A Parsimonious Diffusion Equation for Electricity Demand

E. Tonkes and P. Broadbridge

Delivered to CTAC 2012

Abstract

In both regulated and market-based wholesale electricity markets, the short-term and long-term prediction of power demand is a crucial input to operating a secure and reliable system. Electricity demand fluctuates from hour to hour, day to day and season to season based on the aggregate domestic, commercial and industrial loads attached to the network.

In this paper, we present a simple diffusion equation to model electricity demand in the New South Wales (NSW) region of the Australian National Electricity Market (NEM).

A calibration approach is developed to fit the model parameters. We establish the performance of the model and the model is applied to establish the impacts of the "Earth Hour" events.

Carbon Price Issues for Major Energy Users

Angus Macleod, Presented EUAA, 3 May 2012

Major energy users in Australia will be exposed to financial exposures arising from Carbon schemes in the coming years. Compliance, trading and risk management strategies will take different forms depending on the regulatory environment as the scheme transitions from a fixed price period to a floating price. The inclusions and limitations on fungible securities and projects, such as international linkages and farm in agreements will also impact the practical ways that Carbon exposures, costs and risk can be managed. This workshop works through the Carbon Price issues and linkages with other markets, particularly electricity.

Energy Edge delivers to the EUAA on Clean Energy Future Carbon strategies

Angus Macleod, presented March 2012

This presentation provides gas users with an introduction to Clean Energy Futures regulatory environment. The session discusses how the CEF will work, how permits are acquired, flow through the system and are ultimately surrendered. The arrangements with international permits and volumetric and financial caps and floors are discussed. Finally, a discussion on carbon trading strategies is provided, including pass through contracts, banking, borrowing, CIF schemes, international permit linkages and auction processes.

Challenges of Operating in an Integrated Market Environment

Elliot Tonkes, presented February 2012

The electricity sector in Australia has traditionally operated with production assets linked to long term fuel contracts, and with an exposure to fluctuating market-based wholesale electricity prices. Rapid evolution within the energy sector is currently in progresses which is seeing the electricity sector becoming much more coupled to economy wide influences. The introduction of carbon pricing introduces market-linked pricing on the cost side of generation. The direct fuel costs of generators is also increasingly becoming more market-based with coal and gas costs being linked to international markets, LNG export prices and a more sophisticated domestic network Understanding the market interactions and how to operate, invest and regulate within the new environment create challenges to market participants and policy makers alike. In this seminar, we will discuss the key market influences, and how they will impact on operational behaviour of generators and new entrant decision marking. Issues such as risk management, cashflow management and the value of flexibility will be discussed.

Stochastic models for physical and financial modelling in electricity and water

Elliot Tonkes, presented at AMSI Conference, July 2009

This workshop will provide an overview of some common techniques which have proven useful for modelling the random nature of physical and financial aspects of electricity and water markets.

In electricity and water markets, the uncertainty of supply and demand lends itself to stochastic modelling with a range of approaches. Diffusion models and Poisson processes have been applied to model outcomes such as plant availability, network constraints, consumption levels and market prices. Other techniques such as stratified sampling and multinomial trees provide an alternative method to creating realistic simulations of future outcomes. The workshop will present several stochastic modelling techniques in an introductory style as they apply to electricity and water markets and discuss where they are used and highlight their respective strengths and weaknesses.

Stochastic dynamic programming in electricity and water markets

Elliot Tonkes, presented at AMSI Conference, July 2009

This workshop will provide an introduction to the technique of stochastic dynamic programming, also known as Markov decision processes. This Operations Research technique provides a mechanism to allocate limited resources over time. In this regard, it provides an ideal decision-support tool to manage operations. For example: when to generate if a power station has limited energy supply, when to pump if there are limited hours available for pumping, when to stop pumping if value can be gained from curtailment.

This session will convey the principles of derivative and commercial contract pricing in an uncertain market environment. This will be combined with the SDP in a representative problem to establish how an electricity contract over water pumping can be valued.

