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Article

Forecasting Hydropower with Innovation Diffusion Models: A Cross-Country Analysis

by
Farooq Ahmad
,
Livio Finos
and
Mariangela Guidolin
*
Department of Statistical Sciences, University of Padua, 35121 Padua, Italy
*
Author to whom correspondence should be addressed.
Forecasting 2024, 6(4), 1045-1064; https://doi.org/10.3390/forecast6040052
Submission received: 4 October 2024 / Revised: 10 November 2024 / Accepted: 13 November 2024 / Published: 16 November 2024
(This article belongs to the Section Power and Energy Forecasting)
Browse Figures
Figure 1
<p>Hydroelectricity generation by selected countries.</p> "> Figure 2
<p>American countries: model fits and forecasting.</p> "> Figure 3
<p>European countries: model fits and forecasting.</p> "> Figure 4
<p>Asian and Middle East countries: model fits and forecasting.</p> ">
Versions Notes

Abstract

:
Hydroelectric power is one of the most important renewable energy sources in the world. It currently generates more electricity than all other renewable technologies combined and, according to the International Energy Agency, it is expected to remain the world’s largest source of renewable electricity generation into the 2030s. Thus, despite the increasing focus on more recent energy technologies, such as solar and wind power, it will continue to play a critical role in energy transition. The management of hydropower plants and future planning should be ensured through careful planning based on the suitable forecasting of the future of this energy source. Starting from these considerations, in this paper, we examine the evolution of hydropower with a forecasting analysis for a selected group of countries. We analyze the time-series data of hydropower generation from 1965 to 2023 and apply Innovation Diffusion Models, as well as other models such as Prophet and ARIMA, for comparison. The models are evaluated for different geographical regions, namely the North, South, and Central American countries, the European countries, and the Middle East with Asian countries, to determine their effectiveness in predicting trends in hydropower generation. The models’ accuracy is assessed using Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Mean Absolute Percentage Error (MAPE). Through this analysis, we find that, on average, the GGM outperforms the Prophet and ARIMA models, and is more accurate than the Bass model. This study underscores the critical role of precise forecasting in energy planning and suggests further research to validate these results and explore other factors influencing the future of hydroelectric generation.

1. Introduction

The increase in energy consumption, especially electricity, is directly related to an increase in the world population [1]. This growing demand for energy, if not met sustainably, will harm the environment, particularly due to the air pollution and climate change effects [2] that derive from the burning of fossil fuels such as coal and natural gas [3]. However, fossil fuels, namely coal, oil, and natural gas, still serve as the primary energy sources worldwide for electricity production [4]. To contrast such a reliance on fossil fuels and produce electricity more sustainably, the transition to renewable energy sources has recently accelerated significantly in many countries [5]. Renewable energy includes sources that naturally regenerate, such as hydroelectric, solar, wind, and geothermal energy. With technological advancements, particularly with the initiation of the Fourth Industrial Revolution, the adoption of renewable energy sources is expected to soar. This shift is crucial to alleviating the effects of climate change, greenhouse gas emissions, and increasing energy costs [6].
Hydropower, one of the oldest and most well-researched renewable energy sources, has a history of about 100 years. Despite its longstanding presence, innovation in hydropower remains important, with current efforts centered on enhancing plant flexibility via improvements in turbine design, operational strategies, and digitalization. These innovations are intended to help hydropower plants better address the needs of modern power systems, which encounter more variable energy demands and a growing share of intermittent renewable sources. Reservoir-type hydropower plants, in particular, are well-equipped to offer the necessary emissions-free flexibility for today’s power systems [7]. This critical renewable energy source leverages the power of rivers and reservoirs, playing an essential role in electricity supply, particularly in nations with limited resources [8]. Hydropower is an economical and environmentally friendly option for energy production, providing numerous additional benefits such as flood control, irrigation support, and clean drinking water, alongside electricity generation [9]. These characteristics make hydropower an appealing candidate for increased attention and investment [10]. This explains why hydroelectricity generation rose by nearly 70 TWh (close to 2%) in 2022, reaching 4300 TWh.
Hydropower remains the largest renewable electricity source, producing more than all other renewable technologies combined, and is expected to remain the world’s largest source of renewable electricity generation into the 2030s. Therefore, it will continue to play a critical role in decarbonizing the power system and improving system flexibility. However, it is acknowledged that, without major policy changes, global hydropower expansion is expected to slow down this decade. According to [7], the contraction results from slowdowns in the development of projects in Europe, Latin America, and some Asian countries. However, increasing growth in Asia Pacific, Africa, and the Middle East partly offset these declines. Increasingly erratic rainfall due to climate change is also disrupting hydro production in many parts of the world.
In the Net Zero Emissions by 2050 Scenario, hydropower maintains an average annual generation growth rate of nearly 4% from 2023 to 2030, aiming to supply approximately 5500 TWh of electricity per year. As stated in [7], in the last five years, the average growth rate was less than one-third of what is required, signaling a need for significantly stronger efforts, especially to streamline permitting and ensure project sustainability. Hydropower plants should be recognized as a reliable backbone of the clean power systems of the future and supported accordingly. In many advanced countries, hydropower plants were built between the 60s and the 80s, so nowadays almost half of the global plants are at least 40 years old. This implies that major modernization is required to maintain or improve their performance and increase their flexibility. In addition to renewing major equipment such as turbines and generators, investing in modernization and digitalization can significantly increase plant flexibility, make the plant safer, and resolve environmental and social problems such as inadequate drought management and flood control, depending on the country’s regulations, [7]. In this scenario, planning is crucial for the renovation of hydropower plants because of possible changes in the level of water flows since the plants first became operational.
Starting from this evidence on the central role of hydropower in energy transitions and the importance of planning its future, our study proposes a forecasting analysis on the evolution of hydroelectric data in a selected group of countries. We use the approach of Innovation Diffusion Models [11], whose application to energy growth processes has been developing significantly in recent years. Our paper aims to contribute to the literature on the usage of Innovation Diffusion in energy contexts, by proposing a cross-country analysis of hydropower generation. Specifically, we provide forecasts based on Innovation Diffusion Models and compare their performance with typical time-series models such as the ARIMA models [12] and the more recent Prophet model developed by [13]. The analysis serves the purpose of better understanding the global trends of hydroelectricity, by also providing methodological insight into the forecasting ability of the Innovation Diffusion approach.
The rest of the paper is organized as follows: Section 2 offers a review of the existing literature, setting the stage by discussing previous research and developments relevant to hydropower forecasting. Section 3 outlines the data sources and methods employed in this study. Section 4 presents a detailed analysis of the results, whereas Section 5 is dedicated to a discussion of the overall findings. Finally, Section 6 provides concluding remarks, summarizing the key insights of the study and offering recommendations for future research.

