Short-Term Load Forecasting for Spanish Insular Electric Systems

: In any electric power system, the Transmission System Operator (TSO) requires the use of short-term load forecasting algorithms. These predictions are essential for appropriate planning of the energy resources and optimal coordination for the generation agents. This study focuses on the development of a prediction model to be applied to the ten main Spanish islands: seven insular systems in the Canary Islands, and three systems in the Balearic Islands. An exhaustive analysis is presented concerning both the estimation results and the forecasting accuracy, benchmarked against an alternative prediction software and a set of modiﬁed models. The developed models are currently being used by the Spanish TSO (Red El é ctrica de España, REE) to make hourly one-day-ahead forecasts of the electricity demand of insular systems.


Motivation and Literature Review
Electric power consumption is omnipresent throughout the world. Among the different types of energy, electricity is crucial for human activity at home, at work, in leisure, in commerce and public activities. Today, electricity cannot be stored in large quantities and must be produced at the same time as it is demanded. This implies that the electric systems operator of any country must be prepared to meet future demand without knowing what it will be in advance. Forecasting of power demand plays an essential role in the electric industry, as it provides the basis for making decisions in power system planning and operation. Having good predictions of hourly electricity demand in the next few hours and for the next few days is crucial for the proper functioning of the electricity system with acceptable costs. The cost of electricity is a determining factor in a country's competitiveness. Improving demand forecasts significantly reduces costs in liberalized electricity markets.
The problem of predicting demand with different time horizons is the subject of a continuous process of improvement. The two pillars of progress are the emergence of new statistical forecasting techniques (e.g., deep learning [1]) and the availability of new data. Improved communications allow quick and convenient access to a large amount of information that substantially changes the mathematical model. For example, new forecasting techniques provide up-to-date information on the weather conditions of a region for the next few days at hourly intervals and with high spatial accuracy. Adding this information to power demand prediction models significantly improves their results. The massive arrival of data requires the development of new techniques, and the combination of new data and new techniques has revolutionized the current systems for predicting electricity demand. This has only just begun, great changes are expected in the electricity sector with the development of new electricity storage systems; with the installation of intelligent meters that provide instantaneous where (1) represents a multiple linear regression model with non-stationary and correlated disturbances u t,h . The vectors of explanatory variables x t and z t are assumed to be known and represent the influence of temperature and special days on energy demand, respectively. They are constant throughout all hours of day t, whereas the parameter vectors to be estimated α h and β h are different for each hour. The disturbances u t,h follow an autoregressive integrated moving-average process with weekly seasonality as in (2), where B is the backshift operator such that B k u t,h = u t−k,h ; ∇ = (1 − B) and ∇ 7 = 1 − B 7 are the regular and seasonal difference operators; d and D are the orders of differencing for both operators; φ h (B), θ h (B), Φ h B 7 and Θ h B 7 are polynomials in B and B 7 , of order p, q, P and Q, respectively; and w t,h are independent random variables with zero mean and variance σ 2 h that can be different for each hour of the day.
The estimation of the previous model is made following the Box-Jenkins methodology. The same Reg-ARIMA model with different parameters is valid for 24 h a day, with some exceptions for some hours of the night, where not all parameters are significant. However, in order to simplify the successive updates of the model, it has been decided to maintain the same structure in all the equations. The following sections describe the mathematical model for the influence of temperature and special days, and the hourly update of predictions. More details can be found in [20].

