Hourly Electricity Price Prediction for Electricity Market with High Proportion of Wind and Solar Power

Abstract

In an open electricity market, increased accuracy and real-time availability of electricity price forecasts can help market parties participate effectively in market operations and management. As the penetration of clean energy increases, it brings new challenges to electricity price forecasting. An electricity price forecasting model is constructed in this paper for markets containing a high proportion of wind and solar power, where the scenario with a high coefficient of variation (COV) caused by the high frequency of low electricity prices is particularly concerned. The deep extreme learning machine optimized by the sparrow search algorithm (SSA-DELM) is proposed to make predictions on the model. The results show that wind–load ratio and solar–load ratio are the key input variables for forecasting in power markets with high proportions of wind and solar energy. The SSA-DELM possesses better electricity price forecasting performance in the scenario with a high COV and is more suitable for disordered time series models, which can be confirmed in comparison with LSTM.

1. Introduction

The electric power industry is constantly reshaping under the changing mechanisms and competition, and the prediction of power demand and price has become one of the most important research topics in electrical engineering [1,2]. Forecasting models and methods are widely used in the power industry and have received significant attention. Among them, electricity price forecasting is more difficult. As mentioned in [3,4], its average absolute percentage error (MAPE) can be less than 3%, while the MAPE of electricity price forecasting can be above 7% [5]. Compared to load forecasting, the research on electricity price forecasting is not sufficient. However, it has richer significance in the power market under fierce competition. The frequent fluctuation of the price of electricity increases the risk of market transactions, which affects the development of the power market mechanism, and puts forward higher requirements for all participants. The phenomenon of frequent fluctuations in the price of electricity can be attributed to the characteristics of electricity itself [1]; they are an inelastic demand in the short term, and they have uncertainty in load, power generation side, and transaction process, etc. With the increasing proportion of wind and solar power, their influence on the price of electricity cannot be ignored, while studies on the impact of wind and solar power on electricity price fluctuation are still insufficient.

According to different time scales, electricity price forecasting can be divided into medium and long-term [6], day-ahead [7], hourly [8] and real-time electricity price prediction [9]. Specific to the prediction method of the electricity price, the existing electricity price prediction method can be divided into three categories, i.e., simulation model, game theory model and time-series model, among which the time-series model is most widely used [10,11]. Time-series forecasting methods have roughly gone through two stages of development. In the early stage, statistics-based time-series analysis methods are usually used, such as the autoregressive model (AR), the later improved moving average model (MA), the autoregressive moving average model (RMA) and the autoregressive comprehensive moving average (ARIMA) model) [12,13,14,15]. These methods have the advantages of a small computational cost and fast speed. However, for the electricity price time series with complex nonlinearity, non-steady-state characteristics and uncertainty, the prediction accuracy of the traditional algorithm is normally unsatisfactory [16].

With the rise of artificial intelligence, especially the development of machine learning algorithms, a large number of intelligent algorithms are applied to solve time-series prediction issues, such as support vector machine (SVM), artificial neural network (ANN) [17] and deep learning (DL) [18] algorithms. In recent years, a variety of time series prediction models based on DL have also been widely used, such as the deep multi-layer perceptron (DMLP), recurrent neural network (RNN) [19], long short-term memory network (LSTM) [20,21], deep belief network (DBN), deep extreme learning machine (DELM) [22], etc. In [23], a method based on a generalized extreme learning machine is applied to improve wavelet neural networks to achieve a higher speed and lower computational cost. As shown in [24,25], the deep NNs used to predict electricity prices, convolutional neural networks (CNNs) and long short-term memory networks (LSTMs) are employed. In [26], GRU-TL is uniquely designed for high-frequency (5 min) day-ahead electricity price forecasting on wind farms. However, studies on electricity price forecasting considering wind power and solar power are still insufficient.