2. Background Literature

In the hydroelectric generation forecasting literature, various statistical, econometric, machine learning, and hybrid models have been widely used. These methods differ in methodology, complexity, and performance [14]. The most widely used methods for forecasting hydroelectric generation have been linear time-series models such as ARIMA models and their different extensions [15]. For example, in a study using data from India, the authors used ARIMA models to model and forecast hydroelectric generation, which accounted for more than 60% of global renewable energy [16]. The study used various ARIMA models to analyze historical data from 1971–1972 to 2019–2020. The researchers determined that the ARIMA(1,1,1) model with drift was suitable to forecast the energy demand of the country. In [17], ARIMA models were used with monthly data obtained from the Son La hydroelectric plant in Vietnam, covering the period from January 2015 to December 2019. Similarly, in [18], data from the official site of the Electricity Regulation and Control Agency (ARCONEL) were analyzed for the years 2000 to 2015, focusing on monthly reports on energy production from hydroelectric plants in Ecuador. The results indicated that the ARIMA ( 1 , 1 , 1 ) x ( 0 , 0 , 1 ) 12 model, which incorporated seasonality, best fits the time-series data, allowing for forecasts of energy production for one to twelve months ahead. This model was trained using data from 2000 to 2014 and validated with data from January to December 2015, and its forecast for 2020 suggested an increase in monthly production, with actual values falling within the confidence intervals of the ARIMA model’s predictions with annual seasonality.
In [19], the authors introduced a new gray combination optimization model to forecast China’s hydropower generation, using data from 2000 to 2020. They split the data into a training set (2000 to 2015) and a test set (2016 to 2020). The TDGM (three-parameter discrete grey model) outperformed ARIMA and SVM (Support Vector Machines), achieving a low Mean Relative Forecast Percentage Error (MRFPE). The forecast results indicated that China’s hydropower generation could reach 1687.738 billion kWh by 2025, reflecting a 24.5% increase from 2020. In [20], the authors presented a novel ensemble forecast model to predict medium- to long-term wind and hydroelectric generation, using data from November 2010 to December 2020. The model involved three phases: Phase I combined ARIMA and Bi-LSTM predictions, Phase II incorporated forecasts for seasonal and off-season periods using the Deliberate Search Algorithm (DSA), and Phase III merged the predictions from both phases. The results showed that the Mean Absolute Error (MAE) for wind and hydropower ranged from 1.97% to 5.52% and 2.3% to 6.42%, respectively, while the Root Mean Square Error (RMSE) ranged from 2.7% to 7.8% and from 2.63% to 8.4%, for timeframes from one week to the next year.
In this paper, we propose to use the different approaches of Innovation Diffusion Models to describe and forecast the growth of hydroelectric power. This choice relies on a well-established stream of literature that has employed diffusion models to study the growth of renewable energy sources, to understand the dynamics underlying energy transition by using both univariate (see, for instance, [21]) and bivariate models (see [22]). Recent reviews of this literature may be found for instance in [11,22,23,24]. However, the analyses developed in this literature have typically focused on renewable energy sources like wind and solar ([21]), which are perceived as the most recent and disruptive technologies in the energy transition. Wind and solar power have quite a short history, starting in the mid-1990s, and have been showing an exponential increase during the 2000s, which nonlinear growth models capture well. The literature on wind and solar diffusion has generally focused on explaining their growth as a combination of consumer decisions and the positive effects of incentive measures. In contrast, hydroelectricity has been less studied because it is often seen as a mature technology that poses fewer challenges in both substantial and modeling terms. Through this paper, we aim to question this view, since hydropower projects have longer pre-development, construction, and operational timelines than other renewable energy technologies, and investment risks are higher, requiring specific policy instruments and incentives as well as a longer-term policy perspective and vision [7], for which suitable forecasting is crucial.
The papers employing the Innovation Diffusion approach have tried to describe and explain the growth patterns of renewable energy technologies without paying great attention to their forecasting ability compared to other models. Whereas the focus has been especially placed on the explanatory power of such models, given the good interpretability of the parameters, we take a different perspective here and use Innovation Diffusion Models as a tool to make forecasting analyses.

3. Materials and Methods

In this section, we provide details of the data used in the analysis and the models that have been employed for forecasting. Specifically, we offer a description of the BM, GGM, Prophet, and ARIMA models.

