io (3.10)

where C f is filter capacitor. In order to define inductor current function, voltagereference 3.8 has to be expressed in respect to its derivative.

dv∗f (t)

dt= ωVSMVre f cos(ωVSMt)− jωVSMVre f sin(ωVSMt) (3.11)

Now that reference inductor current is developed, predicted states of inductorcurrent can be express through 3.12

iL(t) = (i f α(t) − ioα(t)

C f) + j(

i f β(t) − ioβ(t)C f

) (3.12)

Similarly to voltage function Gc, result of following equation 3.13 will produce anerror which will be added to cost function.

GL = (i∗Lα − iLα)2 + (i∗Lβ − iLβ)

2 (3.13)

Combining equation 3.10, 3.11 and 3.12 into 3.13 we get inductor current functionwhich will be added to final cost function:

GL = (C f ωVSMV∗f β − i f α + ioα)2 + (C f ωVSMV∗f α + i f β − ioβ)

2 (3.14)

where i f (t) is predicted state using 3.4 and io(t) measured current at commoncoupling point. Now a function can follow inductor reference current as well,which will produce better THD values compared to single objective function.

3.3.3 PR controller

In order to improve power quality, voltage error at particular frequency can be am-plified using resonant controllers and used in final cost function to reduce steady


state error at fundamental frequency, and certain odd harmonic orders to improveoverall THD. Resonant controller can be expressed as :

HPRs =Kerrs

s2 + (nωn)2 (3.15)

where ωn is fundamental frequency, n is odd harmonic order. For practical appli-cation, resonant controllers has to be discretized using impulse invariant methodwhich provides most optimal results stability wise investigated in [34]. Also, Res-onant controllers will take account for delay caused by controller sampling time,Ts

which can negatively effect system stability. Resonant controller in discrete domainz is present in

HPRz = Tscos(nTsωn)− z−1cos((n− 1)Tsωn)

1− 2z−1cos(nTsωn) + z−2 (3.16)

Resonant controller in discrete domain can be written as :

envc = a1n envc[k]− envc[k− 1] + a2n evc[k + 1]− a3n evc[k] (3.17)

where a1n = 2cos(ωnTs), a2n = KnTscos(ωnTs) and a3n = Kncos(ω(n− 1)Ts) Res-onant controller representation in 3.17 will play big role in development of aug-mented state space model presented in section 3.5. Final equation which will beadded to final cost function is as followed in 3.18.

Gerr = ∑ ||(V∗f α −Vf α)2 + (V∗f β −Vf β)

2 ∗ HPRz||2 for n = 1,5,7,11 (3.18)

Now final cost function have a ability to eliminate steady state errors at problematicharmonic orders and improve system THD.

3.3.4 Current overprotection

Important feature which can be implemented into cost function is current con-straints 3.19.

hlim =

0, if |i f | ≤ imax

∞, if |i f | ≥ imax(3.19)

where imax is current constrain value. This will ensure that filter inductor currentdoesn’t exceed constrain value. This is necessary protection in case of error inalgorithm or in malfunction in part of the hardware.

3.4 Simulation of MO-M2PC VSM in islanded operation

Similarly like in previous chapter, VSM is tested in islanded operation. At begin-ning, resistive load will be connected until 0.3 s when inductive load will be addedto determine system transients performance. Simulations will be executed usingfollowing steps.

3.4. Simulation of MO-M2PC VSM in islanded operation 43

1. 0 s - Start of simulation (resistive load)



In order to benchmark and compare MO-M2PC to standard VSM, load values usedin simulations are same as in section 2.4. This will give insight in difference in tran-sient response and system performance between VSM and MO-M2PC. In figure3.3, filter capacitor voltage and filter inductor current are presented. Compared toVSM simulations in section 2.8, it is obvious that MO-M2PC has faster dynamicalresponse at beginning of simulation and during transients, also, even from figure,lower THD is noticeable. Figure 3.4 represents active and reactive power consumed

Figure 3.3: Capacitor voltage and inductor current.

by the load, where compared to simulations in section 2.8, there is no oscillation ofactive power during transients. This means that MO-M2PC has better and fasterpower delivery to load. VSM speed in figure 3.5 has same behaviour as in sec-tion 2.8, therefore, VSM emulates SM behaviour and adjust itself depending onload and power reference. As it can be seen from simulation figures, MO-M2PCprovides better performance and transients for the same investigated parameters.


Figure 3.4: Active and Reactive power.

Figure 3.5: VSM Speed.

3.4.1 Total harmonic distortion

As already explained in section 2.8, THD has important role in distribution sys-tems, and therefore, is important to asses. In figure 3.6, THD for resistive and in-ductive load are presented. THD for both load in 0.15 % which is quite improvedTHD compared to VSM.

3.4. Simulation of MO-M2PC VSM in islanded operation 45

Figure 3.6: THD.

Robustness to LC filter parameters variations

Similarly to 2.8, system has to be investigated to LC parameters variations. Ta-ble 3.1 represents THD of different variations of LC filter parameters. MO-M2PCseems robust to changes in LC filter parameters keeping THD similar for eachtest. It can be determined that only decreasing capacitor value effects current THDnegatively.

Table 3.1: LC parameters value variations.


THD(Resistive load)THD(Inductiveload)

C f + 50% 0.14% 0.14%L f + 50% 0.14% 0.14%C f - 50% 0.35% 0.23%L f - 50% 0.17% 0.16%C f + 50% L f - 50% 0.15% 0.15%C f + 50% L f + 50% 0.14% 0.14%C f - 50% L f + 50% 0.26% 0.19%C f - 50% L f - 50% 0.69% 0.42%


3.4.2 Power sharing of 2 parallel connected VSM

In order to determine transient response of MO-M2PC, islanded simulation of 2parallel connected VSM with load is presented in following section. Simulation isexecuted in following steps.

