Erik Möllerstedt

Out of control because of harmonics-an analysis of the harmonic response of an inverter locomotive

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 challenge.

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 frequencies.

Harmonic modeling of the motor side of an inverter locomotive

An AC-voltage source feeding an electric network results in a periodic excitation of the network. In steady state, all currents and voltages will be periodic with cycle time corresponding to the frequency of the voltage source. If the network is linear, all signals are sinusoidal and the network is solved using traditional methods. If the network contains components with nonlinear or switching dynamics, iterative methods based on harmonic balance are often required to obtain the periodic steady state solution. By linearization of the system around the periodic solution, a linear time periodic model is obtained. This can be used as a local description of the system in the neighborhood of the periodic solution. If only periodic signals are considered, a linearized model 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 periodic disturbances 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 load model for analysis and control of electric distribution networks

Power electronics result in increased harmonic distortion, but can also be used for harmonic mitigation. This requires simple, low order models for nonlinear networks. The Harmonic Norton Equivalent is a load model structure that supports aggregation of nonlinear loads, and makes nonlinear network solving (i.e. simulation) a non-iterative procedure. It is a linearized description of the relation between voltage spectrum and current spectrum. A procedure for obtaining the Equivalents from measurements is presented and shows good agreement with the validation data.

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.