Multivariate constrained robust M‐regression for shaping forward curves in electricity markets

• Author: Tim Verdonck,Peter Leoni,Sven Serneels,Pieter Segaert
• Date: 01 November 2018
• DOI: http://doi.org/10.1002/fut.21958
• Published date: 01 November 2018

Received: 31 January 2017

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Revised: 26 June 2018

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Accepted: 1 July 2018

DOI: 10.1002/fut.21958

RESEARCH ARTICLE

Multivariate constrained robust M‐regression for shaping

forward curves in electricity markets

Peter Leoni

1

|

Pieter Segaert

1

|

Sven Serneels

2

|

Tim Verdonck

1

1

Department of Mathematics, KU Leuven,

Leuven, Belgium

2

Statistics, Machine Learning and

Artificial Intelligence, BASF Corporation,

Tarrytown, New York

Correspondence

Tim Verdonck, Department of

Mathematics, KU Leuven, Celestijnenlaan

200B, 3001 Leuven, Belgium.

Email: tim.verdonck@kuleuven.be

Funding information

Internal Funds KU Leuven, Grant/Award

Number: C16/15/068; Flemish Science

Foundation (FWO), Grant/Award

Number: V414714N

Abstract

In this paper, a multivariate constrained robust M‐regression method is developed to

estimate shaping coefficients for electricity forward prices. An important benefit of

the new method is that model arbitrage can be ruled out at an elementary level, as all

shaping coefficients are treated simultaneously. Moreover, the new method is robust

to outliers, such that the provided results are stable and not sensitive to isolated

sparks or dips in the market. An efficient algorithm is presented to estimate all

shaping coefficients at a low computational cost. To illustrate its good performance,

the method is applied to German electricity prices.

KEYWORDS

electricity market, M‐estimator, outlier, shaping coefficients, trading

JEL CLASSIFICATION

C10, C30, G13

1

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INTRODUCTION

It is a well‐known fact that energy forward and future prices, such as natural gas or electricity prices, are highly

seasonal (Geman, 2005; Huisman, 2009). This is due to various factors such as weather‐dependent supply and demand

and the problem of energy storage (Weron, 2007; Boogert & Dupont, 2008). In contrast to a forward in classical financial

markets, a forward in energy refers to a contract that provides the delivery of the underlying commodity over a fixed

delivery period (Schofield, 2011). This can be anything from a block of 15 min to a full year, depending on the contract.

In Europe, the forward market has developed into a cascading series of prices, whereby the close‐to‐delivery part of

the curve is more densely populated by forward contracts with higher or finer granularity such as days, weekends,

weeks, or months and the further‐from‐delivery forwards are only being traded in the form of quarterly, seasonally, or

yearly contracts.

The problem of transforming prices of traded contracts with a low granular nature into high granularity contracts

has been treated by various authors. The work of Fleten and Lemming (2003) may be seen as the seminal work on this

topic. They propose to obtain an average seasonal shape curve using regression techniques. This seasonal curve can

then be used to estimate prices for finer granularity contracts from higher granularity contracts. For the regression

model, one typically considers dummy variables for the weekly and yearly cycles, as well as information on the cooling

and heating degree days (see, e.g., Hildmann, Kaffe, He, & Andersson, 2012; Kiesel, Paraschiv, & Sætherø, 2018).

However, this regression approach typically induces arbitrage in the model. As a solution to this problem, Fleten and

Lemming (2003) propose to adjust the curves obtained in the previous step by smoothing them and imposing a

nonarbitrage condition.

The work of Fleten and Lemming (2003) motivated various authors to consider more advanced models.

Koekebakker and Os Ådland (2004) and Benth, Koekebakker, and Ollmar (2007) propose a combination of seasonal

J Futures Markets. 2018;38:1391–1406. wileyonlinelibrary.com/journal/fut © 2018 Wiley Periodicals, Inc.

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1391

paths and fourth‐order polynomial splines using maximum smoothness interpolation introduced by Adams and Van

Deventer (1994). These approaches are, however, quite elaborate and more involved to apply in practice. Moreover,

Borak and Weron (2008) reported these methods to be sensitive to model risk. Instead, they propose a dynamic

semiparametric factor model. On the other hand, Caldana, Fusai, and Roncoroni (2016) noted that the algorithm of

Borak and Weron (2008) suffers from underfitting market prices and may fail to account for short‐term periodical

patterns. A comparison between various approaches including works by Fleten and Lemming (2003), Benth et al.

