Improving electricity demand forecasting accuracy: a novel grey-genetic programming approach using GMC(1,N) and residual sign estimation Free

Accurate electricity demand forecasting is critical for ensuring a stable and reliable power supply, especially in developing economies like Cameroon (Tamba et al., 2022). From illuminating homes and powering industries to driving technological advancements and ensuring healthcare access, its steady flow underpins economic development and social progress (Ugembe et al., 2023). Yet, amidst this reliance on a seemingly abundant resource, lurks a hidden challenge: accurately predicting future demand.

For developing economies like Cameroon, grappling with rapid urbanization and burgeoning energy needs, the stakes of inaccurate electricity forecasting are particularly high (Guefano et al., 2021). Power outages ripple through communities, disrupting businesses, jeopardizing essential services, and hindering economic growth (Dieudonne et al., 2022). Conversely, oversupply translates into wasted resources and financial losses. Striking a delicate balance between these extremes necessitates a precise understanding of future electricity consumption patterns.

Traditional forecasting methods, while reliable in certain contexts, often struggle to capture the intricate dance of factors influencing electricity demand in evolving economies (Tamba et al., 2018). Autoregressive integrating moving average (ARIMA) models, for instance, struggle with the inherent non-linearity and complex interdependencies within the data (Quartey-Papafio et al., 2020; Li et al., 2023). Regression models, while adept at capturing linear relationships, may miss the subtle yet influential nuances hidden within (Yildiz et al., 2017).

In this landscape of forecasting uncertainties, grey models appear to offer a glimmer of hope (Xie and Wang, 2017). Tailored for scenarios with limited or incomplete data, they possess an inherent ability to model uncertainties and extract hidden patterns even from the sparsest of datasets (Sapnken et al., 2023a). Their reliance on data generation processes and rolling forecasts makes them particularly well-suited for capturing the dynamic nature of electricity consumption.

However, while grey models excel in extracting insights from limited data, their inherent limitations in capturing intricate non-linear relationships persist. This is where genetic programming (GP) steps onto the stage, a powerful evolutionary algorithm capable of automatically generating non-linear programs that adapt to complex data patterns (Castelli et al., 2015). By harnessing the strengths of both approaches, a novel hybrid grey-genetic model could emerge. This innovative hybrid model seamlessly blends the data-driven insights of grey models with the non-linear adaptability of GP. The convolutional grey multivariate model lays the foundation by capturing temporal dependencies and extracting underlying patterns from the data. GP then enters the fray, sculpting unique non-linear programs that evolve to fit the complexities of electricity consumption data, revealing otherwise hidden relationships between variables.

The GPGMC(1,N) model extends beyond just improving electricity demand forecasting in Cameroon. Its potential value lies in its broader application across various domains. By providing more accurate forecasts, the GPGMC(1,N) model empowers policymakers in developing economies to make informed decisions on infrastructure development, resource allocation, and energy security strategies. This translates to improved energy management for utilities, who can leverage the model for planning, optimizing generation schedules, and minimizing outages. Furthermore, the core principles of GPGMC(1,N) can be adapted to forecast water consumption, traffic patterns, and sales figures in data-scarce environments. This research, by demonstrating the effectiveness of GPGMC(1,N) in Cameroon's electricity sector, paves the way for broader application and adaptation across various sectors in developing economies, ultimately contributing to improved decision-making, resource management, and overall economic development.

In the following sections, we delve deeper into the theoretical underpinnings and practical implementation of this novel model, showcasing its effectiveness in predicting annual electricity consumption in Cameroon and paving the way for a future illuminated by precise foresight.

This study is driven by two overarching objectives, each aimed at illuminating the path towards enhanced electricity forecasting in Cameroon:

1. Develop and implement the GPGMC(1,N) model: The primary objective is to meticulously design and implement the GPGMC(1,N) model, fine-tuning its parameters and adapting it to the specific context of Cameroon's electricity data. This involves defining the population size and operators for GP, selecting appropriate grey model components, and optimizing the training process for enhanced performance.

2. Evaluate the forecasting accuracy and explainability of the GPGMC(1,N): With the model in place, the second objective revolves around rigorously evaluating its forecasting accuracy. We will compare the GPGMC(1,N) predictions to those of established models and assess its effectiveness in capturing temporal trends and seasonal variations. Furthermore, we will delve into the interpretability of the model, identifying the key non-linear relationships and variables that drive its predictions.

In the fast-growing field of electricity forecasting models, the GPGMC(1,N) is a major innovation that has carved out a special place for itself thanks to its characteristics and considerable contributions. The novelties of this study are threefold:

1. Pioneering the convergence of GP and GMC(1,N): To the best of our knowledge, this study marks the first-ever application of a hybridized model combining the strengths of GMC and GP for electricity forecasting. This novel approach transcends the limitations of existing methods, unlocking a new frontier in forecasting accuracy and adaptability.

