# Enhancing Energy System Models Using Better Load Forecasts

Thomas Möbius<EMAIL_ADDRESS>Mira Watermeyer<EMAIL_ADDRESS>Oliver Grothe  Felix Müsgens  BTU Cottbus-Senftenberg, Siemens-Halske-Ring 13,
Cottbus  Karlsruhe Institute of Technology, Kaiserstraße 12, Karlsruhe

###### Abstract

Energy system models require a large amount of technical and economic data,
the quality of which significantly influences the reliability of the results.
Some of the variables on the important data source ENTSO-E transparency
platform, such as transmission system operators’ day-ahead load forecasts, are
known to be biased. These biases and high errors affect the quality of energy
system models. We propose a simple time series model that does not require any
input variables other than the load forecast history to significantly improve
the transmission system operators’ load forecast data on the ENTSO-E
transparency platform in real-time, i.e., we successively improve each
incoming data point. We further present an energy system model developed
specifically for the short-term day-ahead market. We show that the improved
load data as inputs reduce pricing errors of the model, with strong reductions
particularly in times when prices are high and the market is tight.

###### keywords:

Data preprocessing, Day-ahead electricity prices, Energy system modelling

##  1 Introduction

Energy markets are complex and exhibit non-trivial interdependencies, so
decisions from policy and industry stakeholders rely on theoretical models and
other methodological support. Techno-economic energy system models are an
essential tool, as they explain actual developments and provide valuable
insights for future developments based on market fundamentals on the supply
and demand side. They are capable of reflecting structural breaks better than
most other model types. However, they rely on the quality of input data to
provide accurate results. Literature has shown that important input data sets
for energy systems models, in particular load data and wind or solar forecasts
from official sources, often have significant systematic errors (Maciejowska
et al. [2021], Hirth et al. [2018]). These errors can be reduced in a
preprocessing step, where they are considered as an econometric time series.
This preprocessing step can exploit serial structures and predict future
errors, which significantly reduces the errors. We analyse whether using these
improved input data in an energy systems model will improve results.

The contribution of this paper is threefold. First, we develop and provide a
simple time-series model reducing forecast errors of hourly day-ahead load
predictions of transmission system operators (TSOs) in real-time. We focus on
load forecasts because they are the most correlated with the prices of the
day-ahead electricity market and have the most potential for improvement
compared with wind and PV forecasts (see, e.g., Maciejowska et al. [2021]).
One advantage of our approach is that we take publicly available TSO-based
load forecasts as given and thus, in modelling directly their prediction error
as a predictable subject, do not need to develop a complex load forecast
model. On country level, load forecasts are often used to represent the demand
on the day-ahead market clearing.111In all main European markets, wholesale
electricity prices are determined in the day-ahead market clearing one day
before the actual delivery. Thus, load forecasts are central variables for
determining equilibria of demand and supply in energy system models.

Second, we present a fundamental energy system dispatch model called the
_em.power dispatch_ model, developed and calibrated precisely for short-term
use in the day-ahead market. A primary objective of this model is to predict
wholesale electricity prices. Using a rolling window, it consecutively
determines the optimal power plant operation for three consecutive days.
Moreover, the model considers hourly net transfer capacities to limit
electricity transmission across countries and a formulation for medium- and
long-term energy storage. We describe these steps in detail in Section4.

Third, we demonstrate the value of sequentially and continuously improving the
quality of input variables in fundamental energy system models in the
empirical part of the paper. We consider TSO day-ahead load forecasts provided
by one of the most used data sources ENTSO-E Transparency Platform [2021f] and
day-ahead prices forecasted with the energy system model for Germany, one of
the largest and most liquid electricity markets in the world. By capturing and
reflecting systematic biases and autoregressive structures, we reduce the mean
squared error by 26 % compared to the TSO-based load forecast. Therefore,
market participants’ expectations of the day-ahead market clearing can be
better reflected. As a result, the mean squared error of the _em.power
dispatch_ model’s price forecast is reduced by nearly 15 % in hours with high
prices using the improved load forecast compared to using the TSO load
forecast. By demonstrating that energy system models with the improved load
data perform significantly better compared to the TSO data, we provide
valuable insights for many stakeholders in the power sector, particularly
energy system model developers seeking to improve the validity of their
models. Based on these results, we encourage energy system modellers and all
users of fundamental input data to be aware of the predictable structure of
their errors. In particular, stochastic modelling of the errors significantly
reduces the forecast error of input data. It thus improves the quality of
input data as part of sequential data preprocessing in real-time and offers
the possibility to enhance the output of fundamental energy system models.

The remainder of the paper is organised as follows. First, we examine the
literature on energy systems modelling, data quality and time series modelling
in Section2. Section 3 presents the data used in this application. In
Section4, we provide and explain the methodology for the model improving the
load forecasts and the energy system model used to evaluate the impact of the
improved load data. The results are presented in Section5. Finally, a
conclusion is drawn in Section6.

##  2 Literature

Energy system models are widely used in academia, policy-making and industry.
Typically, they determine market equilibria, minimising production costs or
maximising social welfare. A market’s supply and demand sides are equally
essential to derive equilibria. Various models have been developed on the
demand side using time series of load data as an essential input. On the
supply side, models focus on power plants (electricity system models) or gas
production (gas systems). Transmission and distribution infrastructure, i.e.,
connecting supply and demand, can also be included and analysed with energy
system models. A strength of these models is that they can provide valuable
insights into both causes and effects of current and planned developments, as
well as into "what-if" types of analyses. Thus, energy system models are among
the most essential methodologies for a successful energy transition.

Out of a wide range of applications, examples include the determination and
assessment of long-term investment decisions for generation and storage
capacities (e.g., Nahmmacher et al. [2016], Schill and Zerrahn [2018]) or
implications on short-term operational decisions (e.g., Schill et al. [2017]),
transmission expansion planning (e.g., Egerer et al. [2021], Sauma et al.
[2006]), the evaluation of carbon reduction paths (e.g., Vaillancourt et al.
[2017]) and support schemes for renewable energy systems (e.g., Kitzing et al.
[2017]) and the evaluation of interdependencies between energy sectors (e.g.,
Lienert and Lochner [2012] for electricity and gas markets, Heinisch et al.
[2021] transport, electricity and district heating, Koirala et al. [2021]
electricity, hydrogen and methane). Moreover, scholars developed stochastic
models to assess the impact of uncertainty on a power system Riepin et al.
[2021], for example, to quantify the expected costs of ignoring uncertainty of
critical parameters in the electricity and gas sector. Möst and Keles [2010]
provide an overview and classification of stochastic models dealing with
uncertainty in the power sector. With regard to uncertainty, scholars analyse
the effect of risk preferences as well (e.g., Möbius et al. [2021], Ambrosius
et al. [2022]).

In this paper, we focus on the impact of better load forecasts on the
estimation of wholesale electricity prices with fundamental market models.
Estimating wholesale electricity prices is essential for making optimal
economic decisions (e.g., investment and dispatch of various technologies) and
policy decisions (e.g., calculating the implications of a coal phase-out).
Wholesale electricity prices can be forecasted with multiple methodologies,
all with their unique advantages and disadvantages. Energy system models have
advantages, e.g., they perform exceptionally well at structural breaks, are
based on a broad economic theoretical foundation explaining causality, and
provide additional information beyond the forecast. Consequently, much
attention has been paid in the literature to the simulation or prediction of
electricity prices in energy system models. Hirth [2013] simulate electricity
prices to quantify the drop in the market value of variable renewables.
Additionally, Eising et al. [2020] quantify market values for renewables
generating electricity prices in a future power system with the help of an
energy system model. Engelhorn and Möbius [2022] derive market prices assuming
different weather years and quantify weather-specific market values for a
comprehensive database of onshore wind capacities in Germany. Qussous et al.
[2022] use an agent-based model with rule-based bidding strategies to
reproduce spot prices for the German bidding zone.

Market power and strategic behaviour are other applications of wholesale price
forecasts with energy system models. When modelling competitive market prices
and comparing them with actual prices, they were able to point to serious
problems (e.g., Müsgens [2006] and Weigt and von Hirschhausen [2008] for
Germany, and Borenstein et al. [2002] for the United States).