Model for the Temperature's Influence
It is well-known that temperature has a transcendental impact on energy demand. The problem has great complexity that can be solved in several ways [4,21,22]. Any modeling must consider the following aspects: (i) the effect of temperature on demand changes throughout the year, (ii) in large systems it is necessary to take into account the spatial dimension of the problem and (iii) the effect of the temperature is not instantaneous since the temperature of a day can significantly affect the demand of the subsequent days.
In this work, the use of regression-spline techniques [23] has been used to model the nonlinear relationship between demand and temperature. Basically, it involves dividing the temperature range into sections, defined by a sequence of nodes, and each section fits a polynomial. The employed procedure ensures that the polynomials are jointed smoothly in the nodes x * i : i = 1, 2, . . . , r , and the whole function and its first and second derivatives are continuous. The position of the nodes is chosen depending on the shape of the curve to be adjusted. The number of nodes is determined using cross-validation with out-of-sample predictions. In this case, the model implements three nodes, and the regression-spline model for variable x is built using: R(x, x * ) = x * − 1 2 2 − 1 12 x − 1 2 2 − 1 12 where T is the forecasted temperature of the day for the considered insular system. The location of the forecasted weather variable corresponds to the geographic coordinates of the most populated city of each island. The proposed algorithm considers weather forecasts provided by AEMET (State Meteorological Agency in Spain), not actual observed data. The model also considers a non-instantaneous effect of temperature over energy demand: it is considered that the effect temperature has on demand persists through several days after the actual value was reached. The model implements a four-day persistence: the temperature reached during day t can affect the demand of days t, t + 1, t + 2 and t + 3.
Concerning the weather information, the algorithm works as follows: (i) the TSO receives the weather forecasts from the Spanish meteorological agency on a daily basis, (ii) maximum daily temperature forecasts for the most-populated city of each insular system are obtained from the weather forecasts' files, (iii) the obtained temperature information is transformed using the spline-regression model in Equations (3) to (5), (iv) finally, the estimated model in Equations (1) and (2) for each insular system is used to compute the electric load forecasts for each island.

Model for Special Days
Throughout the year there are several weekdays declared as non-working days or public holidays, for cultural, religious or political reasons. For example, there are six national public holidays during the year in Spain. In addition to the national public holidays, there are a few more days that are also holidays but only in certain cities and/or specific regions. The demand profile for these special days differs considerably from regular days, and, in general, the loads of these days are the most difficult to predict [24]. The existence of a public holiday in the middle of the week may also affect demand on the days before and after the holiday. Also, there is empirical evidence that the decrement in demand for a special day changes depending on the day of the week that is celebrated. The characteristics of each special day have been analyzed considering the information collected over several years. The procedure detects identifiable patterns of behavior that can be used to predict changes in demand for future holidays. The procedure requires many parameters in order to consider all the features described in the preceding paragraphs. The method has been validated exhaustively by cross-validation, and improvements in predictions are considerable. Due to the vast number of possible situations, the model requires many regressors to model the effect of special days on demand.
Each public holiday is modeled as the dummy variable Z w,i t where index w represents the day of the week (w ∈ {Su, Mo, Tu, We, Th, Fr, Sa}) and index i stands for the public holiday considered (public holidays are described in Section 3.2). Dummy variable Z w,i t is equal to one if the d-th day corresponds to the i-th public holiday during the w-th day of the week. For example, the national holiday New Year's Day (1st January) is modeled by means of the set of regressors . Additionally, to consider the effect of previous/following days, the previous set of regressors are tripled: one set for the actual festivity day, one set for the festivity's previous day, and the third set for the festivity's next day. These regressors are represented by the vector z t in (1).
Concerning the holidays' information, the algorithm works as follows: (i) every year, at the end of December, the TSO obtains the next year's work calendar based on the information published in the national and regional Official Gazettes, (ii) once the dates of the holidays are known, the binary vector z t in (1) can be computed for all days of the next year, following the previous guidelines to compute Z w,i t , (iii) one the model is estimated at the end of December, the obtained parameters in Equations (1) and (2) allow for the predicting of the effect of each national/regional holiday over the demand of each insular system, (iv) finally, on a daily basis, this holidays effect is used to compute the electric load forecasts for each island.

Hourly Update of the Prediction
Each of the 24 h models are used to make predictions from one day to the next. Prediction errors for adjacent hours (i.e., w t,h and w t,h−1 ) are highly correlated. Considering that the demand data are received by the system every hour, each time a new observation is obtained, this information is used to update the predictions of the successive hours. The error w t,h is partially predicted using the observed errors of the previous hours, w t,h−1 , w t,h−2 , . . . , w t,h−r using the following equation: where parameters δ h,1 , δ h,2 , . . . , δ h,r are estimated by least squares with the residuals of the 24 Reg-ARIMA models. The number r of terms in the equation is three. To simplify the notation, it has been assumed that all hours correspond to the same day t. In the first hours of the day, it will be necessary to use the previous hours that correspond to the previous day. The parameters δ h,1 , δ h,2 , . . . , δ h,r are different for different hours. This update greatly improves predictions for the next few hours and its effect significantly reduces prediction errors up to a 48-h horizon.

