Conclusions
CHAPTER 1 0 CONCL USIONS 1 7
definitive convergence measure, another proposed by Mulvey and Vladimirou [12] ,
and the convergence of various primal and dual variables. Many of these produced similar patterns of convergence.
Due to extremely slow convergence experienced beyond an unpredictable point, a stopping criterion was needed which indicated where convergence appeared to have stopped, since there was no longer a guarantee of convergence to within a pre-specified tolerance. This proved satisfactory an_d allowed the user to determine whether further convergence was possible, or whether the convergence measure used was merely exhibiting temporary non-monotonic behaviour. 'With no way to..-examine the difference in quality between solutions obtained at various points during use of the Progressive Hedging Algorithm, there is little use in taking such an examination further. This does highlight another direction for future research that of investigating the correspondence between the values of various convergence measures and the quality of solutions obtained; this would provide valuable infor m ation on the convergence requirements, as well as possibilities for limiting solution time, in a final implementation.
An examination of the convergence for various values of the Progressive Hedging penalty parameter was carried out, in terms of the convergence measures mentioned above. The results obtained were inconclusive, since the stopping criterion used gave no indication of the quality of solutions
(
or how close to optimallity thesewere
)
, and also because the values of many of the convergence measures depended on the value of the penalty parameter used. Such an investigation would be more appropriate in the context of a full investigation of this particular stochastic ex tension.1 0 . 5 Future D irections
We see the main directions for future research as being able to be encompassed within an investigation which furthers the development of a specific model into a
form which is directly usable by ECNZ. There are three directions that such an investigation could take. The development of a full model for ECNZ would be best served by pursuing these three directions simultaneously, enabling the results of these investigations to be compared and contrasted.
CHAPTER 10. CONCL USIONS 1 72
an examination of the sensitivity in the quality of solutions to changes in the various approximations used; an investigation comparing the various stochastic extensions used in conjunction with appropriately approximated deterministic models arising from the deterministic framework developed here; and, an investigation into the implementation issues arising from the use of the Progressive Hedging Algorithm as a particular stochastic extension.
The development of a specific deterministic m9del requires close consultation with ECNZ. The investigation should be concerned with the sensitivity of the first
week's solution to the use of various approximations. The effects on solution time and solution quality of these approximations also needs to b;-addressed. The
specific deterministic model used will depend, not only on the results of this inves
tigation, but also on the stochastic extension chosen and the physical detail allowed by this extension to ensure computationally tractable models.
An investigation into the various stochastic extensions would need to follow two paths. The first would be to compare solutions and solution times under the
same deterministic base model. The second would be a comparison of the quality
of solutions when the physical systems were tailored so that the solution times were all within some pre-specified bound. The latter investigation would be more useful in terms of the development of a full model, usable by ECNZ; however, it would be difficult to compare solutions obtained via different methods since the level of detail in both the stochastic elements and the physical systems would be different. This means the solutions will need to be compared in terms of how well t hey perform on simulations of the system.
The investigation into the implementation of the Progressive Hedging Algo rithm, as an extension to a deterministic model constructed from the deterministic framework developed here, may seem preemptive vis-a-vis the outcome of the in
vestigation into all of the stochastic extensions. This need not be the case. The investigation into possible stochastic extensions will, due to constraints of time, not be able to "fine-tune" each stochastic extension so as to allow fastest, most efficient, solution time. To ensure that each method is treated with fairness, this means that none of the methods should be tuned to any greater extent than oth
ers. An extensive investigation into a particular stochastic extension will provide
CHAPTER 1 0. CONCL USIONS 1 73
extensions. Also, a scenario approach
(
and more specifically the Progressive Hedg ing Algorithm)
provides the most flexibility for the model as a whole, making such an investigation worthwhile for its own sake.The other future research directions, which have been outlined in this Chapter, while being worthy of further investigation, are not directly relevant to the further
development of a model for use by ECNZ, and so are seen, in terms of the aims of this thesis, as being of secondary importance.
1 0 . 6 D iscussion
The framework developed in this thesis allows flexibility in all aspects of the mod elling of New Zealand's hydro-thermal electricity generation system. This will allow the developers of a full model access to information on the cost
(
in terms of the loss of information)
of approximations made within this framework. The determin istic framework also allows for many different stochastic extensions, so as to allow investigation into the one which best serves the needs of the user.Development of the framework has been taken to a stage which allows future developers a platform upon which to base their investigations. Some of the con clusions about the system and possible modelling extensions provide useful insight for future modellers which will help to direct their investigations towards fruitful
areas.
The framework is fully developed at this point. An investigation into a specific
representation of the physical system will be dependent on the stochastic extension to be used, and the quality of solutions produced by the consequent full stochastic model-whereas a stochastic extension requires a specific representation of the physical system on which to base itself.
Future investigations will require a re-prioritization of aims and intentions from this point, while the development of the framework, for both designing a spe cific model and determining the effects of approximations used within this specific
model, has reached a point of natural conclusion, making this an appropriate point at which to conclude this thesis.
1 74