2. Empowering GMs with non-linear insights: While adept at data-driven forecasting, grey models often struggle to capture the intricate non-linear relationships within electricity consumption data. The GPGMC(1,N) bridges this gap by seamlessly integrating GP's evolutionary search for non-linear patterns, enriching the model's understanding of complex dynamics.

3. Tailoring data-scarce scenarios: Cameroon, like many developing nations, faces challenges in securing extensive historical data for electricity consumption. The novel GPGMC(1,N), built upon the foundation of grey models, thrives in such data-scarce environments, utilizing its robust data generation process to extract valuable insights from limited datasets.

Research is booming in the field of predicting electricity consumption across various sectors and timeframes. As Tamba et al. (2018) point out, these approaches can be broadly categorized into three groups: 1) Statistical models: These rely on established statistical methods like the ARIMA (Tarmanini et al., 2023), XGBoost regression (Wang et al., 2021), adaptive decomposition, and Markov-chain mixture distribution (Munkhammar et al., 2021). Their strengths lie in their interpretability, simplicity, and ease of use (Atalay et al., 2019; Kapoor and Wichitaksorn, 2023). However, they require expert knowledge to find relationships between variables and depend heavily on historical data. 2) Machine learning (ML) models: These utilize advanced algorithms like support vector machines (Haq et al., 2021; Yin et al., 2023), artificial neural networks (Wazirali et al., 2023), LSTM networks (Bilgili and Pinar, 2023), and deep recurrent neural networks (Abdulrahman et al., 2021). They excel at handling complex calculations in electricity consumption forecasting and can deliver impressive results with large datasets. 3) Grey models: These are data-driven methods suitable for situations with limited information. They are particularly useful for short-term predictions and offer an alternative when other models lack sufficient data.

However, predicting electricity consumption becomes increasingly complex with rapid urbanization and industrialization, driven by fundamental changes in the industrial structure (Yin et al., 2023). This highlights the need for continuous research and development of even more sophisticated forecasting methods.

Although the predictive accuracy of ML is well established, it still has some significant weaknesses. For instance, ML forecasts for electricity demand can be data-hungry, prone to overfitting, and offer limited insight into why consumption changes (Sapnken et al., 2023c). This can hinder their performance and usefulness for grid operators. Similarly, statistical models for electricity demand forecasts have difficulty adapting to non-linear changes and require high-quality data, which limits their accuracy in dynamic environments (Ungureanu et al., 2021).

According to Xie and Wang (2017), grey models are very useful for energy forecasts when data is scarce or complex. Their simplicity makes them interpretable and easy to implement, even with limited information, while their robustness allows uncertainty and data gaps to be managed (Qian and Sui, 2021; Lei et al., 2024). Although grey models (GMs) do not always achieve the maximum accuracy of complex models, they are a powerful option for reliable forecasts when data or expertise is scarce.

Given its effectiveness in producing precise predictions for incomplete information systems, the grey system theory introduced by Deng (1982) has found application in various domains, as indicated by Xie and Wang (2017). Just to cite a few, Qian and Sui (2021) proposed an intelligent grey model for forecasting renewable energy demand. Akay and Atak (2007) proposed an adjusted grey prediction model that integrates a new condition and rolling mechanism. Wang et al. (2023) introduced an innovative fractional cumulative operator to devise a structure-adaptive fractional derivative grey prediction model aimed at predicting China's overall energy consumption. There are many other grey models that have been developed and applied in other fields other than energy. Xie and Wang (2017) work summarizes them well.

While simple and data-efficient, univariate grey models (GM(1,1)) raise concerns. They neglect key external factors, leading to inaccurate forecasts when those factors change (Wang et al., 2022b). Sensitive to data imperfections, GM(1,1) models struggle with outliers and gaps, skewing predictions (Qian and Sui, 2021). Their focus on linear relationships misses complex dynamics, limiting accuracy in non-linear scenarios as demonstrated by Sapnken and Tamba (2022). Lacking sophisticated feature engineering, GM(1,1) models may overlook subtle patterns (Zhao et al., 2023). Validating their suitability is challenging due to their simplicity, and overfitting risks producing misleading forecasts. To achieve reliable and accurate predictions, researchers have consider combining them with other models (Guefano et al., 2021), incorporating external factors (Tien, 2012), or restructuring their framework for more flexibility (Wang et al., 2023).

