A genetic programming-based ensemble method for long-term electricity demand forecasting

Introduction

The consumption of electricity is closely tied to a country’s level of economic development, with substantial evidence showing a causal relationship between economic growth and electricity consumption. In developed nations, electricity use is high and consistently increasing, while in developing countries, rapid economic expansion results in volatile macroeconomic variables, causing significant fluctuations in electricity demand. Consequently, accurately forecasting electricity consumption is crucial for effective energy planning, strategy development, and formulating energy policies that consider future economic growth, which is one of the most critical and complex challenges faced by electric utilities (Belke, Dobnik & Dreger, 2011; Kaboli, Selvaraj & Rahim, 2016).

Ethiopia, the second most populous nation in Africa after Nigeria, has a population exceeding 120 million. A large majority of Ethiopians live in rural areas and are primarily engaged in agricultural activities. However, recent years have seen a marked trend toward urbanization as more individuals migrate to cities in search of better economic opportunities. The Ethiopian population is notably young, with a significant portion under the age of 30. This youthful demographic presents both opportunities and challenges: it provides a dynamic source of energy and potential for driving economic growth and social progress, yet simultaneously presents challenges related to unemployment and access to education and healthcare. Ethiopia’s relatively high population growth rate further complicates the government’s efforts to provide adequate services and infrastructure to meet the needs of its growing populace (Embassy of Ethiopia, 2020; Liyew, Ejigu & Habtu, 2024; Mohammed & Yıldırım, 2023; Wikipidia, 2024). The government has implemented policies to attract foreign investment and promote the development of industrial parks and manufacturing sectors. There has been significant growth in the textile, apparel, and leather goods industries. Infrastructure development is another primary focus for the Ethiopian government, with investments in roads, railways, airports, and energy systems. Despite these positive developments, Ethiopia faces challenges such as high levels of poverty, unemployment, and income inequality (Embassy of Ethiopia, 2020; Gebrehiwot, 2021; Oqubay, 2019; Wikipidia, 2024). In recent years, Ethiopia has seen a significant rise in electrical energy demand, increasing by 30% over the past 5 years, driven by its rapidly growing economy and an increasing number of consumers. Despite this rising demand, the current electrical energy supply is inadequate, leaving most households without access to electricity and with limited availability of backup generators. To tackle these challenges, robust and appropriate load forecasting methods must guide energy planning and supply (Embassy of Ethiopia, 2020; Issa, 2024).

There are various studies on long-term load forecasting in the literature. The methods used include fuzzy logic (Ali et al., 2016), particle swarm optimization (PSO) (AlRashidi & EL-Naggar, 2010), genetic algorithms (GA) (Santra & Lin, 2019), artificial neural network (ANN) (Çunkaş & Altun, 2010), simulating annealing (SA), artificial cooperative search algorithm (Kaboli, Selvaraj & Rahim, 2016), genetic programming (GP) (Çunkaş & Taşkiran, 2011; Karabulut, Alkan & Yilmaz, 2008), and multi-objective optimization (Yang et al., 2024b). Some studies use a single estimation method, while others employ multiple methods for comparative analysis. Additionally, several investigations have explored hybrid approaches that combine various techniques to enhance forecasting accuracy. The field of forecasting is inherently dynamic, necessitating continual refinement as the underlying data are perpetually updated, new parameters influencing predictions may emerge, and novel forecasting models are developed. Furthermore, no universally superior method exists for all contexts and datasets; thus, researchers are engaged in an ongoing quest to identify the most effective models tailored to the specific characteristics of the data.

When examining the extant literature on Ethiopia, Gebremeskel, Ahlgren & Beyene (2021) employed the Long-range Energy Alternative Planning (LEAP) system to project the nation’s long-term energy requirements. Their study covers a 33-year planning horizon, from 2018 to 2050, and predicts a significant increase in energy consumption, particularly in electricity demand, driven by economic and population growth. Similarly, Lemma, Fenta & Vadlamudi (2022) developed a neural network-based model for long-term peak load forecasting within the Southern Ethiopian power grid. The model’s input variables include the number of domestic and industrial customers, gross domestic product (GDP), and commercial customers. Degefa (2022) constructed a gray forecasting model to predict Ethiopia’s future energy consumption. The model, trained on actual energy consumption data from 2008 to 2017, was applied to various energy types, including electricity, petroleum products, solid biomass, and total energy consumption. Gebreyohans, Saxena & Kumar (2018) carried out an extensive study on long-term load forecasting for the city of Wolayta Sodo, Ethiopia, employing hybrid models that integrate multivariate linear regression with artificial neural networks, as well as multivariate linear regression with an adaptive neuro-fuzzy inference system. Matewose (2016) focused on energy forecasting for the city of Jimma, Ethiopia. This research utilized monthly and accurate yearly load data obtained from the Jimma distribution system substation as a case study, comparing linear, compound growth, and quadratic regression models.