The main contributions of this paper are as follows:

• A time-sharing electricity price forecasting model is proposed in this paper based on SSA-DELM, considering the impact of wind and solar power on the price of electricity in the form of wind–load and solar–load ratios (wind–load ratio refers to the ratio of wind power to total load, and the same with solar power for solar–load ratio).
• In the scenario with a high coefficient of variation caused by the high frequency of low electricity prices, SSA-DELM has been shown to greatly improve the forecasting effect of the time-sharing electricity price.

The rest of this paper is organized as follows. An electricity price forecasting model (SSA-DElM) is proposed in Section 2; and then, Section 3 conducts an analysis of related factors of electricity price in the case of a high proportion of wind and solar power; in Section 4, a detailed forecasting model is developed and numerically verified. Finally, the conclusions are drawn in Section 5.

2. The Developed SSA-DELM

In this section, deep extreme learning machine optimized by sparrow search algorithm (SSA-DELM) is proposed to improve the performance of the electricity price forecasting model. DELM can also be called multi-layer extreme learning machine (ML-ELM), as it first applies multiple extreme learning machines-automatic encoder (ELM-AE) for unsupervised pre-training and then uses the output weight of ELM-AE to initialize the entire DELM. Compared with other deep methods, DELM has the advantage of a fast training speed. During the pre-training process of ELM-AE, the input layer weights and biases are randomly generated orthogonal random matrices. At the same time, during the unsupervised pre-training of ELM-AE, the least-squares method is used to update the parameters, but only the weight parameters of the output layer are updated, while the weights and biases of the input layer are not updated, which leads to the effect of the DELM affected by the random input weights and random biases of each ELM-AE. As a result, using SSA to optimize random input weights and random biases can significantly improve the network accuracy of DELM. The flow chart and structure are shown in Figure 1 and Figure 2.

2.1. The Structure and Mathematical Expression of DELM

According to the extreme learning machine (ELM) theory, originally proposed in [27,28], in the single-hidden layer feed-forward neural networks (SLFN), the output weight is obtained when the input layer weight and bias constant are randomly determined. ELM has faster training speeds and better generalization performance while maintaining computational accuracy but a drawback of poor robustness. Deep learning is an artificial neural network learning algorithm with multiple hidden layers and multi-layer perceptrons, which realizes the approximation of complex functions and alleviates the local minimization problem of previous multi-layer neural network algorithms [29]. The concept of deep learning and the deep structure of a multi-layer automatic encoder are put forward in [30]. Furthermore, then, a multi-layer extreme learning machine algorithm (ML-ELM) is proposed in [22] by combining the concepts of ELM and deep learning. ML-ELM builds a multi-layer neural network model by stacking the extreme speed learning machine and the automatic encoder algorithm (ELM-AE) that has the advantage of deep learning and a fast learning speed.

Similar to common deep learning models, ML-ELM trains the parameters of each layer through the unsupervised learning method ELM-AE, but ML-ELM does not need to fine-tune the network. In this way, ML-ELM does not need to spend a long time training the network model. It is assumed that the model structure has j input layer neurons, m hidden layer neurons, j output layer neurons and the activation function on the hidden layer is $g\left(x\right)$. ELM-AE can be divided into three different representations based on the hidden layers of the input signal: compressed representation, equal-dimensional representation and sparse representation. The main difference between ELM-AE and the traditional ELM model is that ELM is a supervised algorithm and the output is a tag, and ELM-AE is an unsupervised learning algorithm and its output is consistent with the input. Then, for N training samples, the ELM-AE hidden layer output of ELM-AE can be expressed as Equation (1), and the numerical relationship between the hidden layer output and the output of the neuron output can be expressed as Equation (2).

$$h=g(ax+b),a^{T}a=I,b^{T}b=1$$

(1)

$$h\left(x_i\right)\beta =x_i^T,i=1,2,\dots ,N$$

(2)

