This thesis assesses the voltage unbalance in distribution networks due to load asymmetry or line asymmetry, based on measurement data from a limited number of monitors. The main outcomes of this research are a probabilistic methodology for estimating both momentary and long term unbalance and an optimal monitor placement providing the highest accuracy for the monitored level of unbalance. With increasing numbers of large single-phase loads and distributed generation integrated into the power system, the future distribution network is expected to be more flexible, robust and “smart”. This results in the requirement for high quality of electricity supply to be delivered to customers and is a challenge for the operation of the system. As the unbalance results in excessive heating, accelerated thermal ageing, reduction of efficiency and financial losses, the unbalance should be regulated to be below the statutory limit. Given the fact that unbalance is a long term phenomenon that may not cause any triggering of protection or faulty response of equipment, it can be only determined from available data such as loading levels of a network and the incomplete monitored voltage of a network. Due to limited monitoring in the network and therefore insufficient data, unbalance may be unobservable. This thesis therefore aims to develop a methodology to increase the observability of unbalance in the network in spite of limited monitoring. This research uses Voltage Unbalance Factor (VUF) to quantify the level of unbalance. The first major part investigates the unbalance caused by asymmetrical loadings. By properly identifying the source of unbalance, the basic patterns of propagation of unbalance under possible scenarios are revealed and a methodology of probabilistic estimation of unbalance can be developed accordingly. Seen from the MV level of distribution networks, the loads are usually in the constant power form. Therefore, the variation in the load can be modelled by changing either active power or reactive power or both of them, depending on the data availability. The combinations of daily loading curve at buses and the normally distributed power factors in three phases of loads are used to create an unbalanced condition at the sources. Realistic assumptions of power factors and reasonable categories of types of loads result in realistic modelling of the unbalanced load. The probabilistic VUFs at different buses in the network are calculated and the weak areas in the network are identified using heat maps. The simulation results match the real VUF levels measured in the distribution network. The second part of the thesis explores the influence of asymmetrical lines in addition to the asymmetrical loading on propagation of unbalance. The last part provides a guideline for optimal monitor placement for unbalance. Two methods, manual ranking of buses and automatic optimization using Genetic Algorithm, are proposed. The two methods indicate the same optimal locations for monitor placement in the network. The developed methodologies enable the assessment of unbalance in the network when monitoring is limited and can be applied to real networks to assess the level of unbalance at non-monitored buses.

The integration of distributed generators (DGs) into the distribution grid has exacerbated voltage unbalance issues leading to greater risks of reducing equipment lifetime, equipment damages, and increased ohmic losses. Most approaches to regulating voltage in distribution systems only focus on voltage magnitude and neglect phasor discrepancies and do little to remedy voltage unbalance. To combat this, a novel Optimal Power Flow (OPF) is designed to help operate these resources in a manner that curtails voltage unbalance using the reactive power compensation capabilities of inverters. The OPF was run for a wide variety of loading conditions on a pair of systems using MATLAB and was shown to improve the voltage profile of the system in addition to minimizing losses in most cases. However, it is noted that the OPF loses exactness in highly stressed conditions and is unable to provide meaningful solutions / Master of Science / With the power grid getting greener and smarter by the day, a slew of new challenges arise to overcome. Distributed sources of energy like solar panels and batteries are being added to the grid right from the household level. While they are desirable for reducing our need for traditional sources of energy, the addition of these resources has been shown to cause issues in the quality of the power grid. This is particularly observed at the low-voltage domestic part of the grid where the resources cause issues with the voltage quality. The distribution grid is unbalanced by nature and adding these resources only amplifies this problem. To help mitigate voltage quality issues grid operators are starting to require voltage regulation capabilities from resources to be connected to the grid and a lot of work has been conducted to find the optimal strategies for operating these resources. However, existing paradigms for these sources only focus on fixing the voltage magnitude part of the power quality and neglect phasor relationships. This thesis aims to bridge this gap by developing a method to determine the optimal operation of these resources by using the voltage regulation capability to address both voltage magnitude and voltage unbalance issues in addition to optimal operation.

