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Combination Forecasting of Energy Demand in the UK

September 2018
The Energy Journal 39(01)

DOI:10.5547/01956574.39.SI1.mbar

Authors:

Marco R. Barassi

University of Birmingham

Yuqian Zhao

University of Sussex

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209

Combination Forecasting of Energy Demand in the UK

Marco Barassi* and Yuqian Zhao**

ABSTRACT

In more deregulated markets such as the UK, demand forecasting is vital for the

electric industry as it is used to set electricity generation and purchasing, estab-

lishing electricity prices, load switching and demand response. In this paper we

produce improved short-term forecasts of the demand for energy produced from

ve dierent sources in the UK averaging from a set of 6 univariate and multivar-

iate models. The forecasts are averaged using six dierent weighting functions in-

cluding Simple Model Averaging (SMA), Granger-Ramanathan Model Averaging

(GRMA), Bayesian Model Averaging (BMA), Smoothing Akaike (SAIC), Mal-

lows Weights (MMA) and Jackknife (JMA). Our results show that model averag-

ing gives always a lower Mean Square Forecast Error (MSFE) than the best/opti-

mal models within each class however selected. For example, for Coal, Wind and

Hydro generated Electricity forecasts generated with model averaging, we report a

MSFE about 12% lower than that obtained using the best selected individual mod-

els. Among these, the best individual forecasting models are the Non-Linear Ar-

ticial Neural Networks and the Vector Autoregression and that models selected

by the Jackknife have often superior performance. However, MMA averaged fore-

casts almost always beat the predictions obtained from any of the individual mod-

els however selected, and those generated by other model averaging techniques.

Keywords: Demand for energy; forecasting; model averaging.

https://doi.org/10.5547/01956574.39.SI1.mbar

1. INTRODUCTION

Accurate and rigorous electricity demand modelling and forecasting is extremely import-

ant for energy suppliers, Independent System Operators (ISOs), nancial institutions, and other

participants in electric energy generation, transmission, distribution, and markets. Forecasting the

demand for energy is crucial for planning periodical operations and facility expansion in the elec-

tricity sector at the various levels. For example, models for electric power load forecasting are

essential to the operation and planning of a utility company. At this level, load/demand forecasting

would help an electric utility to make important decisions including purchasing and generating elec-

tric power, load switching, and infrastructure development. On a broader and dierent level, models

and forecasts of a country’s energy demand may provide useful information for the implementation

of specically targeted energy policies. However, obtaining an appropriate forecasting model for

electricity networks is far from being an easy task. In fact, although many modelling and forecasting

methods have been developed, none can be generalized for all demand patterns. This becomes an

even more pressing issue in deregulated markets such as that of the United Kingdom (UK hereafter)

* Corresponding author. Department of Economics, University of Birmingham, Birmingham, B15 2TT, U.K.; E-mail:

m.r.barassi@bham.ac.uk

** Department of statistics and actuarial science, University of Waterloo, ON, Canada; E-mail:yuqian.zhao@uwaterloo.ca

210 / The Energy Journal

where demand patterns have become even more complex. Depending on the available data, their

frequency, the desired nature and detail of forecasting, methodologies to obtain forecasts of the de-

mand for energy can be broadly classied into short-term models, which use traditional time series,

similar day/machine learning approaches and long-term models, which include, end-use, structural

econometric models and time series models with lower frequency data.

In this paper we aim to contribute to the literature on energy demand forecasting by de-

tailing a pseudo out-of-sample combination forecast design which uses high frequency time series

data to obtain improved short-term forecast (up to 1 day) of the demand for energy produced from

ve dierent sources in the UK, averaging from a set of 6 univariate and multivariate models. With

the growing deregulation of the energy industries, obtaining more accurate forecasts, has gained

increasing appeal. The reason is that, since supply and demand experience wider uctuations and

energy prices may increase by a factor of ten or more during peak situations, correct and well timed

demand forecasting has become vitally important for utilities. Short-term demand forecasts help to

estimate load ows and to make decisions that can prevent overloading. Timely implementations of

such decisions would lead to the improvement of network reliability and to the reduced occurrences

of equipment failures and blackouts. On the other hand, demand forecasting is also important for

contract evaluations and evaluations of various sophisticated nancial products on energy pricing

oered by the market.

Short-term forecasting models, which are usually from one hour to one week, use histori-

cal data and play a very important role in the operation of power systems’ operating functions such

energy transaction, unit commitment, security analysis, fuel scheduling and load switching. The

techniques most commonly used include:

• Univariate time series models (Box-Jenkins ARIMA), Holt-Winters exponential smooth-

ing, time series regressions and multivariate time series models such as Vector autore-

gressions (VARs), Bayesian VARs (BVAR) and Factor Augmented VARs (FAVAR)

• Similar day and machine learning approaches including Articial Neural Networks

(ANN), Non-linear Autoregressive Neural Networks (NLANN), Fuzzy Logic, Support

Vector Machines.

Long-term forecasting models play an important role in policy formulation and supply

capacity expansion (incorporate consumer behaviour and characteristics, technology, etc). They in-

clude:

• Time series methods with Lower frequency data. These include both univariate time

series models such as the traditional Box-Jenkins ARIMA, but also Seasonal ARIMA

and models which include fractionally integration such as ARFIMA and also the above

mentioned multivariate time series models (VAR, BVAR, FAVAR) also including cointe-

grated vector error correction models (VECM).

• End-use methods (electricity demand is derived from users’ demand for individual re-

quirements):

• Non-intrusive Load Monitoring Models

• Structural econometric models (Seek to establish the relationship between energy con-

sumption and the factors that inuence it) include:

• Conditional Demand Analysis (Multivariate Regressions, Stochastic Markov Chains)

Earlier attempts of short-term forecasting include Hagan and Behr (1987), Fan and Mc-

Donald (1994), Amjady (2001) and Nogales et al. (2002), who used time series regressions in-

cluding temperature, humidity and past energy consumption to obtain demand projections. More

Combination Forecasting of Energy Demand in the UK / 211

recently, Filik et al. (2011) predicted yearly, weekly and hourly electric energy demand through a

three stage model, showing how short-term predictions are usually more accurate and of immediate

application than medium and long-term ones.

Within the literature on the UK energy system, we nd several studies mostly focussed on

long-term modelling and forecasting. Hunt et al. (2003) applied time series models to forecast the

UK energy demand on a sectoral basis. However, pure time series models have often been criticised

for not considering other important macroeconomic variables as predictors of long-run energy de-

mand. To overcome this issue, Haas and Schipper (1998) use price elasticities, income elasticities

and technical eciency as explanatory variables to forecast energy demand in the UK and other

OECD countries. Similarly, in order to forecast oil, gas, coal and total energy demand in the UK and

Germany, McAvinchey and Yannopoulos (2003) constructed an econometric model incorporating

variables including the price of electricity and technological progress. Cointegration models have

also been used to model or forecast energy demand in the long-run. Among them, Fouquet et al.

(1997) uses a cointegrated VAR to investigate the long run relationship among fuel demand, the real

price level and economic activity and Sadorsky (2009) adopted panel cointegration techniques to

model the long run relationship among GDP per-capita, CO2 per-capita and the demand for renew-

able energy in the G7 countries.

Previous studies on energy demand forecasting can be also classied on the basis of their

parameterisation, that is, whether they employed univariate and/or multivariate (time series) mod-

els. The most widely used univariate forecasting set of models is the ARMA, which can be extended

to ARIMA to account for non-stationarity, SARMA to consider seasonality and ARFIMA to con-

sider fractional integration. Ediger et al. (2006) and Ediger and Akar (2007) employed ARIMA and

SARIMA to forecast fossil fuel in Turkey, using goodness-of-t and information criteria to select

the best forecasting models. However, Sumer et al. (2009) focussed on the importance of capturing

the seasonal eects contained in energy demand and used ARIMA, SARIMA and various regression

models, and found that the regression model with seasonal latent variables provides more accurate

forecast than the class of ARMA models.

In the earlier literature, exponential smoothing also showed reasonable forecasting ability.

Badri et al. (1997) adopted several time-series models including exponential smoothing to forecast

electricity peak-load in the UAE. As an extension of the simple exponential smoothing technique,

Hong (2013) claimed that Holt-Winter smoothing method shows better performance. More recently,

with the introduction of Articial Neural Networks (ANN) and its application in forecasting, many

studies have used ANN-type of models to forecast energy demand, also considering a number of

input variables, such as macroeconomic and environmental variables (Chow and Leung (1996),

Markham and Rakes (1998), Sözen et al. (2005), Ermis et al. (2007)). However, Maia et al. (2006)

showed that a hybrid model of ARMA and ANN model shows better prediction ability of energy

forecast. Similar models have also have been applied by Pao (2006) and Kurban and Filik (2009),

who used Non-linear Autoregressive Neural Networks—(NARNN) to forecast electricity demand

in Taiwan and Turkey, respectively. Geem and Roper (2009) employed the NARNN model to fore-

cast energy demand for South Korea, and they indicated that NARNN produce more accurate pre-

dictions than linear regressions and exponential smoothing.

