Energy management practices: quantitative models for a large consumer in the Italian electricity market

The liberalization of energy market has allowed the development of energy management strategies that aim to reduce the total cost, guaranteeing an adequate service level with a minimum risk. Nowadays, these strategies are developed, above all, by Large Consumers (LC), which have at disposal several levers for the optimization of their energy consumption. In particular, several energy intensive LCs, such as telecommunication companies, could focus on the optimization of their purchasing process in the electricity market. Contract management consists of creating the best portfolio of energy contracts, choosing among flexible bilateral contracts, structured derivative instruments, and bids in the day ahead market. In particular, this thesis focus on a typical midterm electricity portfolio problem faced by an electricity intensive LC in the Italian electricity market. Its aim is to reduce the total cost of electricity contracts, guaranteeing an adequate service level with a minimum risk. Thanks to Multistage Stochastic programming (MSP) approach, a LC may define an adequate procurement policy, composed by an initial move (decisions to be taken in the present) and by a subsequent strategy that rebalances the portfolio during the subsequent stages within time horizon, following the evolution of the underlying uncertainty (Carrion et al., 2007). More specifically, the inference based research reported in this thesis aims at investigating quantitative mathematical models for energy management problems of a LC in the Italian market. All the mathematical models proposed in this thesis are built using real data: all the statistical models for the electricity spot and forward market are estimated using historical prices of the from 2009 to 2014. The internal demand that a LC must satisfy has been defined using historical energy consumption information from a real LC in the telecommunication sector. 1) Integrating multi-stage stochastic programming and machine learning for the evaluation of policies in the electricity portfolio problem In stochastic multi-stage problems, the development of a policy and its assessment on out-of-sample scenarios can be obtained through the adoption of a Rolling Horizon Approach (RHA). This approach requires, at each decision stage, to solve a new scenario-based multi-stage stochastic model, using updated information about the uncertainty and the decision process (Conejo et al., 2010). The first stage optimal solution of this new problem is then implemented, new optimization model is build and so on. Using this approach, the initial move corresponds to the first implemented solution, while the purchasing strategy is composed by the sequential first stage decisions obtained by solving the appropriate multi-stage problems. However, there are no theoretical guarantees about the optimality of the policy obtained, and the computational time required for the policy development and assessment could be very high (Shapiro et al., 2009). An alternative approach, which could be called Nearest Neighbour Approach (NNA), can be based on solving the problem on a single scenario tree, obtaining an optimized decision for each node of the tree. Then, a policy can be defined by simulating an out-of-sample scenario, detecting its nearest scenario in the tree, and applying the corresponding decisions. Nevertheless, Nearest Neighbour Approach produces “unstable” policies, in the sense that their quality is strictly dependent on the starting scenario tree (Thiene & Vial , 2008). Research Question 1 (RQ1): How can a LC define and assess good decision policy for the energy portfolio problem? In this thesis, an innovative approach for the development and the assessment of a LC procurement policy is introduced (Murgia and Sbrilli, 2014). Thanks to the application of machine learning techniques, our approach could develop and assess a non-linear policy in a short time. Defourny et al. (2013) propose a similar approach, called Policy Function Approximation (PFA), which is based on the development of a family of policies, each one obtained starting from the decisions at all stages in a single scenario tree. Then, these policies are ranked thanks to their assessment on out-of-sample scenarios. Differently, the proposed approach, called General Policy Function Approximation (GPFA), is based on the development of a single policy, which is obtained by the application of neural networks techniques to the decisions at all stages in all scenario trees generated. Thanks to the larger amount of data used for the development of the policy, I obtained policies that, when assessed on out-ofsample scenarios, show a lower mean and variance than those obtained through other approaches previously proposed in literature. The application of our approach in electricity procurement problem allows LC management to easily define and evaluate different policies. In particular, LC management could take into account the effect of its risk aversion, thanks to the development of Pareto frontiers that show the trade-off between the expected procurement cost and the level of risk. 2) The impact of the spot price modelling on the electricity portfolio optimization problem SP models for energy portfolio management start from the definition of a set of discrete scenarios that represent the possible future values of the uncertainty. As a consequence, in order to build an adequate portfolio model, it’s necessary to define and estimate an adequate statistical model for the description of the evolution of the electricity spot price. The obtained statistical model should be able to handle the strong uncertainty that generally affects the electricity spot price. In particular, in the long term electricity prices are often characterized by a specific trend, seasonal cycles, and mean reversion, while in the short term they could be affected by weekly and daily cycles, volatility and spikes. Stochastic programming literature provides several models for electricity spot price, which are characterized by a specific way to consider the long term seasonal (LTSC) and the short term component. These components are generally analysed separately so to obtain more detailed and appropriate results (Most and Keles, 2010). The combination of models for LTSC and the stochastic component supports LC in forecasting electricity spot prices, but also in pricing electricity forward contracts. In fact, the price of an electricity forward