Distributed generation (DG) has received much attention recently due to environmental consciousness and rising of the energy efficiency. However, DG interconnecting to low-voltage distribution system may cause voltage variation, and a lot of single-phase DG or single-phase load may result in voltage unbalance. This thesis presents a distributed-STATCOM (D-STATCOM) to alleviate variation of both positive-sequence and negative-sequence voltages at the fundamental frequency. The D-STATCOM operates as susceptance and conductance at the fundamental positive-and negative-sequence frequency, respectively. The susceptance and conductance commands are dynamically tuned according to voltage fluctuation at the installation location. Therefore, the positive-sequence voltage can be restored to the nominal value as well as the negative-sequence voltage can be suppressed to an allowable level. Computer simulations and experimental results verify the effectiveness of the proposed control strategy.

Made available in DSpace on 2014-06-11T19:22:34Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-02-01Bitstream added on 2014-06-13T18:49:38Z : No. of bitstreams: 1 oliveira_jea_me_bauru.pdf: 12636194 bytes, checksum: d216e62aabc6e57242fe0c06923a7c5d (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O desequilíbrio de tensão nos sistemas elétricos pode provocar problemas indesejáveis na operação de equipamentos, principalmente nos motores de indução trifásicos, devido à importância destes motores em ambientes industriais. A utilização de modelos convencionais para a modelagem destes motores operando com tensões desequilibradas impõe resultados imprecisos e a obtenção de modelos adequados apresenta grande complexidade em função das assimetrias e não linearidades. Neste contexto, a utilização de ferramentas inteligentes, mais especificamente, redes neurais artificiais (RNA), reduz substancialmente a tarefa de modelagem, permitindo sua utilização sob condições de assimetrias e não linearidades. Assim, uma bancada de testes foi montada para a aquisição de dados experimentais de um motor de indução trifásico de 1 CV, 4 polos, 220V/380V, tipo gaiola de esquilo. Os dados coletados foram usados para o treinamento e validação de uma RNA que modela a relação entre as tensões, correntes e a potência no eixo. Os resultados experimentais foram comparados com os obtidos com a RNA e com o modelo dinâmico, e constatou-se que a modelagem por meio de RNA é adequada para descrever matematicamente o comportamento de motores de indução trifásicos operando com tensões desequilibradas / Unbalanced voltages in electrical systems can deteriorate the performance of equipments and cause potential safety hazards and be harmful for the respective applications, especially in the three-phase induction motors, the most common energy receivers in industrial. The analysis of three phase induction motors under supply voltage unbalance condictions using the well-known symmetrical components analysis provide inaccurate results, and correct models are laborious, due to the complex nature of voltage unbalance factor like asymmetries and nonlinearities. In this context, the use of intelligent tools, specifically artificial neural networks (ANN), significantly reduces the modeling task and allowing the use under conditions of asymmetries and nonlinearities. Thus, a workbench tests was buit for testing of the 4 pole, 220V/380V, 1 CV squirrel-cage induction motor. Experimental set up for testing were used to ANN's training and validation. The ANN's model showed the relationship between the voltages, currents and shaft power. The results of experimental investigation and computer calculations (ANN and dynamic model) were compared and the results indicate that the ANN is adequate model that makes it possible to mathematically describe an induction motors operating with unbalanced voltage

Existing low-voltage networks may not accommodate high penetrations of low-carbon technologies. The topic of this thesis is unbalance, which if minimised can delay or avoid the constraining of these technologies or the replacing of still-useful network assets. Most of the discussion on unbalance, as seen in the standards and the literature, centres on the effects of voltage unbalance on consumer equipment. Its effects on the network are not equally reported. This thesis recognises fundamental differences between the consumer and network perspectives. It can inform distribution network operators on the interpretation of measurements taken on low-voltage networks and guide research on unbalance due to high penetrations of low-carbon technologies. Much of the work involved simulations of LV networks. Initially, existing 3 x 3 or 5 x 5 approaches to the forward-backward sweep method were thought suitable. After a review of these approaches however, there were doubts as to how accurately they accounted for the shared neutral-earth return path on which the out-of-balance current flows. This led to the derivation of a new 5 x 5 approach using only Kirchhoff s voltage (KVL) and current laws (KCL). Its results are validated thoroughly in the thesis. In addition to satisfying KVL and KCL, they match Matlab SimPowerSystems exactly and are in close agreement with measurements taken on a very unbalanced rural feeder. This thesis also investigates the mitigation of unbalance using the static balancer. This is a transformer with a single interconnected-star winding. It was used in 1930-1950s to correct unbalance. Contributions are made for its possible re-introduction as a retrofit option. They include a model for use in the forward-backward sweep method, validated by laboratory and field measurements, and the quantification of the static balancer s strengths and weaknesses as this can help identify when it should be used.