1. 0 s - Start of simulation (resistive load), 1 VSM is connected

2. Second VSM is connected



Figure 3.7 presents active power sharing of 2 parallel connected VSM. As it can beestablished from figure, active power sharing seems to be accurate and equal. In

Figure 3.7: Active power sharing of 2 parallel connected MO-M2PC VSM.

figure 3.8, reactive power sharing between 2 VSM can be seen.

3.5 Parameter tuning and stability analysis

Recalling from SRF VSM chapter, PID had important role in stability and responseof the system. Therefore, proper tuning of PID controller had to be done in orderto achieve best results. In MPC control, state space model is used to predict systembehaviour, and, therefore, there is no need for PID controller. Standard state spacemodel equation :

ddt[x]= A

[x]︸︷︷︸

Dynamics of system

+ B[u]︸︷︷︸

System responds to inputs

(3.20)

3.5. Parameter tuning and stability analysis 47

Figure 3.8: Reactive power sharing of 2 parallel connected MO-M2PC VSM.

From equation 3.20 it can be noticed that in order to change dynamics of the sys-tem, A matrix has to be modified. Therefore, using pole placement method andadding K gain to each of the state will affect system eigenvalue, and directly changesystem dynamics. Therefore, augmented closed loop state space model has to bedeveloped with introduced K gain value. Cost function from 3.7 can be representedin different form as :

g(|vi| = KiL|eiL[k + 1]|2 + Kvc|evc[k + 1]|2 + |e1vc[k + 1]|2 + ... + |envc[k + 1]|2 (3.21)

where eiL[k + 1] represents error between inductor reference current and predictedinductor current, evc is error between reference capacitor voltage and predictedcapacitor voltage and K are gains that need to be chosen. Since optimum valuesfor viα and viβ are :

dg|vi|dviα

= 0

dg|vi|dviβ

= 0(3.22)

Adding resonant term for fundamental frequency and odd harmonics (5,7,11), so-lution can be found as :

vi = −(K1 IL[k] + K2vc[k] + K3vc[k− 1] + K4e1vc[k] + K5e1vc[k− 1] + K6e5vc[k]

+K7e5vc[k− 1] + K8e7vc[k] + K9e7vc[k− 1] + K10e11vc[k] + K10e11vc[k]

+K11e11vc[k− 1]) + K12 IL[k]∗ + K13vc[k] + K14Vc[k− 1] + K15 Io[k](3.23)

Resulting vector vi is then synthesized in SVPWM in order to produce gate driversignals.


Equation 3.23 can be expressed compact in matrix form as :

vi = −Kxk[k] + Fr[k] + J Io[k] (3.24)

where :K =

[K1 K2 K3 K4 K5 K6 K7 K8 K9 K10 K11

](3.25)

,F =

[K12 K13 K14

](3.26)

,J =

[K15]

(3.27)

, Augmented state vector xa[k] is :

xa =

IL[k]vc[k]

vc[k− 1]e1vc[k]

e1vc[k− 1]e5vc[k]

e5vc[k− 1]e7vc[k]

e7vc[k− 1]e11vc[k]

e11vc[k− 1]

(3.28)

, and vector r[k] consist of current and voltage references:

xa =

I∗L[k]v∗c [k]

v∗c [k− 1]

(3.29)

Open loop augmented state space model can be described with 3.30.

xa[k + 1] = Φaxa[k] + Γavi[k] + EIo[k] + Rr[k] (3.30)


Where,

Φa =

Ad(1, 1) Ad(1, 2) 0 0 0 0 0 0 0 0 0Ad(2, 1) Ad(2, 2) 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 00 −a21 a31 a11 −1 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 00 −a25 a35 0 0 a15 −1 0 0 0 00 0 0 0 0 1 0 0 0 0 00 −a27 a37 0 0 0 0 a17 −1 0 00 0 0 0 0 0 0 1 0 0 00 −a211 a311 0 0 0 0 0 0 a111 −10 0 0 0 0 0 0 0 0 1 0

(3.31)

Γa =[Bd(1, 1) Bd(2, 1) 0 0 0 0 0 0 0 0 0

](3.32)

E =[Bd(1, 2) Bd(2, 2) 0 0 0 0 0 0 0 0 0

](3.33)

R =

0 0 00 0 00 0 00 a21 −a31

0 0 00 a21 −a31

0 0 00 a21 −a31

0 0 00 a21 −a31

0 0 0

(3.34)

Figure 3.9 represents open loop system eigenvalues. Since model eigenvalues are inz(discrete) domain, in order for system to be stable, all eigenvalues has to be insideof unit circle. From figure, it is noticable that all eigenvalues are inside of unitcircle, but since they are arranged close to unity circle, they have slow dynamics.Therefore, gains K has to be added in feedback to form closed loop system, andadjust system eigenvalues. Combining equations 3.24 and 3.30, final augmentedclosed loop state space model is formed as :

xa[k + 1] = (Φa − ΓaK)xa[k] + (ΓaF + R)r[k] (3.35)

Where (Φa − ΓaK)xa[k] eigenvalues are of interest as they provide look at systemdynamics. Therefore, in following discussion, system will be optimized simply


Figure 3.9: Open loop system eigenvalues.