The method to calculate the output weight of ELM-AE is also slightly different from that of the traditional ELM. For the compressed representation and sparse representation, the output weight of ELM-AE can be calculated by Equation (3) [22]:

$$\beta =\left\{\begin{array}{c}{\left(\frac{I}{C}+H^{T}H\right)}^{-1}{H}^{T}X,N⩾m\hfill \\ H^T{\left(\frac{I}{C}+H{H}^T\right)}^{-1}X,N<m\hfill \end{array}\right.$$

(3)

For the equal-dimensional representation of ELM-AE, the output weight $\beta$ is calculated as follows:

$$\beta =H^{-1}X,{\beta }^T\beta =I$$

(4)

ML-ELM applies ELM-AE to train the model layer by layer. The numerical relationship between the output of the i hidden layer and the output of the $\left(i-1\right)$ hidden layer can be expressed by Equation (5).

$$H_i=g\left(H_{i-1}{\beta }_i^T\right)$$

(5)

In the formula, $H_i$ represents the output of the i hidden layer, and when $i=1$, $H_{i-1}$ represents the output of the input layer, which is also the input of the entire network. ${\beta }_i$ represents the weight matrix when ELM-AE trains the $\left(i-1\right)$ hidden layer and the i hidden layer. In this case, the input of ELM-AE is $H_{i-1}$, and the number of neurons in the hidden layer is the same as that on the ML-ELM. The output weight on ML-ELM $\beta$ is obtained by minimizing the regularized cost function of the least-squares estimate. Its structure is shown in Figure 2.

2.2. Mathematical Model of SSA Optimization Algorithm

The internal idea of the swarm intelligence optimization algorithm is to search for the optimal solution distributed in a certain range of solution space by simulating the behavior of some organisms in nature. According to the swarm behavior of ants, wolves and other creatures, many swarm intelligent optimization algorithms have been proposed, such as the common Ant Colony Algorithm (ACO), Gray Wolf Optimization (GWO) and so on. The SSA algorithm proposed in [31] is a new swarm intelligence optimization algorithm inspired by sparrow foraging behavior and anti-predation behavior. The mathematical model of the sparrow search algorithm is established according to the following rules.

Sparrow foraging and anti-predation behavior can be abstracted as a producer-seeker model, and there is an anti-predation early warning mechanism. The main rules are as follows:

1.

Producers usually have a higher level of energy reserve, corresponding to higher fitness, to undertake the task of expanding the search scope and guiding the population to search and forage.

2.

In order to obtain better fitness, the searchers follow the discoverers to look for food. At the same time, in order to improve their predation rate, some searchers will monitor the discoverers to compete for food.

3.

When the whole population is in danger of being preyed on, the producer needs to guide all searchers into the safe area.

4.

During the whole process, any sparrow can become a producer if it finds a better food source, and the proportion of producers in the sparrow population remains constant.

In SSA, the sparrow foraging process is simulated to obtain the solution of the optimization problem. Suppose that there are n sparrows in a D dimensional search space, then the position of the i sparrow in the D dimensional search space is $X_i=\left[x_{i,1},\dots ,x_{i,d},\dots ,x_{i,D}\right]$, where $i=1,2,3,\dots ,n$ is the quantity of sparrow, and the fitness value of the i sparrow is $F_X=f\left(\left[x_{i,1}{x}_{i,2}\dots ,x_{i,D}\right]\right)$.

In SSA, producers with higher fitness give priority to food during the search, which generally accounts for 10–20% of the population. In addition, producers are responsible for finding food and guiding the flow of the entire population. As a result, producers can look for food in different places. According to rules (1) and (4), in each iteration, the position of the producer is updated as follows [31]:

$$X_{i,d}^{t+1}=\left\{\begin{array}{cc}\hfill X_{i,d}^t·exp\left(\frac{-i}{\alpha ·\mathrm{iter}max}\right)& \mathrm{if}{R}_2<ST\\ \hfill X_{i,d}^t+Q·L& \mathrm{if}{R}_2\ge ST\end{array}\right.$$