The objective of this Thesis is to investigate the phenomenon of voltage unbalance on electrical wind power systems. A large part of this work is the literature review of all relative work that has been done so far. This serves first as a guideline to define and measure voltage unbalance and second as a tool to spot open research questions that can inspire future work. A case study is then used to investigate the voltage unbalance at a wind farm in Näsudden, Gotland. Using real-time measurements and a simulation of the power system in MATLAB/Simulink, an evaluation of the propagation of the voltage unbalance from the distribution to the turbine level is carried out. The effect that different transformer connection types have on the propagation is studied through simulations. Many assumptions and simplifications had to be made due to several limiting factors during this work, mainly related to time and data restrictions. The main result shows that when Delta – Wye Grounded and Wye – Wye Grounded transformers are used, the unbalance is halved when it passes to the turbine side. On the other hand, when Wye Grounded – Wye Grounded configuration was used, the unbalance was unaffected. The results also include a comparison of the use of different indices to quantify a voltage unbalance.

Orientador: Paulo José Amaral Serni / Banca: Alessandra Goedtel / Banca: José Alfredo Covolan Ulson / Resumo: O desequilíbrio de tensão nos sistemas elétricos pode provocar problemas indesejáveis na operação de equipamentos, principalmente nos motores de indução trifásicos, devido à importância destes motores em ambientes industriais. A utilização de modelos convencionais para a modelagem destes motores operando com tensões desequilibradas impõe resultados imprecisos e a obtenção de modelos adequados apresenta grande complexidade em função das assimetrias e não linearidades. Neste contexto, a utilização de ferramentas inteligentes, mais especificamente, redes neurais artificiais (RNA), reduz substancialmente a tarefa de modelagem, permitindo sua utilização sob condições de assimetrias e não linearidades. Assim, uma bancada de testes foi montada para a aquisição de dados experimentais de um motor de indução trifásico de 1 CV, 4 polos, 220V/380V, tipo gaiola de esquilo. Os dados coletados foram usados para o treinamento e validação de uma RNA que modela a relação entre as tensões, correntes e a potência no eixo. Os resultados experimentais foram comparados com os obtidos com a RNA e com o modelo dinâmico, e constatou-se que a modelagem por meio de RNA é adequada para descrever matematicamente o comportamento de motores de indução trifásicos operando com tensões desequilibradas / Abstract: Unbalanced voltages in electrical systems can deteriorate the performance of equipments and cause potential safety hazards and be harmful for the respective applications, especially in the three-phase induction motors, the most common energy receivers in industrial. The analysis of three phase induction motors under supply voltage unbalance condictions using the well-known symmetrical components analysis provide inaccurate results, and correct models are laborious, due to the complex nature of voltage unbalance factor like asymmetries and nonlinearities. In this context, the use of intelligent tools, specifically artificial neural networks (ANN), significantly reduces the modeling task and allowing the use under conditions of asymmetries and nonlinearities. Thus, a workbench tests was buit for testing of the 4 pole, 220V/380V, 1 CV squirrel-cage induction motor. Experimental set up for testing were used to ANN's training and validation. The ANN's model showed the relationship between the voltages, currents and shaft power. The results of experimental investigation and computer calculations (ANN and dynamic model) were compared and the results indicate that the ANN is adequate model that makes it possible to mathematically describe an induction motors operating with unbalanced voltage / Mestre