When analyzing energy forecasting studies related to Ethiopia, it is noted that some focus on predictions for a single city or region, some are limited to peak electricity load forecasts, and others consider energy consumption for heating and transportation. Although there are studies estimating Ethiopia’s annual electricity consumption, they are few, and many are not based on updated forecasts. The need to continuously update annual energy forecasts underscores the necessity for new predictions in this area.

Ensemble learning methods have gained significant traction in machine learning over recent years due to their superior predictive performance (Fan et al., 2024; Li et al., 2024; Xiao, Fang & Wang, 2024). These methods combine the predictions of multiple models to produce a more accurate outcome. Various algorithms can serve as base learners, including linear and nonlinear models, support vector machines, ANN, and other traditional machine learning techniques. By aggregating these models, ensemble methods effectively reduce the variance inherent in individual algorithms, thereby enhancing predictive performance accuracy. The initial stage involves generating forecasts using various individual models. The predictions from these models are then assessed, and the good performers are chosen to enhance accuracy in the next stage. In the second stage, further processing is applied to these selected forecasts. This process may include the average, weighted average, or more complex derivatives of the values predicted by the models used in the first stage to refine the final estimates. Ensemble methods facilitate the integration of different forecasting techniques, such as machine learning models, statistical approaches, and optimization strategies, leveraging their combined strengths and performance.

Among ensemble techniques, decision tree-based models such as extreme gradient boosting (XGBoost) (Kumar, Sood & Rawat, 2023) and Random Forest (Hassan et al., 2023) have been extensively utilized. Recent research has investigated the integration of GP to create nonlinear nodes within decision trees, yielding promising improvements in both interpretability and accuracy (Zhang et al., 2023). An Ensemble method employing GP for seasonal precipitation prediction has been proposed (Danandeh Mehr, 2020). This model utilized predictions generated by GP, gene expression programming, and multiple regression methods as inputs for a subsequent GP phase. The results indicated that the proposed model could enhance the forecasting accuracy of the best standalone models by up to 30%. Additionally, ensemble GP has demonstrated superior performance compared to individual methods in addressing the Dynamic Flexible Job Shop Scheduling problem (Xu et al., 2023) and in various classification problems (Meng et al., 2024).

As stated above, the studies aiming at energy forecasting for Ethiopia offer valuable contributions. However, it is essential to note that, to the authors’ knowledge, no existing research has yet utilized various optimization methods to identify which approaches are most efficient for forecasting Ethiopia’s energy needs demand.

This study seeks to answer two fundamental questions:

• 1)

  Is it possible to accurately forecast Ethiopia’s energy demand employing optimization techniques such as GA, PSO, and SA using the population, GDP, and import and export data?

• 2)

  Can the results obtained with GA, PSO, and SA be further improved using a GP-based ensemble method?

In this study, Ethiopia’s electricity demand is estimated using a novel ensemble method based on GP. This model combines the features of various approaches for the nonlinear prediction of complex data, often providing more stability and accuracy than any single model. Electricity demand forecasting is conducted using four methods: GA, PSO, SA, and GP. Their performances are then compared to each other and conventional regression methods. In the first phase, GA, PSO, and SA optimization techniques are applied to linear, quadratic, and exponential regression models to identify the model coefficients. A new model is developed in the second phase, integrating the prediction values obtained from the first phase with the GP method. This model is then used to derive the final prediction values. Future load demands are projected up to 2031 based on two different scenarios. The primary contributions of this study are outlined as follows:

• 1)

  A novel ensemble method based on GP is introduced for long-term load forecasting in Ethiopia, utilizing GA, PSO, SA, and GP algorithms. These optimization algorithms are used to optimize the parameter coefficients of linear, quadratic, and exponential regression models.