These and many other model applications have a dedicated empirical focus.
Thus, the high quality of input data is vital. Concerning load data, a
comprehensive literature review of various methods and models for energy
demand forecasting is given by Suganthi and Samuel [2012], Singh et al.
[2012]. Among others, approaches for standalone load forecasting models are
presented by Weron and Misiorek [2005], Weron [c2006], Naz et al. [2019], Lin
et al. [2018], Rodrigues and Trindade [2018], Wu et al. [2019], Tan et al.
[2010], Chen et al. [1995], Al-Hamadi and Soliman [2004] and Yang et al.
[2013]. For the European electricity sector, data is conveniently gathered and
made publicly available by transmission system operators (TSOs) via the
transparency platform of the ENTSO-E. The platform is a very ambitious and
unique project to provide an extensive data set for electricity markets and is
thus both well-known and widely used. However, it is not without its
shortcomings (see Hirth et al. [2018]). For example, Maciejowska et al. [2021]
analyse of the quality of load data for the Germany-Luxembourg bidding zone.
They detect a bias in TSO load forecasts and develop an alternative load
prediction model that incorporates information from these forecasts to remove
the bias and thus achieve an enhanced load prediction. Cancelo et al. [2008]
analyse the forecast errors of the Spanish day-ahead TSO load forecasts in
detail for serial structures and influences of special days such as Christmas
holidays or New Year’s Eve.

Given a series of load forecasts with forecasting errors that still show a
predictable structure, the method proposed in this paper offers a possibility
to enhance such forecasts. We improve the forecasts by modelling and removing
predictable parts of the errors. Implicitly, Yang et al. [2013] use a similar
step since they remove a structure from their forecasting model (first stage)
in a second stage by a time series approach. However, they rely on neural
networks, while we propose a very simple time series model.

In energy system modelling, such steps can be described as data preprocessing
or, more precisely, continuous data processing and enhancement with subsequent
use. Such continuous data processing is typically not performed for energy
system models. We believe this is a methodological gap in the literature and
aim to bridge it by providing an approach to sequentially improving input data
and sequentially using these continuously improved datasets in an energy
system model. We demonstrate the effectiveness in an empirical application,
focusing on the effect of better load forecasts for electricity price
forecasts derived from energy system models.

##  3 Data

Energy system models require extensive input data to model market equilibria
on both the demand and the supply sides. Since this paper focuses on a day-
ahead time horizon, TSO-based load forecasts published by ENTSO-E may be used
as predictors for the demand side. However, as was pointed out in the
literature section, the quality of these load forecasts is debated and will be
improved in this paper. In Section3.1, we first provide a detailed overview of
the TSO-based load forecast data and forecast errors. Moving to the supply
side of the energy system model, data on techno-economic parameters for
conventional generation, renewables, storage and electricity transmission are
of the utmost importance and are presented in Section3.2.

###  3.1 TSO-based Load Forecast Data

The load data set we use for our analysis contains hourly day-ahead load
forecast data and hourly actual load data from January 1st, 2016, until
December 31st, 2019, for Germany and Luxembourg. It was downloaded from
ENTSO-E Transparency Platform [2021f] in MWh. Missing values were replaced by
the average of the value of the previous week and the week after222There are
1,105 missing values in the hourly TSO load forecast and 38 missing values in
the hourly actual load data.. An illustration of the time series of the actual
load, TSO load forecast and the resulting error, computed as the difference
between actual load and load forecast, is shown in Figure 1.

Figure 1: Actual load and TSO’s day-ahead load forecast in 2017 (left) and
error of TSO’s day-ahead load forecast in 2017 (right).

For the considered years, Table 1 contains descriptive statistics of the TSO
load forecast errors defined as $\epsilon_{t}\coloneqq L_{t}-\hat{L}_{t}$,
meaning actual load minus TSO load forecast. Thus, a positive error states an
underprediction of load.

| all | 2016 | 2017 | 2018 | 2019
---|---|---|---|---|---
Mean | 881.29 | 1,555.38 | 446.50 | 298.60 | 1,222.84
Median | 892.75 | 1,468.00 | 479.50 | 395.63 | 1,195.75
Minimum | -20,358.00 | -7,752.50 | -7,868.50 | -20,358.00 | -7,415.00
Maximum | 12,930.75 | 12,930.75 | 8,392.50 | 9,045.25 | 9,635.25
5%-quantile | -2,477.50 | -1,684.03 | -2,413.75 | -3,180.63 | -2,266.13
95%-quantile | 4,294.40 | 4,894.20 | 3,099.38 | 3,644.75 | 4,811.38
Std. | 2,149.75 | 2,079.19 | 1,746.54 | 2,341.68 | 2,128.55
LB hypothesis | 1.00 | 1.00 | 1.00 | 1.00 | 1.00

Table 1: Descriptive statistics of TSO load forecast errors for the years 2016
to 2019. Except for LB hypothesis, all variables are given in [$MWh$].

The TSO forecast data is mean-biased, as discussed in Maciejowska et al.
[2021]. In our analysis, we find systematic underpredictions with a mean error
of 881.3 MWh across all years and positive mean errors for every year.

However, the absolute level of the error and whether the TSO under- or
overpredicts in its forecasts depend on the day of the week and the hour of
the day. Figure 2 states the averaged hourly forecast errors in a week.
Broadly, we can observe underprediction during weekdays and overprediction on
the weekends, especially on Saturdays. During the day, in the morning and the
evening hours, the error of the TSO day-ahead load forecast is generally
positive and higher than in the other hours of the day. With an average error
of 943.53 MWh at 6 a.m. and 1,180.48 MWh at 7 p.m., the prediction error in
these hours is higher by 7 % (34 %) than the mean error of the entire time
period considered (compare with Table 1). These are the hours when the workday
begins or ends and where production ramps up or down. Although the standard
deviation of the forecast in these hours is not significantly larger than in
the other hours, it appears that the load in these hours is still more
challenging to forecast on average than in the other hours of the day (see
weekday-wise descriptive measurements in Table 5 for more details).

Figure 2: Average weekly pattern of TSO day-ahead load forecast errors from
2016 to 2019.

Finally, we perform Ljung-Box (LB) tests to verify the auto-correlation of the
TSO load prediction errors. The null hypothesis at a 5 % significance level is
rejected for all years, which indicates a strong auto-correlation of the
errors. Comparing the errors with those one hour before (see Figure 3), we can
see a highly linear dependence.

Figure 3: Scatterplot of TSO’s day-ahead load forecast error and TSO’s day-
ahead load forecast error one hour before.

In summary, the load data shows high auto-correlated TSO forecast errors,
which average 1.56 % of the total load’s mean. The mean absolute error of the
TSO load forecast is 1,776 MWh (3.14 % of the total load’s mean). The TSO
forecast errors are biased with some seasonal structures in the bias and are
highly auto-correlated. Hence, autoregressive type models could improve the
TSO load forecast.

###  3.2 Input data for an energy systems model

Aiming to analyse the impact of improved day-ahead load forecasts on the
accuracy of electricity price forecasts, which are derived using an
electricity system model, we develop and parameterise a European electricity
market model with data from January 1st, 2017, until December 31st, 2019. A
meaningful empirical parameterisation of such models requires extensive input
data derived from various sources. To model the demand side, the load data
presented in the previous Section 3.1 is essential. Furthermore, there is
typically an option to shed load during supply scarcities. In our application,
we assume the costs for load shedding to be 3,000 €/MWh.

On the supply side, a number of technologies are available for electricity
generation and storage. Our energy system model distinguishes ten conventional
thermal generation technologies, which form 30 capacity clusters according to
a power plant’s commissioning year. We provide each of the capacity clusters
with different efficiencies, minimum outputs and efficiency losses in part-
load operations, which are derived from Schröder et al. [2013] and Open Power
System Data [2020a]. The capacity, fuel type, generation technology and
commissioning date are derived from ENTSO-E Transparency Platform [2021e] and
Open Power System Data [2020a] and EBC [2021]. For power plants on the German
market, we additionally use data from BNetzA [2021] and UBA [2020]. Fuel
costs, costs for CO2 emissions and the power plant efficiency determine the
variable generation costs of conventional thermal technologies. For fuel
costs, we use daily gas prices that are provided by EEX [2021], monthly coal
prices are taken from Destatis Statistisches Bundesamt [2021], and monthly oil
prices from Destatis Statistisches Bundesamt [2021]. Fuel costs for nuclear,
lignite and waste are derived from ENTSO-E [2018]. These are assumed to be
constant over time. Prices for CO2 certificates are implemented as weekly
values from Sandbag [2020].