Among multivariate time series models, vector autoregressions (VAR) are the probably

the most popular for forecasting purposes. García-Ascanio and Maté (2010) used a VAR to forecast

electric power demand, but they argued that VARs just shows poor predictive ability compared with

more advanced multivariate models. Bayesian VARs—BVAR have often been found to improve

the forecast accuracy of the basic VAR also overcoming the problem of over-tting. Crompton and

212 / The Energy Journal

Wu (2005) forecast coal, oil, gas and hydro energy demand in China through a BVAR; Francis et al.

(2007) used the BVAR to study the relationship between real gross domestic product per capita and

energy demand in Caribbean countries, and obtained forecasts for energy demand in those countries.

Recently, VAR models have been further developed so to include factors extracted from a set of po-

tentially numerous predictors so to obtain a factor augmented vector autoregressive model—FAVAR

(see Chudik and Pesaran (2011) among others). Baumeister et al. (2016) used a FAVAR model to

forecast gasoline price for the US, showing that their predictions are more accurate than those of

standard VARs.

Clearly, there is a wide variety of models and parameterisations among which to choose for

the purpose of obtaining energy demand forecasts, and thus, the choice of which criterion to use in

order to select the optimal forecasting model becomes another important issue (see Pao (2006) and

García-Ascanio and Maté (2010)for comparative studies). Lai et al. (2008) used the mean squared

error—MSE, the mean absolute percentage error—MAPE and the mean squared percentage er-

ror—MSPE to compare the accuracy of rival forecasting models. Moreover, models can be selected

by minimisation of information criteria obtained in-sample for each individual estimated model

(i.e. Akaike Information Criterion—AIC, and Bayesian Information Criterion—BIC ). Within this

stream of literature, Hansen (2007) and Hansen (2008) showed that Mallows’ information criteria

may often select models that providing more precise forecasts, and later Hansen and Racine (2012)

and Hansen (2014) proposed a cross-validation criteria for model selection based on the Jackknife.

Since the seminal article of Bates and Granger (1969) model averaging has been widely

used to produce more accurate forecasts than the individual optimal models. Hendry and Clements

(2004) and Timmermann (2006) showed that simple combinations often give better performance

than more sophisticated approaches. Further, using a frequentist approach, Granger and Ramana-

than (1984) proposed the use of coecient regression methods to determine the magnitude of the

weights of individual models in the averaging process. However, the most popular average method

is Bayesian model averaging (Madigan and Raftery, 1994), where the averaging weights are calcu-

lated based on empirical data and uses the Bayesian Information Criterion of the individual models

as weight in the averaging function. Anderson and Burnham (2002), suggested to use AIC criteria

to replace BIC criteria in model averaging, while Hansen (2007) and Hansen (2008) have continued

this line of research showing that model combination based on Mallows’ criterion, asymptotically

leads to forecasts with the smallest possible mean squared error. Guidolin and Timmermann (2007)

proposed a dierent time varying weight combination scheme where weights have regime switch-

ing dynamics. More recently, Hansen and Racine (2012) proposed a “jackknife model averaging”

(JMA) estimator which selects the weights by minimizing a cross-validation criterion showing that

their method is asymptotically optimal.

In this paper, in order to overcome some of the methodological issues above and produce

improved forecasts of the demand for energy in the UK, we use a forecasting approach based on

model averaging of several popular linear or non-linear, univariate and multivariate forecast models.

Specically, we rst obtain the forecasts from sets of ARMA, Holt-Winters Smoothing (HWS), Non

Linear Autoregressive Neural Networks (NLANN), Vector Autoregressions (VAR), Bayesian VAR

(BVAR), and Factor Augmented VAR (FAVAR) models. Forecasts are generated from each of these

sets of models and within each set they are then compared and selected using rankings based on four

dierent information criteria (AIC, BIC, Mallows’ and Jackknife). The best models as selected by

each of the dierent information criteria within the individual model set and are averaged using six

dierent combination weight metrics including Simple Model Averaging (SMA), Granger-Ramana-

Combination Forecasting of Energy Demand in the UK / 213

than Model Averaging (GRMA), Bayesian Model Averaging (BMA), Smoothing Akaike (SAIC),

Mallows Weights (MMA) and Jackknife (JMA).

The paper is structured as follows: the next section presents the data and explains the

pre-analysis necessary to transform the raw series into workable variables. Section three outlines the

individual forecast models used and the metrics used for selection and weighting purposes. Compar-

isons of the performance of individual models, forecasting methods, information criteria, averaging

functions are discussed in section four. Section ve reports the forecasts of the levels demand for

energy in the UK, a summary concludes.

2. DATA AND PRE-ANALYSIS

2.1 Data

We collected 30 minutes data from National Gridwatch website1 ranging from 00:00:00

of the 21 December 2013 to 00:00:00 of the 21 March 2016, for a total of 39287 observations for

each energy demand process. The data relate to energy demand for the entire UK plus import, minus

export less un-metered sources. Here, we shall assume that supply matches demand at all times. The

UK gridwatch provides energy demand data disaggregated by dierent types of energy sources, and

in this paper, we will concentrate on the most important ve types of sources including coal, nuclear,

combined cycle gas turbines (CCGT hereafter), wind and hydro-power.

Note that, energy providers in the UK dier quite substantially between them for the choice

of fuel sources used to provide energy as can be seen from the table below which lists most of the

utilities in the UK. Note also that, beside the strong deregulation, given the dierences in fuel mix

used by energy suppliers in the UK, accurate short-term forecasting of the demand for energy pro-

duced from the dierent sources becomes even more appealing as it has an important impact on

the prices charged, the choice of production and not least general demand management and load

switching.

In this paper, using the data described above, we obtain 30-minutes to one-day predictions

of the demand for energy as produced from these ve energy sources, that is we obtain forecasts

for the 22 March 2016, at the following times 00:30, 01:00, 02:00, 04:00, 06:00, 08:00, 16:00 and

24:00.

Figure 2 plots the levels of the demand for energy generated from each of the ve sources.

It is noticeable how demand for CCGT and wind sourced energy are relatively more volatile, while

nuclear is more steady. The main reason is that CCGT, though is an ecient way to use gas and

turbines are fast to get online, they use relatively expensive fuel. Thus, these cycling plants will

ramp up and down during the day, and are usually used more during peak hours. As for the wind

turbines, they are expensive but wind is cheap, however, the strength of wind is not constant and it

varies from zero to storm force. This means that wind turbines cannot not produce the same amount

of electricity all the time and there are be times when they produce no electricity at all. On the other

hand, once on, nuclear power stations run at out, the cost of fuel is insignicant and this explains

the much lower volatility in its demand.

Still, regardless of the source, there exist strong“seasonal” components in the data. These

periodicities vary in frequency as seasons, “day-of-the-week” and “hour-of-the-day”. Therefore, it

is necessary to remove these periodic eects prior to analyse the data and obtain the forecasts for

each model of the model sets.

1. http://www.gridwatch.templar.co.uk/

214 / The Energy Journal

Table 1: Fuel Mix of UK Energy Suppliers

Supplier Coal Gas Nuclear Renewable Other CO2 Nuclear Waste

British Gas 2 30 34 33 0.7 0.137 0.0024

Bulb 0 0 0 100 0 0 0

Co-operative Energy 36.6 34.2 13.4 9.8 6 0.493 0.00094

E.On 18.7 32.4 12.8 29 7.1 0.328 0.001

Ecotricity 0 0 0 100 0 0 0

EDF Energy 14.5 8.6 64.3 12.3 0.3 0.167 0.0045

Extra Energy 19 33 13 28 7 0.339 0.0009

First:Utility 18.9 32.7 12.9 28.3 7.2 0.33 0.0009

Flow Energy 18.9 32.7 12.9 28.3 7.2 0.331 0.0009

GnERGY 34 25.6 21.6 16.7 2.1 0.418 0.00151

Good Energy 0 0 0 100 0 0 0

Green Energy UK 0 70.6 0 29.4 0 0.134 0

Green Star Energy 0.1 0.1 0 99.8 0 0.002 0.00001

iSupplyEnergy 38.7 36.2 14.2 4.6 6.3 0.528 0.001

LoCO2 Energy 0 0 0 100 0 0 0

Npower/RWE 16 66 1 16 1 0.408 0.00008

Octopus Energy 1 1 1 97 0 0.013 0.00004

OVO Energy 0 46.9 0 53.1 0 0.183 0

Scottish Power 34 36 3 26 1 0.46 0.0002

So Energy 18.9 32.7 12.9 28.3 7.2 0.331 0.0009

Spark Energy 46.8 27.1 8.4 11.9 5.8 0.579 0.0007

SSE 25 35 7 29 4 0.38 0.00047

Utilita 19 33 13 28 7 0.332 0.00091

UK Average 17 32.3 23.7 24.3 2.5 0.29 0.0017

Source: http://electricityinfo.org/fuel-mix-of-uk-domestic-electricity-suppliers/

Figure 2: The levels of energy demand in UK

Note: This gure plots the level of demand for each energy source.

Combination Forecasting of Energy Demand in the UK / 215

2.2 Removing the Deterministic Components

As highlighted above, energy demand data always contains signicant periodic patterns in

both the short and long-term. In this subsection, we describe the three methods that we shall use to

remove these periodic patterns in turn. In what follows we will rst t an up to 5th degree Cheby-

shev polynomial to remove the season of the year component, a moving average based technique

to remove the day of the week eect and nally a 48th order dierencing to eliminate a day/night

peak/o-peak hour eect.

