1 Introduction

2. Preliminaries

3. Proposed Methodologies

3.2. GWDO-Based Optimizer Model

The previous step of deep learning, using the FCRBM model with ReLU and multivariate autoregressive algorithm, produces a prediction of future electrical energy consumption. The forecast has a minimal error, as determined by the capabilities of the FCRBM model. To enhance accuracy in predicting energy consumption, the FCRBM-based forecaster module’s results are inputted into our suggested GWDO algorithm-based optimization phase.

$$\left\{\begin{array}{c}{x}_{\mathrm{new} }=1 \mathrm{if} \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\left(1\right)⩽\mathrm{s}\mathrm{i}\mathrm{g}(j,i)\\ x_{\mathrm{new} }=0 \mathrm{if} \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\left(1\right)>\mathrm{s}\mathrm{i}\mathrm{g}(j,i)\end{array}\right.$$

(5)

$$v_i=v\mathrm{m}\mathrm{a}\mathrm{x}\times 2\times \left(\mathrm{rand}\left( \mathrm{populationsize} ,n\right)-0.5\right)$$

(6)

The objective of our suggested algorithmic optimization step is to further reduce errors in the predicted energy consumption pattern. Therefore, the optimization phase aims to minimize errors by using an objective function, which is represented by the following model.

$$\underset{R{d}_{th},I{r}_{th},C_i}{\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}}\mathrm{Error}\left(x\right)\forall x\in \left\{h,d\right\}$$

(7)

By integrating the GWDO algorithm into the optimization module, the forecasting error is further reduced, aiming to enhance accuracy and convergence speed by fine-tuning the model’s adjustable parameters. Thus, the optimization phase is intricately linked with the FCRBM-based forecaster to minimize error and enhance forecast accuracy, with MAPE minimization serving as the primary objective function:

$$\mathit{Mini}\mathit{MAPE}\left(j\right)\forall j\in \left\{\mathrm{1,2},3,\dots .\tau \right\}$$

(8)

$$R{d}_{th},I{r}_{th}$$

The terms $R{d}_{th}$ "redundancy threshold,$I{r}_{th},$" "irrelevancy threshold," and "candidates interaction"$C_{ i}$refer to specific concepts. The GWDO method optimizes the suggested phase based on parameters$d_{th}$, $I{r}_{th},$ and${ C}_i$, which are then used in the data preparation phase. The feature selection approach in the data preprocessing step uses optimized values of$d_{th}$, $I{r}_{th}$thresholds for optimum feature selection$C_i$. Integrating the optimization phase with the forecaster phase enhances forecast accuracy, albeit at the expense of a reduced convergence rate.

Equations 9 and 10 define the fitness functions for velocity and position. The position and velocity vectors are updated by comparing random numbers $\left(rand\right(.) \in [\mathrm{0,1}\left]\right)$ with the fitness function$\left(FF\right(.) \in [\mathrm{0,1}\left]\right)$, as outlined in equation 11.

$$FF\left(v\left(i\right)\right)={\displaystyle \frac{\mathrm{MAPE}\left(x_{\mathrm{new} }\left(i\right)\right)}{\mathrm{MAPE}\left(v\left(i\right)\right)+\mathrm{MAPE}\left(x_{\mathrm{new} }\left(i\right)\right)}}$$

(9)

$$FF\left(x_{\mathrm{new} }\left(i\right)\right)={\displaystyle \frac{MAPE\left(v\right(i\left)\right)}{MAPE\left(x_{\mathrm{new} }\left(i\right)\right)+MAPE\left(v\right(i\left)\right)}}$$

(10)

Table 1. Simulation parameters used.

Table 1. Simulation parameters used.

Parameters	Values
Population size	24
Number of decision variables	2
Number of iterations	100
$RT$	3
$g$	0.2
$\alpha$	0.4
$dimMin$	-5
$dimMax$	5
$Vmax$	0.3
$Vmin$	-0.3
$Crossoverrate$	0.9
$mutationrate$	0.1
Learning rate	0.0001
Weight decay	0.0002
Momentum	0.5

If the random number is smaller than the fitness function, the load value will be updated since our objective is to minimize the function.

$$F_{pr}\left(i\right)=\left\{\begin{array}{ll}{v}_n\left(i\right)& \mathrm{if} \mathrm{rand}\left(i\right)⩽FF\left(v\left(i\right)\right)\\ x_{new}^n\left(i\right)& \mathrm{if} \mathrm{rand}\left(i\right)⩽FF\left(x_{\mathrm{new} }\left(i\right)\right)\end{array}\right.$$

