Prediction Accuracy of Stackelberg Game Model of Electricity Price in Smart Grid Power Market Environment

Abstract

With the deepening of power market reform and the increasingly fierce competition in the power market, the accurate prediction of electricity price has become an important demand for power market participants to make scientific decisions, optimize resource allocation, and reduce risks. Electricity price forecast can provide a key reference for the power market, help market participants make wise decisions, promote competition and efficient operation and cope with complex market fluctuations, provide a scientific basis for various entities to optimize resource allocation, reduce risks and improve benefits, and promote the sustainable development of the power industry. This study presents a dynamic retail price prediction method for smart grid based on the Stackelberg game model. Firstly, the correlation test is used to verify the strong correlation between electric load and electricity price. Secondly, the parameters of the Stackelberg model are determined, and the load and electricity price are tested using the white noise test. Finally, by comparing the BP neural network model and quantifying the model parameters, the superiority of the model is verified. The results show that the Stackelberg game model has higher prediction accuracy than the BP neural network model in electricity price prediction.

1. Introduction

Electricity price forecasting can provide critical decision-making support for electricity market participants, including power generators, grid operators, consumers, etc., to help optimize electricity production, distribution, and consumption strategies, so as to effectively manage the balance between power supply and demand, reduce costs, improve economic efficiency, and ensure the stable operation of the power system. Kostrzewski M. et al. [1] proved that the Bayesian stochastic volatility model for day-ahead electricity price probability prediction is better than other non-Bayesian models considering parameter uncertainty and jump and exogenous variables. Florian Ziel et al. [2] made probabilistic predictions of electricity prices in the day-ahead electricity market, extended the X model based on the bid–ask curve of day-ahead electricity, and successfully simulated the realistic electricity price model, especially in the long-term forecast, providing valuable results for detecting the probability of price spikes. Michał Narajewski et al. [3] proposed a new method to predict the hourly electricity price probability by simulating the trajectory of the trading window, and the hybrid model performed well in the German intraday continuous market, which can effectively predict the price distribution in the last 3 h after trading, and the method is applicable to other European continuous markets; the introduction of XBID can also reduce market volatility. Rahul Kumar Agrawal et al. [4] provided a new marginal electricity price prediction model for power positioning that uses a correlation vector machine as the core, combined with extreme gradient enhancement and elastic network regression, to show high accuracy and computational efficiency on real-time data of the New England electricity market, with an average absolute error of 2.6, which is better than other models, and the training time is only 88 s. Kostmann M. et al. [5] found that, despite advanced forecasting techniques, prediction errors may lead to uneconomical consumer participation in the market, and the pricing mechanism needs to be re-examined. Alonso M. et al. [6] used the model averaging method to predict electricity prices by combining multiple price factor models to improve the accuracy of the forecast, especially in the spot price prediction of MIBEL in the Iberian electricity market. Ugurlu U. et al. [7] used a multi-layer gated recurrent unit (GRU) as a new electricity price prediction technique, and experiments showed that the three-layer GRU outperformed other neural network structures and statistical techniques in the Turkish advanced market. Janczura J [8] proposed a new method for the probabilistic prediction of electricity price based on mean point prediction and the expected regression of different models, which was applied to the day-ahead electricity price in the German market, and the results showed that its accuracy was higher than that of the quantile regression mean method, but the variance stability transformation should be applied first. Grothe O. et al. [9] proposed a general method based on the copula technique to generate multivariate probabilistic forecasts that depend on the number of hours based on the univariate point prediction of day-ahead electricity prices, and validated it on five benchmark datasets. Zhang et al. [10] proposed an adaptive hybrid model based on variational decomposition, adaptive particle swarm optimization, seasonal autoregressive integrated moving average, and deep belief network, which empirically showed that it could significantly improve the accuracy and stability of electricity price prediction. Chen et al. [11] proposed an adaptive functional autoregressive (AFAR) prediction model to accurately predict stationary and non-stationary electricity price curves through time-varying operators, and simulation and empirical studies have shown that its performance is better than that of many alternative methods. Zhang R. et al. [12] proposed a hybrid deep learning framework including feature preprocessing, deep learning point prediction, error compensation, and probabilistic prediction modules for day-ahead electricity price forecasting, and the PJM market data verification showed that it has a competitive advantage over other methods. Cai Q. et al. [13] proposed a variable electricity price mechanism considering the deviation rate of wind power, and encouraged wind farms to configure energy storage to reduce output deviation by adjusting the electricity price. The results showed that the mechanism effectively reduced wind curtailment and promoted the coordination between wind power and the power grid.