The process of starting up power plants requires the use of fuel, emits CO2
and leads to material wear in the plant. Data for start-up times, secondary
fuel usage and depreciation are derived from Schröder et al. [2013].

The ability to generate electricity depends not only on the installed capacity
but also on the technical availability of the plants. Therefore, we consider
all scheduled and non-scheduled power plant outages known before the day-ahead
market’s closure. Hourly outages are derived from ENTSO-E Transparency
Platform [2021g].

Since combined heat and power (CHP) plants are used in most electricity
markets, electricity and heat supplies are linked. To account for this
dependency, we provide these units with a must-run condition that ensures
their operation at certain minimum output levels. These output levels are
derived in two steps. First, we determine an hourly heat-demand factor
consisting of a temperature-dependent (spatial heating) and temperature-
independent (warm water and process heat) part. The temperature-dependent heat
demand is generated with heating degree days using mean temperature data from
Open Power System Data [2020b]. We derive the temperature-independent heat
demand using the hourly and daily consumption patterns from Hellwig [2013].
Second, we use the heat-demand factor to allocate annual electricity
generation volumes by CHP plants to single hours. The annual technology-
specific electricity generation by CHP units is taken from European Commission
[2021].

In addition to conventional thermal technologies, we consider renewable energy
sources (RES), energy storage, hydro-reservoirs and run-of-river. Intermittent
RES such as onshore wind, offshore wind and photovoltaics (PV) are implemented
by hourly availability factors that are derived from feed-in forecasts from
ENTSO-E Transparency Platform [2021d]. We do not also improve these forecasts
by sequentially modelling their forecast errors in order to clearly measure
the impact on the quality of the price forecast when we improve the forecast
of the variable that not only offers the greatest potential for improvement
but is also most strongly correlated with day-ahead electricity spot market
prices. Biomass is implemented as base-load as the historic operation is at a
constant level (compare ENTSO-E Transparency Platform [2021a]).

We exclusively consider pumped storage plants (PSP) for energy storage that
actively charge and discharge. The overall turbine capacity of PSPs is made
available by ENTSO-E Transparency Platform [2021e], and the efficiency of a
storage cycle is around 75 % (Schröder et al. [2013]). For PSPs, the energy
storage capacity and the turbine capacity are linked. Assuming an energy-power
factor (epf) of nine, the plant can generate electricity at full load for nine
hours until the storage is empty.

Long-term PSP, as well as hydro-reservoirs, are assigned a variable generation
cost, i.e., the value for water consumption. Using historical electricity
prices from ENTSO-E Transparency Platform [2021b] and the observed generation
and pumping activities in the respective hour from ENTSO-E Transparency
Platform [2021a], a step-wise merit-order for long-term PSP and hydro-
reservoirs is constructed. Run-of-river and mid-term PSP333Note that we call
it mid-term because we focus on the day-ahead market with an hourly
granularity, as opposed to short-term storage with an intra-hourly resolution
closer to time of delivery. are subject to seasonal variations, which we
acknowledge by a monthly availability factor derived from historical
generation data from ENTSO-E Transparency Platform [2021a].

The German electricity market is highly integrated into the European system.
Total interconnector capacity amounts to 27 GW, which is more than 30 % of the
German peak load.444Note that the availability of the interconnectors depends
on various factors (e.g., congestion within a market zone). Both annual
aggregated exports (around 13 % of annual German consumption in 2019) and
imports (around 7 % in 2019) are significant. Hence, we parameterise a Pan-
European electricity market model which includes the bidding zones of most
EU-27 member states 555Bulgaria, Cyprus, Greece, Iceland, Ireland, Malta and
Romania are not included., Norway, Switzerland and the United Kingdom.666Note
that we aggregate the bidding zones of Spain and Portugal to one market,
’Iberian peninsula’, and the bidding zones of Lithuania, Estonia and Latvia to
one market, ’Baltic’. Also, note that we consider the different bidding zones
within countries. However, we aggregate the following zones: in Norway
NO1-NO5, in Sweden SE1-SE3, and in Italy all zones except IT-North. Within
Germany, day-ahead electricity prices are derived following the principle of a
bid-based economic dispatch, neglecting the physical transmission constraints
within the market zone. Since the energy system model has its focuses on the
analyse of day-ahead prices, we follow this approach and treat all of Germany,
plus Luxembourg, as one bidding zone.777Note that the market area, Germany-
Luxembourg-Austria, was split into two market zones (Germany-Luxembourg and
Austria) in 2018. Our model accounts for this fact. Thus, in total, we include
23 different markets in the analysis, which will be referred to as ’nodes’ in
the formal model, connected by net transfer capacities (NTCs). We implement
hourly day-ahead forecasts for NTCs that are made available by ENTSO-E
Transparency Platform [2021c] and JAO Joint Allocation Office [2021].

As the data parameterisation may be interesting for numerous stakeholders but
is difficult and time-consuming to replicate, we publish our input data in the
supplementary material:
github.com/ProKoMoProject/Enhancing-Energy-System-Models-Using-Better-Load-
Forecasts.

##  4 Methodology

In the following, we present our two components to analyse the value of
improved day-ahead load forecasts for electricity price forecasts derived by
an electricity system model: a time series model for the sequential load data
preprocessing and improvement in Section 4.1 and the dispatch market model
that is used to generate price estimators in Section4.2.

###  4.1 Model for load forecast error

To improve load forecasts, we use a well-known time series approach that
achieves a trade-off between performance and complexity. The approach is based
on the idea of forecasting the TSO load forecast error and using this to
enhance the load prediction. Thus, we model the time series of forecast
errors. For this reason, and to obtain a low-parameter model, we do not use
exogenous variables such as feed-in of renewable energy or weather in our
model for forecasting the load forecast error, in contrast to the main load
forecasting methods in the literature, which include temperature and weather
data in particular, e.g., Cancelo et al. [2008], Al-Hamadi and Soliman [2004],
Amjady [2001], Wu et al. [2019], Li et al. [2021]. We propose a purely
endogenous time series approach that can be applied using TSO load forecast
error alone as input data. It is detached from the outgoing model, which in
general already includes exogenous variables. With forecasting the forecast
error, the resulting load prediction $\hat{L_{t}}^{*}$ at time $t$ is then
given by

$\displaystyle\hat{L_{t}}^{*}=\hat{L_{t}}+\hat{\epsilon_{t}},$ (1)

where $\hat{L_{t}}$ is the original TSO load prediction and
$\hat{\epsilon_{t}}$ is our forecasted TSO load prediction error. Thus,
$\hat{L}^{*}$ is an improved load forecast in which we adjust the original
forecast for predictable structure in its error.

For the overall setup, the subindex $t$ will denote consecutive hours. So,
$\hat{L}_{1}$, for instance, is the load forecast for the first hour of the
considered time period and $\hat{L}_{123}$ is the forecast for the hour $123$.
This fits best into the observation process of the actual load data. For
example, in contrast to electricity prices, for which we observe a realisation
of 24 daily hourly prices at the same time, load data can theoretically be
observed hour by hour. For day-ahead electricity prices, alternative
parameterisations such as modelling every day as a 24-dimensional vector, or
using 24 time series each for one hour of the day, would be more appropriate
(see, e.g., Ziel and Weron [2018]).