Cuestas and Gil-Alana (2016), recently showed the ability of Chebyshev polynomials to

t long-term cyclical patterns. Chebyshev polynomials are based on orthogonal cosine functions

of time, such that a linear combination of these functions can exibly approximate most cyclical

patterns. The higher the order of the polynomial, the more non-linear is the cyclical pattern that can

be approximated. Following [12] we dene the polynomial as,

( ) = 2 ( ( 0.5) / ), = 1, 2, , ; = 1, 2, ,

−

P t cos i t n t n i

(1)

where i is the order of Chebyshev polynomial. Specically, when i equals to 0, it gives a linear

constant function with

( )=1

. Since any empirical process yt can be decomposed between a

deterministic and a stochastic part, and if the deterministic term approximated by Chebyshev poly-

nomials, then we can have,

= ( ) , = 1, 2,3, ,

∑



t i in t

y Pt x t

(2)

where xt is assumed to be the stochastic part of the model, and the order of Chebyshev polynomials

is determined by the signicance of parameters θi. The parameter θi can be estimated by,

=1 =1

ˆ=( () ()) ( () ),

−

′

∑∑

nn nt

PtPt Pty

(3)

Finally the de-seasonalised process

y is,

= ( ),

−∑

t t i in

y y Pt

(4)

Because our data set span over 2 years and 3 months, it should contain two complete season

cycles in the demand for each energy source. Figure 3 illustrates the seasonal patterns removed from

the demand of energy obtained from the ve sources. From these plots, we can see that all energy

sources reach their peaks during winter period and fall down during summer except for CCGT. Once

the seasonal long-term cyclical pattern has been removed, it is necessary to remove shorter term

periodic patterns.

In the short-run, there are two more periodic elements which need to be ltered out: week-

day, and peak/o-peak (or day-night) eects. The week-day element can be removed by adopting a

specic moving average method. Since the data frequency is 30 minutes, following [50], we set the

moving average length

equal to 336, which are the number of half-hours in one week. To obtain

the moving average of each point in the series, we need to make sure that data in front and behind

any given data point which we obtain, is of the same length. Hence, we consider

1 = 337+l

obser-

vations for each moving average window. The moving average component is calculated through,

/2 /2

** *

=1 =1

ˆ= ( ),

−+

∑∑

t ti t ti

m yy y

(5)

216 / The Energy Journal

where mt is the moving average term and y*

t is the de-seasonalised series obtained in the rst step.

Hence, the deviation from the moving average is given as,

−

k k lj k lj

wy m

(6)

where

is the index in one moving average window, and j is the number of moving average win-

dows. Thus, the day-of-the-week cyclical component can be obtained as,

ˆ=,−∑

kk i

sw w

(7)

Therefore, the day-of-the-week ltered data is dened as,

=,−

tl t t

y ys

(8)

Figure 4 illustrates the tted weekly component of the demand for energy obtained for each

source. From the plots, it is apparent that some energy sources have a more pronounced day-of-the-

week eect compared to others. Specically, the demand for coal, nuclear and CCGT produced

energy are relatively higher in weekdays as compared with weekend. This pattern, instead, does not

appear so clearly in the demand for energy produced from wind and hydro sources. This is probably

because the rst three energy are the main sources of electricity used in industrial processes and

businesses, and also, the other two are more strongly dependent on weather conditions and cannot

necessarily provide energy in a steady manner. Apart from the week-day eect, it is reasonable to

expect that a peak/o-peak or day/night eect exists too in the energy demand series and this will

also be dealt with in what follows.

After removing the season-of-year and week-day components, we remove the peak/o-

peak and day/night eect by taking the sth = 48 order dierence of each of the demand for energy of

each ω source:

Figure 3: The seasonal component in the demand for energy

Note: This gure plots deterministic seasonal trend tted by Chebyshev polynomials with order 5 for each energy source.

Combination Forecasting of Energy Demand in the UK / 217

* **

,, ,, , ,

ω ωω

−

ts tl t sl

y yy

(9)

After this preliminary analysis, we move forward to the second step of our forecast pro-

cedure where the obtained (now purely stochastic) energy demand series

for each source

will be modelled and forecast by univariate and multivariate time series as well as neural networks

models, as briey described in next section.

3. METHODOLOGY

In this section, we will briey outline the methodology that we shall use to obtain the

forecasts of the demand for all the energy sources. Firstly, we will describe the six types of fore-

cast models going from univariate to multivariate ones; secondly, we will present the four types of

criteria used to select optimal forecasting model within each of the model sets (or )classes). Lastly,

we discuss the model averaging techniques and show they are used to produce improved forecasts.

3.1 Forecast Models

In this subsection, we will introduce the six classes of models which we use to t and

forecast the ltered series. The univariate models are autoregressive moving average (ARMA), Holt

Winter Smoothing (HWS), Non-linear Autoregressive Neural Network (NARNN) model, and the

multivariate models are Vector autoregression (VAR), Bayesian VAR and Factor Augmented VAR

models. Recall that

denotes the type of fuel used as a source of energy.

1. ARMA

The rst forecasting model is the traditional stationary ARMA model. It is the most com-

monly used forecast model also in the energy eld and usually constitutes a benchmark against

Figure 4: The day of the week component in the demand for energy

Notes: This gure plots deterministic weekly trend tted by Weron (2007)’s moving average method with cyclical length

equals to 336. Thus, the weekly trend contains 336 observations, which indicates the cyclical patterns for one week.

218 / The Energy Journal

which other forecasting techniques are compared. Among others, Ediger et al. (2006) and Ediger

and Akar (2007) used ARIMA models to forecast Turkish fossil fuel demand. Also, Sumer et al.

(2009) employed the ARMA class of models to predict electricity demand. In this paper, since the

periodic components and non-stationarity have been removed, we use the following simple AR-

MA(p,q) parameterisation:

,1 , , ,1

=1 =1

ω ω ωω

α β γε ε

+− − − +−

+++

∑∑

th i ti j t j th

(10)

where h is the period forecast ahead. The model is estimated by ordinary least squares and the opti-

mal lag length p and q are determined by information criteria. Once the model has been estimated,

we can use it to predict

,1 , ,

=1 =1

ˆˆ

ω ωω

α β γε

+− − −

∑∑

th i ti j t j

(11)

The optimal forecasting model is selected from the set of ARMA(p,q) with p and

= 1, ,12q

giving a total of 144 models estimated for each energy source.

2. Holt-Winter Smoothing (HWS)

The HWS belongs to the class of exponentially weighted moving average methods. The

model ts the target process using its past smoothed values and gives more weight to the most recent

ones such that it can be expressed as,

, , ,1 ,1

= (1 )( ),

ω ω ωω

αα

−−

+− +

t t tt

s a sb

(12)

Figure 5: The Stochastic Component of Demand for Energy

Note: This gure plots stochastic term after remove deterministic non-linear patterns for each energy source.

Combination Forecasting of Energy Demand in the UK / 219

such that at time t, the actual value of the process is denotes by aω,t, the smoothed estimate is denoted

by ,

and ,

is the trend. In turn, the trend is formulated as,

, , ,1 ,1

= ( ) (1 ) ,

ω ωω ω

ββ

−−

− +−

t tt t

b ss b

(13)

where the parameter

is the trend smoothing parameter. Therefore, the predicted value is obtained

from,

,, ,

ωω ω

tt t

f s ib

(14)

Here we select the smoothing parameter

from

[0.7,0.8,0.9]

while

is selected from

[0.1,0.2,0.3]

(see Hong (2013)). Hence, we consider 9 types of HWS models.

3. Non-linear Auto-Regressive Neural Networks (NARNN)

The restrictions imposed by linear forecast models such as ARMA and HWS, are usu-

ally overcome by adopting more general non-linear forecast models(De Gooijer and Kumar, 1992).

Articial neural networks are a commonly used type of such non-linear models (see Sözen et al.

(2005), Pao (2006) and Kurban and Filik (2009), amongst others), and have been widely used for

the purpose of univariate series forecasting. Although it suers from the criticism of not-so-much

underlying economic foundation, the NARNN model (Chow and Leung (1996) and Markham and

Rakes (1998), amongst others) often provides better forecasting accuracy because it is able to ap-

proximate plenty of functions(Zhang, 2003).

In brief, the NARNN model is a dynamic neural network model which is built on a linear

autoregressive model with feedbacks on several layers. The model regresses current dependent out-

put signal on previous output signals, so that the model equation is dened as follows:

, 1 ,1 ,2 ,

= ( ),

ω ωω ω

+− − − −

+ ++

th t t tp

y fy y y

(15)

where f is a non-linear function, and p is the earliest value of signals considered. Once the model has

completed training and validation, it can be used to forecast in the same fashion:

, , ,1 , 1

ˆ= ( ),

ω ωω ω

+ − −+

+ ++

th t t t p

y fy y y

(16)

An example of the architecture of the NARNN model is shown in Figure 1. In our case, we

set to 10 the number of neurons in the hidden part, and apply a back-propagation method for training

as in Geem and Roper (2009).

The lag length considered in NARNN ranges from 1 to 12, so that there are 12 models for

each energy source.

Figure 1: The Architecture of a NARNN model

Source: http://uk.mathworks.com/help/nnet/ref/narnet.html

220 / The Energy Journal

4. Vector Autoregression (VAR)

The models introduced above are self-forecast models, where the predicted value is mainly

based on the serial correlation of historical data. From the fourth model onward, we forecast the

energy demand processes according to causal relationships. In these models, the energy demand

processes are modelled as a system, such that the predictions for each process would be obtained

from the entire system. The rst causal forecast model is the standard VAR. García-Ascanio and

Maté (2010) used a VAR model to forecast electric power demand in Spain.