(6)

where t represents the current number of iterations, and $X_{i,d}^t$ denotes the value of the i sparrow’s d dimension feature at the time of iteration t. ${\mathrm{iter}}_{\max }$ is the maximum number of iterations, which is a constant, and $\alpha \in (0,1]$ is a uniformly distributed random number. $R_2(R_2\in [0,1])$ and $ST(ST\in [0.5,1.0\left]\right)$ represent the alarm value and the security threshold. Q is a random number with a normal distribution, and L represents a matrix of size $1\times D$, where each element is 1. Q is a random number that obeys normal distribution, and L represents a matrix of size $1\times D$, where each element is 1. When $R_2<ST$, there are no predators around, and producers can search extensively to guide the population to higher fitness. If $R_2⩾ST$, it means that some sparrows have found predators, and all sparrows need to fly quickly to other safe areas.

The rest of the searchers, according to rule (2), as mentioned above, will monitor the producers more frequently. Once they find that the producer has found food, they will immediately leave their current position to compete for food, and the searcher’s location update formula is described as follows:

$$X_{i,j}^{t+1}=\left\{\begin{array}{cc}\hfill Q·exp\left(\frac{x_{\mathrm{worst}}^{\mathrm{t}}-X_{i,j}^t}{i^2}\right)& \mathrm{if}i>n/2\\ \hfill X_P^{t+1}+\left|X_{i,d}^t-X_P^{t+1}\right|·A^{+}·L& \mathrm{otherwise}\end{array}\right.$$

(7)

where $X_P$ is the best position occupied by the producer, and $X_{worst}$ represents the current global worst position. A represents a matrix with a value of $1\times D$, where each internal element is randomly assigned to 1 or $-1$, and $A^{+}=A^T{\left(A{A}^T\right)}^{-1}$. When $i>n/2$, the i searcher has poor fitness and is most likely to starve.

The number of early warning sparrows generally accounts for 10% of the total number of sparrows. The initial positions of these sparrows are randomly generated in this population, and the mathematical model can be described as follows:

$$X_{i,j}^{t+1}==\left\{\begin{array}{cc}\hfill X_{\mathrm{best}}^t+\beta ·\left|X_{i,j}^t-X_{\mathrm{best}}^t\right|& \mathrm{if}{f}_i\ne f_g\\ \hfill X_{i,j}^t+K·\left(\frac{|X_{i,j}^t-X_{\mathrm{worst}}^t|}{\left(f_i-f_w\right)+\epsilon }\right)& \mathrm{if}{f}_i=f_g\end{array}\right.$$

(8)

where $X_{best}$ is the current global optimal location. As a step size control parameter, $\beta$ is the normal distribution of a random number with a mean of 0 and a variance of 1. $K\in [-1,1]$ is a random number that indicates the direction of the sparrow’s movement and is also the step size control coefficient. $f_i$ represents the fitness of the i sparrow, while $f_g$ and $f_w$ are the global best fitness and worst fitness values in the current population. $\epsilon$ is a minimum to avoid zeros in the denominator position. The sparrow is at the edge of the group and is vulnerable to predators when $\epsilon$ indicates that the sparrow is at the edge of the group. $f_i\ne f_g$ shows that sparrows are in the middle of the population, and when they are aware of the danger, they need to be closer to other sparrows.

3. Analysis of Electricity Price

In this section, the factors influencing the volatility of electricity price are explored, applying a data set of the DK1 region in Denmark for 2019–2021 (20 January to 20 March) as a sample, as the Nordic electricity market has the typical characteristics of a high proportion of wind and solar power. With the analysis in this section, significant differences in electricity price fluctuations in 2020 compared to 2019 and 2021 are revealed in two main parts: (1) The coefficient of variation (COV), defined as the ratio of the standard deviation to the mean, in 2020 is far beyond that in 2019 and 2021 due to the high frequency of low and negative electricity prices. (2) The impact of the wind–load ratio and the solar–load ratio on electricity prices plays a decisive role (wind–load ratio refers to the ratio of wind power to total load and the same for solar–load ratio but with solar power).

