# Harmonic Balance Finite Element Method: Applications in Nonlinear Electromagnetics and Power Systems

PrefaceHarmonic Problems in Power SystemsHarmonic Phenomena in Power SystemsSources and Problems of Harmonics in Power SystemsTotal Harmonic Distortion (THD)Definitions of Computational Electromagnetics and IEEE Standards 1597.1 and 1597.2“The Building Block” of the Computational Electromagnetics Model The Sources of EM EnergyThe Geometry of the Model and the Problem SpaceNumerical Computation MethodsHigh-Performance Computation and Visualization (HPCV) in CEMIEEE Standards 1597.1 and 1597.2 for Validation of CEM Computer Modeling and SimulationsHBFEM Used in Nonlinear EM Field Problems and Power SystemsHBFEM for a Nonlinear Magnetic Field With Current DrivenHBFEM for Magnetic Field and Electric Circuit Coupled ProblemsHBFEM for a Nonlinear Magnetic Field with Voltage DrivenHBFEM for a Three-Phase Magnetic Tripler TransformerHBFEM for a Three-Phase High-Speed MotorHBFEM for a DC-Biased 3D Asymmetrical Magnetic Structure SimulationHBFEM for a DC-Biased Problem in HV Power TransformersReferencesNonlinear Electromagnetic Field and Its Harmonic ProblemsHarmonic Problems in Power Systems and Power Supply TransformersNonlinear Electromagnetic FieldHarmonics Problems Generated from Nonlinear Load and Power Electronics DevicesHarmonics in the Time Domain and Frequency DomainA. Time Domain ModelB. Frequency Domain ModelExamples of Harmonic Producing LoadsHarmonics in DC/DC Converter of Isolation TransformerA. Isolation Transformer and ExcitationsB. Full-Bridge Buck-Derived ConverterC. Forward ConverterD. Flyback ConverterE. LLC Resonant ConverterMagnetic TriplerHarmonics in Multi-Pulse Rectifier Transformer-Pulse Rectifier Transformer-Pulse Rectifier TransformerDC-Biased Transformer in High-Voltage DC Power Transmission SystemInvestigation and Suppression of DC Bias PhenomenonCharacteristics of DC Bias Phenomenon and Problems to be SolvedGeomagnetic Disturbance and Geomagnetic Induced Currents (GIC)Geomagnetically Induced Currents in Power SystemsQIC-Induced Harmonic Currents in the TransformerHarmonic Problems in Renewable Energy and Microgrid SystemsPower Electronic Devices - Harmonic Current and Voltage SourcesHarmonic Distortion in Renewable Energy SystemsHarmonics in the Microgrid and EV Charging SystemIEEE Standard 519-2014 [24]Current THD limits at the PCCVoltage THD limits at the PCCHigh-frequency Current Allowance in Low Current Distortion SystemsHarmonic Distortion Evaluations and Controlling HarmonicsReducing Load Harmonic CurrentHarmonic StudiesReferencesHarmonic Balance Methods Used in Computational ElectromagneticsHarmonic Balance Methods Used in Nonlinear Circuit ProblemsTheHarmonic Balance MethodsThe Theory of Harmonic Balance Used in a Nonlinear CircuitCEM for Harmonic Problem Solving in Frequency, Time and Harmonic DomainsComputational Electromagnetics (CEM) Techniques and ValidationTime Periodic Electromagnetic Problems Using the Finite Element Method (FEM)A. Time Domain ApproachB. Frequency Domain ApproachC. Multi-Frequency Domain Approach Using the Harmonic Balance MethodComparison of Time-Periodic Steady-State Nonlinear EM Field Analysis MethodThe Basic Concept of Harmonic Balance in EM FieldsDefinition of Harmonic BalanceHarmonic Balance in EM FieldsA. Nonlinear Magnetic FieldB. Nonlinear Electric FieldNonlinear Medium DescriptionA. Magnetic Medium:B. Electric Medium:Boundary ConditionsA. Dirichlet Boundary Condition:B. Neumann Boundary Condition:The Theory of HB-FEM in Nonlinear Magnetic FieldsA. Current Source-Driven FieldsB. HBFEM FormulationThe Generalized HBFEMHBFEM for Electromagnetic Field and Electric Circuit Coupled ProblemsHBFEM in Voltage Source-Driven Magnetic FieldGeneralized Voltage Source-Driven Magnetic FieldHBFEM for a DC-Biased Problem in High-Voltage Power TransformersDC-Biased Problem in HVDC TransformersHBFEM Model of HVDC TransformerReferencesHBFEM for Nonlinear Magnetic Field ProblemsHBFEM for a Nonlinear Magnetic Field with Current-Driven SourceNumerical Model of Current Source to Magnetic FieldExample of Current-Source Excitation to Nonlinear Magnetic FieldHarmonic Analysis of Switching Mode Transformer Using Voltage-Driven SourceNumerical Model