Furthermore, we decompose the time series into the sum of a seasonal component
and a remaining stochastic component. As we do not observe any trend in the
forecast error data in Section3.1, we do not use the usual trend component of
such decomposition models (see, e.g., Lütkepohl [2005], Hyndman and
Athanasopoulos [2021], Box et al. [2015] for comprehensive introductions into
time series models). Together, the model is

$\displaystyle\epsilon_{t}=SC_{t}+RC_{t},$ (2)

where $\epsilon_{t}$ is the TSO load forecast error, $SC_{t}$ is a seasonal
and $RC_{t}$ is the remaining component at time $t$.

The forecast errors’ average sizes depend on the specific hour of the week
(see Section3.1), so the seasonal component $SC_{t}$ captures a weekly season,
consisting of an average value for each of the 24x7 hours of a week. This
means addressing the hour of the day and the day of the week with a total of
168 dummy variables, as given by

$HoW_{t}^{h,d}=\begin{cases}\text{1,}&\text{if $t$ is the h-th hour of the
d-th day of the week,}\\\ \text{0,}&\text{otherwise.}\end{cases}$

Here $h=1,...,24$ denote the hours of a day and $d=1$ (Monday), …, $7$
(Sunday) the weekdays of a week.

The seasonal component $SC_{t}$ for time $t$ is now defined by Eq. 3 with 4
being the average of TSO forecast errors from the hours of a week from the
time period used to estimate the model (e.g., the last $l_{w}$ hours).

$\displaystyle SC_{t}=\sum_{h=1}^{24}\sum_{d=1}^{7}HoW_{t}^{h,d}\cdot
HS^{h,d},$ (3) $\displaystyle
HS^{h,d}\coloneqq\frac{\sum_{s=t-1}^{t-l_{w}-1}\epsilon_{s}\cdot
HoW_{s}^{h,d}}{\sum_{s=t-1}^{t-l_{w}-1}HoW_{s}^{h,d}},$ (4)

The rest of the time series $RC_{t}=\epsilon_{t}-SC_{t}$ is modelled by the
econometric SARMA $(1,1)$x$(1,1)_{24}$ model given in Eq. 5, i.e., a
(S)easonal (A)uto(R)egressive (M)oving (A)verage model. Here, the value
$RC_{t}$ at hour $t$ depends on its previous value at $t-1$ as well as the
previous model error $\psi_{t-1}.$ Additionally, the model contains a 24-hour
seasonal part which captures stochastic seasonal behaviour in contrast to the
more deterministic seasonal structure filtered by $SC_{t}.$ Formally, the
seasonal part leads to direct effects of all variables lagged by another 24
hours on $RC_{t}$ as given in detail in Eq. (5).

$\displaystyle RC_{t}=$ $\displaystyle\phi_{0}+\phi_{1}\cdot
RC_{t-1}+\phi_{24}\cdot RC_{t-24}-\phi_{1}\phi_{24}\cdot RC_{t-25}$ (5)
$\displaystyle+\omega_{1}\cdot\psi_{t-1}+\omega_{24}\cdot\psi_{t-24}+\omega_{1}\omega_{24}\cdot\psi_{t-25}$
$\displaystyle+\psi_{t},$

where the innovations are assumed to be homoscedastic and normally
distributed, which means $\psi_{t}\sim N(0,\sigma^{2}_{\epsilon})$. Assuming a
normal distribution for the innovations is a simplification and idealisation.

We calibrate and estimate the model on a rolling window. The window length,
denoted by $l_{w}$, is an integer multiple of 24 and thus contains full days
only. The window is also rolled over full days in each step to further reflect
the daily availability of load data and thus the error of the TSO’s load
forecast. In this work, we decide on one window length $l_{w}$ to estimate the
model. Alternatively, one could average multiple models calibrated on
different window lengths, e.g., as proposed in Maciejowska et al. [2021], Ziel
and Weron [2018], Marcjasz et al. [2018]. However, in this paper, where the
simplicity and usability of the model are important considerations, we believe
such an increase in complexity would not be justified.

The estimated model is used to recursively (i.e., on an hour-by-hour basis)
predict the hours of the next day. Since we rely on an autoregressive time
series model, we need load data from the last hours for prediction, which
enter the model as explanatory variables. Although load generation can
theoretically be observed hourly, in practice, the load values of the previous
hours are available with a time lag, meaning they may not be available as
explanatory variables when forecasting the following hours. A solution is to
replace unavailable variables with recursively forecasted variables based on
the last available observations.

To ensure data availability in the sense of a day-ahead forecast at all times,
we only use load observations up to yesterday’s last hour for TSO data as
inputs if we make predictions today for tomorrow. Today’s hours must be
replaced by forecasts based on yesterday. More clearly, let $t=8785$ be the
first hour of January 1st, 2017, for simplicity and let $x$ be the hour of
January 1st from which we forecast the next day’s hours. In the further
course, we assume $x=12$, so we forecast the next day’s hours between 11:00
and 12:00 a.m. today. Depending on availability, real TSO load forecast errors
$\epsilon_{t}$ enter our model or forecasted ones. For hour $t\leq x-12$, we
use the observed real errors ${\epsilon}_{t}$ and the forecasted ones
$\hat{\epsilon}_{t}$ for $t>x-12$. We want to predict the load for the next
day’s 24 hours, thus, $x+13$ to $x+37$. Due to the information delay and
ensuring data availability, we do not indicate the actual load of hours $x-11$
to $x-1$. We also have no information about the hours $x$ to $x+12$ lying in
the future. For this reason, we first estimate the model based on the last
available $l_{w}$ observations (i.e., of hours $x-12-(l_{w}+1)$ to $x-12$.
From that, we predict the errors of the TSO load forecast of the next $48$
hours $x-11$ to $x+37$, i.e., of the hours of January 1st and 2nd, and use the
last $24$ predicted values. Thus, at hours $x+13$ to $x+37,$ for improving the
original load forecasts of the following day. Note that by rolling over the
estimation window daily, we ensure that the prediction of TSO forecast errors
for all load periods of one day is based on the same estimated model.

The proposed model is implemented in MATLAB®. The code, used data and the
generated result are provided on GitHub:
github.com/ProKoMoProject/Enhancing-Energy-System-Models-Using-Better-Load-
Forecasts.

###  4.2 Energy System Model

We develop a new energy system model, the _em.power dispatch_ model, to derive
wholesale day-ahead price forecasts. The model is formulated as a linear
optimisation problem minimising total system costs and includes a detailed
representation of central techno-economic aspects of the European electricity
sector. In particular, the model dispatches various generation technologies to
satisfy electricity demand. In addition to power plant dispatch in Germany,
the model considers international trade between the markets described in 3.2,
electricity production by combined heat and power plants, energy storage and
control power provision. To ensure a linear formulation of such a highly
complex system, we form capacity clusters, parameterised as described in 3.2.
Within each technology cluster, capacity can be started-up and electricity can
be produced in marginal increments (see, e.g., Müsgens [2006]). The advantage
of this approach is twofold. First, computational efforts are reduced. Second,
the marginal of the demand restriction is differentiable at each point and can
thus be interpreted as a wholesale market price estimator. Additionally, the
accuracy of modelling large energy systems, in particular, remains reasonably
high (see Müsgens and Neuhoff [2006]).

Considering all economic and technical restrictions, the model solves the cost
minimisation problem and determines i) the optimal dispatch decision for all
considered infrastructure elements, such as generation technologies, energy
storage and cross-border transmission capacities, and ii) the short-run
marginal system cost that determines the price estimator for the day-ahead
market in hourly resolution.

Furthermore, as our research analyses the impacts on day-ahead price
forecasts, we set up the model to reflect the information available to market
participants on the day before delivery. We thus consider that market
participants do not have perfect foresight for the upcoming days. We achieve
this with a rolling window model that is repeatedly solved and provides
information for 24 day-ahead hours of one "target day" in each model run. To
reduce the problem of starting and ending values, in particular for power
plant start-ups and pump storage plants, each model run includes three days,
as shown in 4. In this setting, the 24 hours of the respective target day are
represented by the second day of the horizon (d+1). This is following the EPEX
spot market organisation, where 24 hourly day-ahead prices are determined at
12 p.m. on the day before delivery (d). In addition to the target day d+1, we
also include the day before (d) and the day after (d+2). Note that we include
a water value to increase the accuracy of seasonal hydro-storage modelling.

