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

1. Introduction

The increase in the world population is directly related to an increase in energy consumption, especially electricity [1,20]. This increased demand for energy, if not met sustainably, can harm the environment, particularly due to air pollution and climate change effects [22] deriving from the burning of fossil fuels such as coal and natural gas [20]. However, fossil fuels, including coal, oil, and natural gas, still serve as primary energy sources worldwide [13]. To contrast this reliance on fossil fuels, interest in renewable energy sources has recently increased significantly worldwide [5]. Renewable energy includes sources that naturally regenerate, such as hydroelectric, solar, wind, and geothermal energy. These sources are far more sustainable and environmentally friendly compared to fossil fuels. 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 for alleviating the effects of climate change, greenhouse gas emissions, and increasing energy costs [18].

Hydropower, one of the oldest and most well-researched renewable energy sources, has a history of nearly 150 years. Despite its longstanding presence, innovation in hydropower remains vibrant, 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 [17]. 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 [28]. 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 [24]. These characteristics make hydropower an appealing and deserving candidate for increased attention and investment [2]. This explains why hydroelectricity generation rose by nearly 70 TWh (close to 2%) in 2022, reaching 4300 TWh. In fact, 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. It is also acknowledged that, without major policy changes, global hydropower expansion is expected to slow down this decade. The contraction results from slowdowns in the development of projects in China, Latin America, and Europe. 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 [17] 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.

Starting from these pieces of evidence, our study aims to analyze the role of hydroelectric data in countries’ energy transition using the approach of Innovation Diffusion Models [9]. The literature on the application of innovation diffusion models for energy growth processes has been developing significantly in recent years and our paper aims to contribute to this direction. We propose a cross-country analysis of hydropower generation, provide some forecasts based on Innovation Diffusion Models [9], and compare their performance with typical time series models such as ARIMA models [16] and Prophet model [29].

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. Section 3 outlines the data sources and methods employed in this study. Section 4 presents a detailed analysis and discussion of the results, interpreting the findings concerning current challenges and opportunities for hydropower. Finally, Section 5 provides concluding remarks, summarizing the key insights of the study and offering recommendations for future research and policy actions to enhance the role of hydropower in sustainable energy systems.

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 [15]. The most widely used methods for forecasting hydroelectric generation have been linear time series models such as ARIMA models and their different extensions [8]. 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 [19]. The study used various ARIMA models to analyze historical data from 1971-1972 to 2019-20. The researchers determined that the ARIMA(1,1,1) model with drift was suitable to forecast the energy demand of the country. In [25], 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 [23], 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 [30], 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). Forecast results indicated that China’s hydropower generation could reach 1687.738 billion kWh by 2025, reflecting a 24.5% increase from 2020. In [21], 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 predict the growth of hydroelectric power. This choice relies on a well-established stream of literature that has employed diffusion models to study renewable energy sources, to understand the dynamics underlying energy transition by using both univariate (see for instance [10]) and bivariate models (see [4]). Recent reviews of this literature may be found for instance in [9], [3], [4], [27].

However, the analyses proposed in the literature have typically focused on renewable energy sources like wind and solar ([10]), whereas hydroelectricity has not been studied in much detail. Moreover, the studies using innovation diffusion models have tried to describe the patterns of growth of these energy technologies, without paying much attention to their forecasting ability compared to other models. In this paper, we try to fill this gap by focusing on hydropower data with a forecasting analysis.

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 Bass, GGM, Prophet models, and ARIMA models.

3.1. Data

The analysis proposed in this paper is based on data from the Energy Institute Statistical Review [6], which covers multiple countries. In North Africa, limited data are in stark contrast to the various start years of hydroelectric projects in different regions, with some countries launching their efforts recently as 2011.

Figure 1 shows hydroelectric generation data for each country considered from 1965 to 2023. The figure indicates that, unlike other countries, Canada and the US have generated a substantial amount of electricity from hydropower, while most countries have maintained similar generation levels. It may be also noticed that India and Norway have significantly increased their hydroelectricity generation, whereas some countries have reduced their reliance on this source.

To analyze the evolution of hydropower in these countries, we have employed a variety of models, starting from diffusion models, namely the Bass model 3.2.1 and GGM 3.2.3, whose performance will be compared with other concurrent models, namely Prophet and ARIMA models.

