The grey prediction model is simple, fast, and efficient. It does not require the sample data to satisfy the normal distribution and independence assumptions, nor does it require a large amount of data and a linear relationship between variables, suitable for various types of data. The modeling mechanism of grey models is one of the leading research elements of grey systems theory [14]. The grey prediction model was constructed in the following steps: firstly, the original data were generated cumulatively to weaken the randomness of the data; secondly, a first-order differential equation model was established for the cumulatively generated series; finally, the relevant parameters in the first-order differential equation model were estimated using the least squares method to obtain the specific formula for the GM(1,1) prediction model.

The grey prediction model obtained by simplification is as follows: (1)xa(k+1)=[xa(1)−u^a^]e−a^k+u^a^

In addition, to improve the prediction accuracy of the grey model, this paper transforms the original series before cumulative generation. It changes the original series into a steadily growing series in an exponential form whenever possible. The original series is assumed to be X={x1,x2,…,xn} and is then modified by adding policy factors to the original series. The expression of the conversion sequence is Xz={x1z,x2z,…,xnz}, as shown in the equation: (2)xiz=1n−i+1(xi+xi+1+…+xn)

Back Propagation (BP) Neural network is a machine learning method, which trains sample data sets to build a model, enabling it to learn and adapt to data rules, can deal with complex problems with nonlinear relationships and adapt to multiple data types.

The CS algorithm has the advantage of improving the local and global search of the algorithm, etc. To enhance the dependence of the BP neural network on the initial weights and thresholds [15]. By applying the CS algorithm to BP neural networks, the initial network weights and points are searched for before the network is trained, and the optimal solution is then assigned to the web for training, which can effectively avoid the problem of BP neural networks falling into local minima due to the initial weights and thresholds, and improve the convergence speed of BP neural networks [16].

In order to improve the performance of the CS algorithm, this paper improves the CS algorithm and proposes an improved cuckoo ICS optimization algorithm. In this paper, the fixed-value Pa in the algorithm is improved by using a random discovery probability instead of a fixed discovery probability so that smaller random probabilities and larger random probabilities occur randomly, thus balancing the algorithm’s ability of global search and local development, improving the convergence speed of the algorithm, and increasing the population diversity.

The improved formulae are shown as follows: (3)Pr=Pamin+(Pamax−Pamin)×rand×τ(1,2)×k (4)Pr∈[0.15,0.55]where Pr is the random probability, Pamin and Pamax are the maximum and minimum values of Pa respect, rand is a random number in the range (0,1), τ(1,2) is a gamma-distributed random number obeying a shape parameter of 1 and a scale parameter of 2, and k is used to control the degree of deviation between the inertia weights and the expected value.

Adding inertia weights to the path and position update formula of cuckoo nest finding to balance the relationship between global search and local search can make the algorithm have strong global search capability in the early stage while having strong local exploitation capability in the later stage, improving the convergence speed and merit finding accuracy of the algorithm: (5)Xi(t+1)=ω(t)Xi(t)+a⊕Levy(λ),i=1,2,…,n (6)ω(t)=rmax−η×(rmax−rmin)×logTmaxt−φ×βwhere t is the current number of iterations; Tmax is the maximum number of iterations; rmin and rmax are the minimum and maximum values of the inertia weights, respectively; η is the log deviation factor; φ is the inertia adjustment factor, and β is the random number of asymmetric distributions between (0,1) generated by the beta distribution.

Multiple linear Regression predicting is a popular and intuitive among many predicting methods. The key to using multiple linear regression predicting is to determine the multiple linear regression equation [17]. When the prediction object y is influenced by more than one factor x1,x2,…,xm, if the correlation between each influencing factor xj(j=1,2,…,m) and y can be approximated linearly, then the prediction can be analyzed by establishing a multiple linear regression model, the general form of the multiple linear regression model is shown in the following equation: (7)y0=b0+b1x1i+b2x2i+…+bmxmi+εi(i=1,2,…,n) (8)εi=yi−y^i(i=1,2,…,n)where bj(j=1,2,…,m) denotes the model regression coefficient; εi denotes the random variable generated by yi in addition to the effect of the independent variable xj(j=1,2,…,m), called the random error.

This paper combines grey predicting with neural networks, making reasonable use of the ability of artificial neural networks to approximate arbitrary functions and the ability of grey predicting methods to predict the overall trend of electricity load changes. Regression predicting combines time series and Regression predicting to enhance predicting accuracy. A combination of multiple load-predicting models was used to obtain a prediction result with the lowest root mean square error.