Insular Models
In this chapter, the initial Reg-ARIMA model currently employed in the Spanish mainland electric system is adapted for the main Spanish insular systems.

Spanish Insular Electric Systems
In this work, the ten most-important Spanish insular systems are studied. Spanish territory includes two archipelagoes: the Canary Islands off the coast of Africa, and the Balearic Islands in the Mediterranean Sea. The Spanish TSO computes the load forecasts for seven islands in the Canary Islands, and three insular systems in the Balearic Islands hourly (https://demanda.ree.es).
The names of the analyzed systems are provided in Table 1, as well as the acronyms used hereinafter throughout this document, their approximate geographical location, their population (2019, Spanish Statistical Office), and the average daily electric consumption. It should be noted that the Energies 2020, 13, 3645 6 of 26 latitudes of the three Balearic systems are quite different than the latitude of the seven Canary systems, significantly affecting the temperature and climate. Concerning the weather data, every day Spanish TSO receives from the State Meteorological Agency of Spain (AEMET) both the daily maximum and the daily minimum forecasted temperatures for the most-populated city of each insular system. Figure 1 depicts the annual evolution of these two weather one-day-ahead forecasted variables.
Energies 2020, 13, x FOR PEER REVIEW 6 of 26 Concerning the weather data, every day Spanish TSO receives from the State Meteorological Agency of Spain (AEMET) both the daily maximum and the daily minimum forecasted temperatures for the most-populated city of each insular system. Figure 1 depicts the annual evolution of these two weather one-day-ahead forecasted variables.
From Figure 1 it can be observed that the temperatures in the Canary Islands (red/blue lines) are more regular and stable than the temperatures in the Balearic Islands (orange/purple lines). Due to this characteristic, it is expected that the use of air conditioning and electric heating machines will have a higher impact on the electric load of the Balearic insular systems. Consequently, a higher dispersion of electric demand in Balearic systems is expected.  Figure 2 provides the hourly electric consumption for the analyzed insular systems. Figure 3 depicts a boxplot of the daily electric load for each insular system during the period between 2016 and 2018. Note that the X-axis is on a logarithmic scale. From Figure 1 it can be observed that the temperatures in the Canary Islands (red/blue lines) are more regular and stable than the temperatures in the Balearic Islands (orange/purple lines). Due to this characteristic, it is expected that the use of air conditioning and electric heating machines will have a higher impact on the electric load of the Balearic insular systems. Consequently, a higher dispersion of electric demand in Balearic systems is expected. Figure 2 provides the hourly electric consumption for the analyzed insular systems. Figure 3 depicts a boxplot of the daily electric load for each insular system during the period between 2016 and 2018. Note that the X-axis is on a logarithmic scale.  •

Electric Consumption Data
The systems with lower demand are "El Hierro" (ELHIE) and "Gomera" (GOMER); • The systems with higher demand are "Mallorca" (MALLO), "Gran Canaria" (GRCAN) and "Tenerife" (TENER); • Balearic systems present higher load variability than Canary systems. For instance, the interquartile range of "Mallorca" (most-populated island in the Balearic Islands) is 2.8 GWh, whereas the interquartile ranges of "Gran Canaria" and "Tenerife" (most-populated islands in the Canary Islands) are 0.7 and 0.8 GWh, respectively. There is a significant difference in the data dispersion, considering the similarity of average daily demands and populations of the three previous systems.