3.1. Data

The analysis presented in this paper is based on data from the Energy Institute Statistical Review [25], which covers multiple countries and ensures a high quality of data. However, for some countries, there were missing values or time series that were too short, preventing an acceptable analysis with the proposed models. For instance, in the African continent, several countries only started operations in 2011 and the available data were not sufficient for reliable modeling. We, therefore, decided to carefully select the countries and regions where data quality and length were sufficient to provide reasonable analyses.
Figure 1 shows hydroelectric generation data for each region considered from 1965 to 2023. Although most countries displayed in the Figure 1 cannot be observed in detail, the illustration is useful to highlight a generally positive trend in the data starting from the 1980s, proving a significant reliance on hydropower across the world. The figure shows that Canada and the US have generated a much larger amount of electricity from hydropower than most countries, which have maintained lower generation amounts. It may be also noticed that India and Norway have significantly increased their hydroelectricity generation over time, whereas some countries have reduced their reliance on this source.
To analyze the evolution of hydropower in these countries, we have employed four different approaches, starting from the Bass model in Section 3.2.1 and the Guseo–Guidolin model in Section 3.2.3, whose performance is compared with other concurrent models, namely the Prophet in Section 3.2.4 and ARIMA models in Section 3.2.5.

3.2. Models

This section is dedicated to a description of the models used for the forecasting analysis.

3.2.1. Bass Model

The Bass model, BM from now on, presents a depiction of the life cycle of an innovation, showing the stages of introduction, growth, maturity, and decline. It is crucial to note that this model was initially developed in the realm of marketing science and aims to demonstrate the evolution of a new product’s growth over time, but then it was discovered that it is also suitable for studying the diffusion of energy technologies [11]. The model operates on the assumption that two primary sources of information influence consumption decisions: external factors such as the media and institutional communication, and internal factors such as imitation and learning from others. One notable advantage of the BM is its ability to effectively explain the initial phase of diffusion, which is attributed to the presence of innovators. There is a significant body of literature on the role of innovators, also known as early adopters [26]. However, it is the BM that explicitly accounts for their role. The BM is formally represented by a first-order differential equation, as follows:
z ′ ( t ) = p + q z ( t ) m [ m − z ( t ) ] , t > 0 .
In Equation (1), z ( t ) is the cumulative number of adoptions at time t, and z ′ ( t ) is defined as the variation over time of adoptions. Parameter m is the market potential, the maximum number of realizable sales within the diffusion, and its value is assumed to be constant throughout the entire process. The residual market, m − z ( t ) , is affected by the coefficients p and q. Parameter p, called the innovation coefficient, represents the effect of the external influence due to media and institutional communication. Parameter q, called the imitation coefficient, is the internal influence, whose effect is modulated by the ratio z ( t ) / m . The parameters p and q are utilized to measure the two distinct classifications of consumers mentioned earlier, namely the innovators and the imitators.
The closed-form solution of the BM equation can be expressed as
z ( t ) = m ( 1 − e − ( p + q ) t ) ( 1 + q p e − ( p + q ) t ) t > 0 .
Three parameters, m, p, and q, define the dynamics of the diffusion process in terms of cumulative sales, z ( t ) , in Equation (2). The market potential m is a scale parameter that enables the modeling of the diffusion process in absolute terms, whereas parameters p and q, as in Equation (2), act on the speed of diffusion. The cumulative process has an s-shaped pattern and approaches saturation level, denoted by the parameter m, at varying rates based on the values of the parameters p and q [11].

3.2.2. Dynamic Market Potential

After a comprehensive examination of the research conducted by [27], it becomes evident that there exists potential to generalize the concept of BM, taking into account the dynamic nature of the market potential. In light of this, it is possible to formulate the variable m ( t ) in a way that accurately depicts the inherent complexities and fluctuations associated with market dynamics
z ′ ( t ) = m ( t ) p + q z ( t ) m ( t ) 1 − z ( t ) m ( t ) + m ′ ( t ) z ( t ) m ( t ) , t > 0 .
Equation (3) characterizes the instantaneous adoptions z ′ ( t ) as a sum of a BM with m ( t ) and a factor m ′ ( t ) z ( t ) / m ( t ) , which allocates a fraction of the market potential variation m ′ ( t ) to z ′ ( t ) , specifically the growth rate z ( t ) / m ( t ) . Within Equation (3), the market potential variation m ′ ( t ) impacts the instantaneous adoptions z ′ ( t ) , which can either be positive and reinforcing if m ( t ) is increasing, or negative if m ( t ) is decreasing. This demonstrates that the adoption of a product receives an additional advantage from an expanding market potential, while a declining market weakens the process. Equation (3) can be conveniently rearranged as follows
z ′ ( t ) m ( t ) − z ( t ) m ′ ( t ) m 2 ( t ) = z ( t ) m ( t ) ′ = p + q z ( t ) m ( t ) 1 − z ( t ) m ( t ) .
The generalization of BM, where the function m ( t ) depends on time, has a closed-form solution
z ( t ) = m ( t ) ( 1 − e − ( p + q ) t ) ( 1 + q p e − ( p + q ) t ) .
Equation (5) demonstrates that m ( t ) can be an independent function that influences the dynamics of the diffusion process, represented by parameters p and q. The specific form of m ( t ) can vary depending on the assumptions made about the market potential. In [27,28], certain structures for m ( t ) have been proposed. The GGM is based on one of these possibilities.

3.2.3. GGM

In [27], authors postulated a specific specification for m ( t ) , under the assumption that the growth of the market potential is contingent upon a communication process regarding the innovation. This process typically precedes the adoption phase and serves the purpose of “building” the market. More precisely, the dynamic market potential m ( t ) is defined according to a structure that resembles a BM, as follows:
m ( t ) = K 1 − e ( p c + q c ) t 1 + q c p c e ( p c + q c ) t .
In Equation (6), the parameters p c and q c govern the communication process. The parameter p c characterizes the behavior of innovative consumers who initiate discussions about the new product, while q c represents the forces that propagate the information, causing it to become “viral”. The parameter K shows the asymptotic behavior of m ( t ) when all informed consumers ultimately become adopters. The Guseo–Guidolin model, GGM from now on, exhibits the following cumulative structure
z ( t ) = K 1 − e ( p c + q c ) t 1 + q c p c e ( p c + q c ) t 1 − e ( p s + q s ) t 1 + q s p s e ( p s + q s ) t , t > 0 .
In Equation (7), the cumulative adoptions, z ( t ) , are depicted as the product of two distinct phases, namely, the communication phase with parameters p c and q c , and the adoption process with parameters p s and q s . Thanks to this specification that added two parameters, the GGM can handle several diffusion patterns, departing from the typical bell-shaped form of the BM, and can describe nonlinear structures in the data more flexibly [27]. This characteristic makes the GGM preferable over the simple BM in several cases.

