** Energy and Price Forecasting**

**5.1 Forecasting Methodology**

**5.1.1 Explanatory Variables**

Section 2.5 explains how an explanatory variable can improve the accuracy of an energy forecast. Both the ARIMAX and NARX forecasting models, are supported with an explanatory variable based on suitable information.

As mentioned in Section 2.5.1, for the explanatory variable to be helpful, it must be related to the time series for forecasting and should include future observations.

5.1.1.1 OMIE Explanatory Variable

In principle, the relation between total electric demand and electric energy price suggests that electricity demand can be an explanatory variable for electricity price forecasting. On the other hand, and as seen in Section 2.3.1, the demand is not the only factor related to energy price. The energy price is built when matching the energy bids and the demand. Therefore, the price requested in the generation bids is as important as the demand and the price offered for it. In the energy market, it is possible to discern between two kinds of generators: manageable and not-manageable. The first kind can generate depending on the price and therefore match in the energy auction whenever certain economic boundaries are met. By contrast, the unmanage-able power plants can generate independently of the selling conditions. These technologies are solar, wind, and nuclear power. The behaviour of the

un-manageable power plants affects the final auction price because their energy bids are close to 0e/MWh, so the energy is purchased. Relatedly, another explanatory variable can be considered: the energy that will be produced regardless of economic boundaries that are subtracted from the total energy demand. This amount of energy is what is left for manageable generators to match and can be defined as ‘Competitive Market’ or ‘Open Market’, and is defined in equation 5.1.

CompetitiveM arket_{h} =Demand_{h}−Solar_{h}−W ind_{h}−N uclear_{h} (5.1)
In this thesis, three possibilities are studied to understand the impact of
an explanatory variable on forecasts:

1. No explanatory variable

2. Demand as an explanatory variable

3. Competitive market as an explanatory variable

A Pearson correlation study is carried out for the two explanatory vari-ables on the historical values for 2016 to determine which one is more suitable to be used in energy price forecasting. The results of the relation between en-ergy price and demand are depicted in Figure 5.1 with a Pearson correlation factor of 0.412.

The results of the relation between energy price and competitive market are depicted in Figure 5.1 with a Pearson correlation factor of 0.719. The relation depicted for Competitive Market and energy price is much higher than the relation with the demand. Even the relation is not perfect and is in fact far from being close to 1 (which means a perfect and direct relation), it is a better support for forecasting for both methods and should improve the forecasting accuracy.

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Figure 5.1: Explanatory variables relation to energy price 5.1.1.2 Solar Energy Explanatory Variable

The energy forecasters may use solar radiation, ambient temperature, and cloudiness as its inputs, whereas power is given as an output [123]. As mentioned before, the relation between irradiation and solar generation is direct for a certain power plant. Therefore, a precise irradiation value is presumed to have an outstanding explanatory variable. In the same way, the cloudiness index has a negative relation with solar energy generation and is hence presumed to be a useful explanatory variable. A generation forecast that is based on cloudiness is deemed to be the most successful method for long-term solar forecasting [96].

The importance for forecasting energy generation lies in the markedly dif-ferent generation between stations that change from a 540MW peak in winter to a 5,600MW peak in summer and achieve a maximum of 12% of the total renewable power injected into the grid. It is also important to understand that solar electricity generation in Spain includes two technologies that can

produce electricity:

Photovoltaic technology generates energy that is influenced only by the solar irradiation on the panel surface at a certain time and is not sup-ported with storage. The installed power of this technology is 4.16GW [124].

The thermo-solar power plants are able to generate electricity through a process that is dependent on solar irradiance and temperature; ad-ditionally, these power plants sometimes have storage systems. The installed power of this technology is 2.3GW [124].

To provide reliable explanatory data for Spain, it is necessary to acquire data from an NWP, which provides the cloudiness index for a certain loca-tion. This information would help to model solar radiation by considering of geographical information. The model is based on a clear-sky radiation calculation and the satellite cloudiness indices in different locations in the country. The calculation of the irradiation for a given location is performed by subtracting the fraction blocked by the clouds from the clear-sky radiation [125, 126]. The calculation of the average irradiation in Spain would be the weighted irradiation with the installed power in the area under study [127].

Figure 5.2 [128] illustrates the fundamental relationship between the
satel-lite observation of the planetary albedo and ground-level irradiance. The
difference between the net incoming irradiance at the top of the atmosphere
(I_{in}T OA) and the net irradiance at the ground level (Ig_{in}) must be equal to
the flux lost (either reflected, scattered, or absorbed).

Figure 5.2: Solar irradiation reaching ground after cloud block

To obtain the irradiation with the developed method, it is necessary to
calculate the extraterrestrial hourly irradiation on a tilted surface (G_{0T}) for
a given point and the tilt angle of a south-oriented surface [129]. This
cal-culation depends upon the location of the plant, the time of the year, and
the slope of the solar collectors. The irradiation on a tilted surface (G_{0T}) is
calculated according to Equation 5.2:

G_{0T} =G_{SC}

1 + 0.033 cos360n 365

cos Θ_{z} (5.2)

WhereG_{SC} is a solar constant (1,367W/m^{2}),n is the day number of the
year, and Θ_{z} is the incidence angle calculated in 5.3:

cos Θ_{z} = cos (φ−α) cosδcosω+ sin (φ−α) sinδ (5.3)
φ is the latitude of the location,αis the slope of the collecting surface, δ
is the declination or angular position of the sun calculated in Equation 5.4,
and ω is the hour angle or the angle of displacement of the sun calculated in
Equation 5.5.

δ= 23.45 sin

360284 +n 365

(5.4)

ω = (h_{s}−12)15 (5.5)

The solar time is denoted by h_{s}. The difference between the solar time
and the standard time is calculated through Equation 5.6, and the result is
given in minutes.