(11)

The problem of load update influencing the random value is addressed by eliminating this influence. Therefore, the comparison is made between the fitness function of the candidate input and the fitness function of the previous one, as shown in equation12. This ensures that the selected load update value maintains a high level of accuracy.

$$F_{pr+1}\left(i\right)=\left\{\begin{array}{ll}{v}_{n+1}\left(i\right)& {\displaystyle \frac{v_n\left(i\right)}{v_n\left(i_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)}}⩽FF\left(v\left(i\right)\right)\\ x_{\mathrm{new} }^{n+1}\left(i\right)& {\displaystyle \frac{x_{\mathrm{new} }^n\left(i\right)}{x_{new}^n\left(i_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)}}⩽FF\left(x_{\mathrm{new} }\left(i\right)\right)\end{array}\right.$$

(12)

4. Experimental Results and Discussions

4.1. Stage One: Electrical Load Forecasting

In this stage, the effectiveness of the FS-FCRBM-GWDO framework, along with benchmark frameworks such as AFC-STLF, Bi-level, MI-mEDE-ANN, and FS-ANN, is evaluated. These benchmarks were selected due to their architectural similarities with the proposed framework. However, the FS-FCRBM-GWDO and the benchmark models have distinct computational challenges, focusing on accuracy, convergence rate, or stability. The FS-FCRBM-GWDO model is tested using real-time hourly energy usage data from the Rwandan power system, covering four years from 2018 to 2021. 80% of the data is used for training the FCRBM model, while the remaining 20% is for testing. The control parameters used in the simulations are consistent across both the proposed and benchmark models, ensuring a fair comparison. The FS-FCRBM-GWDO framework is assessed using two performance metrics: (i) accuracy, measured by mean absolute percentage error (MAPE), variance (σ²), and Pearson correlation coefficient; and (ii) convergence speed, measured by execution time and convergence rate. The variance ( ${\sigma }^2$) is mathematically represented as follows in equation 16.

$${\mathsf{\sigma }}^2={\displaystyle \frac{1}{\tau }}\sum _{t=1}^{\tau }\left(R_t-F_t\right),$$

(16)

The symbol $\tau$indicates the number of timeslots,$R_t$ identifies the actual load, $F_t$ represents the predicted load at time$t$, and ${\mathsf{\sigma }}^2$represents the variance. The accuracy of the performance metrics is computed using the following formula.

$$\mathrm{A}\mathrm{c}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{y}=100-\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}\left(\mathrm{x}\right).$$

(17)

The convergence speed is determined by both the execution time and the convergence rate. The following is a comprehensive description.

1. Execution Time: This metric refers to the duration required for the forecasting model to predict future electrical energy consumption patterns. It is measured in seconds, with faster models having shorter execution times.

2. Convergence Rate: This aspect measures the speed at which the model reaches a specific epoch where performance stabilizes, and the error ceases to decrease significantly with additional epochs. Models with a high convergence rate reach this stabilization point quickly, often at early epochs. Forecasting models are classified as fast if they exhibit minimal execution time and achieve early convergence, indicating efficient performance.

Table 2. GWDO’s Simulation parameters.

Table 2. GWDO’s Simulation parameters.

Control parameters	Value
Number of hidden layers	1
Number of neurons in hidden layer	10
Output layer	1
Number of output neurons	1
Number of epochs	100
Number of iterations	100
Learning rate	0.0019
Momentum	0.6
Initial weight	0.1
Initial bias	0
Max	0.9
Min	0.1
Decision variables	2
Population size	24
Delay of weight	0.0002
Historical load data	4 years
Exogenous parameters	4 years

The learning evaluation process compares a model’s performance on training and testing data over multiple epochs to determine if it is genuinely learning. A poor learning curve, with high variance and bias, indicates overfitting, while a good curve, like that of the FCRBM model, shows low variance and bias with decreasing errors. Initially, the model has a high error rate (MAPE), which decreases with more training, reaching a minimal value, indicating effective learning. Figure 5 illustrates these results.

Figure 4. REG’s dataset with month and year indexes.

Figure 4. REG’s dataset with month and year indexes.