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) 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 found out that it is suitable also for studying the diffusion of energy technologies [9]. The model operates on the assumption that two primary sources of information influence consumption decisions: external factors such as the media and advertising, 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.

$$z^{\prime }\left(t\right)=\left[p+q{\displaystyle \frac{z\left(t\right)}{m}}\right][m-z\left(t\right)],t>0$$

(1)

Where $z\left(t\right)$ is the cumulative number of adoptions at time t, $z^{\prime }\left(t\right)$ is defined as the variation over time of adoptions. 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. $m-z\left(t\right)$ is proportional to the residual market. The residual market is affected by the coefficients p and q. Parameter p, called innovation coefficient, represents the effect of the external influence, due to the mass media communication and advertising. Parameter q, called imitation coefficient, is the internal influence, whose effect is modulated by the ratio $z\left(t\right)/m$. 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 BM solution can be expressed as

$$z\left(t\right)=m{\displaystyle \frac{(1-e^{-(p+q)t})}{(1+\frac{q}{p}{e}^{-(p+q)t})}}t>0$$

(2)

Three parameters m, p, and q define the dynamics of the diffusion process in terms of cumulative sales, or $z\left(t\right)$, 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. $z\left(t\right)$, as given by Equation (2) for a range of parameter values p and q. The cumulative process, as can be seen, has an s-shaped pattern and approaches saturation, denoted by the parameter m, at varying rates based on the values of the parameters p and q [9].

3.2.2. Dynamic Market Potential

After conducting a comprehensive examination of the research conducted by [12], it becomes evident that there exists a 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 $m\left(t\right)$ in a way that accurately depicts the inherent complexities and fluctuations associated with market dynamics.

$$z^{\prime }\left(t\right)=m\left(t\right)\left[p+q{\displaystyle \frac{z\left(t\right)}{m\left(t\right)}}\right]\left[1-{\displaystyle \frac{z\left(t\right)}{m\left(t\right)}}\right]+m^{\prime }\left(t\right){\displaystyle \frac{z\left(t\right)}{m\left(t\right)}},t>0$$

(3)

Equation (3) characterizes the instantaneous adoptions $z^{\prime }\left(t\right)$ as a sum of a BM with $m\left(t\right)$ and a factor $m^{\prime }\left(t\right)z\left(t\right)/m\left(t\right)$, which allocates a fraction of the market potential variation $m^{\prime }\left(t\right)$ to $z^{\prime }\left(t\right)$, specifically the growth rate $z\left(t\right)/m\left(t\right)$. Within Equation (3), the market potential variation $m^{\prime }\left(t\right)$ impacts the instantaneous adoptions $z^{\prime }\left(t\right)$, which can either be positive and reinforcing if $m\left(t\right)$ is increasing, or negative if $m\left(t\right)$ 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.

$${\displaystyle \frac{z^{\prime }\left(t\right)m\left(t\right)-z\left(t\right)m^{\prime }\left(t\right)}{m^2\left(t\right)}}={\left[{\displaystyle \frac{z\left(t\right)}{m\left(t\right)}}\right]}^{\prime }=\left[p+q{\displaystyle \frac{z\left(t\right)}{m\left(t\right)}}\right]\left[1-{\displaystyle \frac{z\left(t\right)}{m\left(t\right)}}\right]$$

(4)

The generalization of BM, where the function $m\left(t\right)$ depends on time, has closed-from-solution,

$$z\left(t\right)=m\left(t\right){\displaystyle \frac{(1-e^{-(p+q)t})}{(1+\frac{q}{p}{e}^{-(p+q)t})}}$$

(5)

Equation (5) clearly demonstrates that $m\left(t\right)$ is an independent function that influences the dynamics of the diffusion process, represented by parameters p and q. The specific form of $m\left(t\right)$ can vary depending on the assumptions made about the market potential. In [12] and [11] certain structures for $m\left(t\right)$ have been proposed. The GGM is based on one of these possibilities.

3.2.3. GGM

In [12] authors postulated a specific specification for $m\left(t\right)$, under the assumption that the growth of the market potential is contingent upon a communication process regarding the new product. This process typically precedes the adoption phase and serves the purpose of “building” the market. More precisely, the dynamic market potential $m\left(t\right)$ is defined according to a structure that resembles a Bass model,

$$m\left(t\right)=K\sqrt{{\displaystyle \frac{1-e^{(p_c+q_c)t}}{1+\frac{q_c}{p_c}{e}^{(p_c+q_c)t}}}}$$

(6)

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 $m\left(t\right)$ when all informed consumers ultimately become adopters. The GGM exhibits the following cumulative structure

$$z\left(t\right)=K\sqrt{{\displaystyle \frac{1-e^{(p_c+q_c)t}}{1+\frac{q_c}{p_c}{e}^{(p_c+q_c)t}}}}{\displaystyle \frac{1-e^{(p_s+q_s)t}}{1+\frac{q_s}{p_s}{e}^{(p_s+q_s)t}}},t>0$$

(7)

In Equation (7), the cumulative adoptions, $z\left(t\right)$, 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 $q_s$.