The grid trading system is shown in Figure 1. This study describes in detail a smart grid retail electricity price forecasting method based on a dynamic Stackelberg model. The model takes into account a two-stage pricing strategy between the power producer and the retailer, as well as between the retailer and the consumer. In order to realize the adaptability and dynamics of electricity price prediction, reinforcement learning technology was used in this study. In addition, this study evaluates the performance and quality of dynamic pricing systems in demand response programs. Simulations of consumers throughout the day show that retail electricity prices are consistently higher than wholesale prices. However, when consumer dissatisfaction due to low consumption rates peaks, retail electricity prices are adjusted to levels close to wholesale prices.

2. Model Building

2.1. Time Series Forecasting Models

2.1.1. Stackelberg Model

When applying time series models for problem analysis, the cumulative autoregressive moving average model (Stackelberg) is a commonly used one [14]. The model structure is relatively simple, can effectively capture the characteristics of electricity price series, is easy to construct, and has high prediction accuracy, which is especially suitable for stationary time series or univariate time series that reach a stationary state after differential processing. The standard form of the Stackelberg model is shown in Equations (1)–(3):

$$\phi (B){(1-B)}^d{Y}_t=\theta (B){\epsilon }_t$$

(1)

$$\phi (B)=1-{\phi }_{1}B-{\phi }_2{B}^2-\cdots -{\phi }_p{B}^p$$

(2)

$$\theta \left(B\right)=1-{\theta }_{1}B-{\theta }_2{B}^2-\cdots -{\theta }_q{B}^q$$

(3)

where B is defined as the lag operator B^pY_t = Y_t−p, and d is the difference order of the stationary sequence. φ₁, φ₂, …, φ_p is the autoregressive coefficient and θ₁, θ₂, …, θq is the moving average coefficient. Y_t is any random time series, and ε_t is a white noise series with an average of 0 and σ² ≠ 0 variance. p is the number of autoregressive terms, which is the highest order of the partial autocorrelation function whose value is not 0; q is the number of moving average terms, which is the highest order of the autocorrelation function with a value of not 0. If d = q = 0, the model is a pure autoregressive model AR(p). If p = d = 0, the model is a pure moving average model MA(q). If d = 0, the model degenerates into an ARMA model and can only be used to analyze stationary sequences. When P = Q = D = 0, the model is a general Stackelberg model.

2.1.2. The Steps to Build the Stackelberg Model

(1)

In order to meet the requirements of the variance homogeneity assumption of the model, the logarithmic transformation of the original load series {x_t} and the electricity price series {y_t} is carried out first.

(2)

The stationarity of the logarithmic transformation sequence {ln y_t} of electricity price series and logarithmic transformation sequence {ln x_t} of load series is tested. If the stationarity of the sequence meets the established criteria, the next steps are continued. If the stationarity of the sequence does not meet the requirements, the logarithmic difference processing should be performed on the sequence and the stationarity test should be re-performed. If the test results show that the sequence still does not meet the stationarity criterion, second-order differential processing should be further performed until the stationarity of the sequence meets the established requirements.

(3)

A Stackelberg model is established for the input sequence after n-order difference {∇ⁿ ln x_t}, as shown in Equation (4).

$${\nabla }^n\ln x_{it}={\displaystyle \frac{{\Theta }_{xi}(B)}{{\Phi }_{xi}(B)}}{\epsilon }_{xit}$$

(4)

(4)

A Stackelberg model is established for the input sequence after nth order difference {∇ⁿ ln y_t}, as shown in Equation (5).

$${\nabla }^n\ln y_{it}={\displaystyle \frac{{\Theta }_{yi}(B)}{{\Phi }_{yi}(B)}}{\epsilon }_{yit}$$

(5)

(5)

The structure of the Stackelberg model is determined using Equation (6) by examining the correlation coefficients of the logarithmic sequences ∇ⁿ ln x_t and ∇ⁿ ln y_t that are stationary after NTH difference.

$$y_t=\mu +{\displaystyle \sum _{i=1}^k\frac{{\Theta }_i(B)}{{\Phi }_i(B)}{B}^{li}{x}_{it}+{\epsilon }_t}$$

(6)

(6)

The residual sequence {ε_t} is fitted as follows:

$${\epsilon }_t={\displaystyle \frac{\Theta (B)}{\Phi (B)}}{a}_t$$

(7)

where {a_t} is a zero-mean white noise sequence.

Figure 2 shows the flowchart for modeling the Stackelberg model.