Figure 6 provides a comparative analysis of the FS-FCRBM-GWDO framework against benchmark models like FS-ANN, AFC-STLF, Bi-level, and MI-mEDE-ANN in predicting day-ahead electrical energy consumption for the REG load data center. Table 3 further details the accuracy of these models, comparing metrics such as mean absolute percentage error (MAPE), variance, and correlation coefficient. The findings clearly indicate that the FS-FCRBM-GWDO architecture offers superior accuracy in forecasting the next day’s electrical energy consumption for Rwanda’s power system. Both the proposed and benchmark models are adept at capturing and adapting to the non-linear patterns present in historical energy consumption time series data.

The proposed hybrid FS-FCRBM-GWDO model uses nonlinear activation functions like tangent hyperbolic (tanh), sigmoidal, and ReLU to predict energy consumption patterns. Unlike benchmark frameworks like FS-ANN, AFC-STLF, Bi-level, and MI-mEDE-ANN, which use sigmoid activation functions, the proposed model uses ReLU and multivariate autoregressive algorithms for rapid convergence and addressing vanishing gradient and overfitting issues. The model’s energy consumption pattern closely aligns with actual data, with a MAPE of 1.10%, outperforming the standard frameworks with values of 2.2%, 2.1%, 3.4%, and 2.6%, respectively. Hence, the results presented in Figure 6 and Table 3 suggested hybrid FS-FCRBM-GWDO model outperforms the standard frameworks in terms of accuracy.

Figure 7 shows a week-long prediction of hourly electrical energy consumption, demonstrating the superior performance of the FS-FCRBM-GWDO model compared to existing models like FS-ANN, AFC-STLF, Bi-level, and MI-mEDE-ANN. The FS-FCRBM-GWDO model achieved a MAPE of 1.18%, significantly outperforming the benchmark models. The model’s accuracy is attributed to the use of a deep learning-based FCRBM with ReLU, a multivariate autoregressive algorithm, and GWDO optimization. Figure 8 and Table 3 show that the FS-FCRBM-GWDO model closely tracks the actual energy consumption curve, ensuring better performance in monthly predictions. Table 4 presents a performance evaluation for the leap year 2018, using MAPE, variance, and correlation coefficient metrics.

FS-FCRBM-GWDO module’s performance in terms of MAPE and convergence speed. Figure 9, Figure 10, and Figure 11 present a statistical assessment of MAPE, execution time, and convergence speed for the proposed FS-FCRBM-GWDO model and benchmark models (FS-ANN, AFC-STLF, MI-mEDE-ANN, and Bi-level). The FS-FCRBM-GWDO model achieves the lowest MAPE of 1.18%, indicating high accuracy, compared to higher MAPE values in the benchmark models of 3.45%, 2.17%, 2.12%, and 2.65%, respectively. However, integrating the optimization module increases execution time from 25 to 95 seconds. The FS-FCRBM-GWDO model balances accuracy and speed by using GWDO for optimization, ReLU activation, a multivariate autoregressive algorithm, deep learning FCRBM, and advanced data preprocessing. Despite longer execution times compared to FS-ANN, the FS-FCRBM-GWDO model offers superior accuracy and efficiency.

Figure 11 illustrates the convergence speed of the proposed hybrid FS-FCRBM-GWDO model compared to benchmark models including FS-ANN, Bi-level, AFC-STLF, and MI-mEDE-ANN, based on 100 iterations. As the number of iterations increases, MAPE decreases for all models. Notably, the proposed model demonstrates rapid convergence, reaching stability around the 10th iteration, indicating its efficient search ability. In contrast, benchmark models such as FS-ANN, Bi-level, AFC-STLF, and MI-mEDE-ANN, converge later, around the 33rd, 29th, 25th, and 21st iterations respectively, showcasing slower convergence rates. This analysis suggests that the proposed GWDO algorithm offers superior performance for optimization in integrated frameworks due to its faster convergence compared to existing benchmark models. The depicted convergence analysis focuses solely on the MAPE performance metric for both proposed and existing models.

Figure 12 compares the proposed hybrid FS-FCRBM-GWDO model to benchmark models, including FS-ANN, Bi-level, AFC-STLF, and MI-mEDE-ANN, regarding the cumulative distribution function (CDF) of error. The FS-FCRBM-GWDO model outperforms the current models in terms of CDF. The FCRBM model, which utilizes deep learning, is capable of providing accurate predictions even in situations characterized by high levels of uncertainty. This is due to the deep layers of the model being able to effectively capture the essential characteristics. Therefore, our suggested FS-FCRBM-GWDO framework is a superior option for distribution system operators to achieve efficient and effective energy management of the smart grid. The FS-FCRBM-GWDO framework, along with other current frameworks like FS-ANN, Bi-level, AFC-ANN, and MI-mEDE-ANN, has been evaluated in terms of computational complexity, execution time, convergence rate, and accuracy.