• 3)

  The accuracy of the proposed models was evaluated using four statistical metrics: mean absolute percentage error (MAPE), mean square error (MSE), root mean square error (RMSE), and correlation coefficient (R²).

• 3)

  Electricity consumption forecasts for 2022–2031 were produced using models under two different scenarios. The first scenario averaged the growth rate of input data from the last 3 years, while the second scenario averaged it over the previous 5 years.

• 4)

  The findings suggest that the proposed ensemble method enhances load forecasting accuracy and efficiency. Regular electricity demand assessments remain essential for Ethiopia.

Materials and Methods

This section provides a comprehensive overview of the dataset, optimization methods, regression models, and error metrics employed in the study, presented in sequence.

Genetic programming-based ensemble method

In this study, we propose an ensemble method for forecasting Ethiopia’s annual total energy consumption. Figure 1 illustrates the proposed GP-based ensemble method, which consists of two stages. In the initial stage, the first prediction values are obtained using various optimization techniques and regression models. In the second stage, these initial predictions are refined with the GP method to produce the final forecast value. The model utilizes a range of inputs, including data related to population, GDP, imports, and exports.

Stage-1

In the first stage of the prediction model, coefficients for various regression models were identified using the GA, PSO, and SA methods. In this stage, estimates were generated through nine different approaches: GA_Linear, GA_Quadratic, GA_Exponential, PSO_Linear, PSO_Quadratic, PSO_Exponential, SA_Linear, and SA_Quadratic SA_Exponential.

SA model

The algorithm was run with the specified parameters listed in Table 2, and the most appropriate coefficients were found for the linear, quadratic, and exponential models.

The model coefficients and full model equation obtained by the SA method are given in Eqs. (14)–(16).

(14) $$Linear:Y=0.0114{\mathrm{X}}_1+0.0775{\mathrm{X}}_2+0.0541{\mathrm{X}}_3+0.1315{\mathrm{X}}_4-0.3191$$

(15) $$\begin{array}{rl}Quadratic:Y=& 0.0004268{\mathrm{X}}_1-0.00055{\mathrm{X}}_2+0.00534994{\mathrm{X}}_3+0.0011499{\mathrm{X}}_4\\ & -0.000579{\mathrm{X}}_1{\mathrm{X}}_2+0.000136{\mathrm{X}}_1{\mathrm{X}}_3+0.002647{\mathrm{X}}_1{\mathrm{X}}_4+0.002549{\mathrm{X}}_2{\mathrm{X}}_3\\ & +0.0010495{\mathrm{X}}_2{\mathrm{X}}_4-0.00105{\mathrm{X}}_3{\mathrm{X}}_4+0.00041{{\mathrm{X}}_1}^2+0.000329{{\mathrm{X}}_2}^2\\ & +0.0021499{{\mathrm{X}}_3}^2-0.005{{\mathrm{X}}_4}^2+0.00502\end{array}$$

(16) $$\begin{array}{rl}Exponential:Y=& -0.4123e^{-0.316{\mathrm{X}}_1}+0.289e^{-0.6552{\mathrm{X}}_2}+0.0921e^{0.1856{\mathrm{X}}_2}\\ & +0.7385e^{0.2832{\mathrm{X}}_4}+0.8932\end{array}$$

Electricity energy consumptions between 2006 and 2021 are calculated using Eqs. (14)–(16) and given in Table 3. According to RMSE values, the values of SA_Linear, SA_Quadratic, and SA_Exponential are 0.416, 0.312, and 0.527, respectively. According to the estimates of SA and regression models, the lowest RMSE value was found with the SA_Quadratic method.

The error values associated with all predictions generated at the conclusion of the first stage are presented in Table 4 and Fig. 2, based on the selected evaluation metrics. The best prediction errors for each error criterion are highlighted in bold in the table. Upon examining the error values in the table and the figure, it is observed that the most successful prediction values among the nine results obtained with the proposed method were achieved using quadratic optimization approaches. Although the PSO_Exponential model appears better than the PSO_Quadratic model in terms of the MAPE error criterion, it exhibits worse prediction errors for the MSE and RMSE criteria. When optimization methods are compared, the PSO_Quadratic model outperformed the GA_Quadratic and SA_Quadratic models, achieving a superior RMSE value of 0.272.