GM(1,N) models can overcome most of the weakness of the basic GM(1,1) model and has a number of strengths, including its ability to handle small sample sizes, to handle non-linear relationships between variables and to deal with missing data (Ye et al., 2024). Nevertheless, the efficacy of the residual series within GM(1,N) is contingent on the prevalence of data points exhibiting consistent signs, a scenario that tends to be infrequent when the number of observations is limited (Min et al., 2012; San Cristóbal et al., 2015). Regrettably, GM(1,N) often associates the progression of the target variable with $(N−1)$ independent variables, considering factors that could influence system variations, as noted by Wang et al. (2022a). This issue was initially highlighted by Tien (2012), who demonstrated that GM(1,N) is essentially a causal model and is generally ineffective for forecasting purposes. Nevertheless, enhanced versions like GMC(1,N) have demonstrated their effectiveness in forecasting, as supported by studies conducted by Wu et al. (2018), Ding and Li (2020) and Yin and Mao (2023).

To improve the accuracy of predicting residual signs in Grey Models (GMs), researchers have explored enhanced residual sign estimators. For instance, Hsu and Chen (2003) introduced an upgraded GM that incorporates both residual ANNs sign estimation and residual modification for electricity consumption forecasts. In a similar vein, Hsu (2003) modified the residual of the GM(1,1) model, employing Markov-chain sign estimation to predict the value of the global integrated circuit industry. Building on these approaches, this study introduces an innovative method that combines residual modification and genetic programming (GP) sign estimation to refine the precision of the residual sign estimator.

GP, a strategy for evolving functions, is effective in performing designated tasks (Gil-Gala et al., 2023). Similar to genetic algorithms (GAs), GP utilizes crossover, mutation, and reproduction rules to find optimal solutions (Ong et al., 2005). What sets GP apart is its ability to perform well without assuming specific relationships between dependent and independent variables, making it particularly suitable for small datasets. GP offers two main advantages: it can derive a mathematical equation through regression analysis, and it can express a mathematical expression using a parse tree technique. Consequently, GP emerges as an efficient tool for achieving optimal residual sign estimation.

None of the aforementioned studies has combined GP and GMC(1,N) despite the benefits they could offer. This research seeks to address a gap in the existing knowledge by integrating residual modification and residual GP sign estimation to enhance a grey prediction model. The motivation behind creating this model lies in two primary reasons. Firstly, when dealing with a limited time-series dataset, employing GP can significantly enhance the precision of predicting residual signs. Secondly, compared to ANNs, GP demonstrates superior accuracy and reliability in constructing forecasting models, as evidenced by previous studies (Ong et al., 2005; Huang et al., 2006). The study utilizes electricity consumption data from Cameroon as an empirical case to showcase the effectiveness of the proposed model. They include:

1. Enhancing policy formulation: With reliable and accurate forecasts provided by the GPGMC(1,N), policymakers can design and implement informed strategies for infrastructure development, resource allocation, and energy security measures. This translates into more efficient management of energy resources, contributing to economic growth and sustainable development.

2. Improving grid stability: Accurate predictions of electricity demand can help optimize grid operations, enabling the anticipation of peak loads and facilitating the allocation of resources to prevent outages and ensure stable power supply. This empowers consumers and industries alike, fostering confidence and promoting economic activity.

3. Paving the way for future models: The success of the GPGMC(1,N) in Cameroon can serve as a springboard for further research and development in the field of grey modelling and hybrid forecasting approaches. This opens doors for exploring the application of similar models in other data-scarce contexts and for diverse forecasting challenges beyond electricity demand.

The remaining sections of this paper are divided as follows: The Section 3 outlines the methodology and principles used; Section 4 presents the simulations carried out, the results obtained and the related discussions; This is followed by Section 5, which reveals the significance of the findings, challenges to be overcome and the opportunities to be seized. Finally, the Section 5 concludes the paper and casts a glance towards future work.

Consider a system of $N$ variables (called sequences) defined as follows:

where $X1(0)$ represents the dependent variable (electricity demand) and $Xi(0)(i=2,...,N)$ represent the independent variables (prices, income, household expenses and number of subscribers). Assume there is a strong correlation between $X1(0)$ and $Xi(0)(i=2,...,N)$⁠. Moreover, we make the assumption that the length of the sequence for each variable $Xi(0)(i=1,...,N)$ is $n$⁠, so:

The GMC model, which involves multivariate grey convolution, relies on the mean sequence and 1-AGO (1st order accumulating generation operator). $Xi(1)(i=1,2,…,N)$ represent 1-AGO sequences of $Xi(0)(i=1,...,N)$ and are established as follows:

1-AGO defined in Eq. (3.1) is identical to the one defined in conventional GM(1,N) model. Eq. (4.1) illustrates the mean sequences generated by successive terms of $Xi(1)$⁠:

$Zi(1)(i=1,2,…,N)$ is defined by Eq. (4.2):

Assuming that $Xi(0)(i=2,...,N)$ and $X1(0)$ are still defined as in Eqs. (1) and (2), respectively, and Eq. (3) still defines the 1-AGO sequences $Xi(1)(i=1,...,N)$⁠. Then, the GMC(1,N) model is given by Eq. (5):

The expression $∑i=2Nβixi(1)(t)$ involves the derivative term represented by $βi$ coefficients, while $−α$ and $u$ are the development coefficient and parameters for GMC(1,N), respectively.