Figure 3.10: Closed loop system eigenvalues - IL gain tuning.

by adding gain factor K. System dynamics will be manipulated starting with KIL.Figure 3.10 illustrates closed loop system eigenvalues plot where gain factor KIL isincreased using 7 steps, each step gain factor KIL is increased by 200. Arrow pointsinto direction where eigenvalues are moving, and as it can be seen, pole settlesaround center of unit circle at gain factor KIL 1400. This is where pole has fastestand most damped properties. Next by adding Kvc gain factor and increasing itin 7 steps by value of 25, new eigenvalues are affected and manipulated. Fromfigure 3.11, it is noticeable how 2 eigenvalues move toward each other, where aftercertain gain factor Kvc eigenvalues become complex which is not desirable, due to


Figure 3.11: Closed loop system eigenvalues - Vc tuning.

lower damping characteristic. Therefore, gain factor Kvc right before poles becomecomplex is chosen which is 149. If gain factor Kvc is continue to be increased above150, system eigenvalues will move outside unit circle, which corresponds to systemgoing unstable. Figure 3.12 illustrates changes on eigenvalues when resonant gainfactor Kvc1 is increased by 1 in 7 steps. As resonant gain factor Kvc1 is increased,

Figure 3.12: Closed loop system zoom at eigenvalues - Resonant gain tuning.

eigenvalues moves closer to unit circle, and therefore, resonant gain factor Kvc1 ischosen to be 1. Same process is used for rest of the resonant terms and final figurewith eigenvalues is illustrated in 3.13. All eigenvalues are inside of unit circle,


Figure 3.13: Final closed loop system eigenvalues.

and eigenvalues are now closer center of circle, which corresponds to faster systemperformance.

3.6 Conclusion

In this chapter, MO-M2PC is developed and implemented in VSM. Simulationsare executed to asses strategy performance in terms of THD, THD when there isparameters uncertainty, transients response when system is introduced to changes.At end MO-M2PC parameters were tuned and stability was analyzed in closedloop system.

Chapter 4

Comparison of simulation results

In this chapter, comparison of VSM with cascade control loops and MO-M2PC sim-ulations presented in sections 2.8 and 3.4 will be discussed and analyzed. Chapterwill be divided into 3 sections where each important factor of comparison will bediscussed and final conclusion will be made. Some results from previous chap-ters may be represented again, for the sake of convenience. Both of simulationsof control strategies are carried out with same important parameters so results arenot effected by parameter tuning. Models are simulated using Simulink, in a goalto compare both control strategies in equal approach. VSM parameters in bothstrategies are same, and both simulation are executed in same way. At beginning,resistive load is connected to islanded operated inverter. At 0.3 s, additional, induc-tive load is connected. As for power sharing evaluation, simulation with 2 parallelconnected VSM is used to obtain results. At 0.1 s, second VSM is connected, whilerest of timeframe is same as in simulation before.

4.1 Total Harmonic Distortion

As power quality has important role in distribution systems, it was necessary toasses both control strategies THD results. THD of output load current was mea-sured using FFT tool in matlab. Results show how VSM with MO-M2PC performbetter than VSM controlled using cascade control loops.

In table 4.1, comparison of THD in case of resistive load is presented for differ-ent variations in LC filter. Results indicate that MO-M2PC is robust to any changesin LC filter values, while Cascade control of VSM is more affected by changes infilter values. It is interesting that worst case scenario for MO-M2PC has a betterTHD value than best case scenario for cascade control VSM.

When inductive load was connected during simulations, THD results are moresimilar than before. MO-M2PC still performed better in power quality than cas-cade control. Based on this simulation results, it can be concluded that MO-M2PC

53

54 Chapter 4. Comparison of simulation results

Table 4.1: Comparison of THD of resistive load.


VSM - Cascade con-trol

VSM-MO-M2PC

C f + 50% 1.10% 0.14%L f + 50% 0.93% 0.14%C f - 50% 1.11% 0.35%L f - 50% 1.52% 0.17%C f + 50% L f - 50% 1.44% 0.15%C f + 50% L f + 50% 1% 0.14%C f - 50% L f + 50% 0.93% 0.16%C f - 50% L f - 50% 1.76% 0.69%

Table 4.2: Comparison of THD of inductive load.



VSM-MO-M2PC

C f + 50% 0.55% 0.14%L f + 50% 0.49% 0.14%C f - 50% 0.46% 0.23%L f - 50% 0.52% 0.16%C f + 50% L f - 50% 0.52% 0.15%C f + 50% L f + 50% 0.55% 0.14%C f - 50% L f + 50% 0.45% 0.19%C f - 50% L f - 50% 0.48% 0.42%

strategy provides better power quality in microgrid islanded operation when pa-rameters can be uncertain.

4.2 Transient response of control strategies

As already explained in previous chapters, transient response is important aspectwhich has to be investigated. Since MO-M2PC combines current and voltage con-trol into one function and loop, faster response and better transients are expected.Transient response was investigated during two important changes applied to sys-tem. At start of simulation (Resistive load connection) and inductive load step.Figure 4.1 represents start of simulation where resistive load is connected. In bothstrategies, slight overshoot is present, but MO-M2PC strategy has faster responseto steady state. This can be seen also from measured active power from inverter atfigure 4.2. As it can be seen from figure, power delivery using MO-M2PC is fast

4.2. Transient response of control strategies 55

Figure 4.1: Transient of cascade control in above and MO-M2PC below figure.

Figure 4.2: Active power of cascade control in above and MO-M2PC below figure.

56 Chapter 4. Comparison of simulation results

and accurate while cascade loop responds slower with some overshoot in powerdelivery. When inductive load is connected at 0.3 s, both strategy respond welland distinctions are hard to notice from simulation figures so it will not be inves-tigated any further. As for Power sharing between 2 parallel connected VSM, bothstrategies provided satisfying results. As expected, power sharing depends moreon VSM output impedance matching rather than inner control loops.