3.2.4. Prophet Model

In [13], the authors proposed an innovative model able to handle a time series with a nonlinear trend, seasonality, and other possible effects appearing in the data. The mathematical components of the Prophet model are defined as
y ( t ) = g ( t ) + s ( t ) + h ( t ) + ϵ t .
In Equation (8), g ( t ) is the trend function that models non-periodic changes in the value of the time series, s ( t ) represents periodic changes (e.g., weekly and yearly seasonality), and h ( t ) represents the effects of holidays that occur on potentially irregular schedules over one or more days. The error term ϵ t represents any idiosyncratic changes that are not accommodated by the model, for which we assume that ϵ t is normally distributed. In this approach, both the nature of the time series (piece-wise trends, multiple seasonality, floating holidays) as well as the challenges involved in forecasting are accounted for. The first one is nonlinear growth denoted by g ( t ) in Equation (8). This sort of growth is typically modeled by the logistic growth model, defined as
g ( t ) = C a + e x p ( − k ( t − M ) ) .
where C is the carrying capacity, k the growth rate, and M an offset parameter. If the trend is linear, then the growth model is defined as
g ( t ) = ( k + a ( t ) T δ ) t + ( M + a ( t ) T γ )
where k is the growth rate, δ has the rate adjustments, M is an offset parameter, and γ j is set to − s j δ j to make the function continuous. With s j , the change points are defined. The seasonality effect is approximated by a standard Fourier series, given as
s ( t ) = ∑ n = 1 N a n cos 2 π n t P + b n sin 2 π n t P .
In Equation (10), N is the count of Fourier components, P shows periods, and a n , b n represents the Fourier coefficients. The above components then amalgamate into an additive model. The effect of h ( t ) on holidays and events that provide predictable shocks often does not follow any specific periodic pattern [13].

3.2.5. Auto-Regressive Integrated Moving Average Model (ARIMA)

One of the traditional statistical techniques that is used frequently for time-series forecasting is the ARIMA model (Auto-Regressive Integrated Moving Average) [12]. To model and predict time-series data, it integrates moving average (MA), differencing (I), and autoregressive (AR) components. ARIMA models accommodate trends and seasonality, as well as linear relationships within the data. A general ARIMA model can be written as
y t ′ = c + ϕ 1 y t − 1 ′ + ⋯ + ϕ p y t − p ′ + θ 1 ε t − 1 + ⋯ + θ q ε t − q + ε t ,
where y t ′ is the differenced series, and the “predictors” on the right-hand side include both lagged values of y t and lagged error terms. We call this an ARIMA( p , d , q ) model, where
  • p = order of the autoregressive part;
  • d = degree of first differencing involved;
  • q = order of the moving average part.
To form more complicated models, the backshift notation is often used. For example, Equation (11) can be written in backshift notation as
( 1 − ϕ 1 B − ⋯ − ϕ p B p ) ( 1 − B ) d y t = c + ( 1 + θ 1 B + ⋯ + θ q B q ) ε t
where ( 1 − ϕ 1 B − ⋯ − ϕ p B p ) is A R ( p ) part, ( 1 − B ) d is d differences, while ( 1 + θ 1 B + ⋯ + θ q B q ) is MA(q) part. Selecting appropriate values for p, d, and q can be difficult, but usually, this is performed by using selection criteria such as the AIC or error measures like the Root Mean Squared Error (RMSE) [12].

3.3. Evaluation Metrics

Three evaluation metrics are considered to measure the performance of the selected models after careful consideration: Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Mean Absolute Percentage Error (MAPE).

3.3.1. Mean Absolute Error (MAE)

MAE works on averaging the squared discrepancies between expected and actual values. The Mean Absolute Error, or MAE, provides a quantitative assessment of prediction accuracy while highlighting the greater errors [12].
MAE = 1 n ∑ i = 1 n ( | y i − y ^ i | )

3.3.2. Root Mean Squared Error (RMSE)

To understand RMSE, it is necessary to first show the formula for Mean Squared Error (MSE). MSE calculates the average of the squared discrepancies between the predicted and actual values. It provides a quantitative assessment of prediction accuracy, with a greater emphasis on larger errors [29].
MSE = 1 n ∑ i = 1 n y i − y ^ i 2
Taking the square root of the MSE gives the Root Mean Squared Error (RMSE):
RMSE = MSE

3.3.3. Mean Absolute Percentage Error (MAPE)

The MAPE averages’ percentage difference between the expected and actual values is measured by MAPE (Mean Absolute Percentage Error), which offers a comparative assessment of predicting accuracy [30].
MAPE = 100 % n ∑ i = 1 n y i − y ^ i y i

4. Results

In the Results section, we evaluate the proposed models’ performance in forecasting hydropower generation data. The datasets analyzed are split into a training set of 53 points (marked in black) and a test set of 6 points (marked in red), allowing us to assess each model’s performance.
Through this analysis, we examine how well each model captures the observed patterns and trends in hydroelectric generation in different regions and periods. We consider metrics such as accuracy, predictive capability, and goodness of fit, to identify the model that most accurately predicts the data. The detailed findings are discussed in the following subsections, clustering the countries according to their geographical position: America, Europe, Asia, and the Middle East.
Further results, based on a training set of 47 data points and a test set of 12 points, are presented in the Appendix A. These confirm the findings presented in this section.