(2007), and Paraschiv, Bunn, and Westgaard (2016) may be found in Kiesel et al. (2018).

The above methods are all nonrobust in the sense that they try to fit an optimal model for all observations.

Therefore, they are highly susceptible to the possible presence of atypical observations or outliers in the data. As the

nonrobust fit of the model is attracted by the outliers, these observations may no longer appear as outliers after the fit.

This effect is known as masking. In the worst possible scenario, the effect of outliers on a nonrobust fit can be so explicit

that regular observations appear to be outlying. This effect is called swamping. Both concepts were illustrated by Davies

and Gather (1993). It is important to note that any detected outlier need not necessarily be an error in the data. Their

presence may reveal that the data are more heterogeneous than assumed. Outliers may also come in clusters, indicating

there are subgroups in the population that behave differently. A robust analysis can thus provide better insights into the

structure of the data and reveal structures in the data that would remain hidden in classical analysis. Extensive

literature exists on detecting outliers and developing methods that are robust to them. An overview may be found in

Rousseeuw and Leroy (2005) and Maronna, Martin, and Yohai (2006).

The problem of robustness for forward curves was also noted by Hildmann, Herzog, Stokic, Cornel, and Andersson (2011)

and Hildmann et al. (2012) who used the least absolute difference (LAD) to obtain a robust estimate of seasonality shape.

However, they need a separate step to solve the arbitrage question, and their methodology focuses on the hourly price forward

curve. Also, Caldana et al. (2016) considered some notions of robustness in their proposed model. As a downside, the outlier

detection rule they propose uses estimated standard deviations which are highly susceptible to outliers on their turn as well.

In this paper, a statistically sound and robust method is proposed that allows to treat both the higher and fine

granular data, as well as the tradeable forward directly, to establish a consistent market pattern. The proposed

methodology applies to any level of granularity. The arbitrage question is incorporated directly in the estimator,

therefore obliterating the need for additional separate steps. Moreover, the methodology is inspired by how the forward

market trades, rather than trying to translate time series on spots like data (high granular) into long‐term patterns,

where true seasonality can be hidden behind noise trends and volatility. Motivated by these considerations, a

multivariate constrained robust M‐regression (MCRM) technique is developed that calculates fast on large data sets and

is relatively easy to explain to practitioners. Note that the method can be seen as a building step in the overall process

and that potentially, filtering techniques can further improve its quality.

In Section 2, the electricity market and its typical properties related to forward prices are briefly described. Section 3

discusses the usage of shaping coefficients in electricity markets. A new method for shaping forward curves in electricity

markets is proposed in Section 4, and the development of an MCRM for these purposes is motivated. In Section 5, the estimator

is described in more detail, and an efficient algorithm is given in pseudocode. The proposed methodology is applied to real data

from the German power market in Section 6 to illustrate its performance. In this section, the proposed method is also compared

to the method of Fleten and Lemming (2003), assessing both the estimation and prediction performance of both methods.

Finally, some conclusions and possible outlook are given in Section 7.

2

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FORWARD PRICES IN ELECTRICITY MARKETS

This paper will focus on the European electricity market (Huisman, 2009; Bunn, 2004), but the proposed methodology

is of course also applicable and relevant in other electricity markets.

Throughout the article, there are three common ways of referring to forward contract prices. The price can be

denoted by FtT T(, , )

12

, where

t

stands for the reference or observation date and

TT

[

,]

12

denotes the delivery period, for

example, T= January 1, 201

3

1and

T= December 31, 2013

.

2

Alternatively, Ftδ(, )denotes the same contract, where

δ

denotes the delivery period as a whole, for example,

Cal‐13

. In some cases, it is easier to use relative delivery periods, for

which the notation Ftρ(,

)

is being reserved, with

ρ

denoting a relative delivery period such as

Y+

1

(1 year forward)

concerning the observation date

t

. From the context, it will always be clear which notation is being used.

An important relationship that one has to understand in electricity markets is how the contracts relate to each other.

Purchasing a baseload contract for delivery in

Q

‐201

2

3(the third quarter of the year 2012) at a forward price of

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LEONI ET AL.