Load forecasting of microgrid based on an adaptive cuckoo search optimization improved neural network

1.   Introduction

Load forecasting is an important component of microgrid control and operation. It is also the basis for optimizing distributed energy sources such as energy storage and photovoltaics. It directly affects the stability and integrity of the entire power system ^[1,2]. Accurate load forecasting has become an important tool for energy market competition ^[3,4,5].

There are various techniques and methods for load forecasting, such as time series analysis, modal decomposition, Fourier analysis, etc. ^[6]. There is serious dynamic uncertainty and nonlinearity in electrical network environment, and the high degree of uncertainty caused by different consumers in load consumption patterns and the highly inconsistent energy consumption patterns of the same consumer need to be considered in load forecasting ^[7], which makes load forecasting in microgrid still a complex problem ^[8]. Traditional data-driven prediction methods have two difficult problems to solve: First, they cannot fully utilize the correlation between measurement data and load data. Second, they do not support extraction patterns independent of fixed pattern lengths ^[9]. Therefore, there are still some significant challenges in load forecasting of microgrid.

In recent years, neural networks have gradually been proven to be an effective prediction method ^[10]. However, the application of neural networks relies on the strategy called empirical risk minimization ^[11]. In this case, the total cost representing the error between the actual value and the predicted value usually has different trends in the training and testing samples, even if the neural network performs well on the training samples, it may still lead to poor performance in new situations. So, some advanced machine learning algorithms such as artificial rabbit optimization, swarm intelligence optimization, balancing composite motion optimization, gradient boosting decision trees algorithms, etc. are often used to improve these conventional prediction methods ^{[12,13,14,15]}.

An improved cuckoo search algorithm was designed to optimize the neural network in this paper. By improving the cuckoo search algorithm, the optimal neural network parameter configuration can be automatically searched, thereby achieving the minimization of load prediction errors in microgrids.

2.   Materials and methods

2.1.   Design of BP neural network

The back propagation (BP) neural network is a multilayer feedforward network trained according to the error BP algorithm, which is widely used in nonlinear approximation problems ^[16]. The structure of BP the neural network is shown in Figure 1.

A BP neural network typically consists of three layers: input layer, hidden layer, and output layer ^[17]. The number of hidden layer nodes in a BP neural network has a significant impact on its prediction accuracy. When the number of nodes is small, the training frequency increases and the training accuracy is not high; when there are many nodes, the training time increases and is easy to overfit. Usually, the number r of hidden layer units can be designed based on experience as ^[18]:

where, p is the number of input layer units; q is the number of output layer units; and d is an integer ranging from 1 to 10.

The main steps of the learning process of the BP neural network are as follows:

Step 1: Initialize the weights and biases of the neural network.

Step 2: Input the learning sample to the network, and obtain the network output by weighting and mapping. The weighted input design for the i^th neuron is:

in which n^(l-1) is the number of neurons in layer l-1, a_j^(l-1) is the output of the j^th neuron in the previous layer, w_ij^(l) is the weight connecting the j^th neuron and the i^th neuron, and b_i^(l) is the bias term in layer l.

The activation function is represented as:

Step 3: Calculate the loss based on the output of BP network and the real labels, that is:

where J is the loss function, m is the number of samples, y_i is the true label, and ${\widehat{y}}_{i}$ represents the predicted value of the neural network.

Step 4: Calculate the gradient of the loss for each weight and bias in the network. This process propagates back from the output layer to the input layer, using the chain rule for gradient calculation.

The error term of the output layer is calculated as follows:

where ${{\delta }_{i}}^{\left(o\right)}$ is the error term of the i^th neuron in the output layer, $\mathit{\nabla} {{a}_{i}}^{\left(o\right)}J$ represents the gradient of the loss function with respect to the output, $\odot$represents element by element multiplication, σ′(z_i^(o)) represents the derivative of the activation function of the output layer, and ${{z}_{i}}^{\left(o\right)}$ represents the weighted input of the neurons in the output layer.