4.1. American Countries

In Figure 2, we illustrate a comparison between the four forecasting models—BM, GGM, Prophet, and ARIMA—using data from American countries spanning the years 1965 to 2023. The figure shows the performance of each model in the countries analyzed. We note that the BM, displayed in green, generally underestimates data during rapid growth periods but performs better in countries like Mexico, where the data has a peak followed by a decline. On the other hand, the GGM, shown in purple, often surpasses the Prophet (blue) and ARIMA (yellow) models, especially in North American countries such as Mexico and the US, but also performs well in Argentina, Ecuador, the Caribbean, and Venezuela. This highlights that the GGM may be better suited for capturing complex trends and regional dynamics. The Prophet model is more accurate in scenarios of continuous growth but tends to misinterpret fluctuating data, resulting in either overestimation or underestimation. This behavior is seen consistently across different countries. As a general observation, we may conclude that the GGM seems to be the best modeling option; however, we need to notice that in countries where a significant jump in the data is observed, with sudden increases or decreases, the GGM model can struggle to accurately predict those points. For instance, as shown in Figure 2, in Ecuador, a sudden increase is observed, and the GGM underestimates the prediction, so in this case, an ARIMA model performs better.
Although these visual insights are useful to provide a first comparison of the models, they must be supported by quantitative evaluations for a thorough assessment. Further analysis using quantitative metrics from the model-fitting results (Section 4.4) is introduced to validate these observations and provide a complete overview of the models’ performance.

4.2. European Countries

Figure 3 shows a comparative analysis of the models applied to data from European countries. We may see that the countries show either a quite smooth and increasing trend, like Iceland, or more constant behavior, like in the case of Germany or France. In the European case, there has been some difficulty in determining the best-performing model. Figure 3 illustrates that almost all models are in good agreement with the data, suggesting some stability in hydroelectric production in these European countries. To better assess the performance of the models, specific countries with more significant deviations have been selected. For instance, in Austria, the GGM and Prophet models exhibit a similar performance, while the BM underestimates the data. In contrast, in countries like Iceland, where hydroelectric production is on the rise, all models tend to overestimate the data, but the GGM shows the closest alignment with the original data among the four models. Visually, the GGM’s performance is better compared to the BM and Prophet model in most countries. However, given some difficulties in determining the best-performing model on a visual basis only, the analysis will benefit from the consideration of more metrics to provide a clearer distinction between models, helping to identify the most precise and reliable model to forecast hydroelectric generation in European countries.

4.3. Asian and Middle East Countries

Figure 4 presents the alignment results of the hydroelectric generation model for Middle Eastern and Asian countries where data are available. Since hydroelectric generation in the Middle Eastern region tends to be lower compared to other regions, we combined the Middle Eastern countries with Asian countries for a more complete analysis. The graphical inspection of the results shows that in Middle Eastern countries, particularly in Iran, all models performed similarly. However, in Iraq and Egypt, the GGM model outperformed the others significantly, while the BM and Prophet occasionally either overestimated or underestimated hydroelectric generation. In contrast, Figure 4 also highlights the fit of the model for hydroelectric generation in Asian countries, including Australia, India, Indonesia, and New Zealand, where substantial investments in hydroelectric projects have led to significant increases over time. Although most models fit the data well, there are noticeable performance differences. For instance, the GGM generally outperforms the concurrent models across most countries, except in the case of Vietnam, where performance varies. Similarly to the Middle Eastern countries, the BM and Prophet sometimes overestimate or underestimate hydroelectric generation. As before, to further refine the analysis, we will evaluate performance metrics to determine which model offers the best predictive ability.

4.4. Evaluation Metrics

Table 1 presents the average performance metrics, namely Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Mean Absolute Percentage Error (MAPE), for the evaluated models. The BM demonstrates an average MAE of approximately 9.7652, indicating that on average the model deviates about 9.7652 units from the actual hydropower generation values. The corresponding average RMSE of 10.456 reflects the typical magnitude of errors, while the MAPE, averaging around 24.525%, represents the average percentage deviation of the model predictions from the actual hydroelectric generation values. However, the GGM exhibits superior performance with an average MAE of approximately 5.1979, indicating smaller deviations from the actual hydroelectric generation values than the BM. The average RMSE of 5.898 suggests smaller errors in magnitude, and the MAPE, averaging 15.498%, indicates a lower average percentage deviation from the actual values. Regarding model performance, the Prophet model shows moderate results, with an average MAE of approximately 5.216, falling between the BM and the GGM. The corresponding average RMSE of 6.0988 represents the typical magnitude of errors, while the MAPE, averaging around 18.796%, represents the average percentage deviation from the actual hydroelectric generation values. The ARIMA model also shows a strong performance, with an average MAE of approximately 5.788. This suggests smaller deviations from the actual values compared to the BM. The average RMSE of 6.7488 indicates the typical error magnitude, and the MAPE, averaging 16.1219%, suggests a lower average percentage deviation from the actual values.
To further validate our findings, model performance has been also evaluated through the Akaike Information Criterion, AIC, showing that the GGM model is lower than that of the other models. This indicates that the GGM model provides a better fit to the data while balancing model complexity, reinforcing its suitability for forecasting in this context. Overall, the GGM demonstrates superior performance compared to the BM, ARIMA, and Prophet models in terms of all the metrics considered, suggesting that it may be the most accurate model for predicting hydropower generation in the considered datasets.
Table 2 presents a comparison of the mean absolute percentage error (MAPE) for the four models. The rows represent the models being evaluated, while the columns denote the models they are compared against. Starting with the Prophet model, which outperformed the BM in 30 of the 43 countries evaluated. Furthermore, Prophet did better than the ARIMA model in 21 countries and surpassed GGM in 17 countries. Next, looking at the ARIMA model, we see that it was outperformed by the Prophet model in 22 countries. The BM outperformed ARIMA in 31 countries, and GGM did better in 17 countries. However, the GGM showed impressive results. It outperformed the Prophet model in 26 countries, the BM in 33 countries, and the ARIMA model in 26 countries. This highlights GGM’s consistent performance across different regions. The BM had more mixed results. It was outperformed by the Prophet model in 13 countries, by the GGM in 10 countries, and by the ARIMA model in 12 countries.
From these results, it is clear that GGM generally has a lower MAPE, which indicates that it provides more accurate forecasts for hydropower data compared to the other models. This suggests that GGM is the most reliable model for predicting hydroelectric generation in various countries, making it a valuable tool for energy planning and forecasting. By examining these comparisons, we can see that, while each model has its strengths, GGM consistently delivers the most accurate predictions. This analysis underscores the importance of selecting the right model for forecasting to ensure effective energy planning and management (the Appendix A provides a detailed breakdown of the results).

