non-convergence) of a sequence gives no indication of how well a single element of t hat sequence approximates the sequence limit. For these reasons the investigation was not taken to its conclusion. In practice, investigation into the effects of the
Electricity Curve approximation may be better served by an empirical study into the quality of solutions obtained.
This is not to say that a full theoretical investigation of this nature is not
important and worthwhile; it just does not fit into t.he framework of this thesis. vVe offer the full investigation into the limit of optimal solutions (when they all exist) of t he approximate formulations as a worthwhile direction for future research, as well as an investigation into the difference between a single approximated formulation and its corresponding unapproximated formulation.
1 0 . 3 Deterministic Implementat ion
Implementation of the deterministic model was discussed. An formative implemen tation was used to isolate many of the difficult approximations, and some of the issues i nvolved in a final implementation were also explored. The formative im plementation used a working model (as described in Chapters 3-5) . This working
model was designed to test the limits of the framework where such limits were seen as important (e.g. the use of piecewise quadratics), to be internally consistent (e.g. all weeks were specified to the same level of detail) , and to provide a level of detail which was at least a level desirable to ECNZ (e.g. the representative geographic
network) . Such a model would be detailed enough to highlight inconsistencies in the framework.
The formative implementation demanded procedures to allow specification of the model being solved (from within the framework provided), and specification of the input data that were required. The continual change of the model and the deterministic framework meant that the solution input needed to be flexible. This made the input structures used unsuitable for use in a final implementation.
Due the the enormity of evaluating solutions directly from the output of MINOS
5 .4 (the solution procedure used), procedures were written to allow solutions to be
viewed using the GUI (graphical user interface) features of Matlab 4. This proved
CHAPTER 1 0. CONCL USIONS
The need for a procedure to determine a feasible (and reasonable) initial so lution, so as to speed up solution times, was highlighted. The inclusion of such a procedure in a formative implementation merely increases the difficulties asso ciated with making amendments to the model, and so was seen as inappropriate during this development phase. Development of such a procedure, therefore, pro vides a direction of future research; however, the development of a corresponding procedure for the stochastic case may supercede t4.is.
1 0 . 4 Stochastic Extension
Allowing future inflows to be uncertain, and future decisions to depend on previous inflows (once they are known) , increases the difficulty of the problem. There are many ways in which the deterministic model developed can be extended to include such uncertainty; several of these were discussed. The necessity of comparing the effectiveness of these methods under similar conditions was also identified. An authoritative comparison would necessarily be extensive, requiring simulation of the system to evaluate various solutions in terms their benefit to the system. Such rigorous testing is well beyond the scope of this thesis, and provides an important direction for future research in development of a full working implementation for use by ECNZ.
For a stochastic model it is much more important to investigate the robustness
and effectiveness of solutions produced. This is often done though simulation of the system, and comparison with current policies and those produced by other methods. Since the focus of this thesis was on the development of a deterministic framework for use as the basis to a full stochastic model (and not on the development of a specific full stochastic model itself), there are many issues and modelling aspects
of a stochastic model which cannot be meaningfully explored here. Instead we provided a brief examination of the feasibility of extending a deterministic model (developed from the framework provided) stochastically, and explored some of the issues which arose in order to "set the scene" for the exhaustive testing and analysis of stochastic extensions, which is seen as an important next phase. This meant that any examination undertaken here could not directly involve investigation of the quality of solutions produced, making any comparison of solutions obtained,
CHAPTER 1 0. CONCL USIONS 1 70
A scenano approach, using Rockafellar and Wets' Progressive Hedging Algo
rithm, was used to illustrate one stochastic extension, and to allow preemptive
investigation of some of the implementation issues in an effort to provide guide lines for the future development of a stochastic extension. This particular stochastic extension was used because i t provides flexibility in the amount of stochastic in formation which can used; this flexibility is bounded only by the solution time of the consequent model (this is, of course, a very sig�ifican t bound). It also does not
limit the formulation of the underlying deterministic model of the system in any, explicit, way.
The modelling issues, generated through the use of the Progressive Hedging Al gorithm, included ideas on the choice of scenarios, the choice of the non-anticipative variables, and a possible method for reducing the solution time through the use of a different decomposition. These issues were not fully addressed computationally, as the benefits they provide need to be evaluated within the context of the solutions they produce.
The convergence of the Progressive Hedging Algorithm is guaranteed only when the subproblems are solved to successively tighter tolerances each iteration under a strict regime. Unfortunately, the large-scale nature of the deterministic model, induced by the detail required in the physical system, means that convergence beyond some fixed tolerance is impossible. Therefore, in theory, convergence was not guaranteed beyond some indeterminant tolerance. In practice, the algorithm did converge. Due to the distance of the solutions to the deterministic subproblems from their respective optimal solutions, scaling was an important consideration.
Better convergence was achieved through the use of a scaled inner product (for the Lagrangian term and the quadratic augmentation), where the variables were scaled relative to their importance to solutions (i.e. in terms of the amount of generation they represent) . This produced a most satisfactory result, and its inclusion should make implementation of a final model more robust.
A brief examination of the effect of the choice of the non-anticipative variable on various convergence measures was made. This showed that the use of storage in this context appeared to allow the most freedom, with the use of generation allowing the least freedom, and the use of release giving slightly more freedom than that given by the use of generation. A variety of convergence measures were investigated, including_ the measure proposed by Rockafellar and Wets' [19] as the