Figure 4: Illustration of the rolling window.

As with the improvement of the load forecast, this approach is repeated
continuously ("rolling window"), once for each day of the observation period.
At each iteration, the input data for d+1 and d+2 are limited to the values
available on day d (i.e. forecasts), so that the incoming day-ahead load
forecast is successively improved and processed in our approach. Correctly
parameterised, our model uses the same data as market participants (e.g.,
energy suppliers, direct marketers, investment banks) when forecasting the
day-ahead prices to optimise their portfolio. Given this day-ahead focus of
our analysis, installed and available capacities are exogenous. The model
endogenously optimises power plant dispatch only.

Our rolling window approach to forecasting hourly prices implies that we
forecast three years with 365 daily model runs each year. As each model run
comprises 72 hourly dispatch decisions with numerous variables in 23 model
regions, the total number of variables is 340 million. In the following, we
present the mathematical formulation of our model. The model is coded in
GAMS888GAMS General Algebraic modelling System (https://www.gams.com/). The
entire code is provided on GitHub: github.com/ProKoMoProject/Enhancing-Energy-
System-Models-Using-Better-Load-Forecasts. A nomenclature containing all
indices, parameters and variables of the energy system model formulation is
provided in B.

The objective function in Eq. 6 minimises total system costs and accounts for
all costs that generation units face in the short-term. We include costs at
full load operation ($vc_{i,n,t}^{FL}$), additional costs for units that
operate at partial load ($vc_{i,n,t}^{ML}-vc_{i,n,t}^{FL}$), and start-up
costs ($sc_{i,n,t}$). Note that we apply a linear formulation of the unit
commitment, and all units have to produce at least a minimum output level.
Additionally, we account for load shedding costs ($voll$) and penalty payments
for curtailing renewables ($curtc$).

Since we apply our model with a rolling window, we consider three days in each
model run. Modelling an additional day before and after the target day seems
appropriate for storages with large energy-to-power ratios, which are
essentially operated on a daily cycle (e.g., the largest German pump storage
facility, Goldisthal, can store enough energy for nine hours of full load
operation). However, other storages (both PSP and seasonal storages without
pumps) have a storage cycle longer than three days. Therefore, we model two
types of PSP, first as mid-term storage that operates a storage cycle within a
three-day horizon, and second as long-term storage that operates a storage
cycle longer than three days. The dispatch of mid-term storage is determined
endogenously, with the exogenous restriction that they both start and end the
cycle with reservoir levels at 30 %. The approach is different for long-term
PSPs, which are assigned a water value ($wv_{stl,n,t}$) that is implemented as
a variable cost factor for electricity generation ($G_{stl,n,t}$) and
consumption ($CL_{stl,n,t}$). We assume that 70 % of the pump storage capacity
is optimised in the medium-term. The remaining 30 % are long-term PSPs.

Compared to pumped storage plants, hydro-reservoirs have a natural water feed-
in and do not perform a pumping process. However, the water budget for
electricity generation is limited according to seasonal inflow volumes.
Therefore, we also apply a water value for electricity generation by hydro-
reservoirs.

$\displaystyle\min TC=$ $\displaystyle\sum_{i,n,t}G_{i,n,t}\cdot
vc_{i,n,t}^{FL}+\sum_{i,n,t}SU_{i,n,t}\cdot sc_{i,n,t}$ (6)
$\displaystyle+\sum_{i,n,t}(P_{i,n,t}^{on}-G_{i,n,t})\cdot(vc_{i,n,t}^{ML}-vc_{i,n,t}^{FL})\cdot
g_{i}^{min}/(1-g_{i}^{min})$ $\displaystyle-\sum_{stl,n,t}CL_{i,n,t}\cdot
wv_{stl,n,t}+\sum_{n,t}SHED_{n,t}\cdot voll$
$\displaystyle+\sum_{n,t}CURT_{res,n,t}\cdot curtc$

Market clearing is ensured by Eq. 7: for all T hours of the given rolling
window, demand ($d_{n,t}$) must equal the sum of generation ($G_{i,n,t}$),
load shedding ($SHED_{n,t}$) and electricity imports ($FLOW_{nn,n,t}$),
reduced by electricity consumption of mid-term energy storage ($CM_{stm,n,t}$)
and long-term energy storage ($CL_{stl,n,t}$) and electricity exports
($FLOW_{n,nn,t}$).

$\displaystyle d_{n,t}=$ $\displaystyle\sum_{i}G_{i,n,t}-\sum_{stm\subset
I}CM_{stm,n,t}-\sum_{stl\subset I}CL_{stl,n,t}+SHED_{n,t}$ (7)
$\displaystyle+\sum_{nn}(FLOW_{nn,n,t}-FLOW_{n,nn,t})$ $\displaystyle\forall
n,nn\in N,t\in T$

The dual variable of the demand constraint Eq. 7 is used as an hourly day-
ahead wholesale electricity price estimator. As we want to analyse how well
these price estimators based on different demand forecasts fit real-world day-
ahead prices, we compare them and compute error measures.

Electricity generation by capacity cluster is limited by an upper and a lower
bound. The upper bound is formalised in Eq. 8 and ensures that electricity
generation does not exceed the running capacity ($P_{i,n,t}^{on}$) in the
cluster. The possible electricity generation by running capacity is further
limited by the reserve for positive control power provision
($PCR_{i,n,bp},SCR_{i,n,bs}^{pos}$). The lower bound is presented in Eq. 9 and
states that running capacities must operate at least at a minimum power level,
including the capacity reserved for negative control power provision
($PCR_{i,n,bp},SCR_{i,n,bs}^{neg}$). Note that primary control power
($PCR_{i,n,bp}$) in Germany is provided synchronously, i.e., a unit has to
provide both positive and negative primary control power. Different products
for positive and negative control power were introduced for secondary control
power. Since fast-reacting units (e.g., hydro- and open-cycle gas turbines)
can be started-up to provide a positive-minute reserve, the effect on the
running capacities is neglected. In addition, we assume that a negative-minute
reserve is provided by multiple market players, not necessarily by power
plants. The hours that belong to bidding blocks are mapped for primary control
power by $bp$ and secondary control power by $bs$.

$\displaystyle G_{i,n,t}\leq P_{i,n,t}^{ON}-PCR_{i,n,bp|t\in
bp}-SCR_{i,n,bs|t\in bs}^{pos}$ (8) $\displaystyle\forall bp\in BP,bs\in
BS,i\in I,n\in N,t\in T$

$\displaystyle P_{i,n,t}^{on}\cdot g_{i}^{min}+PCR_{i,n,bp|t\in
bp}+SCR_{i,n,bs|t\in bs}^{neg}\leq G_{i,n,t}$ (9) $\displaystyle\forall bp\in
BP,bs\in BS,i\in I,n\in N,t\in T$

The running capacity of a power system is limited by the installed capacity
($cap_{i,n,t}$) in combination with either the availability factor
($af_{i,n,t}$) or power plant outages ($out_{i,n,t}$), as shown in Eq. 10. For
thermal generation capacities, we use hourly power plant outages. Renewables
are provided with an hourly availability factor and hydroelectric units with a
monthly availability factor.

$\displaystyle P_{i,n,t}^{on}\leq cap_{i,n,t}\cdot
af_{i,n,t}-out_{i,n,t}\hskip 28.45274pt\forall i\in I,n\in N,t\in T$ (10)

Eq.11 tracks start-up activities ($SU_{i,n,t}$) that increase the running
capacity from one hour to another. Due to the non-negativity condition, start-
ups are either positive or zero.

$\displaystyle P_{i,n,t}^{on}-P_{i,n,t-1}^{on}\leq SU_{i,n,t}\hskip
28.45274pt\forall i\in I,n\in N,t\in T$ (11)

The delta between available feed-in from intermittent renewables and their
actual generation defines the curtailment of renewables ($CURT_{res,n,t}$), as
shown in Eq. 12.