5. Discussion

Our analysis has shown that in highly developed countries, like the US or European ones, hydropower generation levels have not been growing in recent years, while less-developed countries generally have experienced increasing power generation over time. Among the models studied, the GGM proved superior in accurately predicting hydroelectric power generation, effectively capturing complex nonlinear patterns. The ARIMA model performed well in cases with a certain degree of variability in the data, while the BM was more effective with smoother trends. The Prophet model produced a good performance with stable data, and the GGM model consistently outperformed others in cases with small variations in power generation. The performance of the GGM was validated through quantitative metrics such as mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE). On average, the GGM model (5.1979 MAE, 5.8983 RMSE, and 15.4978 MAPE) slightly outperformed the Prophet model (5.2163 MAE, 6.0988 RMSE, and 18.7966 MAPE) and the ARIMA model (5.7881 MAE, 6.7488 RMSE, and 16.1219 MAPE), and was significantly more accurate than the BM (9.7652 MAE, 10.4561 RMSE, and 24.5253 MAPE). In all countries examined, the GGM consistently achieved lower MAPE scores than the other models, demonstrating its reliability and precision in predicting hydroelectric generation. It should be acknowledged that the differences in performance of the considered models are sometimes quite small, and their computational complexity is essentially comparable. Still, the proposed analysis has been useful in proving the possibility of using the GGM as an efficient forecasting tool.

6. Conclusions

The world is currently facing critical challenges from climate change, and the growing demand for reliable and sustainable energy sources and the shift to renewable energy is widely regarded as a primary solution to these urgent issues. Making accurate predictions of renewable energy sources is essential for effective energy planning and management, with hydropower still playing a central role in energy transition. Our research conducted an in-depth analysis of hydropower generation data from a selected group of countries categorized into distinct regions, covering the years 1965 to 2023. This dataset enabled us to evaluate and compare various models for forecasting hydroelectric production patterns in regions including the Americas, Europe, Asia, and the Middle East. Despite significant variations in data generation observed within each region, we were able to capture some regularities and some general insights.
From a methodological point of view, our analysis highlights the GGM’s ability to capture the evolving patterns of hydropower across most analyzed countries. This opens up a new perspective on the use of such models, which have typically been employed for descriptive purposes rather than predictive analysis. Hydropower data present various patterns, whether increasing, flat, or declining, necessitating a flexible modeling structure. The GGM can efficiently capture data nonlinearities. These findings underscore the importance of models like the GGM in enhancing our understanding and forecasting capabilities amid energy transitions. The insights gained can equip policymakers and energy planners to make informed decisions, steering their countries toward sustainable energy futures on a global scale. The further validation of our findings can be achieved through several approaches. First, considering additional data from more recent years would allow us to test the models on updated hydropower generation trends. Extending the timeframe would also enable us to capture more long-term trends, especially in regions where hydropower infrastructure has seen significant development more recently. Applying the models to different regions, particularly areas with distinct socio-economic or environmental conditions, would help assess the generalizability of our approach. Regions that face unique challenges in hydropower generation, such as variable climate conditions or different regulatory frameworks, could offer valuable insights into how adaptable our models are.

Author Contributions

Conceptualization, M.G., F.A. and L.F.; methodology, M.G.; software, M.G.; validation, F.A., M.G. and L.F.; formal analysis, F.A. and M.G.; investigation, M.G. and L.F.; resources, F.A. and M.G.; data curation, F.A.; writing—original draft preparation, F.A. and M.G.; writing—review and editing, M.G. and F.A.; visualization, F.A. and M.G.; supervision, M.G. and L.F.; project administration, F.A., M.G. and L.F. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The data used in this study are available at: https://www.energyinst.org/statistical-review (accessed on 15 May 2023).

Conflicts of Interest

The authors declare no conflicts of interest.