The error term of the hidden layer is calculated layer by layer according to the following formula, that is:

where l is the layer index, W^(l+1) is the weight matrix of the next layer, and T stands for matrix transpose.

Step 5: Update the weights and biases of the network according to the gradient information to reduce the difference between the predicted results and the true values. The weight update law of the output layer is:

The weight update law of the hidden layer is:

where ${w}_{ij}^{\left(o\right)}$ is the mapping weight from the i^th neuron in the hidden layer to the j^th neuron in the output layer, ${w}_{ij}^{\left(l\right)}$ is the mapping weight from the i^th neuron of layer l to the j neuron of layer l + 1, α is the learning law, and k is the number of iterations.

The deviation update law of both the output layer and the hidden layer is designed as:

Step 6: Repeat Steps 2 to 5 until the loss converges or the predetermined number of iterations is reached.

There are a lot of parameters in a neural network, including the number of layers, the number of neurons, the weight, and so on. These parameters have a significant impact on the prediction results. To achieve satisfactory prediction results, it is necessary to optimize these parameters appropriately, which makes the training of neural networks complex and time-consuming.

2.2.   Improvement of cuckoo search algorithm

The cuckoo search (CS) algorithm is a metaheuristic optimization algorithm proposed based on simulating the parasitic breeding behavior of cuckoo ^[19]. In this algorithm, a set of nests is considered as a population, and the eggs in each nest are considered as a feasible solution to the optimization problem ^[20]. During breeding, cuckoos randomly find parasitic nests through Levy flights and lay their own eggs (candidate solution) in the nests, and the parasitic nest with the best egg (highest fitness value) is preserved for the next generation. In order to obtain the best parasitic nest, a nest replacement mechanism is introduced to the CS algorithm to preserves high-quality solutions in each iteration and randomly replace poor solutions. After multiple iterations, the optimal solution will be finally found in a vast solution space ^[21]. Compared with other algorithms, the CS algorithm has advantages such as wide applicability, accurate optimization results, and short calculation time ^[22,23], providing convenience for achieving good global search. At present, the CS algorithm has been applied to scientific and engineering problems such as image processing, feature selection, and fault diagnosis ^[24,25,26]. Some variant cuckoo algorithms based on parameter adaptation, population dividation, and other strategies have also emerged, further enhancing the robustness and adaptability of the algorithm in global search tasks and making some progress in the application of complex optimization problems ^[27]. However, the CS algorithm has certain limitations in terms of search ability, parameter sensitivity, and high-dimensional problems, as well as lacks strict convergence guarantee.

In the conventional CS algorithm, the objective function only utilizes the distance information between anchor nodes and unknown nodes, while the distance information between unknown nodes is not utilized, which to some extent limits its localization accuracy ^[28]. On the other hand, the random step size in the cuckoo algorithm can also lead to problems such as algorithm instability, slow convergence speed, and difficulty in parameter adjustment ^[29,30]. To improve the search performance, an improved CS optimization algorithm that can adjust search direction and search step size in real-time is proposed in this paper. During the search process, the improved CS algorithm automatically calculates the difference between the current position of each nest and the current optimal nest position, and the future search direction is then determined based on this position difference. Meanwhile, the step size adjustment law and position update law based on a time-varying loss function are constructed to realize the adaptive update of the nest positions. When the search performance is good, the step size will automatically decrease to explore the solution space more finely, while when the search performance is poor, the search step size will automatically increase to expand the search range. Through such an adaptive adjustment mechanism, the improved CS algorithm can flexibly adjust the search step size and direction in different stages, thereby ensuring higher search speed and accuracy. The improved CS algorithm only requires adjusting the step size and position update law in each iteration, without involving complex matrix operations or high-dimensional data processing, thus significantly reducing the training time required for the model and allowing for quick load forecasting.