Table 4:

Forecast errors for linear, quadratic, and exponential models at first stage.

Metrics	GA			PSO			SA
	Lin.	Quad.	Exp.	Lin.	Quad.	Exp.	Lin.	Quad.	Exp.
Errors obtained from real data
MSE	0.132	0.097	0.122	0.132	0.074	0.094	0.173	0.098	0.278
RMSE	0.364	0.312	0.349	0.363	0.272	0.307	0.416	0.312	0.527
R2	0.986	0.990	0.988	0.986	0.994	0.994	0.981	0.990	0.970
MAPE	6.16	3.61	5.77	5.95	3.63	3.30	6.94	4.68	6.83
Errors obtained from test data
MSE	0.734	0.695	0.674	0.749	0.665	0.796	1.807	1.214	2.015
RMSE	0.857	0.834	0.821	0.865	0.815	0.892	1.344	1.101	1.419
R2	0.871	0.908	0.881	0.868	0.897	0.875	0.691	0.789	0.701
MAPE	7.85	7.64	8.36	7.89	7.86	8.42	11.77	7.86	11.72

DOI: 10.7717/peerj-cs.2825/table-4

Note:

The best prediction errors for each error criterion are indicated in bold.

Table 4 also presents the error values obtained from using the synthetic data as test data. Synthetic data are also generated to test the model developed using data from 2006 to 2021. The process of synthetic data generation is widely used to enhance datasets and evaluate model performance. In this study, a Gaussian Copula-based method was applied to generate synthetic data. This approach analyzes the probabilistic structure of the original dataset. It accounts for the dependencies among variables to produce new data points with statistical properties similar to those of actual data. As a result, the generated synthetic data serve as an alternative test dataset to assess and validate the model’s performance. Upon examining the results in Table 4, it is observed that quadratic models yield more successful predictions for both real and test data.

As a result, at the end of the first stage, the three best prediction values (GA_Quadratic, PSO_Quadratic, and SA_Quadratic) among these nine results were identified as input values for the second stage. As shown in Table 4, quadratic models consistently yielded lower RMSE, R², and MAPE values compared to linear and exponential models, indicating their ability to capture underlying patterns in the dataset better. Electricity consumption is influenced by various socioeconomic factors, such as GDP growth, population increase, and industrial expansion, which often exhibit nonlinear relationships with energy demand. The flexibility of the quadratic model enables it to accommodate these nonlinearities more effectively than a purely linear approach.

Conventional curve fitting models

In this study, the conventional curve fitting method, a fundamental approach, was selected as the comparison method to evaluate the predictive performance of the proposed GP-based ensemble method. The curve fitting methods were applied to the same input data. Predictions were also generated using classical linear, conventional quadratic, and conventional exponential regression techniques within the traditional curve fitting framework. The results are presented in Table 5. Based on these prediction outcomes, the RMSE error values for classical linear, classical quadratic, and classical exponential regression were found to be 0.362, 0.471, and 0.376, respectively. It is observed that the second-order models in the first stage have lower prediction errors compared to the curve-fitting method. These findings suggest that the chosen optimization methods yield more successful results compared to classical curve fitting models for these data.

Table 5:

Forecast errors for quadratic optimizations and curve-fitting conventional methods.

Metrics	Stage 1 (best results)			Compare methods
	GA_ Quadratic	PSO_ Quadratic	SA_ Quadratic	Curve fitting
				Linear	Quadratic	Exponential
MSE	0.097	0.074	0.098	0.131	0.222	0.141
RMSE	0.312	0.272	0.312	0.362	0.471	0.376
R2	0.990	0.994	0.990	0.986	0.989	0.985
MAPE	3.61	3.63	4.68	5.97	6.74	5.57

DOI: 10.7717/peerj-cs.2825/table-5

Stage-2

In the second stage, the forecasted values of GA_ Quadratic, PSO_ Quadratic, and SA_ Quadratic were incorporated into the GP model. By utilizing the optimal forecast values derived from Stage 1, an equation was formulated through GP, leading to the final forecast output.