4.3 Conclusion

In this section, comparison of two strategies has been done based on simulationsresults. What can be noticed is MO-M2PC superior performance in terms of THDfor both type of load tested. MO-M2PC also shows better performance and fasterresponse when system is affected by sudden changes like load step tested in sim-ulations. It should be also noted is design simplicity in MO-M2PC compared tocascade loop control VSM. Finally a table 4.3 is presented as a summary obtainedfrom simulations results.

Results infer that MO-M2PC is superior control strategy in most of propertiestested.

Table 4.3: Summary of two control strategies simulations results.


VSM-MO-M2PC

Design simplicity - +THD + ++Transients response + ++Stability proof + ++Reference trackingcapability

++ ++

Chapter 5

Experimental Work

In this chapter, the experimental result for MO-M2PC and benchmark strategyin parallel connected inverters microgrids is executed and presented. Since VSMtechnique enables automatic power sharing between multiple inverts without needfor communication links, it is left to determine how MO-M2PC strategy can affectperformance during sudden changes compared to additional benchmark strategy.Therefore, in following sections, MG islanded operation with 2 parallel VSM andrespected control strategies MO-M2PC and FCS-MPC are presented, analyzed andcompared using obtained date from experimental results. Experimental work isobtained through following setup:

• dSpace MicroLabBox with DS1202 PowerPC DualCore 2 GHz processor board

• DS1302 I/O board

• On Semiconductor CMOS level shifter

• Delta Elektronika SM 600-10 dc power supply

• Two 2-level Semikron three-phase VSI

In order to interface with MicroLabBox, the dSPACE software package ControlD-esk is used. dSpace ControlDesk provides the graphical user interface (gui) whichcan be used for changing parameters and monitoring signals, in real-time. Setupschematic is represented in figure 5.1. Delta Elektronika power supply is connectedto two level-three phase Semikron VSI as a dc link source. Two parallel SemikronVSI are connected through LC filter to Load. Inductor current, output current atPCC and capacitor voltage are measured and sampled using dSpace MicroLabBoxwhich is controlled using control desk interface on PC. After control algorithm isprocessed, dSpace MicroLabBox is sending gate signals to two 2 level VSI. Thissetup emulates islanded operation of parallel connected virtual synchronous ma-chines. Goal of experiment is to test two MPC control strategies and their transient

57

58 Chapter 5. Experimental Work

response, power sharing and frequency response. After testing both strategies,results will be analyzed and conclusion will be made.

Figure 5.1: Test Setup schematic.

Figure 5.2: Test Setup.

5.1. Benchmark strategy 59

5.1 Benchmark strategy

Since MO-M2PC inherently better characteristics and advantages over linear cas-cade loop control due to reasons pointed out multiple times in previous chap-ters, different benchmark strategy will be considered. As a benchmark strategyin islanded parallel operation microgrid, MO-M2PC will be tested against anothercommonly used model predictive control FCS-MPC. FCS-MPC is a model predic-tive control method similar to MO-M2PC, which means that in order to developFCS-MPC, state space model of inverter and LC filter has to be developed. Con-verter side current, output current and capacitor voltage are measured. State spacemodel is used to predict variables behaviour until certain horizon of time, andminimization of cost function as a criterion to select optimal solution. Main dif-ference between 2 strategies is FCS-MPC lack of consideration of capacitor deriva-tive into cost function. In second order systems as LC filter, capacitor voltagecannot be instantaneously changed since inductor current cannot instantaneouslychange as well. This problem was mentioned in 3.3.1 and solution was presentedin MO-M2PC. Therefore, FCS-MPC can be represented and explained using fol-lowing cost function 5.1.

CF = Gc + hlim (5.1)

Which consist of capacitor voltage regulation and current limiter. In depth knowl-edge and explanation of sub functions inside of cost function equation can be foundin chapter 3.3.

5.2 Islanded microgrid operation of Parallel VSM

In islanded microgrid operations, for economical and stability wise aspect, activeand reactive power between units should be shared synchronously. Circulatingcurrent among multiple inverters can lead to operational problems and reducereliability in microgrids. Frequency response of VSM is crucial during suddenchanges as it can effect power system stability. Also, a local load in MG requiresacceptable power quality to be delivered without high THD. Therefore, followingaspects will be investigated, starting from FCS-MPC and MO-M2PC power sharing,frequency response and PCC voltage and current under transients. Experimentconsist using following steps :

• Start experiment of 2 VSI in islanded parallel VSM operation using FCS-MPC

• Connection of resistive load (60 ohm)

• Resistive load step (60 to 30 ohm)

• End experiment of 2 VSI in islanded parallel VSM operation using FCS-MPC


• Start experiment of 2 VSI in islanded parallel VSM operation using MO-M2PC

• Connection of resistive load (60 ohm)

• Resistive load step (60 to 30 ohm)

• End experiment of 2 VSI in islanded parallel VSM operation using MO-M2PC

• Interpretation of experimental results

Both strategies are tested under transients using linear step load. Power sharingcapabilities are presented and analyzed. Frequency response of both strategiesare presented and analyzed and PCC voltage and current of both strategies arepresented and analyzed as well. Information obtained should give good insight inMO-M2PC control strategy compared to different strategies. Parameters used inlaboratory setup can be seen in table 5.1.

