The three-phase bridge is the basic switching unit used for the conversion of power from AC to DC and from DC to AC. The valve numbers indicate the sequence of their conduction with reference to the positive sequence of the AC-system phases (R, Y, B). Two series-connected bridges constitute a 12-pulse converter group, the most commonly used configuration in high-voltage and large-power applications. Although the analysis described in this book relates specifically to the 12-pulse converter and a point-to-point DC link, the proposed algorithms can easily be extended to higher pulse converters and multiterminal AC-DC interconnections.

Power-flow analysis is used to determine the steady-state operating characteristics of the power-generation/transmission system for a given set of busbar loads. In this context, steady state means a time-independent condition which implicitly takes into account the final adjustment of the parameters involved in maintaining the specified operating condition at each bus, and the component-related capability constraints. In short-term studies, such as the system response to a change in load specification, the power-flow assessment is made without the assistance of generator-control adjustment. The term quasisteady state is used here when referring to such a condition. An assessment of the power-transfer capability of the DC link also considered in this chapter comes under this category.

In this chapter, a three-phase power-flow and harmonic-converter model are combined and solved using both the sequential and unified Newton's methods. Interaction between the three-phase power flow and a three-phase harmonic converter model has been solved using both decoupled and full Newton methods. The decoupled method displays good convergence if a linearising shunt is present but is slow because the Jacobian matrices need to be recalculated and factorised at every iteration. The full Newton method (with constant Jacobian), is faster and displays more robust convergence. The decoupled method is compatible with any existing three-phase power flow, including a polar fast-decoupled type. A special but simple three-phase power flow is required for the unified method as the converter model must be framed in real variables. Overall, code for the unified method is shorter and less complicated, as there is only one set of sparse storage, mismatch evaluation, convergence checking, etc. The effect of not modelling the power-flow/distortion interaction is a moderate underestimate of unbalance in the power-flow solution, and a large underestimate of distortion at low order noncharacteristic harmonics in the converter model. The AC-DC iterative algorithm can easily be extended into a general purpose model with the capability of including several AC systems, DC systems, possibly with multiple terminals in each system, and all integrated with the power-flow equations. This is essentially a software engineering task, as each half pole contributes a block to the main diagonal of the system Jacobian matrix, with diagonal matrices coupling to other harmonic sources in the same system. The main task is to code the program so as to be versatile, easy to use and modular.

This chapter contains a concise description of the synchronous-generator transient mechanical and electrical response to disturbances in the power system. The modelling of other system components is also discussed with emphasis on the AC-DC converters. The structure of a basic transient-stability program is then described with representation of the DC-link behaviour on the assumption that the link maintains continuous controllability during the disturbance; however, a check for the onset of commutation failure has been included as part of the solution. Prediction of such an event would indicate the limit of applicability of the algorithm and the need to adopt the more detailed stability simulation discussed in Chapter 8.

The generalised Newton-Raphson method is an iterative algorithm for solving a set of simultaneous equations in an equal number of unknowns. At each iteration of the N-R method, the nonlinear problem is approximated by the linear-matrix equation. The linearising approximation can best be visualised in the case of a single-variable problem. The Newton-Raphson algorithm will converge quadratically if the functions have continuous first derivatives in the neighbourhood of the solution, the Jacobian matrix is nonsingular and the initial approximations of x are close to the actual solutions. However, the method is sensitive to the behaviours of the functions f_k(x_m) and, hence, to their formulation; the more linear they are, the more rapidly and reliably Newton's method converges. Nonsmoothness, i.e. humps, in any one of the functions in the region of interest, can cause convergence delays, total failure or misdirection to a nonuseful solution.

This appendix discusses a simplified test system. To explain the mechanism of harmonic interaction, the inverter side of the CIGRE benchmark system is replaced by a constant DC voltage source E.

In this appendix, the numerical integration in AC-DC power system analysis is discussed.