$\displaystyle cap_{res,n,t}\cdot
af_{res,n,t}=G_{res,n,t}+CURT_{res,n,t}\hskip 28.45274pt\forall res\in I,n\in
N,t\in T$ (12)

Some power plants are active in the heat market in addition to the electricity
market. The model thus implements a must-run condition for such units on the
electricity market, which varies over time (e.g., higher in the winter season
due to space heating). Depending on hourly heat demand, Eq. 13 states that the
output of a combined heat and power unit is at least equal to the electricity
generation linked to the heat production ($chp_{i,n,t}$).

$\displaystyle chp_{i,n,t}\leq G_{i,n,t}\hskip 28.45274pt\forall i\in I,n\in
N,t\in T$ (13)

Eq. 13 constraints the cross-border electricity transfer ($FLOW_{n,nn,t}$) by
the net transfer capacity ($ntc_{n,nn,t}$).

$\displaystyle FLOW_{n,nn,t}\leq ntc_{n,nn,t}\hskip 28.45274pt\forall n,nn\in
N,t\in T$ (14)

Eq.15 describes the state of the storage level of a mid-term storage. The
storage level is increased by the generation ($G_{stm,n,t}$) and decreased by
the consumption while charging ($ST_{stm,n,t}^{in}$). The efficiency of an
entire storage cycle ($\eta_{stm}$) is assigned to the charging process.

$\displaystyle
SL_{stm,n,t}=SL_{stm,n,t-1}-G_{stm,n,t}+CM_{stm,n,t}\cdot\eta_{stm}$ (15)
$\displaystyle\forall stm\in I,n\in N,t\in T$

The maximum energy storage capacity ($SL_{stm,n,t}$) of a mid-term storage is
defined by the maximum installed turbine capacity times an energy-power factor
($epf$), as shown in Eq. 16.

$\displaystyle SL_{stm,n,t}\leq cap_{stm,n,t}\cdot epf\hskip 28.45274pt\forall
stm\in I,n\in N,t\in T$ (16)

Eq. 17 restricts the turbine and pumping capacity, where the pumping capacity
is assumed to be lower than the turbine capacity.

$\displaystyle G_{stm,n,t}+1.1\cdot CM_{stm,n,t}\leq cap_{stm,n,t}\hskip
28.45274pt\forall stm\in I,n\in N,t\in T$ (17)

At the beginning and end of each model run, all mid-term storages must be
filled with 30 % of their energy level (Eq. 18 and 19).

$\displaystyle SL_{stm,n,tfirst}=0.3\cdot cap_{stm,n,t}\hskip
28.45274pt\forall stm\in I,n\in N,tfirst\in T$ (18)

$\displaystyle SL_{stm,n,tlast}=0.3\cdot cap_{stm,n,t}\hskip 28.45274pt\forall
stm\in I,n\in N,tlast\in T$ (19)

Long-term storage is not subject to a storage mechanism. However, the
electricity generation and consumption of long-term storage units are also
restricted by the installed capacity of long-term storage by Eq. 20.

$\displaystyle G_{stl,n,t}+CL_{stl,n,t}\leq cap_{stl,n,t}\hskip
28.45274pt\forall stl\in I,n\in N,t\in T$ (20)

Eqs. 21, 22 and 23 ensure the control power provision for primary, positive
secondary and negative secondary control power.

$\displaystyle\sum_{i}PCR_{i,n,bp}=pr_{n}\hskip 28.45274pt\forall bp\in
BP,n\in N$ (21)

$\displaystyle\sum_{i}SCR_{i,n,bs}^{pos}=sr_{n}^{pos}\hskip 28.45274pt\forall
bs\in BS,n\in N$ (22)

$\displaystyle\sum_{i}SCR_{i,n,bs}^{neg}=sr_{n}^{neg}\hskip 28.45274pt\forall
bs\in BS,n\in N$ (23)

The non-negativity constraint is presented in Eq. 24.

$\displaystyle 0\leq$ $\displaystyle
CL_{stl,n,t},CM_{stm,n,t},CURT_{res,n,t},G_{i,n,t},FLOW_{n,nn,t},$ (24)
$\displaystyle
P_{i,n,t}^{on},PCR_{i,n,bp},SCR_{i,n,bs}^{neg},SCR_{i,n,bs}^{pos},SHED_{n,t},SL_{stm,n,t},SU_{i,n,t}$
$\displaystyle\forall bp\in BS,bs\in BS,i\in I,n\in N,t\in T$

We use both models presented alternately. To predict the next day, we first
forecast the load forecast error with the load forecast improvement model and
thus enhance the day-ahead load forecast. As one input data, it enters the
power system model, which estimates the next day’s prices using the presented
approach. This sequence is repeated continuously day by day over the rolling
window for all points in time in our observation period.

##  5 Results

Our paper explores two different methodologies that are combined. It presents
a forecast error improvement model for load forecasts based on data from
ENTSO-E, and it develops the energy system model _em.power dispatch_ which is
built for day-ahead wholesale price forecasts. We present the results
accordingly. First, we show the performance of the model for the load forecast
error using statistical data and different error measures for various time
periods of the enhanced load forecast. Second, we analyse the impact of the
improved forecast on the resulting price estimates of the _em.power dispatch_
model. Therefore, we compare the resulting price estimators generated with the
original TSO load forecast $\hat{L}$ and the enhanced load forecast
$\hat{L}^{*}$ with the actual price observed at the day-ahead market using
several error measures: mean squared error (MSE), root mean squared error
(RMSE), and mean average error (MAE).

###  5.1 Improved Load Data and Achieved Error Reduction

In the following, we quantify the TSO forecast error improvement model
described in Section4.1. Therefore, we compare the improved load forecast
$\hat{L}^{*}$ and the TSO load forecast $\hat{L}$ with actual load data $L$.
For the error improvement model, we use a rolling window width of one year
(i.e., $l_{w}=8760$), which yields the lowest (out of sample) error measures
compared with a width of three months and six months. For this reason, the
prediction of the forecast error, and thus the out-of-sample period, begins on
January 1st, 2017. Table 2 shows the mean and standard deviation of the TSO
load forecast error and the enhanced load forecast error, the error measures
MSE, RMSE and MAE of the TSO load forecast and of the enhanced load forecast,
as well as the percentage improvement.

| year | all | 2017 | 2018 | 2019
---|---|---|---|---|---
Mean | TSO | 655.98 | 446.50 | 298.60 | 1,222.84
| Impr. | -98.89 | -229.06 | -36.32 | -31.29
Std. | TSO | 2,125.72 | 1,746.54 | 2,341.68 | 2,128.55
| Impr. | 1,743.89 | 1,465.74 | 2,043.67 | 1,665.56
| TSO | 4,948,990.80 | 3,249,416.14 | 5,572,010.71 | 6,025,545.56
MSE | Impr. | 3,050,928.19 | 2,200,617.23 | 4,177,415.90 | 2,774,751.43
| % Improvement | 38.35 | 32.28 | 25.03 | 53.95
| TSO | 2,224.63 | 1,802.61 | 2,360.51 | 2,454.70
RMSE | Impr. | 1,746.69 | 1,483.45 | 2,043.87 | 1,665.76
| % Improvement | 21.48 | 17.71 | 13.41 | 32.14
| TSO | 1,691.37 | 1,396.45 | 1,726.67 | 1,951.00
MAE | Impr. | 1,253.95 | 1,106.12 | 1,372.55 | 1,283.17
| % Improvement | 25.86 | 20.79 | 20.51 | 34.23

Table 2: Means, standard deviations and error measures (MSE, RMSE, MAE) for
the original TSO day-ahead load forecast (TSO) and the improved day-ahead load
forecast (Impr.). MSE is given in [$MWh^{2}$], and all other variables in
[$MWh$].

While the load was severely underestimated in the TSO forecast with a mean of
656.0 MWh, it is slightly overestimated in the improved model with -98.9 MWh.
Looking at the individual annual mean values, the high negative value in 2017
is particularly striking. The reason for this is the very strong
underestimation of the TSO load forecast in 2016, with an average deviation of
1555.4 MWh (see Section3.1). The influence of errors from the year 2016 has a
large impact due to the rolling window period of 365 days, especially on the
model estimates of the first days and months of 2017. A shorter window period
of three months sinks the annual mean value of 2017 but has a minor
improvement in error measures (see 6).