3. Data and Parameters of the Stackelberg Model

3.1. Research and Analysis of Input Data

The load data and electricity price data recorded once an hour from 6:00 a.m. on 12 February 2020 to 9:00 a.m. on 16 February 2020 from a certain area of the United States were selected from the PJM electricity market website, and the load x and electricity price y of the two data series entered were studied and analyzed.

Figure 3 shows the relationship between load x and electricity price y.

As can be seen from the above figure, the load and electricity price fluctuate greatly, so the logarithmic transformation of load x and electricity price y is carried out, which can make the change trend of the two series close to their average, respectively, which is conducive to the subsequent study of the data.

Figure 4 shows the relationship between the logarithm of load lnx and the logarithm of electricity price lny.

Figure 5 shows the relationship between the logarithm of the load Δlnx and the logarithm of the electricity price Δlny.

3.2. Correlation Checks

Pearson correlation coefficients and Spearman rank correlation coefficients are two of the most commonly used methods to measure the strength of the linear correlation between two variables. Through the calculation of correlation in MATLAB 2020b, the Pearson correlation coefficient between the logarithm of the hourly electricity price ln y and the logarithm of the load of the sample data lnx is r₁ = 0.7029, which is strongly correlated. The Spearman correlation coefficient between the logarithm of the hourly tariff lny and the logarithm of the load in the sample data lnx is r₂ = 0.6904. It can be seen that there is a strong correlation between the logarithm of the electricity price lny and the logarithm of the load lnx, and the two variables are positively correlated. The above data and the analysis of the Pearson correlation coefficient and the Spearman correlation coefficient verify that, as an exogenous variable, load has a great impact on the change in electricity price.

3.3. Parameters of the Stackelberg Model

The parameters of the Stackelberg model need to meet the stationary and non-white-noise requirements required for time series.

3.3.1. Differential Order d and How It Is Determined

In most cases, the tariff sequences are highly volatile and exhibit the characteristics of non-stationary time series. During the modeling process, it is difficult to directly model the data sequence due to the different randomness of the sequence at different time points, so it is necessary to pre-process the data appropriately. When dealing with non-stationary time series, non-stationary is converted to stationary series using differential techniques. When a non-stationary time series experiences a difference of d, the value of d is the order of the difference if the sequence no longer contains the root of the unit [15].

3.3.2. Methods for Determining the Lag Order p and q

The Box–Jenkins model recognition method is usually used to determine the lag order p and q, that is, the judgment is based on the tailing and truncation of the ACF and PACF of the sample. Then, the final order is determined according to the AIC criteria. In addition, the selection of p and q also needs to meet these two conditions: all parameters must pass the significance test and the AIC value is minimized.

3.3.3. AIC Guidelines

The AIC (Akaike Information Criterion) is a standard for measuring the fitness of statistical models that is based on the concept of entropy and provides a standard for estimating the complexity of the model and fitting the data that meets the requirements.

Typically, it is a weighted function of the number of unknown parameters for fitting accuracy and AIC, as defined in Equation (8).

$$AIC=2k-2\ln (L)$$

(8)

In the formula, k is the number of unknown parameters in the model and L is the likelihood function.

If the difference between the two models is large and the difference occurs in the second term of Equation (8), and when there is no significant difference in the second term of the formula, the first term of the formula has the greatest impact on the value of the AIC, and the model meets the requirements.

Let n be the number of observations and RSS be the sum of the remaining squares; then, AIC becomes Equation (9).

$$AIC=2k+n\ln ({\displaystyle \frac{RSS}{n}})$$

(9)

3.3.4. BIC Guidelines

The Bayesian Information Criterion (BIC) is similar to the AIC and mainly focuses on model selection. In designing the model, the number of parameters was increased, which not only made the model more accurate, but also improved the likelihood function approximation. However, the method is overfitted. In order to solve this problem, both the AIC and BIC models add penalties related to the number of model parameters. The penalty term of the BIC is larger than that of the AIC, and when the number of samples is large, the model accuracy can be greatly reduced, resulting in excessive model complexity. The BIC formula is as shown in Equation (10).

$$BIC=k\ln (n)-2\ln (L)$$

(10)

3.3.5. Stationarity Test and White Noise Test

The stationarity test is the unit root test H₀: β₁ + β₂ + … + β_p = 1. If the sequence is assumed to have at least one unit root, the alternative assumption does not have a unit root. So, if p ≥ α, the null hypothesis cannot be rejected, there is a unit root, and the sequence is non-stationary. So, if p < α, the assumption is not true, there is no unit root, and the sequence is smoothed and valid.

White noise is the simplest stationary time series that is a series of independently distributed normal series in which each time series point is zero and the variance, σ², is normally distributed. White noise is characterized by irregular sequences, repeated oscillations at the mean, and no trend. In model design, the white noise test is commonly used to analyze the residuals.