Table 5.1: Parameters of the laboratory setup

Parameter ValueNominal voltage 100 VNominal frequency 50 HzDC link 300 VFilter inductor,L f 2.4 mHFilter capacitor C f 15 uFResistive load 60 ΩResistive load step 30 Ω

5.2.1 FCS-MPC VSM

In following subsection, FCS-MPC experimental results are presented. On figure5.3, PCC voltage and current can be seen during transients, load step, where bluesine wave represents PCC voltage and purple sine wave is PCC current. It canbe notice fast transition to new value of current. Additionally, harmonics can benoticed, even though from figure THD value cannot be determined. As frequencyresponse of inverter has significant role in power sharing, it is important that fre-quency response have as little oscillation as possible. On figure 5.4, frequencyresponse of FCS-MPC is presented. One can notice small oscillations in frequencyresponse. During load step, frequency responds fast and correctly by decreasingfrequency. It can be concluded that frequency response in FCS-MPC control pro-vides fast, correct behaviour during transients with small oscillations.

Another important factor which can be crucial in multiple inverter islandedoperation is power sharing between inverters. As mentioned earlier, circulating

5.2. Islanded microgrid operation of Parallel VSM 61

Figure 5.3: FCS-MPC PCC voltage and current.

Figure 5.4: FCS-MPC frequency response.

current can lead to operation problems, and therefore, proper and equal powersharing between inverters is required.

Active power sharing between 2 connected inverters can be seen in figure 5.5,where red color represents active power from VSM 1, and blue color active powerfrom VSM 2. Proper active power sharing can be noticed even under transient due


Figure 5.5: FCS-MPC Active power sharing.

to load increase. What can be noticed also is power oscillations, even up to 50W. As one inverter increase power, other inverter decreases. Therefore, it can beinfered that FCS-MPC strategy has a good power sharing accuracy, fast dynamicalresponse with some oscillation in power sharing between 2 inverters.

5.2.2 MO-M2PC VSM

In this subsection, MO-M2PC experimental results are presented. As in case ofFCS-MPC, firstly, inverter PCC voltage and current are presented in figure 5.6,where blue sine wave represents PCC voltage and purple sine wave is PCC current.MO-M2PC PCC voltage and current have good performance during transients.MO-M2PC respond to sudden changes are fast and stable while keeping harmonicdistortion to a minimum. Even though it is impossible to determine THD fromfigure, it is noticeable improvement over FCS-MPC strategy in term of THD.

5.2. Islanded microgrid operation of Parallel VSM 63

Figure 5.6: MO-M2PC PCC voltage and current.

Figure 5.7: MO-M2PC frequency response.

Figure 5.7 show frequency response under MO-M2PC control strategy. As ex-pected, frequency response fit proper expected response of VSM. When load stepis applied, frequency drops fast and accurate. It can also be mentioned, howMO-M2PC seems more accurate with less oscillation in frequency compared toFCS-MPC strategy. This is important since oscillations in frequency can lead sys-tem to become unstable.


Figure 5.8: MO-M2PC Active power sharing

In figure 5.8, MO-M2PC active power sharing can be seen where red colorrepresents active power from VSM 1, and blue color active power from VSM 2.As expected, fast and accurate active power sharing can be determined from fig-ure. Comparing to FCS-MPC control strategy, MO-M2PC provides less oscillationswhen it comes to power sharing. Power sharing between inverters are equal andduring load step, both inverters release power equally and fast.

5.3 Interpretation of experimental results

From experimental results given, following can be determined.

5.3.1 PCC current and voltage regulation

When it comes to PCC voltage and current transient respond to load step, FCS-MPCand MO-M2PC provide fast and accurate response. This is expected since both con-trol strategy operates on same principle of using model predictive control. SinceMO-M2PC includes additional control objective(inductor current) into cost func-tion, it should perform better in terms of THD, explained in section 3.3. This canbe noticed from recorded figures where MO-M2PC seems to provide better results.

5.3.2 Frequency response

When it comes to frequency response, both control strategies provide fast and ac-curate response as expected from VSM. One can notice that in MO-M2PC strategy,

5.3. Interpretation of experimental results 65

frequency response is more stable with less oscillations compared to FCS-MPC.

5.3.3 Power sharing

As power sharing is crucial in islanded operation, both strategy were expectedto perform well in this area. Both strategy had a equal power sharing between2 inverters. MO-M2PC strategy did better in power sharing aspects as there wasless power oscillation than in FCS-MPC control. Sumary of 2 control strategiesused in experiments can be seen from 5.2. Based on experimental results obtained,

Table 5.2: Summary of two control strategies simulations results

VSM - VSM-FCS-MPC

VSM-MO-M2PC

Design simplicity + +THD + ++Transients response ++ ++Stability proof - +Frequency response + ++Power sharing + ++

MO-M2PC provides better performance than FCS-MPC when it operates as VSMin islanded microgrid operation.

Chapter 6

Conclusion

In this project, multi objective model predictive control was implemented intovirtual synchronous machine. In order to validate control strategy, virtual syn-chronous machine with cascade control loops was used as a benchmark and com-parison with MO-M2PC has been made. Additionally, MO-M2PC strategy wasused in experimental work in islanded 2 parallel inverter operation with anothermodel predictive control as a benchmark.