National and Regional Holidays
In this section, the developed model for special days in insular systems is detailed. Concerning the holidays' effect, the mathematical model currently used to predict the electric demand on the Spanish mainland includes a set of regressors for each Spanish national holiday (see Table 2) and an additional regressor ranging from 0 to 1 modeling the percentage of the population affected by   • The systems with lower demand are "El Hierro" (ELHIE) and "Gomera" (GOMER); • The systems with higher demand are "Mallorca" (MALLO), "Gran Canaria" (GRCAN) and "Tenerife" (TENER); • Balearic systems present higher load variability than Canary systems. For instance, the interquartile range of "Mallorca" (most-populated island in the Balearic Islands) is 2.8 GWh, whereas the interquartile ranges of "Gran Canaria" and "Tenerife" (most-populated islands in the Canary Islands) are 0.7 and 0.8 GWh, respectively. There is a significant difference in the data dispersion, considering the similarity of average daily demands and populations of the three previous systems.

National and Regional Holidays
In this section, the developed model for special days in insular systems is detailed. Concerning the holidays' effect, the mathematical model currently used to predict the electric demand on the Spanish mainland includes a set of regressors for each Spanish national holiday (see Table 2) and an additional regressor ranging from 0 to 1 modeling the percentage of the population affected by   • The systems with lower demand are "El Hierro" (ELHIE) and "Gomera" (GOMER); • The systems with higher demand are "Mallorca" (MALLO), "Gran Canaria" (GRCAN) and "Tenerife" (TENER); • Balearic systems present higher load variability than Canary systems. For instance, the interquartile range of "Mallorca" (most-populated island in the Balearic Islands) is 2.8 GWh, whereas the interquartile ranges of "Gran Canaria" and "Tenerife" (most-populated islands in the Canary Islands) are 0.7 and 0.8 GWh, respectively. There is a significant difference in the data dispersion, considering the similarity of average daily demands and populations of the three previous systems.

National and Regional Holidays
In this section, the developed model for special days in insular systems is detailed. Concerning the holidays' effect, the mathematical model currently used to predict the electric demand on the Spanish mainland includes a set of regressors for each Spanish national holiday (see Table 2) and an additional regressor ranging from 0 to 1 modeling the percentage of the population affected by regional holidays. Each special day is modeled considering that: (i) the effect over demand depends on the day of the week and (ii) it influences also the demand of previous and following days. Most of these holidays have a fixed date "month-day". Other holidays (e.g., Good Friday) are movable, i.e., have no fixed date in the Gregorian calendar and may be aligned with moon cycles or other calendars. The model for holidays employed in the forecaster of the Spanish mainland demand, is adapted for the ten insular system. Additionally, for each insular system, the following regional holiday is included: see Table 3.

Insular Temperature Effect
The location of weather stations for electric load forecasting in extensive areas has been studied in state-of-the-art technical literature [25]. The Spanish mainland load forecaster implements the temperature effect using a spline-regression model for the average maximum daily temperature of the ten most representative locations in Spain: Barcelona, Bilbao, Cáceres, Madrid, Málaga, Murcia, Oviedo, Sevilla, Valencia y Zaragoza.
However, considering the comparatively smaller geographical extension of the insular systems, the temperature effect is implemented using just a single local position of the temperature.

