Complementarity modelling of investment in electricity generation capacity
Thesis DisciplineManagement Science
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
Complementarity formulations offer the opportunity to thoroughly investigate and clarify the problem of investment in electricity generation capacity. While complementarity has been traditionally used in the sphere of imperfect competition, we demonstrate it also can play a fundamental role in perfectly competitive situations. We demonstrate that our approach offers richer understanding than the traditional linear programming approach. We attempt no judgement as to the practical benefit of our approach, as the benefits themselves depend on the data of the specific counter-factual used. The disadvantages of our approach are somewhat clearer. We acknowledge that the formulations that result are computationally challenging and that the various standard solution methods available for complementarity problems may not actually represent efficient solution methods for such problems. Nevertheless, we adopt the complementarity framework, as ours is a purely theoretical thesis, designed to explore the potential of a unique solution approach to an oft-solved problem. Complementarity theory provides an environment for developing theoretical formulations that, in many cases, resolve directly from an optimisation problem, but it is also free to include other conditions where required.
After a brief introduction and literature review, Chapter 1 considers the finer detail of traditional solution approaches, including the use of screening curves, linear programming, and a related complementarity problem. Screening curves were traditionally used in the times of central planning to describe the optimal trade-off between technologies in terms of utilisation, and with the addition of a Load Duration Curve, the optimal capacity of each technology. We explore various representations of the LDC and discuss how these interact with investment conditions and how market clearing procedures are viewed in their context. We show, with the use of an example, that LP approaches are incapable of accurately defining key system performance measures such as the Loss of Load Probability, without either “guidance” from the modeller or through the use of a significant number of load classes or slices. Furthermore, we show that supposedly perfectly competitive models produce prices that are either inconsistent with the perfect competition they are predicated on, or inconsistent with the optimal capacity suggested by the model. Our investigation identifies the reason for this deficiency.
Optimal trade-offs have a useful theoretical function, but they also emphasise the nature of the technological choice, and ultimately when the trade-off is with a notional shortage technology, they describe the nature of the total capacity choice. But the screening curve approach quickly succumbs to complexity, and is often replaced by optimisation in the form of linear programming, in which a significant number of constraints could be more easily expressed. Nevertheless, the screening curve concept has some conceptual advantages that we can integrate into the analysis, namely the determination of utilisation levels corresponding to optimal technological trade-offs. Knowing that the traditional LP approach does not accurately reflect the relative timing of investment and operation decisions, or produce solutions that are independent of the LDC definition, we consider the integration of screening curve logic. By way of resolving the downsides of the LP approach, we develop a complementarity formulation that combines the LP solution with the logic of screening curves to derive a problem representation that enables an accurate and consistent solution to the simple problem. In doing so, we make clear that screening curves, per se, are not the motivation, but the vehicle for determining an optimal representation of the system.
Complicating the investor’s decision processes are several technological issues, the relevance of which might vary from market to market, but should be considered. Chapter 3 describes a nonexhaustive range of typical problem extensions that would challenge screening curve analysis and how our basic approach can be adapted to include these. This is important for several reasons. Firstly, it is important we demonstrate the overall extensibility of the approach. Secondly, each extension involves discussion of both the extension itself, which in many cases is represented can be accommodated by additional constraints or altered objectives in the underlying optimisation problem, but also the way in which these extensions impact on the definition of the optimal system representation. In deriving the optimal system representation, we develop duality based pseudo-screening curves to describe optimal trade-offs in particular situations or scenarios. By way of example, we consider the relatively standard fare of cost structure generalisation, capacity inflexibility, energy limits and storage, and finally the formulation and interpretation of configurable technologies as non-linear notional technologies.
In Chapter 4 we refocus on load. We consider demand response in two forms: the short-term demand response that typically requires investment and can be written as a technology, and the wider type of demand response that comes as a result of adjusting consumption patterns and substitutions. By nature, the latter response is based on longer term considerations and, in the spirit of the investment problem, we develop an approach to including this response in a fashion that excludes this form of demand response from behaving as a marginal technology in the electricity market. We then consider reliability using an endogenous augmented LDC formulation. Finally, we present a formulation and investment analysis of intermittent generation based on a chronological load and generation pattern. This case requires the introduction of an additional level of dynamic LDC generation, and the maintenance of a dynamic mapping between the LDC and the chronological load pattern.
No discussion of investment is complete without consideration of risk and uncertainty, and it is therefore important to demonstrate how this can be addressed in our formulation, and what the consequences of risk are. We begin by properly defining these terms before expanding the formulation to consider how risk could be implemented. Our over-arching approach is to develop the framework in accordance with the principles of Ralph & Smeers (2011), including contract market, whose clearance defines the market price of risk endogenously. We distinguish between the perspective of portfolio optimisation based on investor preferences and the perspective of risk constraints using an objective function that combines the expected profits of the firm with a CVaR measure. Uncertainty is presented as a distinct concept. Our presentation focuses on various conjectures and the implications they have for the optimal investment condition.
Throughout the thesis, we use the structure of complementarity models as this is both convenient and the basis of much prior research into similar questions. However, complementarity solution methods per se are not the focus of this research, and we refer the reader to those texts listed for a detailed explanation of the theoretical properties and solution methods associated with complementarity problems. We felt it important to be able to describe the problem within a single framework rather than an ad hoc collection of algorithms, and aim to show that, even though decomposing the problem and using algorithms may be more effective than standard complementarity solvers, complementarity formulations can be implemented for a wide variety of purposes.