For white noise test (Ljung-Box), H₀: ρ₁ = ρ₂ = … = ρ_m, and H₁, at least one is present when ρ_k ≠ 0. The null hypothesis is that the values of the lagged m-order sequences do not interfere with each other and the autocorrelation function is zero; the alternative hypothesis is that there is a correlation between the sequences in the subsequent m-period. So, if p ≥ α, the null hypothesis cannot be rejected and the sequence is white noise. So, if p < α, the null hypothesis is rejected and the sequence is non-white-noise.

In this study, the logarithm of load lnx and the logarithm of electricity price lny are tested to test the stationarity of the two sets of sequences, as shown in Figure 6 and Figure 7.

3.4. Determination of Stackelberg Model Parameters

After accurate calculation, Δlnx white noise test results are shown in

Table 1. Δlnx establishes the results of the white noise test of the residual of the Stackelberg model.

Delay Order	Chi-Square Statistic	p-Value
6	112.962298	0
12	184.238581	0
18	231.174189	0

, Δlny white noise test results are shown in

Table 2. Δlny establishes the results of the white noise test of the Stackelberg model residuals.

Delay Order	Chi-Square Statistic	p-Value
6	6.279953	0.392571
12	27.789030	0.005939
18	33.802861	0.013320

, the p value of the residual white noise of the load logarithm and the price logarithm is less than 0.05. Through the legislative white noise test, the obtained data are in line with the standard. This means that, at a significance level of 0.05, neither the load logarithm nor the price logarithm are white noise, so the relevant parameters are suitable for building the Stackelberg model.

Δlnx Stackelberg prediction model parameters are shown in

Table 3. Δlnx Stackelberg prediction model parameters.

Parameter	Parameter Estimates	t-Value	p-Value
μ	1.294154	0.102799	0.000225244
φ	0.660288	5.35792	0.123236
θ	−1	−8.14426	0.122786

, Δlny Stackelberg prediction model parameters are shown in

Table 4. Δlny Stackelberg prediction model parameters.

Parameter	Parameter Estimates	t-Value	p-Value
μ	−20.2993	−0.00920595	0.000571415
φ	−0.0324844	−0.353057	0.0920089
θ	−1	−13.7489	0.0727328

. The residual series of the load lnx and the logarithmic series of the electricity price lny were fitted, respectively, and the estimate function was used to estimate the parameters of the constructed model and data samples.

From Figure 8, Figure 9, Figure 10 and Figure 11, it can be seen that the model has a high degree of fitting to the electricity price and load, respectively.

The number of interrelationships between the two residual sequences {ɛ_t1} and {ɛ_t2} was calculated, while the number of relationships between sequences {ɛ_t1} and {ɛ_t2} is shown in Figure 12.

Based on the number graph of the correlation between the residuals between the two sequences, the Stackelberg model with load as an exogenous variable was constructed, and the parameters of the model were calculated, as shown in

Table 5. Parameters of the prediction model for Stackelberg electricity prices.

Parameter	Parameter Estimates	t-Value	p-Value
μ	−4.8816	−1.5915	0.1155
φ	−0.7215	−1.5874	0.1124
θ	0.6603	1.3907	0.1643

.

3.5. Prediction Results and Analysis

3.5.1. Stackelberg Model Prediction Results Data Table

The absolute value of error and absolute percentage of error between predicted and actual electricity prices calculated by the Stackelberg model are shown in Figure 13.

According to the data in Figure 13, after excluding the two sets of data that could not be simulated, the remaining valid data total is 98 sets. Among them, there are only two groups with an absolute error value of more than 1, and only two groups with an absolute error percentage of more than 5%. At the same time, there were 32 sets of data with an absolute error percentage of less than 1%. This indicates that the Stackelberg electricity price prediction model constructed in this study has high prediction accuracy and high fitting degree.

Figure 14 shows how the price predicted by the Stackelberg model compares to the actual price.

As can be seen in Figure 14, the predicted values of the Stackelberg model are extremely close to the actual values, which indicates that the model has a high degree of fit and a certain degree of accuracy in predicting electricity prices.

3.5.2. Error Calculation and Analysis of the Prediction Results of the Stackelberg Model

The Stackelberg model was used for simulation, and the error between the predicted and actual electricity prices was analyzed. Through the calculation program, the obtained sum of squares of the residuals is 17.464, the mean square error (MSE) is 0.178, the coefficient of determination R² is 0.984, the mean absolute error (MAE) is 0.341, and the root mean square error (RMSE) is 0.422. After detailed calculations of these error parameters, we found that their values were small, indicating that there was no significant deviation between the data predicted by the model and the real data. Therefore, it can be concluded that the fitting of the Stackelberg model was quite successful and met the expected goals.