Estimation Results
Real data from ten Spanish insular electric systems are used. This section details the settings employed for the case study.
The demand data have been provided by the Spanish TSO (REE). This information is public and can be accessed through the TSO website (https://demanda.ree.es/demanda.html). The parameters of the Reg-ARIMA model are estimated using twelve years of historical data: hourly demand, temperature forecasts, and past holidays calendar for each system. The temperature information has been provided by the Spanish State Meteorology Agency; each insular model implements the maximum daily temperature for the most-populated city. The calendar information on special days for both national and regional public holidays is obtained from the BOE (National Official Gazette) [26].
Concerning the computational specifications, these models have been implemented in MATLAB using a 64-bit eight-core i7 processor (3.6 GHz max.) with 16 GB of RAM. Model estimation requires around two hours of CPU time for each insular system, and the calculation of one-day-ahead predictions requires less than 20 s per system. Note that model estimation should be performed yearly, whereas forecasting computation should be performed every hour.
Once the 24 Reg-ARIMA hourly models in Equations (1) and (2) have been estimated, the residual standard errorσ h is computed and provided in Table 4 and Figure 4. It should be noted that the residual standard errors quantify the differences between the theoretical mathematical model and the actual data for the estimation period. Consequently, lower standard errors imply higher forecasting accuracy.
Energies 2020, 13, x FOR PEER REVIEW 10 of 26 for both national and regional public holidays is obtained from the BOE (National Official Gazette) [26].
Concerning the computational specifications, these models have been implemented in MATLAB using a 64-bit eight-core i7 processor (3.6 GHz max.) with 16 GB of RAM. Model estimation requires around two hours of CPU time for each insular system, and the calculation of one-day-ahead predictions requires less than 20 s per system. Note that model estimation should be performed yearly, whereas forecasting computation should be performed every hour.
Once the 24 Reg-ARIMA hourly models in Equations (1) and (2) have been estimated, the residual standard error is computed and provided in Table 4 and Figure 4. It should be noted that the residual standard errors quantify the differences between the theoretical mathematical model and the actual data for the estimation period. Consequently, lower standard errors imply higher forecasting accuracy.  From Table 4 and Figure 4, it can be observed that "El Hierro" is the insular system with the lowest consumption and highest residual standard errors. On the other hand, it can be observed that the islands with higher electric demand (such as "Tenerife" and "Gran Canaria") present lower residual standard error. Additionally, insular systems with higher temperature variability (i.e., Balearic systems) present a higher standard error during the main sunlight hours (from 11 a.m. to 8 p.m.).  From Table 4 and Figure 4, it can be observed that "El Hierro" is the insular system with the lowest consumption and highest residual standard errors. On the other hand, it can be observed that the islands with higher electric demand (such as "Tenerife" and "Gran Canaria") present lower residual standard error. Additionally, insular systems with higher temperature variability (i.e., Balearic systems) present a higher standard error during the main sunlight hours (from 11 a.m. to 8 p.m.).

Temperature Effect
The temperature influence for each system depends on the geographical location. In Figure 5, the average daily electric consumption versus the maximum daily forecasted temperature for "Mallorca" (Balearic Islands) and "Gran Canaria" (Canary Islands) is depicted. Blue, red and orange dots represent weekdays, Saturdays and Sundays, respectively.

Temperature Effect
The temperature influence for each system depends on the geographical location. In Figure 5, the average daily electric consumption versus the maximum daily forecasted temperature for "Mallorca" (Balearic Islands) and "Gran Canaria" (Canary Islands) is depicted. Blue, red and orange dots represent weekdays, Saturdays and Sundays, respectively.
In Mallorca (left plot in Figure 5), a convex non-linear relationship between demand and temperature is observed. On the other hand, in Gran Canaria (right plot), this relationship is less clear.
The set of parameters measures the thermic effect on electric consumption, based on regression-spline models. Once the Reg-ARIMA model (1) - (2) has been estimated, the computed values for these parameters quantify the temperature effect for each hour.  Figure 6 depicts the effect of temperature on the electric consumption for several hours, for the ten insular systems. As expected, the effect of temperature is more marked on the Balearic Island's systems, whereas the Canary systems do not exhibit a significant nor consistent relationship. Concerning the Balearic systems, it is observed that the thermic effect differs greatly by the time of day: temperature affects demand less at night than during the rest of the day. In Mallorca (left plot in Figure 5), a convex non-linear relationship between demand and temperature is observed. On the other hand, in Gran Canaria (right plot), this relationship is less clear.
The set of parameters α h measures the thermic effect on electric consumption, based on regression-spline models. Once the Reg-ARIMA model (1)- (2) has been estimated, the computed values for these parameters quantify the temperature effect for each hour. Figure 6 depicts the effect of temperature on the electric consumption for several hours, for the ten insular systems. As expected, the effect of temperature is more marked on the Balearic Island's systems, whereas the Canary systems do not exhibit a significant nor consistent relationship. Concerning the Balearic systems, it is observed that the thermic effect differs greatly by the time of day: temperature affects demand less at night than during the rest of the day.
Energies 2020, 13, x FOR PEER REVIEW 12 of 26 Figure 6. Influence of temperature on Spanish insular electricity demand.