A Stochastic Dynamic Programming Problem in Electricity Markets

Elliot Tonkes and Dharma Lesmono (Presented at AMC 2009, Malaysia)

We study a financial and operational problem relating to demand-side management in the Alberta electricity market in Canada. The problem relates to the valuation and derivation of an optimal management strategy to undertake power load curtailment subject to contractual constraints. Our solution applies stochastic dynamic programming based on the system state which includes the electricity spot price and remaining resources. The solution generates the best allocation of curtailment within a calendar year and establishes the optimal exercise boundaries, that is, when to commence and conclude curtailment.

Water Constraints and Electricity

Elliot Tonkes, presented at AMSI Conference, July 2009

This workshop will introduce the relationship between electricity and water in wholesale markets.

The price outcomes of the physical electricity market are driven by the supply-demand balance of power. During periods of drought, there results a shortage of supply due to restrictions on water for hydro power generation, and water needed for cooling in coal-fired power stations. Some experiences of recent history enable us to estimate the degree of dependence of the wholesale electricity price on water supply.

Water supply consumes considerable electricity in its deliver to consumers. The impacts of electricity prices will be discussed to convey how the resources might be managed under a scenario of much higher energy costs.

The valuation of electricity cap contracts

Elliot Tonkes, presented at QANZIAM 2008 Conference, November 2008

This talk introduces an method for establishing the fair value of a cap contract which is an option of interest to players in the wholesale electricity market. The underlying approach is to tie the valuation to visible market signals, chiefly the forward price for swap contracts. It is clear that the Black-Scholes equation cannot be applied directly to electricity spot prices because the pool prices are not governed by regular probability distributions. However, the central limit theorem and a reformulation of the payoff in a Taylor expansion provides an avenue to recycle elements of the Black-Scholes framework.

Contact the author for further details.

The Carbon Challenge

Angus Macleod - Presented at Finance and Treasury Association Conference, November 2007

What is the impact of the recent announcement by the federal government to introduce a Carbon Emissions trading regime? How will Australian corporates create and trade emission reductions in the future and what will be the international implications? What lessons can be learned from offshore experience?

Contact the author for further details.

Applications of the Singular Value Decomposition in Energy Markets

Elliot Tonkes, presented at QANZIAM 2007 Conference, October 2007

Abstract

The Singular Value Decomposition (SVD) is a technique from linear algebra used extensively in data and statistical analysis. In recent times, Energy Edge has applied the SVD as a key element of modelling electricity markets.

The SVD (termed Principal Component Analysis by statisticians) is one of Gene Golub’s favourite techniques: so much so that the number plates of his car are “PROF SVD”.

The SVD is closely related to eigenvalues, being an orthonormal basis created from the eigenvectors of an autocorrelation matrix. The key power of the SVD is to project a high-dimensional data set to a much more manageable set of coordinates. The SVD selects the coordinate frame according to the actual data, in a manner which spans the largest variability in the data. The larger eigenvalues are associated with vector subspaces which are best able to describe variability in the data.

With a much reduced data set, it becomes manageable to perform regression and other standard statistical techniques without the ‘curse of dimensionality’. We use them for:

• demand shapes
• price shapes
• forward curve analysis
• the shape of bid structures

This talk will introduce the SVD and its key properties, and demonstrate how it can be applied to obtain simple models for electricity demand and price.

Carbon Trading and Correlation Complications

Angus Macleod - Presented at Energy Trading Conference, September 2007

There has been a lot or work, papers and presentations in Australia on the possible NET schemes that might be eventually implemented, the implications they may have for GDP, there effectiveness in reducing emissions and even their possible financial impact upon electricity prices and industry in general. However most modelling has been conducted with the benefit of a range of simplifying assumptions and very little focus has been placed on what the introduction of emissions trading in Australia might mean for energy traders, risk managers of utilities and the commercial and industrial sector.

The objective of this presentation is to look at the risk management implications and issues likely to arise for risk managers in the Australian energy, commercial and industrial sector with particular reference to experiences overseas and complications that arise due to correlations between different commodities and the fundamentals that drive their supply and demand.