SolarT ime−StandardT ime= 4(Lst−Lloc) +E+DLS (5.6)
whereL_{st} and L_{loc} are the longitudes for the standard meridian, and the
location, day light savings, and E are a values that are calculated through
Equation 5.7:

E = 229.2(0.000075 + 0.001868 cos (B)−0.032077 sin (B)

−0.014615 cos (2B)−0.04089 sin (2B)) (5.7) Finally, B is calculated according to Equation 5.8.

B = (n−1)360

365 (5.8)

The intensity of the solar radiation that reaches the surface of the earth
decreases with increasing values of the cloudiness index or sky cover [130,
128]. Taking this hypothesis as valid, the method forecasts the irradiance
(I_{f}) for a given location to be dependent on the extraterrestrial hourly
irra-diance (G_{0T}). The method also holds the irradiance to be the complement
to the forecasted cloudiness measured in [0−1] range (N_{f}). The values are
calculated hourly according to Equation 5.9.

I_{f} =G_{0}(1−N_{f}) (5.9)

A similar model was presented by [131], where global irradiance is
ob-tained by adding the cloud’s transmissivity to the model. The value of the
weighted explanatory variable used for Spain is calculated by considering the
forecasted time series, irradiance in the measured places, and the installed
powers in those places. Spain’s weighted averaged irradiance values (I_{E}) are
calculated according to Equation 5.10.

IE = 1

WhereI_{f n}the forecasted irradiance on then location,P_{sn}is the installed
solar power for a given location, andP_{s}is the total solar power accumulated
in Spain that is calculated using Equation 5.11.

P_{s} =

50

X

n=1

P_{sn} (5.11)

As a result, three variables are obtained for the forecast, cloudiness,
ex-traterrestrial irradiation, and irradiation. The calculated national
extrater-restrial irradiance is on a plane that is parallel to the collector aperture, and
the solar national generation in hourly steps indicate a strong relation in
Figure 5.3(a). It is important to bear in mind that the units and factors do
not match for the variables, as the irradiation is measured in [W/m^{2}] and
its maximum value is 1,367W/m^{2}. Solar power is measured in [MW] and its
maximum value is 6,460MW. This relation reaches 0.86 and an R^{2} of 0.741
when measured with the Pearson factor.

In the case of the cloudiness index, the relation with the aggregated energy
generation is less obvious in terms of indexes or graphs in Figure 5.3(b). The
Pearson factor provides a result of 0.09 and an R^{2} of 0.008. Nevertheless,
the relation between cloudiness and solar generation is easily understandable
from the perspective of physics: the existence of clouds blocks the solar
energy generation, as depicted in Figure 5.2. For the irradiance, which is
calculated using the radiation and the cloudiness factor, the relation with
the aggregated energy generation in Figure 5.3(c) reaches the highest value.

The Pearson factor provides a result of 0.90 and R^{2} = 0.816. Therefore, this
time series will be used as an explanatory variable for solar energy forecasts
in further calculations.

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0

Figure 5.3: Explanatory variables relation to solar generation 5.1.1.3 Wind Power Explanatory Variable

The capability of a wind turbine to generate power is directly related with wind speed and direction. In the case of wind direction, it would be useful to understand how the wind turbines shadow each other on a wind farm [132, 133] to improve the calculation of the explanatory variable. To do so, the wind farms’ turbine distribution is required. Unfortunately, this information is not easily accessible. Therefore, the only contribution to the explanatory variable is the wind speed in the location. Equation 5.12 illustrates the power delivered by a wind turbine:

P = 1

2ρAv^{3} (5.12)

Where ρis the air density, A is the area swept by the rotor, and v is the wind speed through the blades.

The value of the weighted explanatory variable used for Spain is calculated
by considering the forecasted time series and the wind speed in the measured
places and the installed powers in those places. Spain’s weighted average
wind speed (v_{E}) values are calculated according to Equation 5.13.

v_{E} = 1

Where v_{f n} n is the forecasted wind speed in the n location, P_{wn} is the
installed wind power for a given location, and P_{w} is the total wind power
accumulated in Spain that is calculated using Equation 5.14.

Pw =

50

X

n=1

Pwn (5.14)

The resulting data provides a time series for wind speed. To study the
correctness of the above-mentioned suppositions, two correlation studies were
carried out on the time series: a Pearson correlation study and an R^{2}
corre-lation study. After wind speed and national wind generation were taken into
consideration, the results were a correlation of 5.4, a Pearson factor of 0.79,
and an R^{2} of 0.63. In the case of wind speed, the same study is made to the
power of three (cubic wind speed, v^{3}) because it is related to wind power, as
seen in Equation 5.12. This study provides a Pearson factor of 0.73 and an
R^{2} of 0.54, which is lower than the initial results. Thus, simple wind speed
should be used to forecast.

0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0

05

1 0 1 5 2 0 2 5 3 0

W i n d S p e e d L i n e a l F i t P e a r s o n = 0 . 7 9

### W in d S p e e d ( m /s ) W i n d G e n e r a t i o n ( M W h )

Figure 5.4: Wind power and speed correlation

5.1.1.4 Thermal Demand Explanatory Variable

Accurate energy demand forecasts can be obtained using simple models that combine weather forecasts with the historical load and weather curves [134, 78]. The demand forecast model could be improved by using a sufficient data series as an explanatory variable that would support the historical demand values [135]. A straight approach to the problem would be to use the ambient temperature as explanatory variable [136]. The relations between the demand and the ambient temperature are inverse with a Pearson value of −0.51 for heating demand and direct with a Pearson value of 0.52 for cooling demand;

both relations are depicted in Figure 5.5.

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Figure 5.5: Thermal demand and temperature correlation