Ce travail est le résultat d'une collaboration entre le laboratoire LAPLACE, la "Seconda Università degli Studi di Napoli" (SUN) et la Société National des Chemins de fer Français SNCF. Le sujet de recherche concerne l'utilisation de dispositifs électroniques de puissance dans les sous stations ferroviaires 25kV/50Hz afin d’améliorer la qualité de l'énergie électrique. Dans le transport ferroviaire, le système d'électrification monophasé 25kV/50Hz est largement diffusé en particulier pour les lignes ferroviaires à grande vitesse. Bien qu'aujourd'hui les systèmes d’alimentation en courant continu soient encore largement utilisés, l'adoption du courant alternatif monophasé offre des avantages économiques pour les infrastructures d'environ 30% en termes d'investissement, d'exploitation et d'entretien. Initialement, compte tenu de la simplicité du circuit, il n'y avait aucune nécessité d'intégrer de l'électronique de puissance dans les sous stations. Toutefois, au cours de la décennie passée, l'intérêt pour ces équipements est apparu car ils peuvent apporter une solution d'optimisation du réseau lorsque le trafic augmente ou lorsqu’une nouvelle sous station est envisagée. Deux principaux types de dispositifs sont installés aujourd'hui sur le réseau ferré français : les compensateurs de puissance réactive et les compensateurs de déséquilibre de tension. Cette thèse présente de nouvelles topologies de compensateurs basées sur le concept d’impédances contrôlées par gradateur MLI. Comparées aux solutions existantes, ces topologies ont des caractéristiques particulièrement intéressantes en termes de pertes dans les semi-conducteurs et de volume des composants réactifs. Le manuscrit contient trois parties principales: La première partie présente le principe de l’électrification en 25kV/50Hz et souligne l’intérêt d’installer des moyens de compensation statique dans les sous stations. Après une description des solutions actuellement utilisées, le concept d’impédance contrôlée par gradateur MLI (CCI : Chopper Controlled Impedance ) est ensuite présenté. La deuxième partie du travail concerne l'utilisation du concept de CCI pour la compensation de puissance réactive. La sous-station SNCF de Revest est considérée comme cas d’étude. Celle-ci est équipée d'un transformateur monophasé de 60MVA dont le primaire est connecté à une ligne de transport 225kV. Deux topologies de compensateur de puissance réactive, basées sur des montages abaisseur ou élévateur de tension sont présentées. Le dimensionnement des gradateurs est effectué sur la base d'une campagne de mesures réalisée à la sous station. Des simulations numériques utilisant des formes d’ondes réelles de courant et de tension sont présentées. Des résultats expérimentaux effectués à la plateforme de test de la SNCF sur un prototype de 1,2MVAR permettent de valider le concept de CCI. La dernière partie du travail concerne le problème du déséquilibre de tension en amont de la sous station. Un circuit de Steinmetz « actif », toujours basée sur des gradateurs MLI, est présenté et étudié. La sous station SNCF d'Evron est alors considérée comme cas étude. Celle-ci comporte un transformateur de 32MVA et est connectée à une ligne de transmission 90kV. Les mesures effectuées sur le site permettent le dimensionnement du compensateur ainsi que l’utilisation des formes d'onde réelles de courant et de tension dans les simulations numériques. Une comparaison avec des solutions classiques basées sur des onduleurs 2 niveaux et 3 niveaux souligne les avantages de la solution proposée. Ainsi, les résultats des calculs et des simulations montrent que l'énergie stockée dans les éléments réactifs est réduite d’un facteur six et que les pertes dans les semi-conducteurs sont réduites de 40%. Des résultats expérimentaux obtenus sur une maquette de 1.5 kVA permettent de valider le principe du circuit de Steinmetz actif. / This work is the result of collaboration between the LAPLACE laboratory, the “Seconda Università degli Studi di Napoli” (SUN) and the French national railways operator SNCF. The research topic treated herein concerns the use of power electronic devices in 25kV/50Hz railways substations to achieve power quality improvements. In railway transportation, single-phase 25kV-50Hz electrification system is widely diffused especially for high-speed railway applications. Although electrified DC systems are still widely applied, the adoption of AC single-phase system offers economical advantages for the infrastructures of about 30% in terms of investment, exploitation and