Optimization Methods for Gas and Power Markets: Theory and Cases

By Enrico Edoli, Stefano Fiorenzani and Tiziano Vargiolu

As power and gas markets are becoming more and more mature and globally competitive, the importance of reaching maximum potential economic efficiency is fundamental in all the sectors of the value chain, from investments selection to asset optimization, trading and sales. Optimization techniques can be used in many different fields of the energy industry, in order to reduce production and financial costs, increase sales revenues and mitigate all kinds of risks potentially affecting the economic margin. For this reason the industry has now focused its attention on the general concept of optimization and to the different techniques (mainly mathematical techniques) to reach it.

Optimization Methods for Gas and Power Markets presents both theoretical elements and practical examples for solving energy optimization issues in gas and power markets. Starting with the theoretical framework and the basic business and economics of power and gas optimization, it quickly moves on to review the mathematical optimization problems inherent to the industry, and their solutions – all supported with examples from the energy sector. Coverage ranges from very long-term (and capital intensive) optimization problems such as investment valuation/diversification to asset (gas and power) optimization/hedging problems, and pure trading decisions.

This book first presents the readers with various examples of optimization problems arising in power and gas markets, then deals withgeneral optimization problems and describes the mathematical tools useful for their solution. The remainder of the book is dedicated to presenting a number of key business cases which apply the proposed techniques to concrete market problems. Topics include static asset optimization, real option evaluation, dynamic optimization of structured products like swing, virtual storage or virtual power plant contracts and optimal trading in intra-day power markets. As the book progresses, so too does the level of mathematical complexity, providing readers with an appreciation of the growing sophistication of even common problems in current market practice.

Optimization Methods for Gas and Power Markets provides a valuable quantitative guide to the technicalities of optimization methodologies in gas and power markets; it is essential reading for practitioners in the energy industry and financial sector who work in trading, quantitative analysis and energy risk modeling.

• Language: English
• Publisher: Palgrave Macmillan
• Release date: Apr 30, 2016
• ISBN: 9781137412973

Book preview

1.1   Classification of optimization problems

In the Preface, we mentioned that optimization problems represent a class of mathematical problems that we cannot consider always homogeneous. Different kinds of problems ask for different representation and solution tools. Hence, a bit of classification may be worthwhile for the sake of clarity.

1.1.1   Linear versus nonlinear problems

The distinction between linear and nonlinear optimization problems has to do with the shape of the objective function and functional constraints. If both constraints and objective functions are linear, we can (obviously) talk about linear optimization problems, while in all the other situations we will have a nonlinear optimization problem.

Linearity strongly impacts on the solution methods we can apply to the problem: for the solution of linear optimization problems, fast solution algorithms, able to reach exactly the optimal solution of the problem, are often available, while in all the other cases we do not have the guarantee of such a simple and fast solution method.

Most of the time, the linear representation of a problem is a helpful simplification we impose on the problem itself in order to make it simpler to solve. Mathematicians often refer to the art of linearization in order to make explicit the fact that making linear a nonlinear problem is an almost esoteric exercise which requires experience and full knowledge of the true problem. We need to carefully understand whether linearization is possible or not and assess the kind of problems simplifying assumptions are carrying on for our specific problem. Nevertheless, when allowed and not distortive, linearization is extremely helpful, especially when our focus is the practical determination of an effective solution (even if not the best one) to our problem and not the formal representation and study of the problem itself.

1.1.2   Deterministic versus stochastic problems

We are not always able to measure or observe the current value of many parameters affecting our target function, mainly because most of the optimization problems we are interested in deal with quantities that are unknown at the time when the optimization is performed, such as the future profit and loss of a financial instrument. Typically, in those cases we are only able to make some assumptions on the unknown parameters or to speculate about their future behavior, but are not able to know the exact value of them. In such cases, we often make probabilistic assumptions on those parameters, based on statistical observations. We say that such optimization problems have a stochastic nature. In more practical terms, also our decisions need to be taken under uncertainty. On the other hand, if we are completely able to measure all the variables that have a direct impact on our optimization problem, the problem is called deterministic.

Real-life decisions are rarely taken under a fully complete information set, i.e., an information set without uncertainty about relevant factors which affect decisions. Most of the time, we are interested in optimizing our behavior with respect to future events that, by definition, are unknown at the current state. Moreover, even when we aim at adjusting our decisions with respect to intrinsically deterministic factors, we are always affected by measurement errors which induce uncertainty (small or big) on the final outcome. As stated for linearization, framing a problem as a deterministic problem is most of the time a kind of simplification made up by the decision-maker, rather than a real case. Deterministic problems are always easier and faster to solve than stochastic ones, but when this simplification is introduced, we need to take into account that we are dealing with an approximation of the true (and more complex) solution.

1.1.3   Static versus dynamic problems

The concepts of static and dynamic relate directly with that of motion through time but, when we consider optimization problems, not all those that have a time dimension must be considered as dynamic problems.