Appendix A

In this appendix, we provide detailed results concerning the four models applied to the selected countries. First, we present error metrics for each of the countries analyzed, whereas in Table A2, we show the results based on data divided into a 47-data-point training set and a 12-data-point test set (80% and 20%). This exercise confirms the findings already shown in Section 4.
Table A1. Overall model results.
Table A1. Overall model results.
CountryModelMAEMSERMSEMAPEAIC
CanadaBM66.08434504.800267.117817.218956.4774
GGM24.4791689.597226.26026.329645.2166
Prophet10.1052129.639011.38592.635435.1885
ARIMA30.10461144.976033.83757.953748.2588
MexicoBM5.245433.53865.791320.278927.0762
GGM5.263832.07535.663519.359526.8085
Prophet5.360147.85836.918022.563329.2095
ARIMA5.393847.46686.889622.672229.1602
USBM41.92501974.402244.434215.439451.5281
GGM16.5125293.978717.14586.198940.1010
Prophet20.1576737.060527.14898.171445.6160
ARIMA16.4345472.996421.74856.650942.9545
ArgentinaBM5.046139.31806.270417.902228.0301
GGM3.556617.14664.140814.364723.0508
Prophet7.478974.98378.659332.573631.9036
ARIMA9.6669114.163710.684741.321834.4258
ChileBM1.49984.54282.13146.957815.0813
GGM2.50579.42303.069712.550419.4589
Prophet3.240914.46733.803616.091322.0314
ARIMA2.624010.93603.307013.187120.3523
ColombiaBM18.4775374.994319.364831.482641.5615
GGM5.511936.92426.07659.313227.6532
Prophet8.051283.53899.140013.445932.5519
ARIMA3.961526.28985.12747.337625.6151
EcuadorBM14.7009220.350914.844260.423238.3713
GGM6.617145.16766.720727.292628.8623
Prophet9.443690.89509.533938.822533.0582
ARIMA1.69903.16181.77816.978812.9069
PeruBM6.293340.78516.386320.493728.2499
GGM3.885617.38794.169912.575223.1346
Prophet5.104527.94495.286316.576625.9814
ARIMA1.39652.80751.67564.640712.1937
VenezuelaBM10.6718157.580512.553115.929336.3596
GGM4.476229.49645.43116.877426.3056
Prophet16.2168266.105416.312724.593739.5034
ARIMA8.455675.96518.715812.818431.9816
Central AmericaBM7.291664.81088.050525.842531.0288
GGM2.519710.19683.193210.138619.9325
Prophet3.863219.74634.443713.657523.8978
ARIMA2.52289.50543.08319.995219.5112
Other CaribbeanBM0.22230.06660.258111.4292−10.2538
GGM0.33750.13470.367018.8181−6.0260
Prophet0.33340.13180.363018.5663−6.1602
ARIMA0.79770.71340.844644.68483.9736
Other South AmericaBM25.6682686.345426.198243.629445.1883
GGM18.9569427.937420.686634.027042.3539
Prophet12.9583209.033514.458023.450838.0550
ARIMA16.9094347.969618.653930.460441.1127
AustriaBM5.446335.40855.950513.762027.4017
GGM1.89156.38062.52605.042717.1196
Prophet2.29388.30022.88106.223818.6977
ARIMA2.918314.08043.75247.977321.8687
Czech RepublicBM0.18600.05840.24179.5313−11.0399
GGM0.18560.05840.24179.5394−11.0404
Prophet0.22880.08930.298812.2814−8.4955
ARIMA0.26230.08700.295012.3346−8.6532
FinlandBM1.36642.78351.66849.088412.1422
GGM1.56613.72131.929110.330013.8845
Prophet1.23841.71711.31048.82869.2438
ARIMA1.21991.96351.40128.965910.0482
FranceBM11.0142149.364812.221518.549436.0383
GGM5.754940.50216.364110.361828.2081
Prophet7.156390.39409.507614.210833.0251
ARIMA5.816360.71727.792111.217930.6374
GermanyBM1.58823.44031.85488.730913.4134
GGM0.97191.05451.02695.16916.3182
Prophet2.27466.06142.462012.392616.8116
ARIMA1.47823.03241.74148.135312.6562
IcelandBM4.214719.20324.382130.515523.7305
GGM1.04311.34891.16147.56077.7955
Prophet0.71110.61020.78125.20423.0361
ARIMA1.18661.53491.23898.64178.5708
NorwayBM22.3395552.394023.503116.297943.8856
GGM10.2276139.341811.80437.358435.6216
Prophet6.574248.92826.99494.866029.3421
ARIMA12.6251214.750014.65449.604338.2168
PolandBM0.33030.13530.367816.2736−6.0011
GGM0.18610.07110.26668.0823−9.8591
Prophet0.21860.06700.258810.8844−10.2163
ARIMA0.23180.07670.276911.5604−9.4049
RomaniaBM4.982627.78205.270929.840125.9463
GGM2.68159.88753.144415.654619.7476
Prophet1.37942.21431.48818.581310.7696
ARIMA1.34743.01701.73708.855912.6256
SlovakiaBM1.46772.30461.518136.330511.0095
GGM0.55770.47810.691414.47851.5718
Prophet0.92691.00941.004723.31556.0560
ARIMA0.34870.19600.44278.9074−3.7771
SpainBM4.837339.57486.290922.259728.0691
GGM9.7004124.012411.136143.146934.9223
Prophet4.871440.54416.367422.507128.2143
ARIMA4.819336.87966.072921.842027.6460
SwitzerlandBM2.61338.56652.92697.583418.8872
GGM3.012510.63483.26118.552320.1848
Prophet1.99748.76272.96026.200219.0230
ARIMA2.319413.56873.68367.312321.6466
TurkeyBM28.4401924.267630.401839.940546.9740
GGM9.9676171.147213.082313.445536.8551
Prophet12.1973268.817816.395715.759039.5642
ARIMA9.3296155.877112.485112.552636.2944
Other EuropeBM14.9316248.981815.779240.162239.1043
GGM5.026738.99236.244412.885827.9802
Prophet5.838051.53897.179114.923729.6540
ARIMA7.425775.99588.717619.216831.9841
IranBM6.984694.00419.695629.890833.2600
GGM6.962291.26239.553130.179333.0824
Prophet6.951486.11859.280031.211132.7343
ARIMA7.009069.73438.350736.936731.4682
IraqBM2.58267.75132.784172.828418.2872
GGM0.81431.02411.012028.50696.1431
Prophet1.34092.50621.583153.152511.5126
ARIMA1.33122.48351.575935.369111.4580
Other Middle EastBM0.49550.35140.592836.5357−0.2754
GGM0.68660.59950.774349.56792.9304
Prophet2.05754.28242.0694141.226014.7271
ARIMA0.33830.14610.382221.0196−5.5398
EgyptBM2.51106.94522.635417.982717.6283
GGM0.64530.61130.78194.56193.0472
Prophet0.63640.60120.77544.48772.9471
ARIMA0.61230.68870.82994.28523.7622
Eastern AfricaBM0.75720.97030.98501.05005.8192
GGM0.67170.94710.97320.93615.6740
Prophet5.329340.72446.38166.823228.2410
ARIMA6.430452.69137.25898.314029.7867
AustraliaBM2.79709.34863.057517.459919.4113
GGM1.70754.11382.028310.488214.4861
Prophet1.01731.33661.15616.51227.7408
ARIMA2.10325.78532.405312.993616.5319
New ZealandBM6.132139.47826.283223.899128.0545
GGM3.364912.41403.523413.055021.1129
Prophet1.04291.22481.10674.04367.2168
ARIMA1.08971.79411.33944.38119.5070
IndiaBM10.8661139.392011.80646.777435.6237
GGM9.4596128.028311.31506.052935.1135
Prophet19.9743508.305322.545612.218943.3865
ARIMA15.4507347.909118.65239.358641.1116
IndonesiaBM8.651879.30158.905135.581632.2395
GGM4.370621.09944.593417.893424.2955
Prophet7.055452.99117.279528.996629.8207
ARIMA4.830526.86435.183119.783125.7448
JapanBM2.14776.72952.59412.746617.4390
GGM2.528710.97703.31323.374420.3748
Prophet6.716452.85917.27048.856329.8058
ARIMA4.017423.62754.86085.346724.9745
MalaysiaBM15.6747249.124315.783754.105639.1077
GGM14.1350201.582214.198048.898937.8372
Prophet6.337740.99086.402421.887028.2801
ARIMA10.8077130.904211.441336.753735.2468
PakistanBM14.2613220.964314.864938.999938.3880
GGM3.140013.72413.70468.577821.7149
Prophet5.514037.97126.162114.764227.8210
ARIMA2.963511.86553.44468.255820.8418
PhilippinesBM1.01491.40411.184911.76028.0363
GGM1.00101.98781.409912.672710.1222
Prophet0.91671.60121.265411.52528.8244
ARIMA0.79701.48951.220510.25868.3906
Republic of Korea BM0.85040.94640.972823.86655.6696
GGM0.43540.27750.526812.3352−1.6910
Prophet0.33810.17940.423610.8917−4.3093
ARIMA0.44770.27400.523512.5721−1.7675
TaiwanBM1.51973.37521.837230.969913.2986
GGM0.96841.44401.201720.35208.2043
Prophet1.01421.23941.113326.38037.2875
ARIMA1.20332.05921.435033.374510.3339
ThailandBM1.27831.95721.399020.305410.0292
GGM0.92971.46801.211618.27188.3033
Prophet0.88211.38971.178917.21847.9747
ARIMA1.44003.06871.751821.388612.7275
VietnamBM34.49851478.513638.451443.937849.7928
GGM24.2945767.567927.705031.229945.8594
Prophet6.664876.64438.75478.113932.0350
ARIMA36.72191513.938538.909447.025349.9348
Table A2. Model comparison based on MAE, RMSE, and MAPE. The results are based on data divided into a 47-data-point training set and a 12-data-point test set.
Table A2. Model comparison based on MAE, RMSE, and MAPE. The results are based on data divided into a 47-data-point training set and a 12-data-point test set.
ModelMAERMSEMAPE
BM12.6590713.6209832.43803
GGM6.612207.4097823.01449
Prophet7.104847.9767424.91147
ARIMA6.881738.0479026.41186