6.1 Brief description of work done

In chapter 1, the motivation behind project was introduced. Main objective wasto develop control strategy inside of VSM which could improve performance andovercome main disadvantages of cascade control loop VSM.In depth modelling of VSM with cascade control loop was done in chapter 2 andit implementation inside of Simulink. This method is used as benchmark con-trol strategy in simulations. Additionally, small signal analysis of developed statespace model was done and eigenvalues behaviour to load changes. System wassimulated and tested in standalone mode to sudden changes as load step. THDwas measured under different parameters uncertainty and analyzed.In chapter 3 MO-M2PC was developed and implemented into VSM. Cost func-tion has been constructed which can follow inductor current, capacitor voltageand minimize error at certain frequencies. MO-M2PC VSM is then simulated inislanded operation to sudden changes. Additionally, THD under different param-eters uncertainty was measured.Chapter 4 introduce comparison of results obtained through simulation. THD andtransient response of MO-M2PC and cascade loop control VSM is compared andanalyzed. At end, conclusion based on simulation results has been drawn forMO-M2PC and cascade control loop VSM.In chapter 5, experimental result is presented. Experiments are obtained through

67

68 Chapter 6. Conclusion

operation of 2 islanded parallel connected inverters. Results presented are ana-lyzed in terms of THD, frequency response and power sharing during transients.In this chapter, brief description of work can be found. Also, future work is pro-posed.

6.2 Future work

As a future work, using same principle of MPC, current controller can be devel-oped in grid feeding inverters.

Bibliography

[1] A. Alfergani et al. “Control strategies in AC microgrid: A brief review”. In:2018 9th International Renewable Energy Congress (IREC). 2018, pp. 1–6. doi:10.1109/IREC.2018.8362575.

[2] H. P. Beck and R. Hesse. “Virtual synchronous machine”. In: 9th Int. Conf.Electrical Power Quality and Utilisation. 2007, pp. 1–6.

[3] M. Begum, L. Li, and J. Zhu. “Distributed control techniques for autonomousAC Microgrid-A brief review”. In: 2017 IEEE Region 10 Humanitarian Technol-ogy Conference (R10-HTC). 2017, pp. 357–362. doi: 10.1109/R10-HTC.2017.8288974.

[4] P. Borazjani et al. “A review on microgrid control techniques”. In: 2014 IEEEInnovative Smart Grid Technologies - Asia (ISGT ASIA). 2014, pp. 749–753. doi:10.1109/ISGT-Asia.2014.6873886.

[5] P. A. Dahono. “A control method to damp oscillation in the input LC fil-ter”. In: 2002 IEEE 33rd Annual IEEE Power Electronics Specialists Conference.Proceedings (Cat. No.02CH37289). Vol. 4. 2002, 1630–1635 vol.4. doi: 10.1109/PSEC.2002.1023044.

[6] S. D’Arco, J.A.Suul, and O.B.Fosso. “A Virtual Synchronous Machine imple-mentation for distributed control of power converters in SmartGrids”. In:Electric Power System Research. Vol. 122. 2015, pp. 189–197.

[7] S. D’Arco and J. A. Suul. “Equivalence of Virtual Synchronous Machines andFrequency-Droops for Converter-Based MicroGrids”. In: IEEE Transactions onSmart Grid 5.1 (2014), pp. 394–395. issn: 1949-3053. doi: 10.1109/TSG.2013.2288000.

[8] S. D’Arco and J. A. Suul. “Virtual synchronous machines — Classification ofimplementations and analysis of equivalence to droop controllers for micro-grids”. In: 2013 IEEE Grenoble Conference. 2013, pp. 1–7. doi: 10.1109/PTC.2013.6652456.

69

https://doi.org/10.1109/IREC.2018.8362575

https://doi.org/10.1109/R10-HTC.2017.8288974

https://doi.org/10.1109/R10-HTC.2017.8288974

https://doi.org/10.1109/ISGT-Asia.2014.6873886

https://doi.org/10.1109/PSEC.2002.1023044

https://doi.org/10.1109/PSEC.2002.1023044

https://doi.org/10.1109/TSG.2013.2288000

https://doi.org/10.1109/TSG.2013.2288000

https://doi.org/10.1109/PTC.2013.6652456

https://doi.org/10.1109/PTC.2013.6652456

70 Bibliography

[9] S. D’Arco, J. A. Suul, and O. B. Fosso. “Automatic Tuning of Cascaded Con-trollers for Power Converters Using Eigenvalue Parametric Sensitivities”. In:IEEE Transactions on Industry Applications 51.2 (2015), pp. 1743–1753. issn:0093-9994. doi: 10.1109/TIA.2014.2354732.

[10] T. Dragicevic. “Model Predictive Control of Power Converters for Robustand Fast Operation of AC Microgrids”. In: IEEE Transactions on Power Elec-tronics 33.7 (2018), pp. 6304–6317. issn: 0885-8993. doi: 10.1109/TPEL.2017.2744986.

[11] Salvatore D’Arco, Jon Are Suul, and Olav B. Fosso. “Small-signal modelingand parametric sensitivity of a virtual synchronous machine in islanded op-eration”. In: International Journal of Electrical Power Energy Systems 72 (2015).The Special Issue for 18th Power Systems Computation Conference., pp. 3–15. issn: 0142-0615. doi: https://doi.org/10.1016/j.ijepes.2015.02.005. url: http://www.sciencedirect.com/science/article/pii/S0142061515000885.

[12] K. Gong, B. Hammer, and U. Konigorski. “Decentralized control approachfor an inverter-based microgrid in islanded mode”. In: 8th IET InternationalConference on Power Electronics, Machines and Drives (PEMD 2016). 2016, pp. 1–6. doi: 10.1049/cp.2016.0297.

[13] J. M. Guerrero, L. Hang, and J. Uceda. “Control of Distributed Uninterrupt-ible Power Supply Systems”. In: IEEE Transactions on Industrial Electronics 55.8(2008), pp. 2845–2859. issn: 0278-0046. doi: 10.1109/TIE.2008.924173.

[14] W. T. Jewell. “Electric Industry Infrastructure for Sustainability: Climate Changeand Energy Storage”. In: 2007 IEEE Power Engineering Society General Meeting.2007, pp. 1–3. doi: 10.1109/PES.2007.385567.