Suppose the value of the loss function corresponding to the k^th iteration is J(k) and the iteration step is S, then the change in the loss function is expressed as:

The adjustment of step S is designed as:

where β represents the adjustment coefficient. If ΔJ > 0, then select β = 1.05 to increase the step size; while if ΔJ < 0, then make β = 0.95 to reduce the step size.

If the position of the i^th nest is P_i and the best position of nests is P_best, then at the k^th iteration, the position difference d_c between the current nest and the best nest can be calculated as follows:

So, the search direction during the k + 1 iteration can be adjusted according to the following equation:

where d is the adjustment of the iterative search direction. Accordingly, the update law of nest position is designed as:

Through the design of the step size adjustment law and position update law, the step size and search direction can be adjusted adaptively in the iterative search process, which is convenient to reach the optimal solution more accurately and quickly, and can also suppress the violent oscillation caused by random step size to the conventional CS process.

The main process of improving the CS algorithm is shown in Figure 2. The optimization process of the improved CS was described as follows:

Setp 1: Initialize parameters such as the number of nests, the maximum number of iterations, the search step size, and the discovery probability.

Setp 2: Evaluate each nest and calculate its fitness value, and select the nest with the highest fitness as the current optimal solution. The highest fitness corresponds to the smallest loss function value.

Setp 3: The bird nest is updated according to the adaptive step adjustment and position update formula to generate a new generation of bird nest positions.

Setp 4: Determine whether the end condition is met. When the maximum number of iterations is reached, the calculation stops and the optimal solution is output. Otherwise, go back to Step 2 and continue the loop iteration.

The improved CS algorithm can enhance the global search ability, improve the accuracy and speed of finding the global optimal solution, and avoid the problem of falling into the local optimal solution because of its self-adaptive ability of search step size.

2.3.   Optimization of neural network based on improved CS

In order to improve the accuracy of load forecasting, the improved CS algorithm was used to optimize the neural network model. A microgrid load forecasting algorithm based on the improved CS algorithm optimized BP neural network was designed. The specific implementation process is as follows:

Step 1: Create BP neural network prediction model.

Step 2: Set the number of nests.

Step 3: Set the maximum number of iterations.

Step 4: Set step parameters such as the initial step size, step decline rate, and minimum allowed step size.

Step 5: Randomly generate the initial locations of nests in the search space.

Step 6: Calculate the fitness of each nest.

Step 7: Find the nest with the best objective function value (that is, the smallest loss function value).

Step 8: Update the parameters of BP neural network. Take the best nest location found in the previous step as the current global best location, and use it to update the weight and deviation of the neural network.

Step 9: Update the location of each nest with step size adaptive adjustment and position adaptive update laws.

Step 10: Check the iterations. If the predetermined maximum number of iterations is reached, the iteration will be stopped. Otherwise, return to Step 6 and continue iterating.

To avoid overfitting in the model, the LASSO (least absolute shrinkage and selection operator) regularization was employed to enhance the generalization ability of the final prediction model ^[31]. By introducing the sum of the absolute values of the model parameters as a penalty term into the loss function, the loss function is transformed to:

where λ is the hyper-parameter that controls the strength of the regularization, and w_ij represents the weights of the corresponding layer in the BP neural network. After processing in this way, not only the generalization ability of the model is enhanced, but also the risk of overfitting the training data is reduced.

3.   Results and discussion

3.1.   Data acquisition and parameter settings

Microgrid is a small power system composed of photovoltaic cell arrays, wind turbines, fuel cells and other distributed energy and energy storage systems, which can operate either grid-connected or in island mode ^[32,33]. The simplified structure of a certain microgrid is shown in Figure 3. The validation of the load forecasting scheme was carried out based on the load data of the microgrid from February 1, 2023 to May 10, 2023, and the sampling interval of sample data was 1 h. The data of the microgrid from February 1, 2023 to May 9, 2023 was selected as the training set for the model, with the 24-hour load data on May 10, 2023 as the test set. This dataset covers operational data under different conditions such as seasons and weather changes, thus reflecting the impact of different operational conditions on load characteristics, which are crucial for the accuracy of load forecasting. Meanwhile, the data samples used in the prediction experiment are sufficient and contain enough data to support effective model training, ensuring the reliability of the prediction results and the generalizability of the model.