of Voltage Source to Magnetic SystemExample of Voltage-Source Excitation to Nonlinear Magnetic FieldThree-Phase Magnetic Frequency Tripler AnalysisMagnetic Frequency TriplerNonlinear Magnetic Material and its Saturation CharacteristicsVoltage Source-Driven Connected to the Magnetic FieldDesign ofHigh-Speed and Hybrid Induction Machine using HBFEMConstruction of High-Speed and Hybrid Induction MachineNumerical Model of High-Speed and Hybrid Induction Machine using HBFEM, Taking Account of Motion EffectNumerical Analysis of High Speed and Hybrid Induction Machine using HBFEMThree-Dimensional Axi-Symmetrical Transformer with DC-Biased ExcitationNumerical Simulation of 3-D Axi-Symmetrical StructureNumerical Analysis of the Three-Dimensional Axi-Symmetrical ModelEddy Current Calculation of DC-Biased Switch Mode TransformerReferencesAdvanced Numerical Approach using HBFEMHBFEM for DC-Biased Problems in HVDC Power TransformersDC Bias Phenomena in HVDCHBFEM for DC-Biased Magnetic FieldA. Basic Theory of HBFEMB. Coupling Between Electric Circuits and the Magnetic FieldC. Epstein Frame-Like Core ModelD. Calculated and Measured ResultsE. Harmonic Analysis of the Magnetizing CurrentF. Harmonic Analysis of the Magnetic FieldG. Harmonic Analysis of Flux DistributionHigh-Voltage DC (HVDC) TransformerDecomposed Algorithm of HBFEMIntroductionDecomposed Harmonic Balanced System EquationMagnetic Field Coupled with Electric CircuitsComputational Procedure Based on the Block Gauss-Seidel AlgorithmPrevious Calculation ProcedureImproved Computational ProcedureComparison Between the Previous Method and the New MethodDC-Biasing Test on the LCM and Computational ResultsB-H Curve of the LCMDC-Biasing TestComputational ResultsAnalysis of the Flux Density and Flux Distribution Under DC Bias ConditionsHBFEM with Fixed-Point TechniqueIntroductionDC-Biasing Magnetization CurveFixed-Point Harmonic-Balanced TheoryElectromagnetic CouplingValidation and DiscussionA. Computational Results and Analysis of the Fixed-Point Harmonic-Balanced MethodB. Discussion on the Selection of the Magnetization Curve in the Computation of the DC-Biasing Magnetic FieldHysteresis Model Based on Neural Network and Consuming FunctionIntroductionHysteresis Model Based on Consuming FunctionHysteresis Loops and SimulationA. Hysteresis Loops Under Sinusoidal ExcitationB. DC-Biasing Hysteresis LoopsHysteresis Model Based on a Neural NetworkSimulation and ValidationAnalysis of Hysteretic Characteristics Under Sinusoidal and DC-Biased ExcitationGlobally Convergent Fixed-Point Harmonic-Balanced MethodHysteretic Characteristic Analysis of the Laminated CoreComputation of the Nonlinear Magnetic Field Based on the Combination of the Two Hysteresis ModelsParallel Computing of HBFEM in Multi-Frequency DomainHBFEM in Multi-Frequency DomainParallel Computing of HBFEMDomain DecompositionReordering and Multi-ColoringLoads Division in Frequency DomainTwo Layers Hybrid ComputingReferencesHBFEM and Its Future ApplicationsHBFEM Model of Three-Phase Power TransformerThree-Phase TransformerNonlinear Magnetic Material and its Saturation CharacteristicsVoltage Source-Driven Model Connected to the Magnetic FieldMagnetic MediumHBFEM Matrix Equations, Taking Account of Extended CircuitsThe Equivalent Circuit of a Three-Phase TransformerMagnetic Model of a Single-Phase Transformer and a Magnetically Controlled Shunt ReactorElectromagnetic Coupling Model of a Single-Phase TransformerSolutions of the Nonlinear Magnetic Circuit Model by the Harmonic Balance MethodMagnetically Controlled Shunt ReactorExperiment and ComputationComputation Taking Account of Hysteresis Effects Based on Fixed-Point ReluctanceFixed-Point ReluctanceComputational Procedure in the Frequency DomainComputational Results and AnalysisHBFEM Modeling of the DC-Biased Transformer in GIC EventGIC Effects on the TransformerGIC Modeling and Harmonic AnalysisQIC Modeling Using HBFEM ModelHBFEM Used in Renewable Energy Systems and MicrogridsHarmonics in Renewable Energy Systems and MicrogridsHarmonic Analysis of the Transformer in Renewable Energy Systems and MicrogridsHarmonic Analysis of the Transformer Using a Voltage Driven SourceHarmonic Analysis of the Transformer Using a Current-Driven SourceReferences