By hypothesizing that the right-hand side (RHS) of Eq. (5) can be expressed as a function $f(t)$ Tien (2012) initially estimates Eq. (5) as the subsequent difference equation.

This involves performing integrals on both sides of Eq. (5) over the range from $(t−1)$ to $t$ and subsequently employing the trapezoid formula for the remaining unspecified terms. Eq. (6) represents a set of linear equations that can be expressed in matrix format as:

where:

The least-squares method can be employed to calculate the parameters $A=[α,β2,...,βN,u]T$⁠, provided that the matrix $BTB$ is invertible.

Furthermore, Tien (2012) employed the starting condition $xˆ1(1)(t=1)=x1(0)(1)$ to derive the time response function ^[1] of the GMC(1,N) model as presented in Eq. (5):

The convolution integral is situated on the right-hand side of Eq. (10), posing challenges in deriving a direct expression. Fortunately, we can employ numerical integrations to estimate the outcome. The trapezoid formula is an easy-to-use method ^[2] that produces accurate time response function, as seen in Eq. (11):

$h(t)$ in Eq. (11) represents the unit step defined as:

Finally, by using inverse 1-AGO, it is possible to determine the predicted value $xˆ1(0)(t)$⁠:

For the sake of brevity, we have moved the analysis of the proof of superiority of the GMC model over the conventional GM(11,N) model to the  appendix, and we have also highlighted the weaknesses of the GMC(1,N) model.

The residual series is the disparity between the target values $x1(0)(t)$ and the anticipated values $xˆ1(0)(t)$⁠. To enhance the forecasting precision of the GMC(1,N) model, it is essential to develop a residual GMC(1,N) model. When original GMC(1,N) and residual GMC(1,N) are combined, the updated forecasted values are obtained. Nevertheless, the effectiveness of the residual series is influenced by the quantity of data points sharing the same sign. If there are fewer than four data points displaying the same sign, it is not feasible to construct the residual GMC(1,N) model.

Hsu and Chen (2003) proposed a forecasting technique using grey models in 2003. Their method combined modifying residuals with estimating their signs using ANNs. While effective, ANNs require large datasets and struggle with justifying the hidden layer complexity. This research, therefore, presents an improved version of the GMC(1,N) model that combines residual modification with estimating signs using GP. This new approach aims to boost the accuracy of predicting residual signs. More details about the proposed model's construction can be found in subsection 3.2.1.

Denote the residual sequence's initial absolute values as $ε(0)$⁠, which is given by,

where,

Using Eqs. (1)–(10), the GMC(1,N) model of $εˆ(0)$ can be created. Eq. (14.1) provides the forecast of accumulated residual series model, from which inverse AGO calculates $ε(0)$ as stated in Eq (14.2),

GP is a method developed by Koza (1994) for data forecasting and grouping, applicable in the realm of computer programs. It has found utility in symbolic regression and the identification of model structures. The key principles—crossover, mutation, and reproduction—share similarities with GAs (Nyathi and Pillay, 2018). Unlike GAs, GP employs a generic parse tree representation rather than the binary logic numbers (0 and 1) of genetic states. Consequently, GP has gained substantial popularity compared to conventional linear forecasting techniques, owing to its adeptness in navigating complex non-linear domains. Additionally, GP is extensively employed in practical scenarios, including forecasting coastal algal blooms, constructing credit scoring models, and simulating rainfall-runoff processes.

The operators $({+,−,log⁡,exp})$⁠, trigonometric functions $({sin⁡,cos⁡,tan})$⁠, and conditional statements (If, then, while) are among GP functions and statements. The GP parse tree in Figure 1 can be used to describe 3 y + x/y as a straightforward example.

Additionally, by combining a generic parse tree with symbolic regression, the GP operation system may provide an ideal prediction function. Figure 2 depicts an example of the crossover operator in GP. Unlike ANNs, GP is versatile in its application across various sample sizes. Moreover, in the process of selecting input variables, GP autonomously identifies the variables that carry the most significance in contributing to the model.