During operation, the initial positions of the 30 nests were randomly generated within the search space. The input of the BP neural network was time and load, i.e., inputsize = 2; and the load of the microgrid was the predicted value, i.e., outputsize = 1. Based on the number of inputs and outputs of the BP network, the number of hidden layer nodes was calculated according to Eq (1) and the number of hidden layer nodes was 3, that is, hidenLayerSizee = 3. The main parameter settings for the CS algorithm were: the nest number was nests = 20; the maximum iteration number was maxGeneration = 200; the initial search step size was initialAlpha = 0.1; the step size decrease rate was alphaDecayRate = 0.05; and the minimum allowable step size was minAlpha = 0.01.

Due to the large difference between load data, the sampling data was processed by Min-Max normalization. Normalization ensures a fair contribution of each feature to the model. By standardizing the dataset, differences between features can be eliminated, the convergence speed of optimization algorithms can be accelerated, model performance and prediction accuracy can be improved, and the instability of numerical calculations can be reduced. The Min-Max normalization is calculated by:

where x is the original feature data and x_s is the data after normalization.

3.2.   Evaluation indexes

The evaluation indexes commonly used in load forecasting of microgrid include mean absolute percentage error (MAPE), mean absolute error (MAE), root mean square error (RMSE) and mean square error (MSE), which can be calculated by:

where N is the total number of samples, y_i is the true value, and ${\widehat{y}}_{i}$ is the predictive value.

3.3.   Comparative analysis

To verify the effectiveness of the proposed improved CS optimized BP neural network (ICS-BP) for microgrid load prediction, simulations were conducted using Matlab simulation software. The results were compared with those obtained from traditional BP, conventional CS optimized BP (CS-BP), chaos sequence improved CS optimized BP (CCS-BP), and particle swarm optimization improved BP (PSO-BP) forecasting methods.

The load forecasting results of microgrids obtained by four different prediction methods are shown in Figure 4. From the predicted results in Figure 4, it can be seen that although the conventional BP neural network prediction method can predict the trend of load changes in microgrids, there was a significant difference between the predicted microgrid load and the actual load value, and the prediction error was relatively large. After using the CS algorithm to improve the BP network, the load prediction values were basically consistent with the actual load values, especially after using the adaptive variable step size ICS algorithm to optimize the parameters of the BP network, the load prediction results obtained were closer to the actual values, and the prediction accuracy was further improved. The prediction performance of the PSO improved BP prediction model was not ideal. The prediction accuracy of the CCS algorithm was higher than that of the standard CS algorithm, but it was still lower compared to the variable step size ICS algorithm.

The absolute error variation curves of sample tracking for three different load forecasting methods are shown in Figure 5.

From the error curves shown in Figure 5, it can be observed that when the load changed very little, the prediction errors of the five forecasting methods were relatively small. However, when encountering severe load fluctuations, the BP forecasting model brought large prediction errors. The ICS algorithm significantly reduces the prediction errors generated by the BP network. The load forecasting process based on the ICS-BP prediction model produced the smallest prediction error and there was no significant fluctuation in prediction error, which proves that the ICS-BP prediction model has good adaptability and can effectively resist the influence of external disturbances such as load changes. In contrast, when the traditional CS algorithm was used to improved the BP network, although it reduced load prediction errors to some extent, the error curve of CS-BP had significant fluctuations, indicating that the CS-BP algorithm shows unstable performance in microgrid load forecasting. The prediction error of the CCS-BP algorithm has also been reduced, while the prediction error of PSO-BP was also relatively large, which was close to that of the simple BP network, indicating that PSO has little effect on improving the prediction performance of BP network.

