An Aggregated Approach to Harmonic Modelling of Loads in Power Distribution Networks

[abstract missing

Frequency-Domain Analysis of Linear Time-Periodic Systems

In this paper, we study convergence of truncated representations of the frequency-response operator of a linear time-periodic system. The frequency-response operator is frequently called the harmonic transfer function. We introduce the concepts of input, output, and skew roll-off. These concepts are related to the decay rates of elements in the harmonic transfer function. A system with high input and output roll-off may be well approximated by a low-dimensional matrix function. A system with high skew roll-off may be represented by an operator with only few diagonals. Furthermore, the roll-off rates are shown to be determined by certain properties of Taylor and Fourier expansions of the periodic systems. Finally, we clarify the connections between the different methods for computing the harmonic transfer function that are suggested in the literature

A Harmonic Transfer Function Model for a Diode Converter Train

A method for analysis of electric networks with nonlinear and switching components is presented. The method is based on linearization around the nominal AC voltage, which results in linear time periodic (LTP) models. For nonlinear and switching components, there is coupling between different frequencies, which may cause stability and resonance problems. The models capture this coupling and can thus be used for small signal stability and robustness analysis. A short introduction to transfer functions for LTP systems is given. To illustrate the method, an LTP model for the Adtranz locomotive Re 4/4 is derived. The system consists of an AC-side with a transformer, and a DC-side with a DC motor and a smoothing choke. The AC-side and the DC-side are connected by a diode bridge rectifier. The model clearly shows the coupling between frequencie

Dynamic Analysis of Harmonics in Electrical Systems

Frequency domain analysis and design of power systems is complicated in the presence of harmonics, switching dynamics, nonlinearities, unbalances, and for systems with mixed ac/dc dynamics. The reason is that linearization of the system does not lead to a time invariant system, but a system with periodically time varying dynamics, which implies that there is coupling between different frequencies. Often one has to rely on simplifying assumptions and simulation. The thesis uses linear time periodic (LTP) models to analyze power systems. The harmonic transfer function (HTF) for LTP systems is introduced. Using the HTF, the system can be treated as an infinitely dimensional linear time invariant system, which means that the system, under certain convergence conditions, can be analyzed using the well developed theory for LTI systems. The thesis contains four papers with power system applications. Paper I describes the modeling and analysis of networks including components with switching dynamics, such as diodes and thyristors. An algorithm for parameter estimation from experimental data is presented. Papers II and III treats modeling and analysis of single-phase railway systems. The modeling of the locomotives is performed in collaboration with industry. Paper IV treats analysis and control aspects of a converter for grid connection of a micro-turbine used for distributed power generation. This is a three-phase application done in collaboration with the industry

Out of Control Because of Harmonics - An Analysis of Harmonic Response of an Inverter Train

Presents a method to use linear analysis to capture the frequency coupling of nonlinear and time-varying components. System stability is analyzed by connecting the harmonic transfer functions of the different component models. This facilitates an object-oriented approach to modeling, which supports reuse of models. An analysis of the complete railway system is, of course, difficult. Several locomotives can be moving along the power distribution line at the same time, and depending on the distance between them, the interaction changes. The power consumption also changes, depending on operating modes. During normal operation, energy is consumed from the network, but as modern locomotives use electrical braking, the power flow changes direction during deceleration, and energy is delivered back to the grid. The inverter trains are not passive systems. The converters are controlled with only limited system knowledge (local measurements of currents and voltages), making analysis and control design an even bigger challeng

Harmonic Modeling of the Motor Side of an Inverter Locomotive

An AC-voltage source feeding an electric network results ina periodic excitation of the network. In steady state, allcurrents and voltages will be periodic with cycle time corresponding to thefrequency of the voltage source. If the network is linear, all signalsare sinusoidal and the network is solved using traditional methods.If the network contains components with nonlinear or switchingdynamics, iterative methods based on harmonic balance are oftenrequired to obtain the periodic steady state solution.By linearization of the system around the periodic solution, a lineartime periodic model is obtained. This can be used as a localdescription of the system in the neighborhood of the periodicsolution. If only periodic signals are considered, a linearizedmodel can be represented by a matrix, called the Harmonic Transfer Matrix(HTM).The method is applied to the motor side of a modern inverter train.Via the HTM, the steady state response to constant or periodicdisturbances or changes in reference values can be obtained

Periodic Modelling of Power Systems

This paper treats modelling of power systems with converters in alinear time-periodic framework.A power converter is a nonlinear switching device connecting an ACsystem to a DC system. The converter generates harmonics that mightcause instabilities in systems of this kind. About a nominal periodictrajectory the power converter is well described by a periodic gainmatrix, whereas the power grids often can be described by lineartime-invariant models. Put together they form a linear time-periodicmodel. It is also shown in this paper how Integral Quadratic Constraints may beused for robustness analysis. To conclude an inverter locomotive ismodeled with the described techniques

A Simple Model for Harmonics in Electrical Distribution Networks

A modularized approach to modeling of harmonics in electrical distribution networks at steady state is presented. It is based on harmonic balance and exploits that the loads in a distribution network are connected in parallel in such a way that their operating conditions are approximately known in advance (e.g. 230 V, 50 Hz) and the harmonic distortion of the voltage is limited. The model for a component is given by a linear relation between the Fourier coefficients of the deviations from nominal current and voltage. The linear relationship implies that aggregation of loads and network solving are a matter of solving linear equation systems. This leads to fast calculations without convergence problems