In this study, GP is employed instead of ANN sign estimation for predicting the sign of the residual, deviating from the approach taken by Hsu and Chen (2003) in developing a residual sign estimator. A forecasting equation can be obtained when the GP model is utilized, in addition to being able to build a forecasting model for a limited data set. Symmetric mean absolute percentage error (SMAPE) (Kim and Kim, 2016) can be used as the objective function to lower the GP forecasting error.

where $et=xˆ1(0)(t)−x1(0)(t)$⁠, $n$ stands for the length of the test dataset and $xˆ1(0)(t)$ and $x1(0)(t)$⁠, respectively, refer to the projected load value and actual load value of the test data. Given that SMAPE is theoretically bound by a number of conditions, including parameter value range and original data, it was chosen as the objective function.

This research tackles predicting the ups and downs (positive and negative signs) of residual series using a two-step GP model. It’s like a coin toss: heads (positive) or tails (negative). First, the model introduces a “dummy variable” called $ζ(t)$⁠. It's like a flag that indicates whether the residual for a specific year (⁠$t$⁠) is positive (⁠$ζ(t)=1$⁠) or negative (⁠$ζ(t)=0$⁠). Next, the model predicts the future sign of the residual (⁠$ζ(t+1)$⁠) based on the past two signs (⁠$ζ(t−1)$⁠) and $ζ(t)$⁠). It's like making an educated guess about the next coin toss based on the previous two. The model's “ $brain$” (the GP parameters) are listed in Table 1. In essence, this two-stage GP model uses the past to predict the future fluctuations in residuals, ultimately revealing the ups and downs in the data.

The $t$ ^th year residual's sign, $σ(t)$⁠, can be written as follows:

Thus, with Eqs. (1)–(16), the proposed forecasting approach, referred to as GPGM(1,N) (Multivariate grey convolution model powered with chaos-based multigene genetic programming), may be generated as:

The flowchart of the proposed prediction model is shown in Figure 3.

The accuracy of energy predictions plays a crucial role in managing and planning electrical grids. To evaluate how well predicted energy values (denoted as $xˆ1(0)(t)$⁠) match actual values (denoted as $(x1(0)(t)$⁠), statisticians use various criteria. Four common ones are:

• • Mean Absolute Error (MAE) (Eq. (18)): This measures the average absolute difference between predicted and actual values (de Myttenaere et al., 2016). A lower MAE indicates better accuracy, as it means predictions are closer to reality on average.

• • Mean Squared Deviation (MSD) (Eq. (19)): Similar to MAE, MSD calculates the average squared difference between predictions and actual values (Shi et al., 2022). However, it penalizes larger errors more heavily than smaller ones, emphasizing the importance of accurate forecasts for extreme values.

• • Root Mean Squared Error (RMSE) (Eq. (20)): This is simply the square root of MSD (Karunasingha, 2022). It provides the error in the same units as the original data, making it easier to interpret.

• • Coefficient of Determination ($R2$) (Eq. (21)): This measures the proportion of variance in the actual values that can be explained by the predicted values (Taira et al., 2023). A value of 1.0 indicates perfect prediction, while 0.0 implies no correlation. $R2$ helps assess how well the model captures the underlying trends in the data.

By analysing these different criteria, researchers and engineers can gain valuable insights into the strengths and weaknesses of their energy forecasting models. They can then use this information to refine their models and improve the accuracy of their predictions, leading to more efficient and reliable energy systems.

Equations (18)-(21) introduce key criteria for analysing energy prediction accuracy. $x1(0)(t)$ represents actual consumption, $x̅1(0)(t)$ is the average of $x1(0)(t)$⁠. While MAE, MSD, and RMSE are common statistics for evaluating energy forecasts, their interpretations differ.

MAE and RMSE, focusing on the average prediction error, provide similar insights but in different units. Both range from 0 to infinity and don't differentiate between overestimations and underestimations. However, the square root nature of RMSE gives it a unique characteristic: it prioritizes large errors heavily. This makes RMSE particularly valuable when minimizing significant discrepancies is crucial. Interestingly, MAE can be used to set boundaries for RMSE:

1. $MAE≤RMSE$⁠: This guarantees that RMSE will always be greater than or equal to MAE. If all prediction errors are equal, they coincide (⁠$RMSE=MAE$⁠).

2. $MAE∙√k≥RMSE$⁠: This inequality reveals the biggest potential difference between the two metrics, occurring when all prediction errors stem from a single sample. In this scenario, RMSE for that sample can be as high as MAE multiplied by the square root of the sample size (⁠$k$⁠).

In essence, these relationships highlight the trade-offs between MAE and RMSE. While MAE offers a straightforward average error, RMSE emphasizes larger discrepancies, making it a valuable tool for situations where minimizing significant errors is paramount.

This section dives into the simulation results obtained with GPGMC(1,N) model and some alternatives. The modelling processed follows three steps. First, we meticulously acquire data on electricity demand and relevant factors. This data undergoes cleaning and transformation to ensure compatibility with the GPGMC(1,N) model. After training the model with historical data, we fine-tune its parameters to optimize performance. Finally, we obtain future values of influencing factors by applying a simple regression of each factor over time because the models do not directly predict the influencing factors.