To determine the parameters of the GP model, including the terminal set, function set, and other elements related to the evolutionary process, we conducted preliminary experiments. At the start of these runs, we added more functions to the function set, including minimum, maximum, square-root, and inverse functions, and employed a relatively wide range of constants in the terminal set. The GP with these settings rarely performed well, but it tended to evolve complex functions. In other words, the GP produced well-performing simple functions when simple functions and a relatively narrow range of constant values were used in its function and terminal sets, respectively. At the end of the preliminary runs, the parameters for the GP optimization method were set as in Table 2. This model is obtained with GP.

(17) $$\mathrm{F}={\mathrm{Y}}_1+0.1-{\displaystyle \frac{0.1}{-{\mathrm{Y}}_1+{\left({\mathrm{Y}}_2-{\mathrm{Y}}_1+2\right)}^2+1}}$$where F is the forecast value, $Y_1$ the forecast values of PSO_Quadratic, ${\mathrm{Y}}_2$ the forecast values of SA_Quadratic. According to the formula obtained with GP, the forecast values of GA_Quadratic are not used. In this case, the model might have chosen only those inputs that demonstrate the highest predictive accuracy or contribute most significantly to the target variable.

The error values corresponding to the optimal results from Stage 1, as well as the error values of the final predictions derived from Stage 2 of the proposed method, are presented in Table 6 and Fig. 3. According to all error metrics, the predictions resulting from Stage 2 are the most accurate. The estimation errors obtained in Stage 2 were found to be 0.039, 0.198, 0.995, and 2.83 for MSE, RMSE, R², and MAPE respectively. Furthermore, the final prediction results obtained by the proposed ensemble method outperform those obtained in Stage 1 across all metrics.

Table 6:

Error comparison for quadratic models and proposed method.

Metrics	Stage 1 (best results)			Stage 2
	GA_ Quadratic	PSO_ Quadratic	SA_ Quadratic	Proposed ensemble method
MSE	0.097	0.074	0.098	0.039
RMSE	0.312	0.272	0.312	0.198
R2	0.990	0.994	0.990	0.995
MAPE	3.61	3.63	4.68	2.83

DOI: 10.7717/peerj-cs.2825/table-6

Note:

The best prediction errors for each error criterion are indicated in bold.

When the model’s performance is examined on an annual basis, the error values are relatively low in some years. The model can make predictions with higher accuracy, especially in periods where the data set does not show rapid change. On the other hand, there has been a slight increase in errors during periods of rapid economic growth or sudden changes. This situation indicates that predictions can be more difficult in years of radical change.

Parameters sensitivity analysis and confidence interval

Parameters sensitivity analysis

In Table 7 and Fig. 4, the correlation relationships between the input parameters (X1, X2, X3, X4) and the output variable (Y) are analyzed based on the results of the Monte Carlo analysis. The standard deviation and variance values express the width and variability of the distribution of each parameter. As can be seen in the table and figure, parameter X1 is a factor that significantly affects the output (Y) of the system, as it exhibits both high correlation coefficients and a balanced variance. Therefore, X1 should be regarded as a priority variable in model improvement and optimization processes. The low correlation coefficients and variability of X2 and X4 indicate that the effect of these parameters on Y is weak. X3 holds some importance in terms of quadratic relationships, but it is low in other types of relationships and variability. This parameter can be considered as a second priority. The input variables X1, X2, X3, and X4 denote population, GDP, imports, and exports, respectively.

Table 7:

The correlations and variances between the input parameters.

Input parameters	Correlation			Standart deviation	Variation
	Linear	Quadratic	Exponential
X1	0.97524	0.6439	0.9936	11.958	143.012
X2	0.0049	−0.0627	0.0253	27.069	732.775
X3	0.0259	0.2711	−0.0054	4.57	20.886
X4	−0.1716	−0.1116	−0.0366	1.78	3.171

DOI: 10.7717/peerj-cs.2825/table-7

Confidence interval

Figure 5 shows the confidence interval analysis of the GP-based ensemble method. The estimated consumption data and the actual consumption data are within the 95% confidence interval limits. Considering the strong correlation between the estimated consumption values and the actual consumption values, it is possible to accurately estimate the consumption values with the intended GP-based ensemble method.