maintenance. In early ages, due to its very simple diagram, there was no necessity to integrate power electronics in substations. However, for the last decade, the interest in power electronic equipments raised since they can provide the solution for network optimization when traffic increases or when a difficulty is foreseen for a substation implementation. Two types of devices are implemented today on the French Railway Network: Reactive Power compensators and Voltage Unbalance compensators. This thesis presents an investigation into new topologies based on the concept of “Chopper Controlled Impedances”(CCI). Compared to existing solutions, the new topologies show interesting features in terms of semi-conductor losses reduction and volume of reactive components. The manuscript is developed through three main parts: Firstly, the French railways system is introduced and the interest in installing power electronic compensators in substations is highlighted. After a brief description of currently used solutions, the CCI concept is presented: the use of Pulse Width Modulated AC Choppers allows achieving structures which behave as variable impedances. In the second part, the use of CCI structures in reactive power compensation is investigated. The SNCF substation of Revest is under study. It is equipped by a 60MVA single phase transformer with the primary side connected to a 225kV transmission line. Based on the step-down or step-up functioning mode of CCIs, two topologies of reactive power compensator are presented. The converter design is developed on the base of a measurement campaign carried out at the substation. Numerical simulations using real current and voltage waveforms are presented. Finally, experimental results carried out at the SNCF test platform on a 1.2MVAR prototype are shown. In the last part, the problem of voltage unbalance is treated. Using the concept of CCI, the feasibility of an active Steinmetz circuit based on AC choppers is explored. As a case study, the substation of Evron is considered. It is a 32MVA substation connected to a 90kV transmission line. Measurements carried out on the substation site allow the compensator design and the possibility to consider real waveforms for current and voltage in numerical simulations. A comparison with classical solution based on two levels VSI and three levels NPC-VSI highlights the advantages of the proposed solution. Calculation and simulation results show that the stored energy in reactive elements is reduced by a factor six whereas the semiconductor losses are 40% lower. Experimental results obtained on a scaled demonstrator (1.5 kVA) validate the principle of the active Steinmetz circuit.

Die Zunahme unsymmetrisch betriebener Kundenanlagen mit hoher Betriebsdauer und -strömen in Niederspannungsnetzen führt neben der stärkeren Belastung der Betriebsmittel und des Spannungsbandes zu einer Erhöhung der Spannungsunsymmetrie. Diese Arbeit untersucht diese Beeinflussungen anhand von Wiederspannungsnetzsimulationen bei verschiedener Durchdringung von Elektrofahrzeugen und Photovoltaikanalgen. Dazu werden anhand von Labor- und Netzmessungen probabilistische Lastmodelle für Haushalte, Photovoltaikanalgen und Elektrofahrzeuge entwickelt, welche die unsymmetrische Betriebsweise über einen Tag berücksichtigen. Die Auswirkungen auf die Spannungsunsymmetrie werden anhand des Verhältnisses zwischen Gegen- zu Mitsystemspannung und die Stromunsymmetrie anhand von unsymmetrischen Leistungsanteilen beschrieben. Neben der Analyse der Auswirkungen der unsymmetrisch betriebenen Kundenanlagen auf die erwähnten Kenngrößen werden mögliche Maßnahmen zur Reduzierung des Einflusses vorgestellt und durch Simulationen geprüft. Anhand der durchgeführten Betrachtungen und Simulationen wird ein Niederspannungsäquivalent abgeleitet. Mit diesem können Profile für die unsymmetrischen Leistungsanteile bestimmt werden, mit denen die Sternpunktbelastung und der Einfluss auf die Unsymmetrie im übergeordneten Netz abgeschätzt werden kann.:1 Einführung 1 1.1 Stand der Technik 1 1.2 Ziel der Arbeit 2 1.3 Struktur der Arbeit 3 2 Grundlagen 4 2.1 Elektroenergiequalität und EMV Koordinierung 4 2.2 Allgemeine Bewertungsgrößen 8 2.2.1 Gleichphasigkeitsindizes 8 2.2.2 Qualitätsreserve 9 2.2.3 Quantil 9 