We built all the models and fed them training data as shown in Figure 3. For the BPNN model, we stuck with the default settings. The effectiveness of these models can significantly differ based on various parameters; however, identifying the optimal ones for forecasting can be a time-consuming process. Therefore, we compare them in a realistic setting where future outcomes are unknown, and the evaluation resources are limited.

We ran each model through 10 simulations and fine-tuned them to fit the data. Then, we gave them a 50-run stability test. The simulations were executed using Matlab R2021a on a personal computer equipped with 8 gigabytes of RAM and an AMD Ryzen 3 3200U processor with Radeon Vega Mobile Gfx operating at 2.60 Hz.

We used a combination of data sources, including: 1) Electricity consumption (in GWh) and subscriber data. These data came directly from ENEO-Cameroon (2023), the national electricity company; 2) Economic data: Information on real income per capita (FCFA/habitant), price of electricity (FCFA/kWh) and final household expenditure (FCFA) were drawn from World Bank (2021) indicators. All these datasets are shown in Table 2 (Source: ENEO-Cameroon (2023)).

To ensure accurate results, we divided the data into two sets: 1) Training set, which comprises 70% of the overall dataset (that is from 2001 to 2013). This data was used to build and train the model; 2) Validation set, made up of the remaining 30% (from 2014 to 2019). The splitting ratio 70:30 was chosen in order to achieve a balance between training the model and having enough data for validation. Ultimately, the validated model was utilized to project electricity consumption for the timeframe extending from 2020 to 2030.

A rigorous comparison was conducted between the developed GPGMC(1,N) and some competing grey models. ARIMA and BPNN were employed as benchmark models representing alternative non-grey approaches. Additionally, a linear regression (LR) model was implemented to validate the consistency of independent variables with reality. The LR model's analysis of variable signs aligns with real-world expectations. Positive values across final household expenses, number of subscribers and real per capita income reveal a direct link between electricity consumption and these factors (see Table 3). For instance, rising income translates to higher appliance spending, ultimately driving up household electricity use. Similarly, the sign of the price is in line with the economic theory of supply and demand, according to which a rise in the price of electricity would result in a fall in demand.

Based on the established metrics of MAE, MAD, and RMSE, where lower values indicate superior performance, the GPGMC(1,N) model emerged as the most effective predictor of annual electricity consumption in Cameroon. Table 4 shows that the combination of GP and GMC(1,N) is more effective in forecasting electricity demand. This is why the predictive curve of the GPGMC(1,N) fits the real data almost perfectly (see Figure 4). The results equally show that while GPGMC(1,N) exhibited a slight margin over ARIMA, BPNN, and competing grey models, it’s training phase required a longer duration (1263 ms). When the residuals of each model are plotted on the same graph, it is clear that those of the GPGMC(1,N) model are by far the smallest for the entire data set (see Figure 5). This is further proof that the GPGMC(1,N) predictive curve fits the real data well. The graphical representation of the absolute percentage errors (APEs) generated by each model reinforces this last proof of the superiority of the GPGMC(1,N). Indeed, as Figure 6 shows, the APEs of predictions obtained with the GPGMC(1,N) model are the lowest of all in each forecasting period.

Comprehensive analysis revealed that GPGMC(1,N) secured the first position in terms of overall efficiency, exceeding all alternative models. Conversely, GM(1,N)-VAR(p) and BPNN demonstrated the lowest performance, although they compensated for this by requiring minimal training time. Table 4 clearly identifies the most efficient predictive model for each performance metric, with GPGMC(1,N) consistently occupying the top positions. The reasons why GPGMC(1,N) take more time for training is because GP evolves entire programs or structures. This makes fitness evaluation more complex and computationally expensive, as the entire program needs to be executed and its output evaluated against the desired outcome. Additionally, GP can grow in size and complexity as they evolve, further increasing the computational cost of evaluation. Despite this apparent weakness, the new model nevertheless manages to produce excellent results when all the validation criteria are considered.

The stability of the GPGMC(1,N) model is a crucial aspect of its performance. To ensure reliability, we employed a rigorous testing procedure that involved multiple simulations and fine-tuning:

1. Step 1: Multiple simulations (10 runs): We repeated the training process for the GPGMC(1,N) model and all competing models ten separate times. This helped account for the inherent variability present in training algorithms. Each run utilized a random split of the data between the training and validation sets.

2. Step 2: Fine-tuning for optimal performance: Following each training run, we fine-tuned the model parameters to achieve the best possible performance on the validation set. This fine-tuning process helped ensure the models were optimized for the specific characteristics of the electricity demand data in Cameroon.