Holidays Effect
The effect of holidays on electric demand is modeled by means of the set of parameters in Equation (1). Once these parameters are computed, its estimated values quantify the percentual reduction of electric consumption for every hour of each special day. For example, the effect of Christmas Day is considered with seven parameters, one for each day of the week. Another seven parameters are used for the previous day (December 24th) and seven more for the following day

Holidays Effect
The effect of holidays on electric demand is modeled by means of the set of parameters β h in Equation (1). Once these parameters are computed, its estimated values quantify the percentual reduction of electric consumption for every hour of each special day. For example, the effect of Christmas Day is considered with seven parameters, one for each day of the week. Another seven parameters are used for the previous day (December 24th) and seven more for the following day (December 26th). Consequently, 7 × 3 = 21 parameters are required to model the Christmas Day effect for each hour (See Section 2.2 for more details). Figure 7 shows the effect of Christmas Day on Monday and the two adjacent days (in columns) depending on the insular system analyzed (in rows). Each point corresponds to a parameter, the abscissa indicates the hour of the day, and the ordinate value represents the estimated value. Each value measures the load reduction (in percentage) of the holiday compared with a normal day. When Christmas Day is celebrated, a significant decrease in demand occurs relative to what would be expected for a working day. This plot can also be generated for all the National/Regional Holidays, and for all the days in the week (for reasons of space limitations, all these plots are not provided in this document). Green dots correspond to no-significant effects (p-value > 0.05), orange dots correspond to significant effects (p-value between 0.0001 and 0.05), and red dots correspond to very significant effects (p-value < 0.0001). The maximum reduction is observed at 9 a.m. and 10 a.m., when demand may decrease up to 30-40%. In some systems (e.g., Mallorca and Menorca), there is a second reduction of demand at 4 p.m. and 5 p.m;  For Balearic systems, there is a significant effect on demand on Tuesday, December 26th. This is due to the celebration of the regional holiday "St Stephen's Day". For Canary systems, the effect over demand on Tuesday is negligible.
The effect of public holidays on hourly electric consumption for each insular system, measured as percentual increment, is provided in Table 5. This table quantifies the average hourly descend of electricity demand for each national or regional holiday that occurs on a weekday.  Figure 7, it can be observed:

•
There is a significant reduction (higher than 10%) of electric consumption during Christmas Day on Monday from 7 a.m. until 11 p.m; • The maximum reduction is observed at 9 a.m. and 10 a.m., when demand may decrease up to 30-40%. In some systems (e.g., Mallorca and Menorca), there is a second reduction of demand at 4 p.m. and 5 p.m; • For Balearic systems, there is a significant effect on demand on Tuesday, December 26th. This is due to the celebration of the regional holiday "St Stephen's Day". For Canary systems, the effect over demand on Tuesday is negligible.
The effect of public holidays on hourly electric consumption for each insular system, measured as percentual increment, is provided in Table 5. This table quantifies the average hourly descend of electricity demand for each national or regional holiday that occurs on a weekday.

Accuracy Performance
In order to evaluate the accuracy of the developed insular models, two forecasting situations are analyzed: (1) predictions made at noon for 24 h of the next day, and (2) predictions made at 10 a.m. for 24 h of the next day. In both cases, electric load forecasts are computed from June 1st, 2018 until June 30th, 2019 (i.e., 13 months). Consequently, the performances of the different insular models are tested using the 395 days of the considered period. The performance metric employed in this work is the Root Mean Square Percentage Errors (RMSPE) for each hour h, computed as: where y t,h andŷ t,h are the observed and forecasted demands for hour h of day t, respectively; and n is the number of days for the analyzed time period.