The main comparison data of prediction accuracy and speed of five different prediction methods are shown in Table 1. In the table, t_c represents the computational time of the prediction algorithm.

From the error indicators presented in Table 1, it is evident that among the five forecasting methods, the ICS optimized BP network exhibited lowest prediction error compared to the BP network, the conventional CS optimized BP network, the CCS optimized BP, and the PSO optimized BP network, while the BP network has the largest error among the five prediction methods. The MAPE of ICS-BP load forecasting decreased by 52.3, 32.8, 12.3, and 42.3%, respectively, compared to conventional BP, CS-BP, CCS-BP, and PSO-BP. Meanwhile, the RMSE, MAE, and MSE indicators of the ICS-BP model were 75.6, 70.6, and 94.0% lower than those of the traditional BP model, respectively, fully demonstrating the excellent load forecasting performance of ICS-BP. In addition, the MAPE value of the ICS-BP forecasting method can be considered as approximately zero, indicating that the improved forecasting model is very close to a perfect prediction model.

Analyzing the computational time of the five prediction methods in Table 1, it can be found that the computational time of the ICS-BP model was slightly longer than that of BP, but was much shorter than the CS-BP, CCS-BP, and PSO-BP models, which indicates that the introduction of ICS increased the complexity of the calculation, but the impact was not too obvious. The computational time of CS-BP, CCS-BP, and PSO-BP was much longer than that of BP and ICS-BP, indicating that the introduction of CS, CCS, and PSO has a great impact on the computing speed of the BP prediction network. The ICS-BP prediction model can guarantee the real-time performance of load forecasting.

The comparison data between predicted values and true values are shown in Table 2.

From the data in Table 2, it is evident that there were significant differences in the absolute error values between the predicted and actual values under the five different forecasting methods. The maximum errors generated by the ICS-BP, CS-BP, CCS-BP, PSO-BP, and conventional BP network were 15.78, 76.5, 38.1, 89.22, and 93.55 kW, respectively. The maximum prediction error of the ICS-BP network was 83.1, 79.4, 58.7, and 82.4% lower than that of conventional BP, CS-BP, CCS-BP and PSO-BP, respectively. The ICS optimized BP network model achieved more accurate and reliable results in microgrid load forecasting.

4.   Conclusions

The BP neural network optimized by the ICS algorithm can effectively improve the accuracy of load forecasting in microgrid. The adaptive step size adjustment law and adaptive position update law are designed for the CS algorithm, which solves the problems of slow search speed and iterative oscillation caused by random step size of the conventional CS algorithm, and improves the search accuracy and speed of the CS algorithm. The ICS algorithm applied to the parameter optimization of the BP neural network solves the problems such as slow training speed and difficult parameter optimization of conventional BP network, and significantly improves the accuracy and reliability of load forecasting in microgrid. Compared with the traditional BP network, the conventional CS, the chaos sequence, and the PSO optimized BP network, the MAPE value of the ICS algorithm optimized BP network decreased by 52.3, 32.8, 12.3, and 42.3%, respectively. Additionally, the other indicators of the ICS algorithm optimized BP network are also significantly better than those of the other three methods. Compared with the traditional BP network, the RMSE, MAE, and MSE of ICS-BP decreased by 75.6, 70.6, and 94.0%, respectively, and the computational speed of ICS-BP was 42.4, 55.6, and 34.0% faster than PSO-BP, CCS-BP, and CS-BP, respectively. The proposed ICS-BP pediction model possesses good robustness and generalization capability, making it suitable for various load forecasting scenarios. Future studies can expand the application of this model to solve a wider range of process prediction problems and to promote its application in the actual microgrid systems.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the Intercollegiate Cooperation Project of Colleges and Universities in Liaoning Province (Grant [2020] 28), and the Key Project of Liaoning Provincial Department of Education (LJKZZ20220057).

Conflict of interest

The authors declare there is no conflict of interest.