2.3 Betriebsmittelbelastung 10 2.4 Verlustleistung und -energie 10 2.5 Langsame Spannungsänderung 10 2.6 Unsymmetrie-Kenngrößen 13 2.6.1 Spannungsunsymmetrie 16 2.6.2 Stromunsymmetrie 18 2.6.3 Unsymmetrische Leistung 20 3 Einflussfaktoren auf die Unsymmetrie 24 3.1 Übergeordnetes Netz 24 3.2 Transformator 25 3.3 Leitung 27 3.4 Erdung 28 3.5 Kundenanlagen 33 3.5.1 Anschluss der Kundenanlagen 33 3.5.2 Statisches Verhalten hinsichtlich Spannungs- und Frequenzabhängigkeit 34 3.5.3 Analytisches Modell zur vereinfachten Abschätzung der Spannungsunsymmetrie 35 3.6 Zusammenfassende Bewertung der Einflussfaktoren 40 3.7 Maßnahmen zur Reduzierung der Unsymmetrie 41 3.7.1 Verringerung der Gegensystemspannung des übergeordneten Netzes 43 3.7.2 Verringerung der wirksamen Gegen- bzw. Nullsystemimpedanz am Verknüpfungspunkt 43 3.7.3 Verringerung des Gegen- bzw. Nullsystemstroms der anzuschließenden Kundenanlage 43 3.7.4 Erhöhung der unsymmetrischen Lastimpedanz parallel betriebener Anlagen 44 3.7.5 Beeinflussung des Phasenwinkels des Gegen- bzw. Nullsystemstroms 44 3.8 Auswahl des Messorts zur Bestimmung der höchsten Spannungsunsymmetrie 47 4 Simulationskonzept und -modelle 49 4.1 Auswahl an Kundenanlagen 49 4.1.1 Erzeugungsanlagen im Niederspannungsnetz 49 4.1.2 Elektrofahrzeuge 51 4.2 Simulationsablauf 52 4.2.1 Wahl eines Simulationsszenarios und eines Simulationsnetzes 53 4.2.2 Installation von Photovoltaikanlagen und Ladepunkten für Elektrofahrzeuge 54 4.2.3 Festlegung von zu simulierender Zeitdauer und Mittelungsintervall 54 4.2.4 Lastflussberechnung je Zeitschritt 55 4.3 Stochastische Beschreibung der gleichzeitig ladenden Elektrofahrzeuge je Außenleiter – zentrales Laden 55 4.4 Simulationsmodelle - dezentrales Laden 57 4.4.1 Übergeordnetes Netz 57 4.4.2 Betriebsmittel des Niederspannungsnetzes 60 4.4.3 Kundenanlagen 62 5 Simulationsergebnisse 72 5.1 Zentrales Laden 72 5.1.1 Methodik 72 5.1.2 Auslastung der Betriebsmittel 73 5.1.3 Leitungsverluste 73 5.1.4 Unsymmetrischer Leistungsanteil 74 5.1.5 Spannungsunsymmetrie und Spannungsdifferenz 75 5.2 Dezentrales Laden 77 5.2.1 Methodik 77 5.2.2 Auslastung der Betriebsmittel 78 5.2.3 Leitungsverluste 79 5.2.4 Spannungsdifferenz 80 5.2.5 Unsymmetrischer Leistungsanteil 82 5.2.6 Spannungsunsymmetrie 83 5.2.7 Bewertung möglicher Maßnahmen zur Reduzierung der Spannungsunsymmetrie 89 5.2.8 Einfluss unsymmetrischer Koppelimpedanzen auf die Spannungsunsymmetrie 92 5.3 Resümee und Handlungsempfehlungen 94 6 Niederspannungsäquivalent für unsymmetrische Leistungsanteile 97 6.1 Lastgang der unsymmetrischen Leistungsanteile 97 6.2 Geräteklassenabhängiger unsymmetrischer Leistungsanteil 99 6.2.1 Unsymmetrischer Leistungsanteil Haushaltslasten 99 6.2.2 Unsymmetrischer Leistungsanteil Elektrofahrzeuge 99 6.2.3 Unsymmetrische Leistungsanteile PV-Anlagen 103 6.3 Überlagerung der Zeitverläufe 104 6.4 Beispiel 105 7 Zusammenfassung, Schlussfolgerungen und Ausblick 107 Literaturverzeichnis 110 Anhang 118

An AC system’s voltage unbalance by a fundamental frequency negative sequence system is usually the main cause for the emission of non-characteristic harmonics by current source converters as used in conventional HVDC systems. This emission takes place on both sides of each 12-pulse converter. On the DC side mainly a 2nd harmonic voltage appears driving a 2nd harmonic current. The magnitude of this harmonic current can exceed the magnitudes of the characteristic harmonics even if no low order resonance exists. Further non-characteristic harmonics generated by the converter under such unbalanced supply voltage conditions have frequencies with a frequency distance to the characteristic harmonics of 2 times the fundamental frequency. The main technical drawbacks are the unintended coupling between both AC systems and the risk of thyristor over-stresses by DC current discontinuities at low power transfer levels. On both AC sides the largest 2 non-characteristic current harmonics generated by a 12-pulse HVDC converter under unbalanced supply voltage conditions are a negative sequence system of the fundamental harmonic and a positive sequence system of the 3rd harmonic. Also on the AC sides further harmonics are emitted by the converter with a order number distance of 2 to the orders of the characteristic