The standard deviation of the improved load forecast is lower than the
standard deviation of the TSO load forecast across all years.

The error measures MSE and MAE given in Table 2 show a significant improvement
of the load forecast. With an RMSE of 2,224.6 MWh, we achieve a 21.48 %
improvement over the TSO load forecast for the period from January 1st, 2017,
to December 31st, 2019. The most considerable improvement can be observed in
2019 with 32.14 %. A breakdown of the improvement among the components
(seasonal and remaining) of the model shows that both the seasonal and
remaining components account for a large share of the improvement, and neither
component dominates.

Maciejowska et al. [2021] also improve the TSO load forecast. From October
1st, 2016, to September 30th, 2019, they achieve an enhancement in RMSE over
365 days from a minimum of 23.71 % to a maximum of 34.38 %. Comparing both,
achieving a slightly higher improvement also means using a multivariate
modelling framework with six different rolling window widths, and consequently
six model estimates and six point forecasts for each hour of the forecast
period. Our approach is intended to allow a user with less modelling expertise
and computational capacity to enhance the commonly used TSO forecast of load.
With a less complex, univariate model, we still achieve substantial
improvement and thus achieve error measurements that are comparable with error
measurements in the literature, e.g., Do et al. [2016], Ziel [2018], Li et al.
[2021].

To better attribute and understand the effect of load improvement on price, we
also determine the percentage improvement in MSE for the hours of a day, and
the days of the week, as shown in Figure 5. The observed daytime and weekday
structures in the TSO load forecast error are also evident in the improvement.
During the day, hours 2 through 5 and 16 through 20 achieve the most
considerable percentage improvement. Weekdays can be improved more than
weekends; Tuesdays and Wednesdays show an especially strong improvement. In
the TSO load forecast, these are the hours and days that have the largest mean
error. Therefore, hours and days that have a sizeable mean error are the ones
that have the most potential for improvement. Enhancing the load forecast by
reducing this error is the primary goal of modelling and predicting the error
of the TSO load forecast.

Figure 5: Average percentage MSE improvement for the day-ahead load forecast
for each hour of a day (left) and for each weekday (right).

###  5.2 Impact of Improved Load Data on an Energy System Model

In the previous Section5.1, we proved that with a relatively straightforward
approach, the ENTSO-E load data can be significantly improved. Thus, this
approach is particularly suitable for energy system modellers to enhance
critical input data. In the following, we quantify the impact of the improved
load forecast on day-ahead wholesale price forecasts based on the _em.power
dispatch_ model. To do this, we run the model twice, first using the original
TSO-based load forecasts $\hat{L}$ and second, using the improved load
forecasts $\hat{L}^{*}$ presented in Section5.1. For both cases, we derive
estimates of the day-ahead wholesale prices and calculate error measures
comparing the results to actual observed day-ahead prices.

Using the improved load data set, we see an overall reduction in the error of
the price estimator. For the entire time horizon, Table 3 states a reduction
of the MSE by 1.75 %, the RMSE by 0.88 % and the MAE by 0.42 %.

Comparing our results with those of other models in the same modelling class,
we find that our model generates very good price estimates. Qussous et al.
[2022], for example, report an MAE of 9.44 €$/MWh$ for 2017, 8.88 €$/MWh$ for
2018 and 6.69 €$/MWh$ for 2019.

| all years | 2017 | 2018 | 2019
---|---|---|---|---
| Impr. | Orig | Impr | Orig | Impr | Orig | Impr | Orig
[1] MSE | 89.15 | 90.73 | 133.13 | 135.29 | 72.47 | 73.22 | 61.84 | 63.70
[2] RMSE | 9.44 | 9.53 | 11.54 | 11.63 | 8.51 | 8.56 | 7.86 | 7.98
[3] MAE | 5.94 | 5.96 | 6.75 | 6.80 | 5.98 | 6.04 | 5.09 | 5.28
Reduction [1] | 1.75% |  | 1.62% |  | 1.04% |  | 3.01% |
Reduction [2] | 0.88% |  | 0.81% |  | 0.52% |  | 1.49% |
Reduction [3] | 0.42% |  | 0.85% |  | 1.02% |  | 3.65% |

Table 3: Error measures for the price estimator of the _em.power dispatch_
model comparing the improved load forecasts (Impr.) by original load forecasts
(Orig), given in [$MWh^{2}$] for MSE, in [$MWh$] for RMSE and MAE.

Table 3 further shows disaggregated error measures by year. It can be seen
that an improvement in the error measure is achieved in all three years.
However, the magnitude of this improvement varies; the relative error
reduction is largest in 2019 and smallest in 2018. This observation correlates
with the magnitude of the annual improvement in the load forecast, shown in
Figure 5.

Furthermore, we analysed whether the improvement of the load estimator and the
price estimator correlate with the hour of the day. Figure 6 shows the average
percentage improvement of the MSE of the day-ahead load prediction per hour of
the day (left) and of the day-ahead price estimators (right). It can be seen
that an hour’s load and hour’s price improvement do not correlate. Depending
on the respective hour of the day, improvement of load prediction seems to
have a different impact on the resulting price estimator.

The reasoning for this discrepancy is two-fold: i) the model is more sensitive
in one hour than in another hour, depending on the respective position in the
merit order, and ii) an improvement in the load forecast in one hour may
affect another hour due to temporal interdependencies such as storage
operation and unit commitment decisions.

Figure 6: Average percentage MSE improvement of day-ahead load prediction
(left) and day-ahead price estimators (right) for each hour of a day.

Having shown that the impact of better load forecasts on price forecasts
derived in an energy system model is positive on average but varies between
hours, we now examine the extent of error reduction at different points in
time, starting with differentiation between high (Peak) and low demand (Off-
Peak) periods. Figure 7 states the error reduction of the price estimator and
that of the load forecast for the entire time period and time categories peak,
off-peak, weekdays and weekend days. The most considerable error reduction of
the price estimator is observed in peak hours and on weekdays in general. In
the hours between 8 p.m. and 8 a.m. as well as on weekends, the effect on the
price estimator is relatively low. On weekends, this observation correlates
with the improvement of the load data, both of which are at their minimum.
However, in off-peak hours, the impact on the price estimator is negligible,
despite the great improvement in the load forecast.

As such, the model benefits significantly from improved load input data during
peak hours and in total on weekdays, where demand and price levels are
generally higher than off-peak hours and especially on weekends.

Figure 7: Percentage error reduction of the price estimator and the load in
different time periods.

Based on the observation that price forecasts improve more during peak periods
than in off-peak periods, we analyse the relation between wholesale price and
forecast improvement. Figure 8 shows the improvement of the price estimator
for five different price segments where electricity prices are equally
separated in 20 % quantiles based on their level. The first quantile (q1)
represents the lowest 20 % quantile and the last quantile (q5) the highest 20
% quantile of electricity prices of the respective year between 2017 and 2019.

It can be seen that the error reduction of the price estimator is most
relevant in hours with high and medium prices. Overall, the largest
improvement can be observed in 2018 and 2019 with an MSE reduction of nearly
15 %, here at times with the 60-80 % highest prices. In contrast, the improved
load forecast data does not lead to a better price estimator in low-price
periods. In all years, we even observe an increasing error in these price
ranges. In summary, the improved load forecast is most beneficial for the
model in the hours when the market equilibrium is found on the right side in
the merit order, i.e., where changes or errors in the demand have the highest
price impact.

Figure 8: Relative error reduction of the price estimator in different price
segments of the respective year from 2017 - 2019, starting with the lowest 20
% quantile of electricity prices (q1) to the highest 20 % quantile (q5).

Hence, our analysis shows that the price forecasts are generally better when
a) demand is high and b) prices are high. As traded volumes (in monetary
terms) are the product of prices and volumes, it is interesting to note that
price forecast improvement is highest when it matters the most.

##  6 Conclusion

This paper discusses data preprocessing in the context of fundamental energy
system models. We present a simple time series model to improve the TSO-based
load forecast data provided by ENTSO-E. The model captures and removes
systematic biases and autoregressive structures present in the load forecast
errors. Since the model is applied to observed forecast errors rather than to
the load data itself and does not include load-specific external variables, it
can be easily transferred to the preprocessing of other quantities of
interest.