Further evaluation of the FS-FCRBM-GWDO framework and existing frameworks, such as FS-ANN, Bi-level, AFC-ANN, and MI-mEDE-ANN is presented in Table 5. This evaluation encompasses computational complexity, execution time, convergence rate, and accuracy metrics. Based on simulation results, performance analysis, and discussions, it is concluded that the proposed hybrid FS-FCRBM-GWDO model surpasses benchmark models in terms of convergence rate, accuracy, computational complexity, and execution time aspects.

3.2. Electricity Cost Evaluation Using RTPS and CPPS under OTI

By scheduling smart home appliances using anticipated RTPS, the proposed GmEDE algorithm efficiently reduces electricity cost when compared to modified Evolutionary Differential Evolution (mEDE) and Grey Wolf Optimization (GWO). The program optimizes the transition of appliances from on-peak to off-peak timeslots by coordinating pricing schemes with patterns of energy use. The suggested GmEDE-based expenses reduce demand peaks and energy prices compared to both GWO and mEDE. Simulations demonstrate that by arranging smart home equipment in the best possible way, the suggested GmEDE algorithm continuously lowers power bills.

The proposed GmEDE-based framework outperforms both GWO and mEDE in terms of reducing peaks in demand and electricity costs. Figure 18-A shows that unscheduled loads result in high demand peaks, resulting in high prices during specific hours. Figure 18-B shows that GWO shows higher costs at the beginning timeslots, while GmEDE maintains minimum costs throughout the 24 hours. Figure 18-C shows that the proposed GmEDE algorithm consistently reduces electricity costs by optimally scheduling smart home appliances.

The electricity cost profile for a 15-minute OTI shows that the forecast CPPS remains constant except during critical peak hours. The maximum peak in an unscheduled load scenario is 181.55 cents, but when smart home appliances are scheduled, it reduces to 83.07 cents. The 30-minute OTI has similar costs, but no peaks emerge except at the starting time of the day. The proposed GmEDE algorithm significantly reduces the unscheduled appliance electricity cost from 766.8 cents to 203.46 cents for the 60-minute OTI. The overall unscheduled cost is reduced from 1300.891 cents to 1085.91 cents when smart home appliances are scheduled using the GmEDE algorithm.

Additionally, the overall electricity cost reduction for 30 and 60 minutes OTI is depicted in Figure 22. A brief comparison of electricity cost under forecasted RTPS and CPPS for 15, 30, and 60 minutes OTI is provided in Table 8. In summary, the proposed framework optimally schedules smart home appliances, leading to reduced overall aggregated electricity costs for residents compared to mEDE and GWO under forecasted RTPS and CPPS.

In Figure 20 and Figure 21, the results analysis shows that the proposed GWDO algorithm outperforms other heuristic techniques (GA, BPSO, WDO) and unscheduled load in optimizing energy consumption and reducing electricity costs. Without RESs and ESS, GWDO reduces peak power consumption by 35.16% compared to 32.96% for GA, 31.86% for BPSO, and 33.51% for WDO. With RESs, GWDO achieves a 28.39% reduction in peak power consumption, outperforming GA (24.69%), BPSO (30.86%), and WDO (32%). Additionally, GWDO provides the lowest electricity costs, peaking at 0.49 cents/kWh compared to GA (0.9 cents/kWh), BPSO (0.6 cents/kWh), and WDO (0.55 cents/kWh), demonstrating the most stable and optimal profiles across scenarios.

Figure 20. Energy consumption profile of users: (a) Without RESs and ESS; (b) With RESs; (c) With RESs and ESS.

Figure 20. Energy consumption profile of users: (a) Without RESs and ESS; (b) With RESs; (c) With RESs and ESS.

Figure 21. Electricity cost profile: (a) Without RESs and ESS; (b) With RESs; (c) With RESs and ESS.

Figure 21. Electricity cost profile: (a) Without RESs and ESS; (b) With RESs; (c) With RESs and ESS.

Figure 22. Aggregated electricity cost evaluation under forecast RTPS and CPPS.

Figure 22. Aggregated electricity cost evaluation under forecast RTPS and CPPS.

6. Conclusions

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