Results

Two scenarios were developed to assess the impact of increases in future input parameter variables on electricity energy consumption. In Scenario 1, the growth rates of the last 3 years are averaged, reflecting a short-term trend that captures more recent dynamics and possible fluctuations. Scenario 2, on the other hand, provides a broader and more fluid historical perspective that considers medium-term patterns, using average growth rates over the last 5 years.

Discussion

This section discusses the implications of the results obtained in the ‘Results’ section and future forecasts of the ensemble method. Figure 6 illustrates the future predicted values using quadratic methods in Stage 1 for both scenarios. Based on the forecast values derived from these methods, the SA_Quadratic method yields comparable results in both scenarios. For the year 2031, electricity consumption is projected to be approximately 38 TWh. According to the PSO_Quadratic method, electricity consumption for 2031 is estimated to be around 30 TWh, although somewhat similar predictions are obtained in both scenarios. The GA_Quadratic model provides significantly different results in each scenario compared to the other methods. In Scenario 1, the GA_Quadratic result was about 40 TWh for 2031, while in Scenario 2, the electricity consumption was around 14 TWh for the same year. When all these results are analyzed collectively, the quadratic models that produced the best results in Stage 1 yielded quite different prediction values when considered independently. This substantial discrepancy is unacceptable for an issue like annual energy consumption, which influences the rate at which the country’s energy infrastructure expands. Therefore, rather than relying on individual methods in the first stage, it is more appropriate to utilize the proposed GP-based ensemble method, which models the data from 2006 to 2021 with the lowest error, for forecasting future outcomes years.

The input values in Scenario 1 and Scenario 2 were updated and applied to the ensemble method proposed in this study. The forecasts for the ensemble method are shown in Table 9. According to this forecast, Ethiopia’s annual electricity consumption between 2022 and 2031 ranges from 12.45 to 30.17 TWh for Scenario 1 and 12.64–30.78 TWh for Scenario 2. According to the rates of change of the input variables determined in Scenario 1 and Scenario 2, electricity consumption is expected to be higher in Scenario 2 than in Scenario 1. While electricity consumption in 2021 was 11.58 TWh, it can be seen that the consumption value in both scenarios will exceed 30 TWh. According to these values, about three times more electricity will be consumed in 2031 compared to 2021. This increase in electricity consumption will require the expansion of generation resources, transmission and distribution lines, and the commissioning of new infrastructure.

The findings of this study have significant implications for Ethiopia’s energy infrastructure planning. The expected tripling of electricity consumption by 2031 highlights the need for strategic investments in power generation, transmission, and distribution systems. To meet this increasing demand, Ethiopia must prioritize expanding the capacity of renewable sources such as hydro, solar, and wind power while also focusing on reducing losses and enhancing reliability through the modernization of the existing grid. Encouraging public-private partnerships is advisable to establish a robust energy infrastructure that can satisfy Ethiopia’s future energy requirements and bolster economic growth. Furthermore, incorporating the proposed community forecasting model into policy planning can enhance decision-making processes and facilitate optimal decision-making planning. This article also has some limitations. The number of samples in the data set is very small (16), which partly weakens the generalization ability of the model. Another limitation is that it is impossible to validate the forecasting performance of the model in advance for future years when the input parameters of the model change unpredictably depending on many factors. These limitations should be considered when interpreting the results. Despite these limitations, the findings show that ensemble methods can improve forecast accuracy and provide a projection for energy planning.

Conclusions

This study presents an ensemble method based on GP that integrates SA, GA, and PSO algorithms to enhance long-term load forecasting. The model was tested on historical data and synthetic data, demonstrating its potential to produce reasonable electricity consumption estimates. After the model’s accuracy was validated, Ethiopia’s total annual electricity consumption was estimated up to 2031 based on two scenarios. According to the forecast for electric power consumption in the coming years, electricity usage in 2031 is expected to increase by about threefold compared to 2021 in both case scenarios.

This article highlights the advantages of the proposed ensemble method, which combines SA, GA, PSO, and GP algorithms, supported by a series of statistical methods tests. Although the model shows accuracy based on historical data and synthetic data, the small dataset size limits the model’s generalization ability. Future research should focus on expanding the dataset, including additional influential variables, and using appropriate validation techniques to improve model reliability. Despite these limitations, the findings suggest that ensemble methods show promise in improving long-term electricity demand forecasting and provide a helpful tool for policymakers and energy planners.

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