A system of

composed by endogenous variables

1,t

, 2,

, ..., ,

, where

refers to the

demand for energy from each fuel source such that here

=5k

. Thus, the VAR model with lag length

p can be formulated as,

+− −

Φ+

∑

th i ti t

(17)

where each

is a

×KK

coecient matrix, and

is a K-dimensional vector of errors terms with

mean vector zero and diagonal variance covariance matrix

. We estimate each VAR by maximum

likelihood (MLE). Then, we use the in-sample estimation results and iterate forward to obtain the

out-of-sample predictions.

−

+−

∑

th i ti

(18)

Compared with univariate forecast models, the advantage of VAR models (same for all VAR-type

of models which will follow) is that they provide predictions not only based on historical tting of

individual process but also by means of lags of other endogenous variables in the system. Note that,

in our in-sample estimation and model selection, the maximum lag length considered will be

= 12p

5. Bayesian Vector Autoregression (BVAR)

While it is common to use VARs to obtain forecasts, it has also been argued that VARs esti-

mated by Bayesian methods would provide better forecast with more parsimonious models because

standard VARs often incur in over-tting problems (Spencer, 1993). Compared with standard esti-

mation, the BVAR2 treats model’s parameters as random variables, and applies Bayesian estimation

imposing restrictions on the dynamics of the parameters according to a specic type of prior. Based

on this assumption, the coecients on longer lagged variables are more likely to be near zeros, re-

sulting in a more parsimonious estimation. Indeed, we still use the model showed in Equation (17),

however, with the prior adopted being the Minnesota Prior (Del Negro and Schorfheide, 2004). In

the VAR system, there are K equations, and each one can be expressed as,

,1 , , ,

=1 =1

φε

+− −

⋅+

∑∑

it h i j jt i it

(19)

In this case, the prior about coecients are captured in the prior density function

()

. Then, using

Bayesian theory, the estimators are obtained by the posterior density functions

(|)

i j it

,, ,

( | )( )

( | )= ,

()

φφ

it ij ij

i j it

gy g

gy gy

(20)

2. For an application see Crompton andWu (2005) who applied BVAR model to predict energy consumption in China.

Combination Forecasting of Energy Demand in the UK / 221

and the predictions of yi,t can be obtained from following,

, ,,

=0 =1

−

+−

⋅

∑∑

it h i j jt i

(21)

where again, the maximum lag length considered will be

= 12p

6. Factor Augmented Vector Autoregression (FAVAR)

Last, we use a factor augmented VAR. The FAVAR model has been widely applied to

large data especially in macroeconomics (Bernanke and Boivin (2003) and Bernanke et al. (2004)).

Chudik and Pesaran (2011) claim that a less parameterised VAR model augmented with factors will

not lose any relevant information and would often produce better forecasts than standard VARs. In

the energy related literature, among others, Baumeister et al. (2016) have adopted VAR, BVAR and

FAVAR models to predict gasoline price in US market.

The FAVAR aim at modelling a system

with N variables and assume a subset

which contains M variables, and the dynamics of

are driven by unobservable forces in

. These

unobservable forces are factors extracted from

, containing most of the relevant information. The

system can thus be formulated as follows:

Λ ⋅ +Λ ⋅ +

t tt

x Fy

(22)

where Λf is

×NK

coecient matrix for K factors, Λy is a

×NM

coecient matrix, and

is a

1×N

vector of error terms.

In this paper, we classify the energy demand processes into two groups, one group is the

objective observed process for a specic source

, while the other group is made up of the energy

processes obtained by other sources from which one factor is extracted. We denote this groups data

. The FAVAR model can be written in state-space form comprising two equations: the observa-

tion equation and the state equation. In the observation equation, the number K of factors

, where

=1K

in this paper, can be extracted from the variables in

through principal components. Thus,

the state equation is,

,1 , ,1

ω ωω

+− − +−

Φ+

∑

th i ti th

(23)

where

+−







. Again,

is a coecient matrix, the dimension of which depends on the

number of factors extracted from

, and if there is only one factor, the coecient matrix will be

22×

. Therefore, the objective process

can be predicted by one

that has been estimated,

ˆ=,

ωω

−

+−

∑

th i ti

(24)

where

ˆ=







. For each objective process, the predictions are obtained through the causal

relationship with factors extracted from the remaining processes. In this paper, we extract one factor

from remaining four energy series. For each energy source, the optimal model is selected by consid-

ering lag length

from 1 to 12.

222 / The Energy Journal

3.2 IMS and DMS forecast methods

For all the models in each class, we construct forecasts of the demand for energy using both

iterated multi-step (IMS) and direct multiple steps (DMS) methods. The IMS method provides h

steps ahead predictions through a one step ahead predictor

iterated forward h times. In each

iteration, we estimate the Equation 25 below using the training sample, and then forecast one period

ahead for the out-of-sample through Equation 26.

11 1

= ( ) , = 1, , 6,

Θ+ 

t it t

y yi

(25)

1 11

= = ( ),

t it

fy y

(26)

where



is the i th model, and Θ1 is a set of parameters in i th model.

The DMS method predicts h steps ahead through forecasting

+h th

directly. This is

achieved by using the estimated Equation 27 using the training sample, and then using Equation 28

to predict

for the out-of-sample.

= ( ) , = 1, , 6,

Θ+ 

th h i t th

y yi

(27)

= = ( ),

h th h i t

fy y

(28)

where Θh is a set of parameters in i th model for DMS.

According to Marcellino et al. (2006), the IMS method should provide lower forecast error

once the one-period ahead model is well specied. However, the DMS method is relatively more

robust to misspecication in the forecast model. Thus, in general, DMS has been often preferred to

IMS in empirical studies. As there is no strong evidence to support a clear cut choice between IMS

and DMS, we shall obtain forecasts with both methods and will compare their respective forecasting

ability.

3.3 Model Selection

For each model discussed above, we will consider dierent parameter settings and lag

lengths, and assume that the best forecasting model exists among those considered here. The op-

timal or best model will be chosen in-sample by means of dierent information criteria including

Akaike Information Criterion (AIC), Beyesian Information Criterion (BIC), Mallows’ Information

Criterion (MIC) (Hansen (2007) and Hansen (2008)) and the Jackknife (JKC), a cross-validation

criterion suggested by Hansen and Racine (2012) and Hansen (2014). Each information criterion is

computed using the in-sample tted error

ˆˆ

()= ()

−

t tt

m y ym

. Hence, the estimated tted error vari-

ance equals to

1ˆ

ˆ()= ()

σε

∑

, where

is the number of observations in-sample.

The AIC and BIC information criteria reward for lower tted errors but penalize for higher

number of parameters estimated, so that

ˆ()

is the estimated error variance of model

, and the

number of parameters in each estimated model is denoted as

()km

. The AIC and BIC can be respec-

tively expressed as:

= ( ( )) 2 ( ),

⋅+AIC n ln m k m

(29)

Combination Forecasting of Energy Demand in the UK / 223

= ( ( )) ( ) ( ),

⋅ +⋅BIC n ln m k m ln n

(30)

where n is the total number of observations in-sample.

The Mallows’ information criterion uses the estimated mean squared errors,

ˆ ˆˆ

=( ())( ()) 2 () (),

′

− − +⋅ ⋅

tt tt

MIC y y m y y m m k m

(31)

where

ˆ()

is the tted value of

from model

and

ˆ()

and

()km

are dened as above. Last,

we use the Jackknife, a cross-validation criteria. To use this cross-validation method, we obtain a

leave-one-out estimator for each in-sample point for every model, and then obtain a cross-validation

tted error



through the following equation.

−



mi i im

e yy

(32)

where

−im

is the leave-one-out one step estimate of

based on the estimated parameters from the

remaining observations. Then, the expression for the Jackknife (JKC) can be formulated as,

⋅

∑



JKC n

(33)

Note that, for each of the criterion used, the optimal/best forecasting model will be the one

which in each class/set of models minimizes the information criterion. In case of equal value of the

information criterion for two dierent models within the same class, the more parsimonious model

will be preferred.

3.4 Model Averaging

In this subsection, we briey outline the model averaging methods which we shall use

to improve the accuracy of our forecasts. We denote the prediction from i th model as

()

for

1≤≤iJ

, and the average prediction

is dened as

= ( ),

∑

t it

f wf i

(34)

where

()

is obtained from the forecasting model class/set



, and

is the weight attached to the

individual

()

obtained from the

candidate forecasts. At this point, the major issue in model av-

eraging becomes how to specify the weights

as dierent weighting functions are likely to provide

dierent levels of forecasting accuracy.

3.4.1 Simple Model Averaging (SMA)

The simple model averaging provides an equally weighted average of the predictions of

all the best forecast models in each class, such that the weights in simple model averaging are just

. The SMA forecast is,

= ( ),

∑

f fi

(35)

Note that SMA is known to improve the accuracy of forecasts as long as the model candidates are

well specied. However, once some of the candidates are not well specied, the accuracy of aver-

aged prediction will signicantly decrease.