Forecasting Situation 1: Predictions Made at Noon
In this first situation, electric load forecasts are computed at midnight of day d, generating the 24 h load predictions for day d + 1. It should be noted that these 24 values are one-step-ahead predictions due to the use of 24 h Reg-ARIMA models. Table 6 provides the global RMSPE and the hourly RMSPE for the ten insular models.
From Table 6 it is observed that the insular model with higher forecasting error coincides with the less-populated island ("El Hierro"); whereas the model with the lowest prediction error corresponds with the most-populated island ("Tenerife"). On the other hand, insular electric systems with higher temperature variation along the year (i.e., islands from the Balearic archipelago), usually obtains less accurate forecasts.
These deductions can be graphically visualized by means of the scatter plot between the following variables: population, average consumption and forecasting error. From Figure 8 it is observed that population and electricity consumption are strongly linearly correlated (R 2 = 0.979), whereas the average yearly error is negatively correlated with population and consumption. Additionally, it is observed that the forecasting error is augmented for systems with higher temperature ranges (Balearic Islands).
for 24 h of the next day. In both cases, electric load forecasts are computed from June 1st, 2018 until June 30th, 2019 (i.e., 13 months). Consequently, the performances of the different insular models are tested using the 395 days of the considered period. The performance metric employed in this work is the Root Mean Square Percentage Errors (RMSPE) for each hour h, computed as: where , and , are the observed and forecasted demands for hour h of day t, respectively; and n is the number of days for the analyzed time period.

Forecasting Situation 1: Predictions Made at Noon
In this first situation, electric load forecasts are computed at midnight of day d, generating the 24 h load predictions for day + 1. It should be noted that these 24 values are one-step-ahead predictions due to the use of 24 h Reg-ARIMA models. Table 6 provides the global RMSPE and the hourly RMSPE for the ten insular models. From Table 6 it is observed that the insular model with higher forecasting error coincides with the less-populated island ("El Hierro"); whereas the model with the lowest prediction error corresponds with the most-populated island ("Tenerife"). On the other hand, insular electric systems with higher temperature variation along the year (i.e., islands from the Balearic archipelago), usually obtains less accurate forecasts.
These deductions can be graphically visualized by means of the scatter plot between the following variables: population, average consumption and forecasting error. From Figure 8 it is observed that population and electricity consumption are strongly linearly correlated ( = 0.979), whereas the average yearly error is negatively correlated with population and consumption. Additionally, it is observed that the forecasting error is augmented for systems with higher temperature ranges (Balearic Islands).  The Spanish electricity market is organized as a "day ahead market": each day prior to 10 a.m., all the energy bids for the following day must be presented by the agents to the Market Operator. Thus, for planning reasons, each day d at 10 a.m. the Spanish TSO obtains the 24 h electricity forecasts for the day d + 1. It should be noted that ten values correspond to one-step-ahead predictions, and fourteen correspond to two-step-ahead predictions.
In order to compare the accuracy of predictions, the Spanish TSO provided reference forecasts from the currently working forecasting software at their facilities. Thus, in this second case study: (i) 24 h forecasts for day d + 1 are computed each day d at 10 a.m., and (ii) the accuracy results are compared with another forecasting model. The reference model is based on a hybrid algorithm that combines neural networks and time series models used by the Spanish TSO in its daily predictions. For reasons of space and confidentiality, no details are provided. The aim of this article is to describe how prediction errors vary depending on the size and climate characteristics of the electricity systems. The effects described in this paper do not depend so much on the prediction algorithm used as on the characteristics of the electrical system itself. Tables 7 and 8 provide the global, hourly and monthly RMSPEs for the developed insular models (labeled as "final") and the reference forecaster outputs (labeled as "refer"). Additionally, Tables 9 and 10 provide the percentual improvement for the global, hourly, and monthly RMSPEs for the proposed insular models.
To check if the proposed algorithm provides forecasts with significantly lower errors than the reference predictions, the Diebold-Mariano inference test [27,28] has been used for each system, computing the DM test statistic and its p-value. If the p-value is lower than 0.05, as usual, it can be stated that the accuracy of the proposed method is higher than the reference forecaster. Tables 7-11 show the following: • The proposed insular models outperform the reference forecaster. For the Balearic systems the global RMSPE improvement ranges from 22 to 30%; whereas the improvement for the Canary Islands is minor: from 2 to 8% (Gran Canaria); • Comparing the prediction errors using the Diebold-Mariano test, it is observed that the proposed method provides statistically significant lower errors than the reference forecaster for all insular systems except Tenerife and Palma (with a 95% confidence level); • Concerning the Balearic Islands, the proposed method provides a higher improvement during night and morning hours: from 8 p.m. to 11 a.m. For the three considered systems, the average hourly improvement during these hours is 32% (ranging from 22 to 47%). During the rest of the day, the average hourly improvement is 19% (ranging from 13 to 28%); • Concerning the Canary Islands, the proposed method does not provide a significant enhancement from 7 a.m. to 10 p.m. (the average hourly improvement is 1%) for the seven insular systems. During the rest of the day, the average hourly accuracy increment is 9%, ranging from -1 to 20%; • In general terms, the monthly prediction errors are higher in winter months (November, December and January), and lower in summer: June and July; • For systems with higher temperature variation (i.e., Balearic systems), both forecasters present higher hourly error from 11 a.m. to 10 p.m. On the other hand, Canary systems exhibit a more regular hourly error, for both forecasting models.   To analyze the effect of temperature and holidays on prediction accuracy, the results obtained have been compared with two simplified versions of the model:

