“As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality”

Albert Einstein

There is an “urban myth” about a romance between Alfred Nobel’s wife and a famous Swedish mathematician named Mittag-Leffler with Nobel reputed to proclaim: “This tribe of bastards will not get a single penny out of me!” The truth of the matter is that Alfred Nobel was never married.

The Nobel Prize is awarded for the discovery or achievement that has “done mankind the greatest good.” When mathematics is not applied to explain the result of physical, biological or economic experiments, then it is all just a game played between “beautiful minds”.

The aim of this digression is to point out the importance of studying the physical world before attempting to model it. Returning to our original question, the purpose of mathematical modelling is to help explain the physical world. Thus before any electricity market models are discussed, let’s first analyse and understand the underlying Electrical Grid.

The Grid is a giant Automatic Control System (ACS) where multiple switches, fuses and peaking generators constantly support the supply-demand balance. They also ensure the frequency of 50Hz (±0.2%)[1] in the network (otherwise your electric shaver might spin a bit fast for you!). This ACS is governed by the so-called ‘Optimisation of Power Flow’ (OPF) algorithm. The inputs to this algorithm are designated voltages and currents in every line of a circuit as well as the load in every system node (or bus), which is a connection point for a large consumer or a generator. By minimizing the rather unfortunately named ‘Social Welfare Function’, the OPF algorithm outputs the price in every node and the dispatch schedule for every generator in a network[2].

Just to give you a fleur of the overall complexity, we present a sample of the 30-bus grid[3] (Fig.1). A similar circuit diagram, but with hundreds of nodes, flashes like a Christmas tree on the wall of NEMMCO’s Control Centre at Carlingford, NSW.

Fig1. 30- Bus system. Bars represent nodes, blue circles represent generators, the arrows represent scheduled loads.

The peculiar fact is that even though such an eye catching display is continuously blinking before NEMMCO’s collective eyes, the OPF is not used there as the price determination mechanism. In contrast to our Trans-Tasman neighbors who price electricity in every single node of their 244-node grid[4], NEMMCO has introduced a simplified five-region model with only five Regional Reference Prices (RRP) to be determined (Fig 2). In order to mimic the OPF, myriad of loss factors and special approximations were introduced into NEMMCO’s Linear Programming Engine (called either NEMDE or SPD depending on the NEMMCO document’s author).

Fig. 2 National Electricity Market physical model

As a result, NEMDE operates as the so-called ‘Greedy Algorithm’[5], which works as follows.

In order to satisfy demand, subject to system constraints, it fully ‘consumes’ every bid submitted to the pool (sorting them by creating bid-price couple and then ordering these couples by arranging price bands in ascending order). It accumulates these bids until the demand is met, so that the last (partly dispatched) bid of the marginal generator will define the RRP. This procedure is performed every five minutes and averaging the bids over six five-minute intervals yields the half-hourly market price.

Graphically, the price formation could be visualised by a crossover of the vertical ‘demand’ line with a regional five-minute bid stack, which is comprised from bids that are stacked in order of increase of respective prices. (Fig. 3)

Fig. 3. Bid stack evolution across one day. Yellow line represents one generator (ACIL Tasman in isolation). Vertical line represents regional demand as at 6:00.

In times of moderate demand the flatter part of the bid stack is being dispatched and a small incremental change in demand will cause a correspondingly small change in price. When the demand is higher, the action happens on a steep part (or, what is even worse, inflection points) of the bid stack. In this case even a minuscule change in demand will trigger a jump in price. This part of the bid stack is particularly vulnerable for gaming by generators (still bidding “in good faith”[7]).

So, once again we are left with the question: what is the most adequate (i.e., reflecting the properties of the underlying Grid) and practical framework for Risk Management? I will tackle this in a future article.