We define an optimization problem as dynamic only when it requires decisions to be taken through time, and the sequence of optimal decisions are not independent of each other. In other terms, a dynamic optimization problem is when decisions or actions taken at a certain moment can potentially affect those we could take in subsequent moments.

On the other hand, a static optimization problem is a problem where the sequence of decisions, even if spread through a certain time frame, are temporally independent. In a static problem we will select the current optimal choice without considering the impact it may have on future choices, or better we will not consider co-dependency because we assume decisions are independent of each other.

The reader should note that considering the natural dynamicity and co-dependency of a certain optimization problem is not equivalent to the repetition several times of a static exercise through time. Considering the natural co-dependency that many optimization problems have is really a complex task. It is often complex to represent realistically the decisions sequence (framing the optimization problem), and even more complex to solve it somehow.

In real life, we are almost always asked to take decisions whose effects will perpetuate in the future (short-term future to long-term future), and each action we engage in will modify peremptorily the set of our future choices, opening some opportunities and closing some others. Decisions in business, and in energy markets in particular, do not differ. Hence, many of the problems we will face while searching for the best result will be most of the time complex (nonlinear), surrounded by uncertainty (stochastic) and extremely dynamic. Basically, we are always in the worst situation! However, not in all the situations where we are asked to take a business decision do we need the maximum potential accuracy. In some situations we just need an indication, while in some other situations a precise decision support is necessary.

Typically, the longer the time horizon in which to observe the effects of our business decision, the more we are inclined to simplify the optimization problem we are facing, ignoring also the impacts that associated decisions will have on other problems of shorter-term horizon. This sounds really like a paradox, because it implies that the more the problem is important for our business, the more we base our decisions on human emotions instead of adopting more formal approaches. On the other hand, it is true that the longer the time horizon of the problem, the higher the complexity of the decisions and the uncertainty which affects relevant variables. Hence, for those problems, a formal framing and solving of the true problem would be impractical and simplifications are necessary in order to derive at least an indication of what is best to do.

In the remaining part of the chapter we will propose some examples of business optimization problems which are typical of modern power and gas sectors. These problems will span from investment decisions to trading strategies, with the scope of correctly orienting the reader towards the correct framing of different classes of optimization problems, and leaving the solution methods for the subsequent chapters.

1.2   Optimal portfolio selection among different investment alternatives

In modern financial theory, portfolio selection is historically associated with the seminal work of Markowitz (1952). Markowitz’s problem can be summarized in the following way: an investor can allocate its capital into N assets and must decide (in advance with respect to the investment operation) how much to invest in each single asset, trying to balance the expected return and the risk of the portfolio. Markowitz’s problem deals with financial assets; hence, these are infinitely divisible assets that can be liquidated at any time in the market. An asset’s return is basically the proportional increment of value that the asset faces during the holding period. The portfolio’s risk is univocally measured by the variance of its return.

Markowitz’s problem is a single period static optimization problem that can be formalized as follows

where is the expected return of the assets, is the variance-covariance matrix among the assets, and are the amounts invested in every asset. The expected return of the portfolio is given by rTθ and the variance of the portfolio by θT Σθ, that is the quantity we want to minimize given a fixed rate of return ; the total amount invested in each single asset must add up to the initial amount w0.

Sometimes, the risk-minimizing approach of the classical Markowitz’s problem is replaced by a more practical risk-adjusted performance maximization of the following type

without substantial changes to the nature of the optimization problem.

When our scope is that of selecting optimal investment alternatives among industrial initiatives (real assets) rather than financial assets, the nature of our optimization problem changes a bit:

•   First of all, we cannot assume that real assets are infinitely divisible, and most of the time we have to decide among a digital alternative of investing a certain amount into a certain economic initiative or not.

•   Secondly, real asset investments cannot be considered as liquid investments. As we decide to invest, we will assume holding the asset for a certain time period. Early exit is always an option to evaluate, but is not the main element of the investment decision.

Hence, investment return is not typically measured and expressed by the proportional value change of the asset over a certain time period but rather by some synthetic indicator of the net cash flow produced by the investment during the holding period as, for example, the Internal Rate of Return (IRR). The IRR of a certain investment or project is defined as the discount rate that makes the Net Present Value (NPV) of the project equal to zero. IRR is thus a time-weighted rate of return.