224 / The Energy Journal

3.4.2 Granger-Ramanathan Model Averaging (GRMA)

Granger and Ramanathan (1984) proposed a model average weighting based on coe-

cients of the regression model. The regression is made of an average forecast

regressed on the

candidate predictions

()

and is formulated as:

= ()

ββ ε

∑

t it t

f fi

Granger and Ramanathan (1984) impose three constraints on the coecients of this re-

gression. First, the intercept coecient is equal to zero,

; second, the coecients on each

candidate prediction should be non-negative,

≥

for all i; last, the sum of coecients of the re-

gression must be equal to one,

∑

. Having specied these constraints, they used the estimated

coecients as the weights for averaging, that is

= ( ),

∑

t it

f fi

(36)

3.4.3 Bayesian Model Averaging (BMA)

Bayesian model averaging assumes that there always exists at least one well-specied

model among all candidate models and therefore one should give more weight to well or better spec-

ied candidates while less weight is attached to the rest of the models. The probability of candidate

models to be well-specied is giving as a prior, and then the Bayesian posterior probability can be

calculated conditional on real data. These Bayesian posterior probabilities for each candidate model

are the weights for averaging all potential models. As the prior probability that any model is a well

specied model is not known for each candidate model, the weights can be approximated by using

the Bayesian information criteria.

( ( ))

()

−

∑

exp BIC

Thus, the averaged forecast is given by:

= ( ),

∑

t it

f wf i

(37)

3.4.4 Other Model Averaging Functions

Anderson and Burnham (2002), proposed to replace BIC in the weighting function with

AIC, this resulting into smoothed AIC (SAIC) weighting function. In this case the weights

will

be:

( ( ))

()

−

∑

exp AIC

and the SAIC model averaging (AMA) forecasts are given by:

Combination Forecasting of Energy Demand in the UK / 225

= ( ),

∑

t it

f wfi

(38)

Similarly, as suggested by Hansen (2007), Hansen (2008) and (Hansen and Racine (2012), Hansen

(2014) we can use Mallows’ information criteria (MIC) or the Jackknife cross-validation criteria

(JKC) to replace the BIC, thus obtaining weights

and

. such that the two weighting functions

will respectively be,

( ( ))

()

−

∑

exp MIC

with the Mallows’ model averaging (MMA) forecast being,

= ( ),

∑

t it

f wfi

(39)

and

( ( ))

()

−

∑

exp JKC

while the Jackknife model averaging (JMA) forecast is,

= ( ),

∑

t it

f wf i

(40)

4. RESULTS AND COMPARISON OF THE FORECASTS

In this section, we provide our results and a comparison between forecast of dierent time

horizons according to the various criteria. Because of the heavy computational burden, we shall

consider static forecasts obtained with both IMS and DMS methods. Our forecast methods can be

extended to a dynamic type by constructing recursive or rolling out-of-sample forecasts iteratively,

however, this would be made at the expense of an even heavier computation burden and it is not

done here.

In the following, we report results of, and comparisons between dierent forecast models,

forecast methods, forecast model selection criteria and eventually model averaging methods. The

accuracy of these predictors is measured by Mean Squared Forecast Error (MSFE hereafter), and

model averaging predictions are statistically tested by using tests provided by Diebold and Mariano

(1995). We predict demand for each energy source with forecast horizons of 30 minutes, 1 hour, 2

hours, 4 hours, 8 hours, 12 hours, 18 hours and 24 hours.

4.1 Comparison based on Forecast Error

For any forecast model i, if

()

is the predicted value of objective

, and the forecast error

ˆ()

is expressed as,

226 / The Energy Journal

ˆ()= (),

−

t tt

i y fi

(41)

Thus, the estimated forecast error variance is,

1ˆ

ˆ()= () ,

σε

∑

(42)

where m is the number of out-of-sample predicted points. This paper uses the MSFE to measure the

accuracy of predictions.

()=[ ()],−

MSFE m E y f m

(43)

Table 2 displays the MSFE of an average between IMS and DMS forecasts of the best

model of each model class. There are several points to note. In terms of the model selection criteria,

we notice that the best/optimal models suggested by the JKC tends to often produce more precise

forecasts in terms of lower forecast error. Comparing dierent classes of models, NARNN and VAR

appear to forecast better than the others model classes. Within the ve energy sources, we can see

that it is easier to obtain accurate predictions of the demand for nuclear energy. This result is fairly

expected as nuclear plants once on-line run at-out and also conrms the preliminary analysis above

which showed less noise in nuclear energy demand process. On the contrary, CCGT and wind en-

ergy demands are relatively more dicult to predict, and display larger values of the MSFE.

Table 3 reports the MSFE of the dierent model averaging forecasts. Although the DMS

method reports lower MSFE in 16 out of the 30 cases, it is still hard to conclude whether IMS beats

DMS or the reverse, because each method dominates the other depending on the source of energy

Table 2: MSFE of Individual Models

ARMA HW NARNN VAR BVAR FAVAR

*Coal AIC 0.0428 1.3233 0.0284 0.0304 0.0438 0.0274

BIC 0.0428 1.3233 0.0284 0.0305 0.0436 0.0334

MIC 0.0428 1.3706 0.0284 0.0327 0.0429 0.0397

JKC 0.0289 1.3233 0.0295 0.0304 0.0438 0.0274

*Nuclear AIC 0.0239 0.0005 0.0005 0.0004 0.0013 0.0007

BIC 0.0239 0.0005 0.0004 0.0004 0.0012 0.0016

MIC 0.0239 0.0005 0.0005 0.0004 0.0012 0.0007

JKC 0.0004 0.0005 0.0005 0.0004 0.0013 0.0007

*CCGT AIC 0.1278 0.1527 0.0971 0.0935 0.1378 0.1017

BIC 0.1278 0.1527 0.0963 0.0954 0.1404 0.1013

MIC 0.1278 0.1565 0.1032 0.0955 0.1480 0.0962

JKC 0.1083 0.1527 0.0971 0.0935 0.1378 0.1017

*Wind AIC 0.1564 0.1201 0.0442 0.0909 0.1254 0.0835

BIC 0.1742 0.1201 0.0545 0.0866 0.1384 0.4522

MIC 0.1564 0.0843 0.0442 0.0860 0.1898 0.0835

JKC 0.1097 0.1201 0.0442 0.0909 0.1254 0.0835

Hydro-Power AIC 0.0508 0.7546 0.0591 0.0574 0.0483 0.0576

BIC 0.0508 0.7546 0.0571 0.0553 0.0490 0.0586

MIC 0.0487 0.6072 0.0591 0.0547 0.0523 0.0588

JKC 0.1790 0.7546 0.0591 0.0574 0.0573 0.0576

Note: This table reports the mean square forecast error of the best individual forecast models according to dierent information

criteria. The MSFE averages those of IMS and DMS predictions from all forecast horizons.

Combination Forecasting of Energy Demand in the UK / 227

for which demand is forecast. Among the six types of model averaging methods, generally the

BMA, MMA are superior to the others. Lastly but most importantly, from Table 3, we can see that

there always exists a model averaging method which gives a lower MSFE than the best/optimal

model from any class as from Table 2.

4.2 IMS vs DMS

In this subsection, we specically compare the forecasting ability of IMS and DMS. Figure

6 shows the plots of the MSFEs obtained from the two forecasting methods. Generally, it seems

preferable to use DMS, particularly if using ARMA and FAVAR models to forecast a longer horizon.

An exception is the case of BVAR for which the IMS beats the DMS in generating forecasts of the

demand for energy produced by coal, ccgt, wind and hydro-power.

In more detail, the ARMA model with DMS method produces better forecasts for coal,

nuclear and hydro-power. With regard to CCGT and wind sources, the FAVAR shows better fore-

casting ability, but with IMS in one instance and with DMS in another. Also, as a general result,

we observe that the predictions are more accurate in the beginning and at the end of the forecast

Table 3: MSFE of Model Averaging Methods

SMA GRMA AMA BMA MMA JMA

*Coal IMS 0.0440 0.0440 0.0297 0.0297 0.0231 0.0340

DMS 0.0496 0.0330 0.0268 0.0268 0.0255 0.0430

*Nuclear IMS 0.0013 0.00022 0.0005 0.0004 0.00022 0.0007

DMS 0.00038 0.00063 0.00043 0.00047 0.00036 0.00036

*CCGT IMS 0.1004 0.1275 0.0932 0.0927 0.0988 0.0996

DMS 0.1113 0.1191 0.1017 0.1004 0.1175 0.1086

*Wind IMS 0.0805 0.2896 0.0429 0.0457 0.0821 0.0679

DMS 0.0508 0.1366 0.0390 0.0423 0.0536 0.0584

Hydro-Power IMS 0.0606 0.0958 0.0500 0.0474 0.0449 0.0543

DMS 0.0635 0.0426 0.0508 0.0508 0.0483 0.0556

Note: This table documents the mean square forecast error of the model average methods. The MSFE averages predictors

from all forecast horizons.

Figure 6: MSFE of IMS and DMS

Notes: The gure compares IMS and DMS forecasts precision for six classes of forecast models. The y axis is the value of

MSFE, and the x axis is the forward prediction steps.

228 / The Energy Journal

horizon, as shown by the MSFE, which is always low in forecasting 30 minutes, 1 hour and 1 day

ahead, but grows to higher levels in the mid-term, indicating that serial correlation or cross-serial

correlation (in multivariate models) is stronger in the short and long-term but weaker in the medium

term (relative to the frequency of the data). Therefore, for empirical purposes, forecasting with DMS

is advised for short and longer terms.

