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The equivalent harrhonic impedances associated with time-variant devices such as FACTs and HVdc converters are in many cases phase dependent. The second rank tensor is proposed as a means of performing t h e nodal analysis of networks incorporating phase dependent admittances. A useful geometric interpretation of phase dependency is developed, and applied t o t h e calculation of the harmonic phase dependent impedance of a HVdc converter on t h e ac side.

The steady state equations that describe the converter and dc system in the harmonic domain are solved by means of Newton's method, in a manner suitable for embedding in an iterative harmonic analysis of the complete power system. The solution includes the interaction of the converter with the dc system, and the effect of variation in the firing and end of commutation angles caused by ac voltage and dc current harmonics. The convergence of Newton's method is investigated, and methods for accelerating the solution are implemented. Finally, the solution obtained is validated by means of time domain simulation. J. Arrillaga FIEEE. commutating current dc side current (sum of phasors) commutating reactance number of harmonics to be analysed dc voltage (sum of phasors) commutating voltage (sum of phasors) thyristor forward voltage drop dc shunt admittance instantaneous alpha order (sum of phasors) average firing order timing reference for i'th firing angle of firing angle of end of commutation Le + L b -Q t = the i'th sampling function (sum of phasors) W

When calculating non-characteristic harmonics generated by an HVde converter, unbalance in the ac system at fundamental frequency and all harmonies must be taken into account. This entails that a full three-phase load Row be part of the solution, as there is considerable distortion interaction hetween the load flow and the converter. To date, this has been achieved by iterating sequentially between the load flow and a harmonic converter model, in a type of deeoupled method which is likely t o suffer poor convergence. The decoupled method is implemented here, and convergence compared with a unified Newton solution of the load flow and converter models. Finally, the effect of not modelling the distortion interaction between the two systems is investigated.

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