contract is related to the expected spot prices in its delivery period. Because the delivery period of a forward contract is generally equal or higher than one month, the correct forecasting of LTSC represents a more crucial, but also more difficult, issue for a LC. So, the analysis of the statistical performance of the forecasting models for electricity spot prices, and especially for LTSC, could increase the LCs awareness in the development of their electricity procurement policies. Nevertheless, the agreement about the performance of these forecasting models for electricity spot prices is not unanimous in the literature, given that they could produce very different results, even when analysing the same historical price series. Moreover, the forecast performance of these models in a given market and in a given period can be evaluated beyond doubt only ex-post. The choice of an inadequate model could lead a LC to implement an electricity procurement policy based on incorrect forecasts of forward and spot prices, which could induce LC to a too low (or too high) hedging level and, consequently, to high financial losses. The choice of the most appropriate among a set of candidate forecasting models for an uncertain process could be affected by the well-known model uncertainty, which has been widely analysed in finance literature, especially in the last decades. Research Question 2 (RQ2): Which is the impact of the choice of the spot price statistical model on the developed decision policy? In order to test the impact of model uncertainty on a LC procurement policy, I performed a set of experiments, each defined by two main parameters: LTSC models, chosen among the most reliable models developed in the literature and LC risk aversion, chosen among four different levels. Given a level of LC risk aversion, I varied the level of market risk and the LTSC models used for spot price forecasting. So, for each configuration i, I defined a set of scenarios wi and infer a decision policy fi. Besides, for each configuration i, I computed the optimal value of the cost functional θi. Starting from these values, I built a comparison matrix, whose element (i, j) represents the Cross Value of the Stohastic Solution (CVSS). This indicator measures the relative distance from the value of the cost functional, obtained by evaluating the decision policy fi on the validation scenarios wj, and LBi. This value indicates the loss that a LC faces when its strategy has been developed by using a configuration i, while the price uncertainty is instead described by a configuration j. When the configuration i and j differ for the choice of LTSC model (the level of risk premium), CVSS could support a LC to evaluate the impact of the alternative LTSC models (the alternative levels of risk premium) on its procurement policy. In this way, a LC could detect and choose the configuration that minimises the maximum regret, which could be considered as the most robust. Results show that a wrong estimation of the LTSC of the statistical model may have a very strong impact on the procurement policy. I aim to deepen this insight by increasing the number of LTSC models under analysis. Finally, I aim to evaluate also the impact of the level of LC risk aversion, by comparing the results obtained in the four levels under investigation. 3) The Approximate Second Order Stochastic Dominance criteria for Long term contract management In the technical literature about mathematical models in energy finance, the risk aversion of the decision maker is introduced using the so-called mean risk (MR) framework. This approach requires that the performance of the purchasing strategy should be evaluated on the basis of two factors: the expected value of the costs to be incurred for energy purchasing and a risk functional directly proportional to the variability of the chosen strategy. In current practice, one of the most widely used functional risk is the Conditional Value at Risk at a given confidence levelα For the problem under consideration, the CVaRα corresponds to the expected value of the costs incurred in the (1 − α)% of the worst scenarios. Intuitively, this value quantifies the costs in a subset of extreme scenarios, that identifies the right tail of the distribution of the costs. The popularity of CVaR comes from its computational tractability: in fact, that this indicator can be determined as the solution of a convex optimization problem, then can be easily included in a linear programming model. Hence, mean-risk based models are very convenient from the computational point of view. However, in the mean-risk approach the risk aversion of the decision maker is simply included in the model through a parameter that weighs the risk component: the higher the weight associated with risk the greater the risk aversion of the decision maker. Despite its intuitiveness, this approach may underestimate the complexity of the decisionmaking process. On the contrary, Second-order Stochastic dominance (SSD) has been widely recognized as a sounder criteria of choice, due to its close relation with the utility theory. In particular, thanks to SSD it’s possible to express the preference of any risk-averse decision maker without explicitly specify a utility function. Using the SSD criteria it’s possible to improve MR solutions. Indeed, SSD-efficient solutions are not dominated by MR-efficient solutions with respect to the expected value and show a lover CVaRα for all the possible confidence levels, i.e less risky. This underlines the importance of choosing SSD efficient solutions. Research Question 3 (RQ3): How can a LC find contract portfolios that are (approximately) SSD efficient? In this thesis, I propose a new model for the optimization of the energy portfolio of a LC based on the concept of SDD . In particular, exploiting the link between second order stochastic dominance and conditional value at risk it is possible to define a multi-objective approach to find SSD-efficient solutions. Since the resulting model turns out to be intractable from the computational point of view, I propose an approximation numerically tractable, called Approximate Second Order Stochastic Dominance (ASSD). The proposed model is applied to a realistic case study regarding the long-term management of energy contracts portfolio of a LC in the Italian market. The case study highlights the difference of the obtained portfolio with respect to the MR solutions. Results shows that the proposed model may be useful for the risk assessment of the decision maker in a real case.