Figure 7: Comparison among Information Criteria

Notes: The gure compares model selection information criteria for six classes of forecast models. The y axis is the value of

MSFE, and the x axis is the forward prediction steps.

Combination Forecasting of Energy Demand in the UK / 229

4.3 The Comparison of Information Criteria

In section 3.3 we outlined four types of information criteria used to selecting the optimal

in-sample forecast model. Here we compare their performance out-of-sample, computing the MSFE

of the up-to-one-day forecasts of the demand for energy produced from the best models of each set

as selected by each information criteria. Figure 9 illustrates he MSFE of the optimal forecast models

in each class as selected by AIC, BIC, MIC and JKC respectively.

In brief, BIC and MIC consistently suggest the same optimal forecast model, while the

AIC and JKC are more likely to indicate similar forecast model. For all energy sources, the MSFE

obtained from the forecasting model suggested by AIC and JKC almost never underperform to the

counterpart suggested by BIC and MIC. Particularly in each sector, the optimal models suggested

by these four information criteria are more or less the same for HW, NARNN, VAR and BVAR

models, and the only exception is the BVAR model for wind-produced energy. In term of ARMA

and FAVAR models, the AIC and JKC select better out-of-sample forecasting model, except for

the short horizon forecasts generated by ARMA model in the coal-produced energy demand series,

and relatively longer horizons by ARMA for hydro-power. Now, since each combination method is

averaging the optimal models from six types of forecasting candidates, the fact that AIC and JKC

are selecting models that give lower MSFE, it is reasonable to expect that using these criteria in

weighting function for model averaging will also give more accurate predictions. Below we exam-

ine the issue in more detail.

4.4 Comparison of Model Averaging Methods

In this sub-section, we compare the forecasting performance of the dierent model averag-

ing methods, namely: SMA, GRMA, AMA, BMA, MMA and JMA. Figure 8 displays the MSFEs

obtained out-of-sample for the various model averaging methods. The multi-steps predictions are

computed with both IMS and DMS. Consistent with the results displayed in Table 3 and Figure 6

and already discussed, the DMS method provides slightly more accurate forecasts than the IMS.

Within the IMS-based forecasts, we nd that, overall, SMA and AMA produce more ac-

curate predictions for nuclear, CCGT and wind for most of the forecast horizons, while MMA is

superior in generating predictions for coal and hydro. In more detail, the AMA, BMA and MMA

methods produce good forecasts albeit none of them clearly dominated the others. Also, the JMA

generates better forecasts for coal but loses eciency as the forecast horizon approaches the 24

hours; the BMA method produces better predictive ability for nuclear and CCGT, while MMA is

best for hydro-power sourced energy; the AMA provides better forecasts for the demand of wind-

sourced energy. On another hand, if one used DMS, the AMA, BMA, MMA and JMA show very

similar forecast abilities with slightly dierences for the demand produced by nuclear, CCGT and

hydro-power sources. Although the performance of the model averaging methods in forecasting the

demand for energy obtained using coal and hydro-power are not much dierent regardless of the

weighting functions used, the MMA and AMA produce slightly better predictions for the demand of

energy sourced from coal and wind, respectively.

In more depth, in the case of coal-based energy, the GRMA predictors obtain lower fore-

cast error in the beginning of the predicted period and at its end, while JMA often predictions are

superior in the middle of the forecast period. For nuclear-sourced energy, the BMA consistently

outperform to other model averaging methods in the accuracy of its prediction. For CCGT, all

model averaging techniques perform more or less the same, and again, BMA is the one that slightly

230 / The Energy Journal

more accurate than the others. Regarding the demand for wind-sourced energy, the MMA provides

the most accurate predictions in forecasting the longer term (one-day ahead), but AMA generally

outperforms the others. Lastly, considering hydro-power produced energy, using IMS, the MMA

shows the best forecasting ability while GRMA performs poorly. However, using DMS, both MMA

and GRMA produce more accurate forecasts compared with rest of the model averaging methods.

Next, we use the Diebold-Mariano (DM) and the Wilcoxon’s sign-ranked (Sign) tests by

(Diebold and Mariano, 1995) to test the prediction equivalence of AMA, BMA, MMA and JMA

(given the relatively poorer performance of SMA and GRMA). Table 4 displays the pair-wise results

of both tests for IMS and DMS, respectively. Generally, both the DM and the Sign test provide sim-

ilar results, expect for a few cases which concern the predictions obtained by IMS. On Table4, for a

pair of ordered forecasts obtained by weight functions x and y, a positive (negative) and statistically

signicant value of the statistic would imply superiority (inferiority) of the predictions obtained

with weighting function y over those obtained if using weighting function x. The results can be

Figure 8: Comparison among Model Averaging Methods

Notes: The gure compares six types of forecast model averaging methods within six types of forecast models. The y axis is

the value of MSFE, and th x axis is the forward prediction steps.

Combination Forecasting of Energy Demand in the UK / 231

Table 4: Prediction Equivalent Tests on Model Average Predictors

IMS

Coal Nuclear CCGT Wind Hydro

DM_test Sign_test DM_test Sign_test DM_test Sign_test DM_test Sign_test DM_test Sign_test

AMA vs BMA -0.78(0.42) -0.42(0.67) -0.83(0.34) -1.40(0.18) 1.42(0.16) 2.10(0.04) 0.76(0.42) 1.52(0.14) -0.93(0.35) 1.12(0.26)

AMA vs MMA 3.62(0.00) 2.52(0.01) -0.48(0.63) 1.68(0.09) 1.35(0.18) 2.10(0.04) 1.56(0.12) 2.10(0.04) 1.93(0.05) 2.10(0.04)

AMA vs JMA -1.65(0.10) 1.12(0.26) 1.74(0.08) 2.52(0.01) -0.33(0.74) 1.68(0.09) 0.50(0.62) 2.10(0.04) -0.97(0.33) 1.12(0.26)

BMA vs MMA 3.63(0.00) 2.52(0.01) -0.27(0.78) 2.38(0.02) -0.56(0.58) 1.68(0.09) -1.21(0.23) 1.12(0.26) 5.36(0.00) 2.52(0.01)

BMA vs JMA -1.65(0.10) 1.12(0.26) 2.76(0.01) 2.52(0.01) -1.00(0.32) 1.12(0.26) -1.89(0.06) 0.42(0.67) -0.91(0.36) 1.12(0.26)

MMA vs JMA -2.77(0.01) -2.52(0.01) 0.71(0.48) 1.12(0.26) -1.91(0.06) -0.42(0.68) -1.35(0.18) 0.42(0.67) -1.20(0.23) 1.12(0.26)

DMS

Coal Nuclear CCGT Wind Hydro

DM_test Sign_test DM_test Sign_test DM_test Sign_test DM_test Sign_test DM_test Sign_test

AMA vs BMA -0.38(0.70) 2.10(0.04) 0.59(0.55) 0.42(0.67) -0.59(0.55) 0.42(0.67) 1.57(0.12) 2.10(0.04) -0.53(0.59) 1.12(0.26)

AMA vs MMA 6.12(0.00) 2.52(0.01) -2.22(0.03) -0.42(0.67) 2.54(0.01) 2.10(0.04) 2.31(0.02) 2.38(0.02) 0.91(0.36) 2.10(0.04)

AMA vs JMA -2.01(0.04) -0.42(0.67) -2.15(0.03) -1.40(0.18) -0.74(0.46) 1.12(0.26) 2.94(0.00) 2.52(0.01) -2.33(0.02)

-0.42(0.67)

BMA vs MMA 6.13(0.00) 2.52(0.01) -2.20(0.03) 0.42(0.67) 2.57(0.01) 2.10(0.04) 1.99(0.05) 2.38(0.02) 1.84(0.07) 2.38(0.02)

BMA vs JMA -2.01(0.04) -0.42(0.67) -2.07(0.04) -1.40(0.18) -0.57(0.57) 1.68(0.09) 2.83(0.00) 2.38(0.02) -2.35(0.02)

-0.42(0.67)

MMA vs JMA -4.03(0.00) -2.52(0.01) -0.91(0.36) 1.12(0.26) -3.46(0.00) -1.40(0.18) -0.88(0.38) 1.12(0.26) -2.79(0.01) -0.42(0.67)

Notes: This table reports the prediction equivalence results for pair-wise model averaging methods which are using the IMS and DMS. DM_test refers to the Diebold and Mariano test, and

Sign_test refers to the Wilcoxon’s signed-rank test. Both tests use the critical values of standard normal distributions.

232 / The Energy Journal

roughly summarized as: AMA and BMA provide statistically equivalent forecasts for all energy

sectors regardless of whether the forecasts are obtained by means of IMS or DMS methods. When

using IMS, the MMA forecasts are, in general, found statistically superior to the AMA/BMA except

for Nuclear Energy where contrasting results are found. Also, JMA-produced forecasts, although of-

ten equivalent to AMA and BMA, are found in a few cases to be even slightly superior to them. For

DMS forecasts, the MMA again outperforms all others methods while the JMA loses its slight su-

periority compared to the AMA and BMA forecasts and it is actually outperformed when producing

forecasts for coal, nuclear and hydro generated energy. Finally, for both the IMS and DMS methods,

MMA obtained forecasts are found consistently superior to those given by the JMA weight function.