Denmark has the highest proportion of wind power generation. By the end of 2016, wind energy accounted for 42% of the total electricity generation, and solar power accounted for 6.9%. Wind power accounted for 46.8% in 2019, and solar power accounted for 4%. In 2020, 50% of the electricity consumption of the Danish power sector comes from variable renewable energy (VRE). In some periods, the electricity generated by variable renewable energy will even exceed the total electricity demand [32]. In this case, the fluctuation of electricity prices is more uncertain, and the prediction of electricity prices is more difficult.

Figure 3 displays the histogram of the frequency distribution of electricity prices in the first half of 2019–2021.

Table 1. Power Data Statistics in Danish DK1 (2019–2021).

Time	Mean			Variant Coefficient			Wind-and-Solar–Load Ratio
	2019	2020	2021	2019	2020	2021	2019	2020	2021
0:00	34.13	12.75	36.75	0.50	0.78	0.37	-	0.95	0.62
1:00	33.18	12.03	35.71	0.55	0.80	0.37	-	1.00	0.63
2:00	33.33	11.87	34.25	0.54	0.79	0.40	-	0.97	0.64
3:00	34.32	12.66	34.37	0.52	0.72	0.40	-	0.93	0.62
4:00	36.71	13.91	38.29	0.46	0.67	0.32	-	0.88	0.60
5:00	42.05	19.23	45.71	0.40	0.62	0.30	-	0.76	0.54
6:00	48.21	23.89	53.98	0.38	0.57	0.32	-	0.69	0.49
7:00	49.99	26.34	57.44	0.38	0.58	0.36	-	0.66	0.46
8:00	48.44	24.88	55.43	0.39	0.57	0.36	-	0.65	0.44
9:00	46.58	22.69	51.49	0.39	0.58	0.34	-	0.62	0.43
10:00	45.44	21.68	49.70	0.39	0.55	0.35	-	0.64	0.41
11:00	44.12	19.91	47.20	0.38	0.57	0.35	-	0.66	0.41
12:00	43.10	18.59	44.35	0.38	0.61	0.37	-	0.65	0.41
13:00	42.65	18.28	43.20	0.39	0.62	0.38	-	0.68	0.44
14:00	43.40	18.71	44.98	0.39	0.59	0.34	-	0.70	0.47
15:00	45.43	21.07	48.22	0.36	0.54	0.30	-	0.68	0.49
16:00	50.66	25.98	56.43	0.30	0.48	0.25	-	0.66	0.47
17:00	52.51	28.48	63.05	0.24	0.53	0.29	-	0.67	0.47
18:00	49.42	25.47	60.16	0.20	0.57	0.28	-	0.71	0.49
19:00	45.42	20.42	51.54	0.21	0.57	0.25	-	0.76	0.52
20:00	42.27	18.55	46.84	0.29	0.58	0.24	-	0.80	0.56
21:00	41.27	17.52	45.32	0.30	0.57	0.23	-	0.83	0.57
22:00	37.29	15.57	40.50	0.39	0.62	0.27	-	0.90	0.60
23:00	35.28	13.71	38.82	0.46	0.73	0.33	-	0.94	0.63