3. Step 3: Stability testing with 50 runs: After the training and fine-tuning phases, we conducted a 50-run stability test. In each run, the already optimized model was used to forecast electricity consumption on the validation set (data from 2014 to 2019). By analysing the variation in forecasting performance across these 50 runs, we were able to assess the model's stability.

The results of the stability tests are presented in Table 5. These results suggest the GPGMC(1,N) model demonstrates greater stability compared to the competing models. The GPGMC(1,N) model has a lower average MAE (0.0125) compared to the competing models, indicating its forecasts are generally closer to the actual values on average. Moreover, the standard deviation of MAE for GPGMC(1,N) (0.0010) is also the lowest, suggesting less variation in forecast accuracy across the 50 runs. Finally, the maximum forecast error ^[3] for GPGMC(1,N) (2.50) is also the lowest, implying a lower probability of significant deviations from actual values in any single test run.

Having confirmed the validity and stability of the GPGMC(1,N) model, we utilized it to predict the electricity demand in Cameroon for the duration spanning 2020 to 2030. The outcomes of the forecast are detailed in Table 6. It can be seen that electricity demand in Cameroon will rise from 7659.9 GWh in 2020–10797.5 GWh in 2030, which represents a record level of demand for the country. It is therefore necessary for public authorities and policy makers to take measures in order to meet this demand over the coming years.

The results of this study raise challenges to ensure reliable and affordable electricity access for all, but also open a window of opportunity to drive economic growth and foster sustainable development. Major challenges, opportunities and implication for policy are discussed in this section.

Cameroon is a developing country with a growing population and a booming economy, with a population growth rate of 2.6% and GDP growth of 3.54%. This demographic and economic growth is leading to an increase in demand for electricity. Recent studies have revealed that demand for electricity in Cameroon is expected to continue to increase over the next years (Guefano et al., 2021; Dieudonne et al., 2022; Sapnken et al., 2023b). This increase in electricity demand presents a number of challenges for the Cameroonian government. Firstly, it will require significant investment in electricity generation, transmission and distribution infrastructure. This includes the expansion of existing power plants, the construction of new power stations (potentially from renewable energy sources) and the modernization of the electricity network to cope with the increased load (SND30, 2020).

Secondly, the growth in demand for electricity could jeopardize the country's energy security. Cameroon currently relies heavily on hydroelectricity, with more than 60% of its (Tamba et al., 2022), which is a renewable energy source but can be vulnerable to droughts and climate change. Diversifying the energy mix with more resilient sources, such as solar and wind power, can improve the country's energy security and reduce its dependence on imported fossil fuels (Sapnken et al., 2024).

Finally, the rising cost of electricity can have a negative impact on the lives of Cameroon citizens, particularly those on low incomes. It is important to put in place policies to mitigate this impact, such as subsidies targeted at low-income households or programmes to promote energy efficiency (Jacques Fotso et al., 2023).

Despite these challenges, the growth in demand for electricity also presents opportunities for Cameroon. Firstly, increasing access to electricity can support economic growth and job creation (Ayuketah et al., 2023). Businesses and industries need electricity to run, and access to electricity can stimulate productivity and innovation. Secondly, investment in renewable energy and energy efficiency can create new jobs in sectors such as installation, maintenance and research. These jobs can be particularly beneficial for young people and women. Finally, the transition to renewable energy sources can help to protect the environment (Kouer and Meukam, 2023). Renewable energies do not produce greenhouse gases, which can help combat climate change.

To meet the challenges and seize the opportunities presented by the growth in demand for electricity, the Cameroonian government needs to take a number of policy measures. These include.

1. The development of a long-term national energy plan that sets out strategies to meet future demand, diversify the energy mix and ensure affordability.

2. The implementation of regulatory frameworks that encourage investment in renewable energy and energy efficiency, attract private sector participation and ensure fair competition in the energy sector.

3. The implementation of demand-side management programmes that encourage consumers and businesses to adopt energy-saving practices and technologies, reducing peak demand and improving grid stability.

4. Developing social safety net programmes to protect vulnerable populations from the impact of rising electricity costs.

By taking proactive steps to address the challenges and opportunities presented by growing electricity demand, Cameroon can ensure a sustainable and prosperous future for its citizens.

A new hybrid grey prediction approach powered by genetic programming, abbreviated GPGMC(1,N), has been proposed in this paper. This approach combines the advantages of grey prediction models, which are capable of handling limited and non-normal data, with the advantages of genetic programming, which can be used to improve forecast accuracy. More specifically, GP allows the signs of the forecast residuals to be estimated more rigorously. This allows the GMC(1,N) model, on the one hand, to better capture long-term trends in the data and, on the other hand, to better capture short-term variations in the data.