To analyse the effect of enhanced load forecasts on electricity system models,
we feed the improved load forecast data into the _em.power dispatch_ model.
The model is used to generate price estimates for the German day-ahead
electricity market, and we present the structure, assumptions, and
optimisation equations of the model in detail. Concerning the effect of
sequentially preprocessed inputs, we find that the benefits of sequentially
improved load forecasts strongly depend on the respective price level, with
more extensive benefits for higher price levels. This is a universal result in
line with fundamental theory since in merit order markets, the impact of load
changes on price changes increases with the overall level. We find that in
phases of relatively high prices, as in 2018 and 2019, the continuous and
sequential, i.e., day-by-day, load data preprocessing leads to an average
reduction of _em.power dispatch_ ’s prices forecast mean squared error by
nearly 15 %. Hence, as the value of traded energy is the product of prices and
volumes, our analysis shows that forecasts are generally better when a) demand
is high and b) prices are high, i.e., when it matters the most.

Based on these findings, we recommend energy system modellers to carefully
analyse not only the structure and equations of their models but also the
quality of input data. This paper demonstrated in the empirical setting of the
German wholesale electrictiy market that input data can be improved
significantly and that these improvements can be achieved with very simple
time-series models. Furthermore, we demonstrate that results of the energy
system model benefit from the improved input data.

##  7 Data availability

Datasets related to this article and a source code for the entire project are
available in a public GitHub repository. On
github.com/ProKoMoProject/Enhancing-Energy-System-Models-Using-Better-Load-
Forecasts. you find code and data for the time series model improving the load
forecasts as well as code and data for the energy system model. The codes
reproduce the benchmarks from the paper.

## Acknowledgements

The work was supported by the German Federal Ministry of Economic Affairs and
Climate Action through the research project "ProKoMo - Better price forecasts
in the energy sector by combining fundamental and stochastic models" within
the Systems Analysis Research Network of the 6th energy research program.

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##  Appendix A Descriptive statistics for various time resolutions

Day | Mean | Median | Minimum | Maximum | 5%-Q. | 95%-Q. | Std.
---|---|---|---|---|---|---|---
Mon. | 1,517.71 | 1,436.63 | -6,277.00 | 12,930.75 | -1,895.75 | 5,111.45 | 2,159.26
Tue. | 1,685.17 | 1,581.50 | -4,867.25 | 11,469.00 | -1,473.05 | 5,126.95 | 1,984.32
Wed. | 1,601.81 | 1,516.00 | -7,868.50 | 9,053.75 | -1,242.65 | 4,667.90 | 1,874.89
Thu. | 1,361.43 | 1,365.25 | -17,543.50 | 9,520.25 | -1,372.88 | 4,330.47 | 1,896.08
Fri. | 944.31 | 990.75 | -10,596.88 | 9,772.25 | -2,060.85 | 3,911.48 | 1,892.38
Sat. | -102.26 | -43.63 | -7,661.75 | 7,809.00 | -2,967.80 | 2,380.13 | 1,696.76
Sun. | -378.19 | -340.13 | -7,752.50 | 7,522.00 | -3,378.53 | 2,458.15 | 1,859.72
Table 4: Weekday wise averaged descriptive statistics of TSO load forecast errors for the years 2016 to 2019. All variables are given in [$MWh$]. Hour | Mean | Median | Minimum | Maximum | 5%-Q. | 95%-Q. | Std.
---|---|---|---|---|---|---|---
1 | 699.64 | 742.75 | -5,710.00 | 6,594.75 | -2,270.28 | 3,509.15 | 1,713.19
2 | 652.07 | 696.75 | -6,302.25 | 6,701.50 | -2,290.58 | 3,460.69 | 1,711.79
3 | 646.41 | 657.75 | -6,558.25 | 6,882.75 | -2,246.04 | 3,535.18 | 1,739.51
4 | 6,90.28 | 687.50 | -8,539.00 | 6,888.25 | -2,269.38 | 3,635.30 | 1,760.22
5 | 855.45 | 852.75 | -15,353.75 | 7,617.25 | -2,298.63 | 3,959.94 | 1,991.04
6 | 943.53 | 1,032.50 | -17,543.50 | 9,113.00 | -2,658.48 | 4,335.81 | 2,295.79
7 | 946.49 | 1,075.75 | -20,358.00 | 11,469.00 | -2,896.14 | 4,688.48 | 2,515.40
8 | 912.14 | 1,011.00 | -19,681.75 | 11,152.25 | -2,795.23 | 4,529.93 | 2,461.27
9 | 908.80 | 959.00 | -16,540.25 | 11,658.25 | -2,692.56 | 4,594.01 | 2,387.68
10 | 902.67 | 900.00 | -15,234.63 | 12,930.75 | -2,836.34 | 4,707.63 | 2,417.03
11 | 916.34 | 926.75 | -14,149.88 | 10,484.25 | -2,791.68 | 4,646.26 | 2,401.02
12 | 923.30 | 951.00 | -14,261.88 | 10,471.25 | -2,818.44 | 4,782.05 | 2,414.55
13 | 907.81 | 982.00 | -15,839.88 | 9,777.25 | -2,980.00 | 4,740.98 | 2,446.60
14 | 785.04 | 882.75 | -15,116.63 | 9,306.75 | -2,963.74 | 4,549.70 | 2,412.79
15 | 718.73 | 807.50 | -15,220.38 | 9,510.50 | -2,933.24 | 4,477.88 | 2,396.13
16 | 900.33 | 895.50 | -14,434.38 | 9,520.25 | -2,777.48 | 4,756.91 | 2,323.86
17 | 1,080.73 | 1,051.50 | -13,068.00 | 9,951.25 | -2,526.88 | 4,832.60 | 2,285.24
18 | 1,172.53 | 1,128.00 | -12,133.38 | 10,094.50 | -2,110.33 | 4,819.08 | 2,140.34
19 | 1,180.48 | 1,161.25 | -11,652.75 | 9,409.75 | -1,985.76 | 4,610.81 | 2,078.72
20 | 1,061.99 | 1,071.00 | -9,940.75 | 8,430.50 | -1,956.98 | 4,248.70 | 1,978.27
21 | 875.01 | 909.25 | -8,443.13 | 7,422.00 | -2,121.79 | 3,943.60 | 1,888.30
22 | 877.85 | 872.75 | -6,668.50 | 6,932.25 | -2,032.00 | 3,853.51 | 1,821.24
23 | 863.47 | 806.00 | -5,642.38 | 6,947.25 | -1,928.34 | 3,706.63 | 1,726.26
24 | 729.88 | 763.25 | -5,607.88 | 7,011.50 | -2,262.01 | 3,560.98 | 1,701.36
Table 5: Hourly averaged descriptive statistics of TSO load forecast errors
for the years 2016 to 2019. All variables are given in [$MWh$].

##  Appendix B Nomenclature

##  Appendix C Error measures for different rolling window lengths

| all | 2017 | 2018 | 2019
---|---|---|---|---
MSE | 4,948,990.80 | 2,272,350.07 | 4,575,637.58 | 3,133,176.51
RMSE | 2,224.63 | 1,507.43 | 2,139.07 | 1,770.08
MAE | 1,691.37 | 1,132.86 | 1,462.48 | 1,377.82
Table 6: Error measures (MSE, RMSE, MAE) for the the improved day ahead load forecast with a rolling window length of three month. MSE is given in [$MWh^{2}$], RMSE and MAE in [$MWh$]. | all | 2017 | 2018 | 2019
---|---|---|---|---
MSE | 4,016,010.14 | 4,896,546.88 | 4,355,814.17 | 2,795,669.38
RMSE | 2,004.00 | 2,212.81 | 2,087.06 | 1,672.03
MAE | 1,304.12 | 1,200.16 | 1,411.57 | 1,300.63
Table 7: Error measures (MSE, RMSE, MAE) for the the improved day ahead load
forecast with a rolling window length of six month. MSE is given in
[$MWh^{2}$], RMSE and MAE in [$MWh$].