1. Energy Edge who we are and our E3 International Alliance Energy Edge is a company that specialises in commodity market risk management services and business activities. It utilises the unique skill sets of its human resources to help clients meet their need for increasingly sophisticated risk management, trading, market, commercial and corporate finance related issues in energy, environmental and other commodity markets.

   Energy Edge and E3 International have formed an alliance that allows the parties to bring together their unique set of environmental economics, commercial, trading and risk management skills with the objective of delivering a higher quality and more comprehensive service in this area than is currently available in Australia. E3 International and Energy Edge have developed a full Carbon risk management service to be provided to Australian utilities, industry and financial institutions. This service is enterprise wide covering the full range of Carbon risk management services from direct and indirect exposure identification and quantification to the establishment of policies, procedures, systems and skills necessary for the active risk management of Carbon related risks via the trading of Emission Allowances and/or Certified Emission Reductions.

2. First rule of commodity trading and risk management – Understand the nature of the commodity, its key drivers and supply and demand fundamentals. Relative to electricity, Carbon markets have the benefit of being storable, bankable and increasingly internationally tradable through the growing CER market. Storability and bankability means that forward prices should efficiently reflect the value of the commodity at different points in time through effective arbitrage trading and the concept of cash and carry.

Water and Energy markets – what are the connections?

Angus Macleod (Presented at Energy Users Association of Australia Conference, 18 April 2007)

In the last few years the east coast experienced a 1 in 100 year drought. On top of that came the perfect storm or more precisely the absence of one. The NSW/VIC alps have experienced worst snow fall in decades. Qld 2006/2007 summer rainfall had a 98% probability of exceeding actual results.

Nobody could really have foreseen the severity of the last 12 months.

• Not state governments or federal governments
• Not industry
• The Bureau and other private forecasters did not (40–60% chance of above average rainfall)
• The market certainly did not!

This seminar explores the connection between water supply, electricity prices and risk management. It is intended to give energy users an insight into the issues and risks which are becoming apparent.

Derivative pricing and trading strategies

Elliot Tonkes - Presented at AMSI Conference on the Mathematics of Electricity Supply and Pricing, April 2007

Objective: The objective of the workshop is:

1. To provide electricity market representatives with an overview of certain mathematical techniques which are applicable to solve several real-world industry problems
2. To provide mathematicians without significant industry knowledge with an overview of the energy markets, exposure to some relevant problems and to demonstrate mathematics to solve those problems.

The main objective of this workshop is to cover the derivative markets. The talk will demonstrate several pricing techniques, concentrating on pragmatic approaches. Time will be spent illustrating the main classes of derivatives that are traded in the markets, and the commercial drivers behind the contracts.

As the main volume of derivatives are held for risk management purposes, the workshop covers the construction of a portfolio of derivatives and trading strategies to manage the derivatives and to achieve certain portfolio risk-return objectives.

As the derivatives are written on a complex underlying physical model, a substantial introduction to the Australian physical National Electricity Market is included.

Part 1. The physical Markets

• How the NEM works
• Models of price and demand dynamics:
  1. stochastic differential equation models
  2. dispatch models
  3. randomized models based on historical sampling
  4. Principal Components Analysis
• Drivers of price outcomes and volatility
  1. Demand
  2. Transmission
  3. Outages
  4. Bidding behaviour
  5. Regulatory environment
• Mathematical techniques to estimate revenues or volume weighted average prices (inner products of volume and price stochastic processes)

Part 2. Derivatives and derivative pricing

A description and motivation of industry-standard derivatives, presentation of pricing methods and concepts of volatility:

• Swaps
• Options on swaps
• Asians
• Caps and floors
• SRA contracts

Part 3. Exotic derivatives and trading strategies

• Delta hedging in the real world
• Physical plant valued as a real option
• Contracts with nominations
• Development of a dynamic programming method to price and manage a curtailment contract

Positive Flow Clamping

Energy Edge provided to Queensland Generators and submitted to AEMC, December 2007

A consulting report provided by Energy Edge to a collection of Queensland power generators. The report is displayed at the AEMC website.

Future of the QGS

Energy Edge provided to QGGF, September 2008

A consulting report provided by Energy Edge to the QGGF to assist with submissions in the CPRS review and displayed at the Climate Change website..