5. FORECASTING THE DEMAND FOR ENERGY IN LEVELS

In this section, we nally obtain forecast of the levels of the energy demand for the ve

energy sources on 22 March 2016. Specically, we re-combine the IMS and DMS predictions ob-

tained through all the model averaging methods, with the deterministic/periodic terms captured

by Equations 4, 8 and 9, therefore obtaining forecasts for the level of the UK demand for energy

produced by the ve sources considered. Figure 9 illustrates the predicted and actual values for

coal, nuclear, CCGT, wind and hydro-power, respectively. Treating the forecasts from the individual

forecast models as bench-marks, the gures compare the predictions obtained from model averaging

to that of the individual forecasting models obtained by means of both IMS and DMS.

Among six bench-mark forecast models, the HWS model is the worst performing one, and

the NARNN is the best and actually shows a performances close to that of model averaging meth-

ods. However, it is important to re-iterate that there always exists a model averaging forecast that

can beat the forecasts obtained from a bench-mark model. Another notable fact is that the prediction

become more accurate as the forecast horizon approaches the 24 hours, and also that DMS outper-

form IMS in longer-term forecasting.

In more detail, the model averaging predictions for coal, CCGT, wind and hydro-power,

remain relatively accurate after adding the periodic and deterministic components. This is not the

case for the demand for energy nuclear-sourced, where in fact, forecasts of the level is less accurate

due to the inaccuracy in tting the deterministics.

Using the IMS method, predictions become more and more accurate reaching the 1 day

horizon. For the coal-sourced energy, to predict the shorter-term (30 minutes—4 hours), the GRMA

method produces the most precise forecasts, while MMA becomes superior in forecasting the longer

horizon (6–24 hours). For nuclear energy, MMA and JMA allow us to obtain better predictions than

others. All model averaging methods give similar forecasting of the CCGT-sourced energy. AMA

predictions outperform the other forecasting methods for the the demand of wind-fuelled energy,

and lastly, in the hydro-power sector, the MMA method again shows best forecasting ability than

other model averaging methods.

6. CONCLUSIONS

In this paper we have produced more accurate short-term forecasts of the demand for en-

ergy in the UK using a forecasting approach based on model averaging of several popular linear or

non-linear, univariate and multivariate forecast models. Specically, we used an algorithm that once

obtained the forecasts from sets of ARMA, Holt-Winters, Non Linear Autoregressive Neural Net-

works, Vector Autoregressions, Bayesian VAR and Factor Augmented VAR models selects the best

Combination Forecasting of Energy Demand in the UK / 233

forecasting model from each model-set according to four dierent information criteria (AIC, BIC,

Mallows’ and Jackknife). The best models as selected by each of the dierent information criteria

within each model set are then averaged using six dierent combination weight metrics includ-

ing Simple Model Averaging, Granger-Ramanathan Model Averaging, Bayesian Model Averaging,

Akaike Model Averaging, Mallows Weights and the Jackknife.

Our results conrm the merits of combination forecasting as a superior forecasting strat-

egy. Among the single forecasting models, NARNN and VAR forecast are superior in terms of lower

MSFE whilst HWS perform worst. Unexpectedly, DMS forecasts outperforms those obtained by

IMS in terms of accuracy. For all energy sources, the MSFE obtained from the forecasting model

selected by AIC and JKC almost never underperform compared to their counterparts suggested by

BIC and MIC. Among the six types of model averaging methods, generally the BMA, MMA are

superior to the others. Lastly but most importantly, there always exists a model averaging method

which gives a lower MSFE than the best/optimal models within each class however selected.

Figure 9: The Model Averaging vs Benchmark Models

Notes: The gure reports six types of model averaging forecasts from six sets of forecast models. The y axis is the value, and

the x axis is the forward prediction steps. We do not plot the results of HW model considering its poor performances.

234 / The Energy Journal

As highlighted above, accurate forecasts are a precious resource for demand response, and/

or load management. With timely and accurate prediction of demand, load management programs

facilitate system load balancing by avoiding peak occurrences. On the other hand, they can also

be crucial for demand response, which has been gaining prominence in recent years as an eec-

tive and inexpensive tool for reducing overall utility peak demand while improving system-wide

energy eciency. Through the curtailment of electricity consumed by end-users during periods of

high demand or electricity grid instability, demand response technology addresses unexpected vari-

ances in electricity supply and demand levels. When wholesale electricity market prices are high or

when overall grid system reliability is compromised, demand response programs oer incentives

to end-users in order to aect time of use, instantaneous demand level, and/or aggregate electricity

consumption.

Given accurate forecasts with low error such as those we obtained using model averaging,

it is theoretically possible for network management to, for example, temporarily curtail a portion

of the network load in some areas of a city whenever approaching a predetermined peak in demand

in others areas. It is thus necessary for the network management to determine an acceptable peak

demand load maximum for the various areas. The real-time energy monitoring system provided

by smart metering together with the model averaging forecast signals an upcoming breach of the

predetermined peak demand load maximum. Thus, curtailment policy shreds unnecessary loads

during these events in order to control overall peak loading and prevent an unwanted peak demand

occurrence. Clearly, accurate model averaging forecasts would be particularly useful for ecient

and cost-eective peak demand energy management across city municipalities and other large en-

ergy end-users. In this case there would be added benet not only to the electric utility provider but

also to the environment through ecient and reduced power generation capacity. Such reduction

and ecient usage of power generation would undoubtedly contribute to the energy sustainability

of local municipalities and their communities.

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Combination Forecasting of Energy Demand in the UK / 237

APPENDIX: PREDICTION EQUIVALENT TESTS

Diebold and Mariano (1995) proposed statistical tests to compare the forecasting errors

from pair-wise models. In the present paper, we introduce two types of tests: Diebold and Mariano

asymptotic test and Wilcoxon’s signed-rank test. These two tests are aiming to distinguish the null

that

0, ,

: [ ( )] = [ ( )]

it jt

H Ege Ege

versus,

1, ,

: [ ( )] [ ( )]≠

it jt

H Ege Ege

where

()

is a forecasting loss function on model i. Also, dene that the loss dierential series

[ ( ) ( )]≡−

t it j t

d ge ge

for model i and j. Thus, hypothesis can also be understood as

[ ]=0

The rst test used is Diebold and Mariano asymptotic test, which is under mild assumption

that

is a covariance stationary and short memory series. Then, we have,

1= (0,1)

2 (0)



(44)

where

= [ ( ) ( )]−

∑

it jt

d ge ge

, and the variance term,

( 1)

= ( 1)

ˆˆ

2 (0) = ( ) ( )

()

π γτ

−

−−

∑

where

()

IST

is the lag window and

()ST

is the truncation lag. Noted that

( )=0

()

IST

for

−h

as the h-step-ahead forecast errors are

1−h

dependent at most.

ˆ( ) = ( )( )

γτ

−

−−

∑

d dd d

As the Diebold and Mariano Test is a two-side test, it not only tests the equivalence, but also pro-

vides superior and inferior comparisons. A case that the statistic

falls outside of the right(left)-

hand condence interval implies the forecasting error

,it

(

,jt

) is greater than

,jt

(

,it

) with a measur-

able function

()⋅g

, thus, the predictor

,it

( ,

) is less accurate than ,

(

,it

The second test introduced is Wilcoxon’s signed-rank test. The test statistics follows a

standard normal distribution under the assumption that loss dierential series

is independent

identically distributed (i.i.d). Since we compare predictors for dierent forecasting horizons, the

is reasonable to be i.i.d.

( 1)

= (0,1)

( 1)(2 1)

−



TT T

(45)

where

= () ( )

∑

a tt

S I d rank d

where

( )=1

and it equals to 0 otherwise. The

()⋅rank

is the Wilcoxon’s rank operator.

Wilcoxon’s signed-rank test can also compare the superiority-inferiority through the sign.

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Jun 2022
ENERG POLICY

Selecting a suitable energy demand forecasting method is challenging due to the complex interplay of long-term trends, short-term seasonalities, and uncertainties. This paper compares four time-series models performance to predict total and peak monthly energy demand in India. Indian's Central Energy Authority's (CEA) existing trend-based model is used as a baseline against (i) Seasonal Auto-Regressive Integrated Moving Average (SARIMA), (ii) Long Short Term Memory Recurrent Neural Network (LSTM RNN) and (iii) Facebook (Fb) Prophet models. Using 108 months of training data to predict 24 months of unseen data, the CEA model performs well in predicting monthly total energy demand with low root-mean square error (RMSE 4.23 GWh) and mean absolute percentage error (MAPE, 3.4%), but significantly under predicts monthly peak energy demand (RMSE 13.31 GW, MAPE 7.2%). In contrast, Fb Prophet performs well for monthly total (RMSE 4.23 GWh, MAPE 3.3%) and peak demand (RMSE 6.51 GW, MAPE 3.01%). SARIMA and LSTM RNN have higher prediction errors than CEA and Fb Prophet. Thus, Fb Prophet is selected to develop future energy forecasts from 2019 to 2024, suggesting that India's annual total and peak energy demand will likely increase at an annual growth rate of 3.9% and 4.5%, respectively.