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Proposed approach without the temperature effect, where the thermal information is not included in the model; • Proposed approach with a basic model for holiday effect, which considers only the influence of holidays on demand depending on the hour of the day, assuming a similar response from Monday to Friday for all festivities and neglecting the weekend effect. Thus, just a set of 24 regressors has been employed. Figure 9 provides the hourly RMSPE for the previous methods, using the same forecast settings described in Section 4.3.2.
Energies 2020, 13, x FOR PEER REVIEW 24 of 26 Figure 9. Case study 3: hourly RMSE for the prediction period, compared with alternative models. Table 12 provides the global percentual improvement of the proposed method compared with the alternative modified forecasters, for the ten insular systems.   Table 12 provides the global percentual improvement of the proposed method compared with the alternative modified forecasters, for the ten insular systems. Concerning the temperature effect: it can be observed that the model that does not consider the temperature effect provides a higher-error forecast, especially during sun hours, in Balearic systems (Mallorca, Menorca, and Ibiza-Formentera) and the Canary systems with a higher population (Tenerife, Gran Canaria, Lanzarote and Fuerteventura). The forecasting accuracy for smaller systems (El Hierro, La Gomera and La Palma) does not exhibit a significant variation in the case of temperature removal.
Concerning the holidays' effects: the proposed model of holidays provides a significant improvement compared with the forecaster with a basic model for holidays for Tenerife and Gran Canaria during the whole day, whereas the effect in the rest of the systems can be observed between 7 a.m. and 11 a.m. For smaller systems (El Hierro and La Gomera) the differences are almost negligible due to their large hourly errors.

Discussion
In any electric power system, the load forecasting plays a key role in the appropriate management and efficient administration of energy resources. A lack of prediction accuracy leads to an increment in operating costs.
In this work, a mathematical Reg-ARIMA model is developed to predict the electric consumption for ten insular systems located in the two Spanish archipelagos: the Canary and Balearic Islands. Once the ten models have been estimated, the temperature's influence and the effect of special days have been quantified and analyzed.

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The performance of the proposed algorithm has been tested using real data from the years 2018 and 2019, and the forecasting accuracy has been compared with alternative modified models and another prediction software currently used at the Spanish TSO facilities. The developed models prove to be more accurate, robust, and reliable.

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The percentual improvement of accuracy of the proposed method for each Balearic system ranges from 22 to 30%; whereas the average improvement for the Canary Islands is 11% (compared with the reference model).

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The proposed method provides a higher improvement during the night and morning hours: from 8 p.m. to 11 a.m. in the Balearic systems. For the three considered systems, the average hourly improvement during these hours is 32% (ranging from 22 to 47%).

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The temperature effect is very significant in Balearic systems. On the other hand, the festivities effect is significant in the most-populated islands: Gran Canaria, Tenerife and Mallorca.
Future work will focus on including additional weather variables in the developed insular models (e.g., solar radiation, wind speed and direction, cloudiness, etc.); and forecasting the electric consumption of Spanish mainland regional systems.
Author Contributions: J.J. conceived and designed the experiments; E.C. and J.J. analyzed the data, E.C. performed the experiments, E.C. and J.J. wrote the paper. All authors have read and agreed to the published version of the manuscript.
Funding: This work has been funded by Red Eléctrica de España (REE) as an R&D project.