[15] A. Jäger-Waldau et al. “Photovoltaics in Europe after the Paris Agreement”.In: 2018 IEEE 7th World Conference on Photovoltaic Energy Conversion (WCPEC)(A Joint Conference of 45th IEEE PVSC, 28th PVSEC 34th EU PVSEC). 2018,pp. 3835–3837. doi: 10.1109/PVSC.2018.8547634.

[16] V. Kaura and V. Blasko. “Operation of a phase locked loop system underdistorted utility conditions”. In: IEEE Transactions on Industry Applications 33.1(1997), pp. 58–63. issn: 0093-9994. doi: 10.1109/28.567077.

[17] V. Khatana and R. Bhimasingu. “Review on Three-Phase PLLs for Grid In-tegration of Renewable Energy Sources”. In: 2017 14th IEEE India CouncilInternational Conference (INDICON). 2017, pp. 1–6. doi: 10.1109/INDICON.2017.8488071.

[18] A. Kumar and D. Chatterjee. “A survey on space vector pulse width modu-lation technique for a two-level inverter”. In: 2017 National Power ElectronicsConference (NPEC). 2017, pp. 78–83. doi: 10.1109/NPEC.2017.8310438.

https://doi.org/10.1109/TIA.2014.2354732

https://doi.org/10.1109/TPEL.2017.2744986


https://doi.org/https://doi.org/10.1016/j.ijepes.2015.02.005

https://doi.org/https://doi.org/10.1016/j.ijepes.2015.02.005

http://www.sciencedirect.com/science/article/pii/S0142061515000885

http://www.sciencedirect.com/science/article/pii/S0142061515000885

https://doi.org/10.1049/cp.2016.0297

https://doi.org/10.1109/TIE.2008.924173

https://doi.org/10.1109/PES.2007.385567

https://doi.org/10.1109/PVSC.2018.8547634

https://doi.org/10.1109/28.567077

https://doi.org/10.1109/INDICON.2017.8488071

https://doi.org/10.1109/INDICON.2017.8488071

https://doi.org/10.1109/NPEC.2017.8310438

Bibliography 71

[19] F. Luo et al. “A generalized droop-control scheme for decentralized controlof inverter-interfaced microgrids”. In: 2013 IEEE International Symposium onCircuits and Systems (ISCAS2013). 2013, pp. 1320–1323. doi: 10.1109/ISCAS.2013.6572097.

[20] M. Malinowski and S. Bernet. “A Simple Voltage Sensorless Active DampingScheme for Three-Phase PWM Converters With anLCLFilter”. In: IEEE Trans-actions on Industrial Electronics 55.4 (2008), pp. 1876–1880. issn: 0278-0046. doi:10.1109/TIE.2008.917066.

[21] M. Malinowski, M. P. Kazmierkowski, and S. Bernet. “New simple activedamping of resonance in three-phase PWM converter with LCL filter”. In:2005 IEEE International Conference on Industrial Technology. 2005, pp. 861–865.doi: 10.1109/ICIT.2005.1600756.

[22] L. Meng et al. “Microgrid central controller development and hierarchicalcontrol implementation in the intelligent microgrid lab of Aalborg Univer-sity”. In: 2015 IEEE Applied Power Electronics Conference and Exposition (APEC).2015, pp. 2585–2592. doi: 10.1109/APEC.2015.7104716.

[23] A. Micallef et al. “Performance comparison for virtual impedance techniquesused in droop controlled islanded microgrids”. In: 2016 International Sympo-sium on Power Electronics, Electrical Drives, Automation and Motion (SPEEDAM).2016, pp. 695–700. doi: 10.1109/SPEEDAM.2016.7526013.

[24] P. Pinson et al. “Towards fully renewable energy systems: Experience andtrends in Denmark”. In: CSEE Journal of Power and Energy Systems 3.1 (2017),pp. 26–35. issn: 2096-0042. doi: 10.17775/CSEEJPES.2017.0005.

[25] D. Reigosa et al. “Analysis of grid connected converters using a feed-forwarddisturbance decoupling current control”. In: The 2010 International Power Elec-tronics Conference - ECCE ASIA -. 2010, pp. 2976–2983. doi: 10.1109/IPEC.2010.5543717.

[26] Ihab S. Mohamed et al. “Implementation of model predictive control forthree-phase inverter with output LC filter on eZdsp F28335 Kit using HILsimulation”. In: International Journal of Modelling Identification and Control 25(June 2016), pp. 301–312. doi: 10.1504/IJMIC.2016.076830.

[27] S. K. Sahoo, A. K. Sinha, and N. K. Kishore. “Control Techniques in AC, DC,and Hybrid AC–DC Microgrid: A Review”. In: IEEE Journal of Emerging andSelected Topics in Power Electronics 6.2 (2018), pp. 738–759. issn: 2168-6777.doi: 10.1109/JESTPE.2017.2786588.

[28] H. Suyanto and R. Irawati. “Study trends and challenges of the developmentof microgrids”. In: 2017 15th International Conference on Quality in Research(QiR) : International Symposium on Electrical and Computer Engineering. 2017,pp. 383–387. doi: 10.1109/QIR.2017.8168516.

https://doi.org/10.1109/ISCAS.2013.6572097

https://doi.org/10.1109/ISCAS.2013.6572097

https://doi.org/10.1109/TIE.2008.917066

https://doi.org/10.1109/ICIT.2005.1600756

https://doi.org/10.1109/APEC.2015.7104716

https://doi.org/10.1109/SPEEDAM.2016.7526013

https://doi.org/10.17775/CSEEJPES.2017.0005

https://doi.org/10.1109/IPEC.2010.5543717

https://doi.org/10.1109/IPEC.2010.5543717

https://doi.org/10.1504/IJMIC.2016.076830

https://doi.org/10.1109/JESTPE.2017.2786588

https://doi.org/10.1109/QIR.2017.8168516

72 Bibliography

[29] G. G. Talapur et al. “Combined Droop and Master-Slave Method for LoadSharing in Stand-alone AC Microgrid”. In: IECON 2018 - 44th Annual Con-ference of the IEEE Industrial Electronics Society. 2018, pp. 1705–1710. doi: 10.1109/IECON.2018.8591708.