Inside the Crystal Ball: New Approaches to Predicting the Gasoline Price at the Pump

Article

Full-text available

Jan 2015

Although there is much interest in the future retail price of gasoline among consumers, industry analysts, and policymakers, it is widely believed that changes in the price of gasoline are essentially unforecastable given publicly available information. We explore a range of new forecasting approaches for the retail price of gasoline and compare their accuracy with the no-change forecast. Our key finding is that substantial reductions in the mean-squared prediction error (MSPE) of gasoline price forecasts are feasible in real time at horizons up to two years, as are substantial increases in directional accuracy. The most accurate individual model is a VAR(1) model for real retail gasoline and Brent crude oil prices. Even greater reductions in MSPEs are possible by constructing a pooled forecast that assigns equal weight to five of the most successful forecasting models. Pooled forecasts have lower MSPE than the EIA gasoline price forecasts and the gasoline price expectations in the Michigan Survey of Consumers. We also show that as much as 39% of the decline in gas prices between June and December 2014 was predictable.

Testing for long memory in the presence of non-linear deterministic trends with Chebyshev polynomials

Article

Full-text available

Jun 2015
STUD NONLINEAR DYN E

This paper examines the interaction between non-linear deterministic trends and long run dependence by means of employing Chebyshev time polynomials and assuming that the detrended series displays long memory with the pole or singularity in the spectrum occurring at one or more possibly non-zero frequencies. The combination of the non-linear structure with the long memory framework produces a model which is linear in parameters and therefore it permits the estimation of the deterministic terms by standard OLS-GLS methods. Moreover, the orthogonality property of Chebyshev’s polynomials makes them especially attractive to approximate non-linear structures of data. We present a procedure which allows us to test (possibly fractional) orders of integration at various frequencies in the presence of the Chebyshev trends with no effect on the standard limit distribution of the method. Several Monte Carlo experiments are conducted and the results indicate that the method performs well. An empirical application, using data of real exchange rates is also carried out at the end of the article.

Priors from General Equilibrium Models for VARS*

Article

May 2004

This article uses a simple New Keynesian dynamic stochastic general equilibrium model as a prior for a vector autoregression, and shows that the resulting model is competitive with standard benchmarks in terms of forecasting, and can be used for policy analysis.

Intelligent Energy Demand Forecasting

Book

Jan 2013

Wei-Chiang Hong

As industrial, commercial, and residential demands increase and with the rise of privatization and deregulation of the electric energy industry around the world, it is necessary to improve the performance of electric operational management. Intelligent Energy Demand Forecasting offers approaches and methods to calculate optimal electric energy allocation to reach equilibrium of the supply and demand. Evolutionary algorithms and intelligent analytical tools to improve energy demand forecasting accuracy are explored and explained in relation to existing methods. To provide clearer picture of how these hybridized evolutionary algorithms and intelligent analytical tools are processed, Intelligent Energy Demand Forecasting emphasizes on improving the drawbacks of existing algorithms. Written for researchers, postgraduates, and lecturers, Intelligent Energy Demand Forecasting helps to develop the skills and methods to provide more accurate energy demand forecasting by employing novel hybridized evolutionary algorithms and intelligent analytical tools.

Next day load forecasting using artificial neural network models with autoregression and weighted frequency bin blocks

Article

Apr 2009
INT J INNOV COMPUT I

In this study, two different hybrid approaches based on Artificial Neural Network (ANN) models with Autoregressive (AR) method and Weighted Frequency Bin Blocks (WFBB), are used for next day load forecasting. To compare with the hybrid approaches and conventional models, the next day load forecasting is also performed by using AR and ANN models, separately. In the first hybrid approach, ANN model with AR method, the results of the AR method applied to all data taken from Turkish Electric Power Company and Electricity Generation Company, is used as an only additional input for ANN model. In this approach, the ANN structure has two layers composed of 49 and 24 neurons for input and output layers, respectively. In the second hybrid approach, ANN model with WFBB, the results obtained from WFBB are used for all inputs in the ANN model. In this approach, input and output layers in the ANN structure are composed of 48 and 24 neurons, respectively. Feed Forward Back Propagation (FFBP) is chosen for all neural network models in this study. The forecasting results obtained from AR, ANN and the two hybrid models are compared to each other in the sense of root mean square error (RMSE). It is observed that the RMSE values for the hybrid approaches are smaller than the conventional models. Then, the hybrid models forecast better than the conventional models.

Avoiding Pitfalls When Using Information-Theoretic Methods

Article

Jul 2002

We offer suggestions to avoid misuse of information-theoretic methods in wildlife laboratory and field studies. Our suggestions relate to basic science issues and the need to ask deeper questions (4 problems are noted), errors in the way that analytical methods are used (7 problems), and outright mistakes seen commonly in the published literature (5 problems). We assume that readers are familiar with the information-theoretic approaches and provide several examples of misuse. Any method can be misused-our purpose here is to suggest constructive ways to avoid misuse.

Nonparametric Sieve Regression: Least Squares, Averaging Least Squares, and Cross-Validation

Article

Bruce E. Hansen

This chapter concerns selection and combination of nonparametric sieve regression estimators. We review the concepts of series and sieve approximations, introduce least-squares estimates of sieve approximations, and measure the accuracy of the estimators by integrated mean-squared error (IMSE). We show that the critical issue in applications is selection of the order of the sieve, as the IMSE greatly varies across the choice. We introduce the cross-validation criterion as an estimator of mean-squared forecast error (MSFE) and IMSE. We extend the current optimality theory, by showing that cross-validation selection is asymptotically IMSE equivalent to the infeasible best sieve approximation. We also introduce weighted averages of sieve regression estimators. Averaging estimators have lower IMSE than selection estimators. Following Hansen and Racine (2012) we introduce a cross-validation (or jackknife) criterion for the weight vector, and recommend selection of the weights by minimizing this criterion. The resulting jackknife model averaging (JMA) estimator is a fea-sible averaging sieve estimator. We show that the JMA estimator is optimal in the sense that it is asymptotically IMSE equivalent to the infeasible optimal weighted average sieve estimator. While computation of the JMA weights is a simple application of quadratic programming, we also introduce a simple algorithm which closely approximates the JMA solution without the need for quadratic programming. Research supported by the National Science Foundation. I thank Chu-An Liu for excellent research assistance, Guido Kuersteiner and Ryo Okui for pointing out an error in Hansen and Racine (2012), and the referee and editor for helpful comments and suggestions.

Model Selection and Accounting for Model Uncertainty in Graphics Models Using Occam's Window

Article

Dec 1994

We consider the problem of model selection and accounting for model uncertainty in high-dimensional contingency tables, motivated by expert system applications. The approach most used currently is a stepwise strategy guided by tests based on approximate asymptotic P-values leading to the selection of a single model; inference is then conditional on the selected model. The sampling properties of such a strategy are complex, and the failure to take account of model uncertainty leads to underestimation of uncertainty about quantities of interest. In principle, a panacea is provided by the standard Bayesian formalism which averages the posterior distributions of the quantity of interest under each of the models, weighted by their posterior model probabilities. Furthermore, this approach is optimal in the sense of maximising predictive ability. However, this has not been used in practice because computing the posterior model probabilities is hard and the number of models is very large (often greater than 1011). We argue that the standard Bayesian formalism is unsatisfactory and we propose an alternative Bayesian approach that, we contend, takes full account of the true model uncertainty byaveraging overamuch smaller set of models. An efficient search algorithm is developed for nding these models. We consider two classes of graphical models that arise in expert systems: the recursive causal models and the decomposable

A novel modeling approach for hourly forecasting of long-term electricity demand

Article

Jan 2011
ENERG CONVERS MANAGE

In this study, a novel mathematical method is proposed for modeling and forecasting electric energy demand. The method is capable of making long-term forecasts. However, unlike other long-term forecasting models, the proposed method produces hourly results with improved accuracy. The model is constructed and verified using 26-year-long real-life load data (4 years with hourly resolution) obtained from the Turkish Electric Power Company. The overall method consists of a nested combination of three sub-sections for modeling. The first section is the coarse level for modeling variations of yearly average loads. The second section refines this structure by modeling weekly residual load variations within a year. The final section reaches to the hourly resolution by modeling variations within a week, using a novel 2-D mathematical representation at this resolution. The adoptions of such nested forecasting methodology together with the proposed 2-D representation for hourly load constitute the novelties of this work. The major advantage of the proposed approach is that it enables the possibility of making short-, medium-, and long-term hourly load forecasting within a single framework. Several mathematical functions are applied as models at each level of the nested system for achieving the minimal forecasting error. Proposed model functions with their corresponding forecasting accuracies are presented in terms of root mean square error (RMSE) and mean absolute percentage error (MAPE).

Short-term hourly load forecasting using time-series modeling with peak load estimation capability

Article

Aug 2001

Nima Amjady

This paper presents a new time series modeling for short term load forecasting, which can model the valuable experiences of the expert operators. This approach can accurately forecast the hourly loads of weekdays, as well as, of weekends and public holidays. It is shown that the proposed method can provide more accurate results than the conventional techniques, such as artificial neural networks or Box-Jenkins models. In addition to hourly loads, daily peak load is an important problem for dispatching centers of a power network. Most of the common load forecasting approaches do not consider this problem. It is shown that the proposed method can exactly forecast the daily peak load of a power system. Obtained results from extensive testing on the Iran's power system network confirm the validity of the developed approach

Combination Forecasting of Energy Demand in the UK

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