To be continued…

[1] Guide To Ancillary Services in the National Electricity Market, available from www.nemmco.com.au

[2] Evaluation of Power Systems Congestions Using Nodal Price Analysis, G. C. Stamtsis, J. Christiansen, I. Erlich, available from http://www.uni-duisburg.de/FB9/EAN/downloads/papers/article_for_meps02.pdf

[3] Taken from THE USES AND MISUSES OF OPF IN CONGESTION MANAGEMENT, presentation by George Gross University of Illinois at Urbana-Champaign Seminar “Electric Utilities Restructuring” Institut d’Electricité Montefiore Université de Liège December 8, 1999 © Copyright George Gross, 1999

[4] http://en.wikipedia.org/wiki/New_Zealand_Electricity_Market

[5] http://en.wikipedia.org/wiki/Greedy_algorithm

[6] From Ho King Calvin Kwok, PhD Thesis Proposal, UNSW, April 2004

[7] Clause 3.8.6 (f): “…prices specified for each price band being offered must increase monotonically with an increase in available MWs”

“First weigh the considerations, then take the risks”.
Field marshal Helmuth von Moltke

The next morning there is a blackout across the Australian East Coast …A sad but possible California-style scenario.

The recent Basel II Accord has established a relatively rigid framework of requirements for Risk Management for the banking and eventually, investment industry. Nothing even close to this Good Risk Management Practices yet exists for the Energy Sector.

The key factor in any risk management procedure is portfolio simulation and stress testing and there are several essential differences between the underlying Electricity Market and any other Financial Market that make the blind employment of the latter methodologies almost meaningless.

First of all the Electricity Market is driven physically rather than sentimentally. You may like or dislike Telstra shares, but you will never hesitate to put lights on when it is dark.

No matter what, supply should always meet demand, otherwise the physical grid will ‘protest’ and shut itself down. Only shareholders protested when trading of shares of OneTel was suspended.

When the demand in electricity is low, it is economically rational for the generator to continue supplying a minimum amount of energy and pay the penalty to NEMMCO rather than shutting ‘steamers’ down completely (plus starting and ramping them again with the rise in demand). This effectively creates a negative spot price, which turns most of the Log Normal models (nicely fitting the equity market) into useless mathematical exercises.

Certainly, numerous work-arounds have been developed in order to keep the familiar financial methodologies alive, but most of them turn into a miniskirt – by covering your knees you might expose something else… Ultimately, by employing familiar and widely accepted Financial Market methodologies to the Electricity Markets, you are increasing risk exposure instead of mitigating it.

Performing the mathematical modelling of electricity spot prices reminds me of a Russian saying: the further you go-the scarier it gets. A distinctly discrete set of half-hourly price readings with multiple spikes cannot be represented by a single-factor Geo-Brownian process. The combination of such a process with Poisson-like jump diffusion (and, finally, moving into a multifactor space) helps a lot, until you start fitting the parameters for your model and ask yourself a question: how do I distinguish in the discrete world a low-magnitude jump from the outliers of the Normal Distribution?

So, what would be the perfect framework? I believe, that in the absence of adverse events like plant outages (or another NSW Treasury Green Paper on Energy), there is only one independent stochastic driver – the system load. Even this is not purely stochastic, having a strong deterministic weather-correlated multi-cyclic component.

Therefore, to determine the future state of the market we need a deterministic Forward Curve together with the statistically defined Confidence Interval.

By incorporating the Forward Load Curve into the NEMMCO’s Linear Program Optimisation Algorithm (SCADA) one can resolve for the marginal generator and therefore, get the Market Price. All the necessary equations for this Algorithm are on NEMMCO’s website (just try to interpret them…).

This is, potentially, the ‘Unified Risk Management Theory’ (type of the Unified Field Theory which should explain every event in Nature by the same set of equations) that would answer all the questions. The first component of such a theory should be a reliable forecasting and simulating mechanism for the system load. The prerequisite for the latter is a reliable and ‘easy-to-handle’ statistical distribution.

As we could expect, the distribution of the raw load data (picture below) looks extremely ‘unfriendly’. It is a hunchback-type, fat-tailed, lumpy beast that would scare anyone who tries to approach it with the traditional analytical weaponry.

Therefore, somehow, the data should be transformed and massaged (or vice versa) providing at the end something symmetric, smooth and far more attractive.

Put simply, the ultimate goal is to obtain one single ‘nice’ analytical formula, with the minimum fitting parameters, which should not require speculations such as: this is a jump and this is just a giant step forward. Most likely, it must have fat tails, whose nature should inherently reflect the properties of the underlying Electricity Grid and indicate the dominant type of generation.

The only remaining question is: how, in the world, are we going to implement these requirements in a practical framework for risk management? We will tackle this question in part 2 of this article.