harmonics. However, in practical AC system operation special attention has to be paid to the 3rd harmonic distortion level, in particular when low order resonance appears between the system impedance and the impedance of the converter station AC filters. In order to avoid the above mentioned problems, large smoothing reactors and sometimes large blocking filters are installed on the DC side and the voltage distortion on the AC sides is reduced by AC filters. However, these filters require an expensive high component rating if they are tuned to the 2nd or 3rd harmonic respectively. The work shows that a modification of the valve firing can reduce the levels of the 2nd and 3rd harmonic without investment into additional primary equipment. Furthermore, this offers the chance to reduce the minimum power transfer level since also the risk of an intermittent DC current can be reduced. A corresponding algorithm and a control strategy are proposed. However, the calculation of an appropriate firing pattern requires a detailed modelling of the processes within the converters, especially the formation of the harmonics and the harmonic transfer between AC and DC sides. The work proposes a component vector model for the calculation of the harmonics. This model assumes that each harmonic consists of a first component representing the ideal conversion process, a 2nd component representing the impact of different commutation angles and in the case of the modified firing a 3rd component considering the impact of the intended non-equidistant firing. The work shows, that the harmonic component vectors resulting from voltage unbalance and from firing modulation can be treated separately and superimposed linearly. The calculation of the harmonic component vectors is performed applying the method of switching functions. For the consideration of the commutation and firing angle differences the modelling of switching functions based on differential impulses is proposed. However, especially an accurate representation of the above mentioned 2nd component vector requires a correct calculation of the commutation angles and their valve-specific differences. The investigations of this work have revealed that the conventional method of calculating the commutation angles – assuming an ideal smoothed DC current - may not produce results of sufficient accuracy. This is especially true in the case of a high ripple of the DC current, e.g. smoothed with a small smoothing reactor. A small smoothing reactor is typical for HVDC back-to-back applications. Therefore a new analytical method for the calculation of the commutation angles has been developed which in particular considers the typical pulse form of the DC current and additionally the impacts of the voltage unbalance and of the proposed modification of the firing on the ripple shape of the DC current. Moreover, as this analytical method requires the instantaneous values of the DC current at the instants of valve firing, a further analytical method for the calculation of these discrete current values has been developed. The equations are valid under the same conditions as the new ones for calculation of the commutation angles, i.e. resistive-inductive AC system fundamental frequency impedances, any degree of DC current smoothing between ideal smoothing and a ripple at the limit for current discontinuities. Symmetrical conditions, supply voltage unbalances and non-equidistant firing as proposed are applied. It is shown that, using this method, also the discrete values of the DC current at the end of the commutation intervals can be determined. In practice one of these discrete current values indicates the minimum value during one period of the fundamental frequency. This offers the chance for a more exact analytical determination of the limit for the appearance of DC current discontinuities. For typical parameters of a back-to-back installation the new methods and the new analytical equations have been compared with simulation results showing excellent correlation for typical voltage unbalances of not more than 1...2% and firing angle differences of not more than 2.5°. This verification is performed for the harmonics, the commutation angles and the discrete values of the DC current at the firing instants as well.