The hybrid GPGMC(1,N) approach was applied to real electricity demand data in Cameroon. The results showed that this combination is able to improve the accuracy of electricity demand forecasts compared to traditional grey prediction models. The results have important implications for research and practice in electricity demand forecasting. They suggest that hybrid approaches that combine the advantages of different types of prediction models can be used to improve forecast accuracy. Future research could explore the use of the approach for electricity demand forecasting in other contexts. It could also explore the use of other optimization techniques, such as ML, to further improve forecast accuracy.

The success of GPGMC(1,N) in electricity forecasting could also open doors for its application in diverse fields with limited data and volatile fluctuations. Potential areas include water demand forecasting for efficient water resource management, traffic congestion prediction for optimized traffic flow control, and even stock market trend analysis for informed investment decisions. The data-sparseness resilience and the adaptive learning capabilities of GP make GPGMC(1,N) a promising tool for enhancing forecasting accuracy across various domains.

We are particularly grateful to Mr David Horgan for his help in troubleshooting the algorithm implemented in Section 3. His expertise in Python/Matlab programming was crucial in resolving the performance issues and ensuring reliable results.

This work was supported by the Natural Science Foundation of Sichuan Province (No. 2023NSFSC0428), the Central Government Funds of Guiding Local Scientific and Technological Development (No. 2023ZYD0004), the Sichuan National Applied Mathematics Center open fund (No. 2024-KFJJ-01–01), and the Chengdu Science and Technology Project (No. 2024-YF05-00323-SN).

According to the results derived in Section 3.1, we can draw two major analyses, which are as follows:

• • If the grey system is a single-variable model, meaning when $N$ equals 1, the second component on the right-hand side of Eq. (5) diminishes, leading to a simplified model:

This particular differential equation corresponds to the image equation of the traditional GM(1,1) model. By fixing $f(τ)$ to a fixed value $u$⁠, we are able to derive the following from Eq. (10):

This serves as the time response function for the GM(1,1) model. The parameters $α$ and $u$ can still be determined through the least-squares method outlined in Eq. (7). However, the matrix $B$ is altered, leading to:

• • If the model parameter $u=0$⁠, the GMC(1,N) model presented by Eq. (5) reduces to Eq. (A.2).

The essential formula for the GM(1,N) model is represented by Eq. (13). In the earlier studies conducted by Liu and Lin (2011, pages 107–147), it was established that the RHS of Eq. (A.2) remains a constant. The temporal response function of the multivariate GM(1,N) model can be accurately derived from the traditional GM(1,1) model as:

Unlike the GMC(1,N) model, the GM(1,N) model features one less parameter in the matrix $A∼=[α,β2,...,βN]T$⁠. The values of these parameters can also be determined using the least-squares method. B also demonstrates a distinction, namely:

The preceding analysis unmistakably indicates that GMC(1,N) outperforms the conventional GM(1,N) model in the specified domains:

1. The configuration of the GMC(1,N) model is consistent with the classical GM(1,1), while the standard GM(1,N) cannot be transformed into the GM(1,1) model.

2. Within the traditional GM(1,N) model, the inaccurate treatment of the driving term as a constant is evident, leading to a flawed time response function for the GM(1,N) model.

3. When calculating $A=[α,β2,...,βN,u]T$⁠, the GM(1,N) model is more appropriate. In contrast, the GMC(1,N) model incorporates the summation of values from the second column to the $N$ ^th column of $B$⁠, while the conventional GM(1,N) model considers these terms as fixed constants.

However, the GMC(1,N) model still has at least three shortcomings, which we mention below:

1. In order to estimate $A=[α,β2,...,βN,u]T$ from the GMC(1,N) model, the difference equation (Eq. (6)) is solved using the least squares method. The time response function, however, is produced using Eq. (10) (the first-order differential equation). Eq. (6) and Eq. (10) are roughly equivalent, but they are fundamentally different. Due to this parameter mismatch, the GMC(1,N) model may become unstable.

2. Similar to the conventional GM(1,N) model, the GMC(1,N) model also functions as a factor and state model. Nevertheless, the simplicity of the GMC(1,N) model's structure remains a limitation. While several modifications have been introduced to enhance the model's structure, there has been a lack of investigation into the linear influence of time $t$ on its performance. This gap in research may contribute to the model's suboptimal prediction accuracy.

3. Finally, a significant portion of studies has overlooked the possibility that the input data might include irrelevant information, thereby causing overfitting or underfitting issues in the prediction stage.

The previous analysis lead to the conclusion that the GMC(1,N) model still has flaws related to parameter estimation mismatch and that it is too basic to handle systems found in real-world settings. These glaring flaws motivate to develop an efficient GMC(1,N) model that fixes them.