3.2.4. Prophet Model

In [29] the authors proposed an innovative model able to handle time series with nonlinear trend, seasonality, and other possible effects appearing in the data. The mathematical components of the Prophet model are defined as,

$$y\left(t\right)=g\left(t\right)+s\left(t\right)+h\left(t\right)+{ϵ}_t$$

(8)

In Equation (8) $g\left(t\right)$ is the trend function that models non-periodic changes in the value of the time series, $s\left(t\right)$ represents periodic changes (e.g., weekly and yearly seasonality), and $h\left(t\right)$ 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 $g\left(t\right)$ in Equation (8). This sort of growth is typically modeled by the logistic growth model, defined as

$$g\left(t\right)={\displaystyle \frac{C}{a+exp(-k(t-m\left)\right)}}$$

(9)

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

$$g\left(t\right)=(k+a{\left(t\right)}^T\delta )t+(m+a{\left(t\right)}^T\gamma )$$

Whereas before k is the growth rate, $\delta$ has the rate adjustments, m is the offset parameter, and ${\gamma }_j$ is set to $-s_j{\delta }_j$ to make the function continuous. $s_j$ is defined as the change points. The seasonality effect is approximated by a standard Fourier series given as,

$$s\left(t\right)=\sum _{n=1}^N\left[a_{n}cos\left({\displaystyle \frac{2\pi nt}{P}}\right)+b_{n}sin\left({\displaystyle \frac{2\pi nt}{P}}\right)\right]$$

(10)

In Equation (10), N is the count of Fourier components, P shows periods, and $a_n$, $b_n$ represents Fourier coefficients. The above components then amalgamate into an additive model. The effect of $h\left(t\right)$ on holidays and events that provide predictable shocks often does not follow any specific periodic pattern [29].

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) [16]. 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

$$y_t^{\prime }=c+{\varphi }_1{y}_{t-1}^{\prime }+\cdots +{\varphi }_p{y}_{t-p}^{\prime }+{\theta }_1{\epsilon }_{t-1}+\cdots +{\theta }_q{\epsilon }_{t-q}+{\epsilon }_t,$$

(11)

where $y_t^{\prime }$ 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.
• $\epsilon =$ is called the white nose and is assumed to be independent and identically distributed variables sampled from a normal distribution with zero mean.

To form more complicated models, the backshift notation is often used. For example, Equation (11) can be written in backshift notation as

$$(1-{\varphi }_{1}B-\cdots -{\varphi }_p{B}^p){(1-B)}^d{y}_t=c+(1+{\theta }_{1}B+\cdots +{\theta }_q{B}^q){\epsilon }_t$$

(12)

Where $(1-{\varphi }_{1}B-\cdots -{\varphi }_p{B}^p)$ is $AR\left(p\right)$ part, ${(1-B)}^d$ is d differences, while $(1+{\theta }_{1}B+\cdots +{\theta }_q{B}^q)$ 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) [16].

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 [16].

$$\mathrm{MAE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left(\right|y_i-{\widehat{y}}_i\left|\right)$$

(13)

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 [7].

$$\mathrm{MSE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2$$

(14)

Taking the square root of the MSE gives the Root Mean Squared Error (RMSE):

$$\mathrm{RMSE}=\sqrt{\mathrm{MSE}}$$

(15)

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 [14].

$$\mathrm{MAPE}={\displaystyle \frac{100\%}{n}}\sum _{i=1}^n\left|{\displaystyle \frac{y_i-{\widehat{y}}_i}{y_i}}\right|$$

(16)

4. Results

In the results section, we evaluate the effectiveness of the models proposed in predicting hydropower generation data. This evaluation examines how well each model captures the observed patterns and trends in hydroelectric generation in different regions and periods. We consider metrics such as goodness of fit, accuracy, and predictive capability to identify the model that most accurately represents the data. The detailed findings are presented in the following subsections.

4.1. American Countries

In Figure 2, we illustrate a comparison between four forecasting models—Bass, Prophet, ARIMA, and GGM using data from American countries spanning the years 1965 to 2023. This dataset is split into a training set of 52 points (marked in black) and a test set of 7 points (marked in red), allowing us to assess each model’s performance across various contexts. The figure demonstrates the performance of each model in the countries analyzed. We may notice that the Bass model, displayed in green, generally underestimates data during rapid growth periods but performs well in countries like Mexico, where the data has a peak followed by a decline. Conversely, 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. 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 datasets. The Bass model once again underestimates during rapid growth but performs well in regions like the Caribbean, Chile, and Argentina, where data peaks followed by a decline. In contrast, the GGM consistently outperforms the Prophet and ARIMA models, particularly in Argentina, Ecuador, the Caribbean, and Venezuela. Although these visual insights are valuable, they must be supported by quantitative evaluations for a thorough assessment. Further analysis using quantitative metrics from the model-fitting results section is necessary to validate these observations and provide a complete evaluation of the models’ performance.