Electricity infrastructure need not be so costly

Elliot Tonkes, Australian Financial Review Lies and Statistics Column, 7 October 2006, pp 63

Abstract

This article considers how the growth of air conditioners and other peaking demand loads are adding to the infrastructure costs to deliver reliability in the electricity industry. We discuss how domestic demand side management can be effective in enhancing economic allocation of resources and reducing financial risk in the utilities sector.

Volatility conventions in electricity markets

Angus Macleod and Elliot Tonkes

Abstract

The concept of volatility in the electricity markets is often discussed and analysed. This article presents a review on participants' interpretations of volatility in the market and illustrates the confusion that can arise from by quoting volatility from conventional calculations. The paper presents a set of proposed volatility metrics that will be useful as reference numbers in pricing derivatives in the market, quoting implied volatilities and to be utilised in market risk management.

A Longstaff and Schwartz Approach to the Early Election Problem

E. Tonkes and D. Lesmono

To appear in Advances in Decision Science

Abstract

This article is devoted to an application of a novel method in optimal stopping problems using the method of Longstaff and Schwartz. The derivation of the optimal exercise boundary holds strong similarities with the American option valuation problem from mathematical finance. A seminal technique refined by Longstaff and Schwartz provided a method to estimate the exercise boundary of American options using a Monte Carlo method and a least squares objective. In this article we modify the basic technique to establish the optimal exercise boundary for calling a political election. Several innovative adaptations are required to make the method work with the additional complexity in the electoral problem. The transfer of Monte-Carlo methods from finance to determine the optimal exercise of real-options appears to be a new approach.

A Note on the Existence and Uniqueness of a Bounded Mean-Reverting Process

D Lesmono, P Pollett, E Tonkes, K Burrage

Jounal of the Indonesian Mathematical Society, April 2009

Abstract

We study a stochastic differential equation (SDE) describing a class of meanreverting diffusions on a bounded interval. The drift coefficient is not continuous near the boundaries. Nor does it satisfy either of the usual Lipschitz or linear growth conditions. We characterize the boundary behaviour, identifying two possibilities: entrance boundary and regular boundary. In the case of an entrance boundary we establish existence and uniqueness of the solution to the SDE.

Economically Optimal Water Management Strategies

Elliot Tonkes and Dharma Lesmono

Abstract

This paper considers optimal water management strategies in the presence of uncertainty (rainfall and consumption). The solution to an optimal active management program is determined with stochastic dynamic programming.

Opportunistic timing and manipulation in Australian Federal elections

Dharma Lesmono, Elliot Tonkes & Kevin Burrage

European Journal of Operational Research (192(2), 2009)

Abstract

In many Parliamentary Systems, election timing is an important decision made by governments in order to maximize their expected remaining life in power. Governments can also introduce policy or economic actions to enhance their popular standing and thus their chance of being re-elected. On the other hand, an oppositions’ natural objective is to gain power, and they will also apply controls through their own policies to reduce the governments’ chance of being re-elected. In this paper we employ a dynamic programming approach to determine the optimal timing for governments and oppositions to best utilize their limited resources. At each decision branch, the optimal control is interpreted as a Nash-Cournot equilibrium of a zero-sum political game which, in certain states, admits mixed strategy solutions. We perform a case study on the Australian Federal Election for House of Representatives.

A moment matching approach to the valuation of a volume weighted average price option.

A.W. Stace

International Journal of Theoretical and Applied Finance (10), pp 95-110, 2007

Abstract

In this paper we develop a method to find the price of several options whose payoff depends on a volume weighted average price (VWAP). Fixed and floating strike VWAP, together with digital VWAP contracts are considered. Throughout we assume that the stock follows a geometric Brownian motion and the rate of trades evolves as a mean reverting process. It is assumed that the VWAP at the final time has a lognormal distribution. The parameters of the approximating lognormal distribution are obtained by matching the first two moments of the volume weighted average price with a lognormal process. A price is then obtained for the fixed strike and digital options when the market price of risk is a constant. We concentrate on the price for calls, prices for puts can be obtained in an analogous manner.

Visit the journal article here