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Figure 1. Hydroelectricity generation by selected countries.
Figure 1. Hydroelectricity generation by selected countries.
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Figure 2. American countries: model fits and forecasting.
Figure 2. American countries: model fits and forecasting.
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Figure 3. European countries: model fits and forecasting.
Figure 3. European countries: model fits and forecasting.
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Figure 4. Asian and Middle East countries: model fits and forecasting.
Figure 4. Asian and Middle East countries: model fits and forecasting.
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Table 1. Model comparison based on MAE, RMSE, and MAPE.
Table 1. Model comparison based on MAE, RMSE, and MAPE.
ModelMAERMSEMAPEAIC
BM9.7652910.4560724.5253424.88533
GGM5.197935.8983715.4978619.59935
Prophet5.216276.0934018.7653720.74633
ARIMA5.788096.7488716.1219220.40277
Table 2. Model comparison based on MAPE. Each cell shows the number of cases where the model in the row outperformed the model in the column.
Table 2. Model comparison based on MAPE. Each cell shows the number of cases where the model in the row outperformed the model in the column.
ProphetARIMABMGGM
Prophet0213118
ARIMA2203211
BM121108
GGM2532350
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Ahmad, F.; Finos, L.; Guidolin, M. Forecasting Hydropower with Innovation Diffusion Models: A Cross-Country Analysis. Forecasting 2024, 6, 1045-1064. https://doi.org/10.3390/forecast6040052

AMA Style

Ahmad F, Finos L, Guidolin M. Forecasting Hydropower with Innovation Diffusion Models: A Cross-Country Analysis. Forecasting. 2024; 6(4):1045-1064. https://doi.org/10.3390/forecast6040052

Chicago/Turabian Style

Ahmad, Farooq, Livio Finos, and Mariangela Guidolin. 2024. "Forecasting Hydropower with Innovation Diffusion Models: A Cross-Country Analysis" Forecasting 6, no. 4: 1045-1064. https://doi.org/10.3390/forecast6040052

APA Style

Ahmad, F., Finos, L., & Guidolin, M. (2024). Forecasting Hydropower with Innovation Diffusion Models: A Cross-Country Analysis. Forecasting, 6(4), 1045-1064. https://doi.org/10.3390/forecast6040052

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