[30] A. Toliyat and A. Kwasinski. “Energy storage sizing for effective primary andsecondary control of low-inertia microgrids”. In: 2015 IEEE 6th InternationalSymposium on Power Electronics for Distributed Generation Systems (PEDG).2015, pp. 1–7. doi: 10.1109/PEDG.2015.7223077.

[31] H. Wang. “Microgrid generation planning considering renewable energy tar-get”. In: 2016 IEEE International Conference on Power and Energy (PECon). 2016,pp. 356–360. doi: 10.1109/PECON.2016.7951587.

[32] L. Wang. “Carbon Tax Policy and Technological Innovation for Low-CarbonEmission”. In: 2011 International Conference on Management and Service Science.2011, pp. 1–4. doi: 10.1109/ICMSS.2011.5998649.

[33] T. Wu et al. “A Review for Control Strategies in Microgrid”. In: 2018 37thChinese Control Conference (CCC). 2018, pp. 30–35. doi: 10.23919/ChiCC.2018.8482549.

[34] A. G. Yepes et al. “Effects of Discretization Methods on the Performance ofResonant Controllers”. In: IEEE Transactions on Power Electronics 25.7 (2010),pp. 1692–1712. issn: 0885-8993. doi: 10.1109/TPEL.2010.2041256.

[35] P. Zhang et al. “Hierarchical Management of Load and Coordinated Con-trol of Energy in Microgrid with Multiple Distributed Generations”. In: 20185th IEEE International Conference on Cloud Computing and Intelligence Systems(CCIS). 2018, pp. 1013–1017. doi: 10.1109/CCIS.2018.8691142.

[36] W. Zhao and G. Chen. “Comparison of active and passive damping methodsfor application in high power active power filter with LCL-filter”. In: 2009 In-ternational Conference on Sustainable Power Generation and Supply. 2009, pp. 1–6. doi: 10.1109/SUPERGEN.2009.5347992.

https://doi.org/10.1109/IECON.2018.8591708

https://doi.org/10.1109/IECON.2018.8591708

https://doi.org/10.1109/PEDG.2015.7223077

https://doi.org/10.1109/PECON.2016.7951587

https://doi.org/10.1109/ICMSS.2011.5998649

https://doi.org/10.23919/ChiCC.2018.8482549

https://doi.org/10.23919/ChiCC.2018.8482549


https://doi.org/10.1109/CCIS.2018.8691142

https://doi.org/10.1109/SUPERGEN.2009.5347992

Appendix A

Direct-quadrature-zero transforma-tion

Three phase signals from abc reference can be transformed into 2 signal rotationalreference frame using park inveriant transformation. This transformation allowscontrol of three AC signals as a 2 DC signal simplify complexity of three phasesystems. d

q0

=23

cos(θ) cos(θ − 2π3 ) cos(θ + 2π

3 )

−sin(θ) −sin(θ − 2π3 ) −sin(θ + 2π

3 )12

12

12

abc

(A.1)

The inverse transformation from the dq0 frame to the natural abc frame is given inA.2. a

bc

=23

cos(θ) −sin(θ) 1cos(θ − 2π

3 )) −sin(θ − 2π3 ) 1

cos(θ + 2π3 )) −sin(θ + 2π

3 ) 1

dq0

(A.2)

73

Appendix B

Clarke transformation

Three phase signals from abc reference frame can be transformed into stationaryorthogonal using following transformation B.1 where Kα,β is transformation matrix.This simplifies three phase system into 2 quantities, necessary for SVPWM.

ζ = Kα,βζabc

Kα,β =23

[1 − 1

2 − 12

0√

32 −

√3

2

](B.1)

Inverse clarke transformation is given by B. fa

fbfc

=

1 0 1− 1

2

√3

2 1− 1

2 −√

32 1

fα

fβ

f0

(B.2)

75

Appendix C

Linear VSM State space model

State space model presented in appendix is developed from [11] using differentparameters implemented into Matlab. Linear full state space model of virtual syn-chronous machine can be represented as :[

∆x1

∆x2

]=

[A11 A12

A21 A22

] [∆x1

∆x2

]+ B∆u (C.1)

Where, matrix A11 is presented in C.1, A12 in C.2, A21 in C.3, A22 in C.4 andmatrix B in C.5.

Where state space model is constructed using following full system equationsC.6.

Figure C.1: Matrix A11

77

78 Appendix C. Linear VSM State space model



79


Figure C.5: Matrix B

80 Appendix C. Linear VSM State space model

Figure C.6: Full system equations

Appendix D

Active and reactive power calcula-tions

Three phase signal can be transformed into 2 signal rotational reference frameusing park inverient transformation, and then active and reactive power can becalculated using following equations:

P =32(Vd Id + Vq Iq + 2V0 IO) (D.1)

P =32(Vq Id −Vd Iq) (D.2)

81

Documents

Implementation of Multi Objective - Modulated Model ... · Generally, basic control in Alternate current (AC) MG can be divided into master-slave control [29], hierarchical control