In addition, the Stackelberg model takes into account the exogenous variable of load in the construction process, which greatly enhances the flexibility of the model. The model overcomes the limitation that traditional time series models often ignore other variables, thereby improving the accuracy of forecasting.

4. Electricity Price Prediction Model for BP Neural Network Model

4.1. BP Neural Networks

A BP neural network, or backpropagation neural network, is a multi-layered feedforward neural network. It receives data through the input layer, undergoes a nonlinear transformation of the hidden layer, and, finally, outputs the result through the output layer. In the training process, the backpropagation algorithm adjusts the weights of each layer to make the output gradually approximate the real value so as to realize the approximation of complex functions and pattern recognition. In this study, the electricity price prediction model of the BP neural network is compared with the Stackelberg model [16].

4.1.1. BP Neural Network Model Prediction Results

The BP neural network model is used to predict the electricity price, and Figure 15 shows the absolute value of the error between the actual value of the electricity price and the BP neural networks and the absolute value of the error percentage between the two models.

According to the data in Figure 15, among the 98 sets of data, only 18 sets of data have an absolute error value less than or equal to 1 between the prediction results and the actual value of the BP neural network model; however, there are three groups of data with an absolute error value of more than 10. In terms of the percentage error between the predicted value and the actual value, only 31 sets of data had an absolute error percentage of less than 10%; in addition, there was a group of data with an absolute error percentage of more than 100%. In contrast, the Stackelberg model had a prediction of a percentage error of just over 5% in absolute terms. Therefore, compared with the BP neural network model, the prediction error of the Stackelberg model is smaller, and its fitting degree is higher.

4.1.2. Comparison of Forecast Results

Figure 16 shows the image comparison between the prediction result of the BP neural network model and the actual value.

From Figure 16, it can be seen that the fitting degree of the model is low and the error is large.

Figure 17 shows the image comparison of the predicted electrical value and the actual value of the BP model and the Stackelberg model.

As can be seen from Figure 17, the Stackelberg model fits more closely than the BPNN model.

4.2. Error Analysis

After simulating the BP neural network model, the error analysis of the predicted and actual electrical values was performed. Through calculation, the residual sum of squares between the two data sets was 2528.064, with a mean square error of 25.797, a coefficient of determination of 0.555, a mean absolute error of 3.557, and a root mean square error of 5.079. These error parameter values are generally high, indicating that the model has large errors. To further understand the error distribution, these parameters were compared with the Stackelberg model, and the detailed results are shown in

Table 6. Comparison of error parameters of the Stackelberg model with the BP neural network model.

	Stackelberg Model	BP Neural Network Model
Sum of squares of residuals SSE	17.464	2528.064
Mean square error MSE	0.1782	25.797
Coefficient of determination R2	0.984	0.555
Mean absolute error MAE	0.341	3.557
Root mean square error RMSE	0.422	5.079

.

According to the analysis results in Table 6, compared with the Stackelberg model, the neural network BP method shows large errors and biases in electricity price prediction, the difference between the predicted value and the actual value is more significant, and the image of the prediction result is more disorganized. Although the calculation process of the neural network BP method is relatively simple, the fitting error of the model is large, so, compared with the neural network BP method, the Stackelberg model is more suitable for predicting this group of multivariable electricity price data.

After the in-depth study of BP neural network related data, it can be concluded that the model has certain advantages when processing the calculation and prediction tasks of large amounts of data. However, the insufficient sample data size in this study is one of the main reasons for the large error when using the BP neural network model for prediction.

5. Conclusions

Electricity price forecasting is crucial in power market operation by helping market participants cope with fluctuations and optimize supply scheduling, providing operators with optimal resource allocation decisions to improve efficiency, guiding power trading, reducing transaction costs, improving benefits, assisting energy management to improve utilization efficiency, and providing consumers with low-cost electricity guidance.

In this study, a dynamic retail electricity price prediction method of smart grid based on the Stackelberg game model is proposed. By comparing the predicted electricity price and the actual electricity price, and calculating the error parameters, the validity of the model in electricity price prediction is verified. At the same time, the proposed method is compared with the traditional BP neural network model. The results show that the Stackelberg game model has a higher goodness of fit and better robustness and prediction accuracy in the field of electricity price prediction. It provides a stable and reliable method for electricity price forecasting, which is conducive to the stable operation of the electricity market and the decision making of suppliers and consumers.