Different investment opportunities are characterized by different initial investments (different absolute amount, but also different time allocation), different time horizon and cash flows and different exposure to economic risk factors. In this different environment, the optimal portfolio selection can be reformulated using the concept of IRR. The IRR is a rate of return used to measure and compare the profitability and desirability of the investment. The higher a project’s IRR, the more desirable it is to undertake the project: assuming all projects require the same amount of up-front investment, the project with the highest IRR would be considered the best and undertaken first. The IRR follows from the NPV as a function of the rate of return: a rate of return for which this function is zero is an internal rate of return. In formulas, if we define the NPV as a function of the interest rate r, initial capital required to enter the investment W and the sequence of cash flow

then the IRR for such an investment is the rate r* solution of the equation

Let us now consider M different investments i = 1,…,M, each one characterized by

•   Yi(t): the net cash flow stream produced by the i-th investment alternative at time t = 0,…,N;

•   Yi: the whole sequence of net cash flows over the time horizon [0, T], i.e. Yi = ;

•   Wi: the initial capital investment required to buy the i-th investment project;

•    the duration of each investment

then by defining Ni as a binary variable indicating if the i-th project has been selected for our investment or not, and given

•   w0: the maximum initial capital that can be invested;

•   d0: the maximum desired duration of the investment;

we can formulate the optimal selection problem as follows:

Obviously, the one presented above is just an example of an optimal portfolio selection problem. Specific adjustments/integration to the problem presented above should be produced in order to consider pre-existing investments, for instance.

A typical application of Markowitz’s type mean-variance portfolio selection problem in the energy sector is traditionally associated with the so-called optimal generation fuel mix. Basically, this problem aims at identifying the optimal (in the mean-variance Markowitz’s sense) portfolio of different generation technologies. The first recognized application in the field of energy utilities was done in Bar-Lev and Katz (1976). By using Markowitz’s diversification, the authors derive an efficient frontier for a fossil fuel mix, and they investigate various investment opportunities by the variation of cost and risk assumptions. Awerbuch and Berger (2003) apply mean-variance theory for the first time on generation portfolios in liberalized power markets and put their focus on the generation portfolio of the European Union (EU-15).

The mean-variance problem applied to optimal generation fuel mix determination can be formalized exactly as a standard Markowitz’s problem as done in Equation 1.1, but with some adaptation to the assets/technology return measure: the vector θ now contains the portfolio’s weights expressed as percentage of i-th asset/technology over the global output. r is again the vector of assets expected returns over the single period considered, but given the specific character of the energy sector and especially energy utilities, it necessitates a new definition. Generation technologies, in contrast to the stock market, are real assets described by many different parameters. Differences among these parameters are connected to the energy market mechanism and its uncertainties and risks. To capture all elements, the annual expected return is usually suggested as measures of return in this kind of problem.

The annual expected return per unit of electricity production (€/MWh) can be estimated by historical cash flows, considering all relevant revenues and cost sources. In general, the cash flow CF produced by a generation asset per unit of electricity produced can be represented as

where Rp represents revenues from electricity sales, Cf the fuel cost, and CO&M other costs linked to emissions certificates and operation and maintenance cost, Cc capital cost and δ the capital depreciation.

The solution of Markovitz’s problem presented so far allows for the determination of the efficient technology frontier that can then be used to compare the optimal technology allocation with the current situation of a specific company, a region or a country. Obviously, the optimal generation fuel mix can be complicated in order to better consider dynamic investment processes and impact of new technologies (see Madlener and Glensk (2010, November) for more details).

1.3   Energy asset optimization

Massive use of optimization techniques is definitely required when energy assets (power plants, transmission lines or pipelines, gas storages or energy production fields) are the object of our attention. Energy assets are complex industrial plants, and their optimal use should properly consider this complexity together with the complexity of the energy markets they are surrounded by. Energy asset optimization is required (or implicitly assumed) in many different valuation problems typical of the energy industry, from long-term investment valuation to short-term operational optimization. In this section we will propose typical asset optimization problems for power and gas industries.

1.3.1   Generation asset investment valuation with real option methodology

The problem of the evaluation of investment opportunities related to a certain energy asset, or among comparable alternatives, consistent with a pre-defined investment strategy, is a problem which logically follows from those considered in Section 1.2.

Most of the time we are very keen to consider and optimize flexibilities embedded in energy assets we already hold in our portfolio, but we are not keen in the same way in considering and optimizing the value of flexibilities embedded in energy investment projects, especially those flexibilities that are not directly related to the asset structure but to the investment structure (financing, early exit, transformation of the original project, etc.). The methodology of real options can be applied in a simple way in order to bring out project flexibility value and then selecting not necessarily the best asset, but the best project.

Real options’ valuation applies option valuation techniques to capital budgeting decisions. A real option is the right, but not the obligation, to undertake certain business initiatives, such as deferring, abandoning, expanding, staging, or contracting a capital investment project. For example, the opportunity to invest in the expansion of a power plant, or alternatively to sell the plant itself, are real call or put options, respectively. Real options are generally distinguished from conventional financial options in that they are not typically traded as securities, and do not usually involve decisions on an underlying asset that is traded as a financial security.

Real options analysis, as a discipline, extends from its application in corporate finance to decision-making under uncertainty in general, adapting the techniques developed for financial options to real-life decisions. The flexibility available to management relate generically to project size, project timing, and the operation of