This paper uses the inverse of the variance method to assign greater weights to the more accurate prediction values. The weighted average value is the final value to improve the prediction results. Take the three predicting methods selected in this paper as an example. The expressions for calculating the weight wi are derived as follows: (9)wi=ei−1∑i=1mei−1

f1, f2, f3 do the three predicting methods obtain the predicted values, respectively, w1, w2 and w3 are the weighting coefficients, and w1+w2+w3=1. fc is the weighted average, e1, e2, e3 and ec denote the squared prediction error corresponding to f1, f2, f3 and fc. The specific expression is shown in the following equation: (10)fc=w1f1+w2f2+w3f3 (11)ec=w1e1+w2e2+w3e3 (12)ei=(y−y^i)2

The grey regression neural network prediction process used in this paper is shown in Fig. 2.

To further improve the prediction accuracy, this paper also introduces the Informer model, which adopts a stacking strategy and fuses it with the grey regression neural network model to form a new combined prediction model. The Informer model is a sequence prediction model based on the self-attention mechanism [18]. It is mainly used in time series prediction and natural language processing tasks. The model is an extension and improvement of the Transformer model, allowing it to capture long-distance dependency information in the time series through the multi-layer self-attention mechanism and improve the prediction accuracy [19].

The key feature of the Informer model is the introduction of a new attention mechanism called the Globally Attentive Temporal Mechanism (GATM), which uses the Globally Attentive Temporal LSTM [20]. Unlike the traditional self-attention mechanism, the Globally Attentive Temporal LSTM computes the attention of the entire historical sequence at each time step, thus capturing the global dependencies in the sequence more effectively [21]. Additionally, the Informer model incorporates a structure of Long Short-Term Memories (LSTMs) to extract long-term and short-term patterns in the sequence data. By utilizing both global attention and LSTMs in the model, the Informer model effectively models the multi-scale and multi-granularity features of time series data [22].

The stacking strategy utilizes a meta-machine learning model to combine different base learning models, aiming to enhance the overall accuracy of generalization [23]. The strategy involves combining multiple basic predictive models and learning their respective predictions to generate a final prediction. This is achieved by using a meta-model, often a simple linear model, to integrate these predictions [24]. By combining the strengths of different models, the stacking strategy enhances prediction accuracy, overcomes the limitations of individual models, and improves the generalization ability of the overall model [25].

The investment in energy storage plants and the extended operating period require consideration of the necessary costs for operation and maintenance, end-of-life recycling, etc., throughout the plant’s life cycle [26]. They established a cost model for energy storage plants and optimized the allocation of storage plants to maximize annual net returns. (1)

Energy storage investment costs: the cost of energy storage is mainly used to purchase batteries and other equipment at the beginning of the project and requires a one-time investment in start-up capital, mainly for the purchase of battery capacity and battery power equipment required for energy storage power plants [27]. The formula for calculating the average annual investment cost of energy storage can be expressed as:(13)C1=α(r,Y)(CP×Pe+Ce×Ee) (14)α(r,Y)=r(1+r)Y(1+r)Y−1where CP and Ce are the cost per unit of power and capacity of the energy storage plant, respectively, Pe and Ee are the power and ability of the energy storage plant, respectively, Y is the operational life of the energy storage, and r is the base discount rate. α(r,Y) It is the equal annual value of the capital regression coefficient and n is the entire life cycle of the investment project.

Failure of energy storage can severely affect the capacity allocation of the storage device. If an energy storage device fails, greater storage capacity may be required to cope with unexpected situations or more extended periods of power supply. An energy storage failure may result in the energy storage station being unable to charge and discharge as planned, leading to an unstable energy supply. Additional energy storage capacity may be required to compensate for this instability to ensure a reliable supply. Energy storage failures may require higher maintenance and repair costs. If the capacity of the energy storage device is not configured correctly, it may result in increased time and resources required for maintenance and repair.

Battery failure rate refers to the ratio of failed cells to the total number of individual cells in a lithium iron phosphate energy storage cell during the evaluation period [29]. The calculation formula is as follows: (20)FR=ILBLBwhere FR is the rate of a cell failure, ILB is the number of cells that failed during the evaluation period, and LB is the total number of cells in the electric storage unit.

Battery storage power stations are used for peak and valley reduction so that they can be charged at low load times and discharged at peak load times, which not only effectively reduces the peak-to-valley difference between day and night, smoothes the load, relieves the peak load on the transmission and distribution line expansion requirements, but also allows for direct profits from “low storage and high generation” through the time-sharing tariff system [30]. Therefore, the capacity of energy storage plants can be configured by their peak-shaving effect on the load curve.

This paper constructs a model of the effect of peak shaving as shown below: (21)S=λ1Pmax−PminPmax×ω+λ21n∑i=1n(Pi−P¯)2where S is an evaluation indicator reflecting the effect of peak shaving, Pmax and Pmin are the peak and trough values of the load, respectively, Pi denotes the load value of the predicted data at the moment i, P¯ denotes the average value of the predicted load data, ω is the balance parameter, and λ1, λ2 are the weighting factors satisfying λ1+λ2=1.