4.2. European Countries

Figure 3 shows a comparative analysis of the models applied to data from European countries. In the European case, it is evident the difficulty in determining the best-performing model due to the striking similarity in the hydroelectric generation patterns in these developed nations. This uniformity in hydroelectric power output causes all models to closely fit the data. Figure 3 illustrates that almost all models are in close agreement with the data, suggesting stability and minimal fluctuations 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 similar performance, while the Bass model underestimates the data. In contrast, in countries like Iceland, where hydroelectric production is on the rise, all models overestimate the data. However, the GGM shows the closest alignment with the original data among the four models. Visually, the GGM’s performance is notably better compared to the Bass and Prophet models in most countries. This analysis will include 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 is available. Since hydroelectric generation in the Middle Eastern region tends to be lower compared to other regions, we combined the Middle Eastern countries for a more coherent analysis. The results show 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 Bass and Prophet models occasionally either overestimated or underestimated hydroelectric generation. To further refine this analysis, we will evaluate performance metrics to determine which model offers the best predictive ability.

In contrast, Figure 4 also highlights the fit of the model for hydroelectric generation in Asian countries, including Australia, 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 Prophet and Bass models across most countries, except Vietnam, where performance varies. Similarly to the Middle Eastern countries, the Bass and Prophet models sometimes overestimate or underestimate hydroelectric generation. To complete this analysis, performance metrics will be considered to identify the best model in terms of prediction 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 Bass model 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 compared to the Bass model. 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 Bass Model 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 strong performance, with an average MAE of approximately 5.788. This suggests smaller deviations from the actual values compared to the Bass model. 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. Overall, the GGM demonstrates superior performance compared to the Bass, ARIMA, and Prophet models in terms of average error metrics, suggesting that it may be the most accurate model for predicting hydropower generation in the evaluated dataset.

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, it outperformed the Bass model 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 Bass Model 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 Bass model in 33 countries, and the ARIMA model in 26 countries. This highlights GGM’s consistent performance across different regions. The Bass Model 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 provides a detailed breakdown of the results).

5. Conclusions

The global community is currently grappling with critical challenges posed by climate change and the increasing demand for reliable, sustainable energy sources. The shift to renewable energy is widely considered a primary solution to these urgent issues. As a result, accurately predicting the rise of renewable energy sources has become essential for effective energy planning and management. Among these, hydroelectric still plays a central role.

Our research conducted an in-depth analysis of hydropower generation data from 43 countries, which we categorized into five distinct regions, covering the years 1965 to 2023. This extensive data set allowed us to evaluate and compare various models for forecasting hydroelectric production patterns in different geographic regions, including the Americas, Africa, Europe, Asia, and the Middle East. Significant variations in data generation were observed within each region. To address this, we used a train-test data split, using 53 data points for training and 6 points for testing, out of a total of 59.

In highly developed countries, maintaining consistent levels of power generation is challenging due to various factors, while in less developed countries, power generation tends to increase over time. From our graphical analysis, we observed notable changes in power generation in certain countries. The ARIMA model performed well in these scenarios, while the Bass model was more effective with increasing datasets. The Prophet model generated clear graphs with stable data, and the GGM model outperformed the others in cases with small variations in power generation, whether increasing or decreasing.

Among the models studied, the GGM consistently proved superior in accurately predicting hydroelectric power generation. It effectively captured complex nonlinear patterns in the data, and its performance was validated through quantitative metrics such as mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE). On average, across all countries, the GGM model (5.1979 MAE, 5.8983 RMSE, and 15.4978 MAPE) slightly outperformed the Prophet (5.2163 MAE, 6.0988 RMSE, and 18.7966 MAPE) and ARIMA (5.7881 MAE, 6.7488 RMSE, and 16.1219 MAPE) models and was significantly more accurate than the Bass Model (9.7652 MAE, 10.4561 RMSE, and 24.5253 MAPE). In all countries examined, the GGM consistently achieved lower MAPE scores than other models, demonstrating its greater reliability and precision in the prediction of hydroelectric generation.

Although the Bass model showed notable success in specific countries with distinctive data trends, such as Chile and regions in eastern Africa, the GGM outperformed all other models in a broader range of regions and conditions.

In summary, the findings of our analysis show an interesting ability of the GGM to capture the evolutionary patterns of hydropower in most analyzed countries. This opens a new perspective on the use of this kind of model, which has been generally employed with descriptive purposes [9], rather than predictive. In this sense, hydropower data have offered the possibility to deal with a variety of patterns, with a growing trend, with a flat, or, in some cases, with a declining behavior for which a sufficiently flexible structure is necessary. Whereas the Bass Model does not have this flexibility, the GGM can capture the nonlinearities of the data efficiently. These findings highlight the critical importance of models like the GGM in enhancing our understanding and forecasting capabilities in the context of energy transitions. The insights gained from these models can equip policymakers and energy planners to make more informed decisions, steering their countries toward sustainable energy futures on a global scale.