shows the general statistics of the electricity market in the DK1 region of Denmark in the first half of 2019–2021. As can be seen from Figure 3, in the first half of 2019, the overall electricity price is in the range of [35, 50]. Then, the range of the 2020 price is obviously shifted to the left, and the overall price is in the range of [0, 35]. The number of moments with a negative price is higher than that of the previous year. The price range of the first half of 2021 is obviously shifted to the right, and the overall price is in the range of [35, 65], and it can be observed that the frequency of zero price and negative price periods has greatly increased in the three years. The frequency of a negative electricity price in the first half of 2020 was 3.55%, an increase of 11.29% compared to 3.19% in the first half of 2019. As mentioned above, this is due to the fact that the generation of variable renewable energy exceeds the total electricity demand in some periods. It can be found in Table 1 that the daily average maximum price generally occurs in the morning (6:00–9:00) and evening (16:00–22:00), while the daily average minimum price appears in the early morning (00:00–05:00). The lowest electricity price is often negative or even lower than −10. In the past three years, the statistical values of the standard deviation of the data are all above 10, which proves that the electricity price fluctuates violently, and the COV is mostly in the range of 0.3–0.7. It is proven that the discretization of data is large, and the variability is strong. Among them, the COV in 2020 is higher than that in the other two years. The effect of electricity price fluctuation affected by core variables is obvious. Accordingly, the work of the paper is devoted to solving electricity price forecasting problems with characteristics of a high COV.

Previous studies have shown that load is an external variable that has a significant impact on electricity prices. Figure 4 is a scatter plot of the electricity price and load, including wind and solar, in 2020. It can be seen in Figure 4 (left) that the load and electricity prices exhibit a certain positive correlation, and the high-peak electricity price is prone to a large load. Additionally, the region of the medium–high electricity price corresponds to the load, and it is concluded that the load cannot easily and accurately predict the trend of the electricity price. In Figure 4 (center), there is a negative correlation between the load and the wind load, and the moment of the low electricity price is often the time of the high wind load. In Figure 4 (right), the scatter diagram of electricity price has no apparent correlation, but when a load of solar increases, the probability of a low electricity price will also increase, and the coefficient of variation of the electricity price will decrease significantly.

The Pearson correlation coefficient is often used to show the correlation between electricity price and all kinds of loads. In the 2020 data set, the correlation coefficients between electricity price and total load, wind load, wind–load ratio, wind-and-solar–load ratio are 0.27, −0.45, −0.46, −0.52 and −0.52. Compared to the total load, wind load and wind-and-solar load, the negative correlation between wind–load ratio, wind-and-solar–load ratio and electricity price is more obvious, that is, the COV is often higher when there is a higher wind–load ratio and wind-and-solar–load ratio. It can also be found that the correlation coefficient difference between wind–load ratio and wind–solar ratio is tiny because in the Nordic electricity market, Denmark’s DK1 region, the wind power load is much larger than the solar load, while in the period with more solar output during the day, the higher solar–load ratio will lead to an increase in the fluctuation range of the coefficient of variation of electricity price, for example, from 11:00 to 14:00 in 2020, when the wind–load ratio is similar. The coefficient of variation fluctuates less at times with a higher solar–load ratio, and a similar situation can be observed in 2021 data. Therefore, considering the influence of wind and solar power on electricity price, the wind-and-solar–load ratio is introduced as the core external variable of electricity price fluctuation.

4. Electricity Price Model Design and Experiment Analysis

The work in this section has two parts. One is to construct an hourly price forecasting model by selecting different input variables. Due to the difference in the time sequence of electricity prices, the single-value forecasting method is adopted to establish a single-value forecasting model for 24 periods. The comparison of the models demonstrates the validity of the wind–load ratio and solar–load ratios as input variables. Secondly, it is observed that BP, DELM and SSA-DELM perform better than LSTM in the scenario with a high COV (in the year 2020). In terms of index RMSE, SSA-DELM has a 65.9% improvement over the commonly used LSTM.

4.1. Design of Electricity Price Model

There are three models with different input variables presented below:

(1) Model A: Time-series model considering hourly total load.

The input variable of the model is the hourly total load of each period $L_1$–$L_{24}$. In the course of the experiment, in order to explore the influence of the continuous period of load, such input variables, such as the average load ${\overline{L}}_1$–${\overline{L}}_{24}$ for each period in the previous 7 days and the load forecast for the previous hour are also included. In the test results for the 23rd period, the root mean square error (RMSE) in the same forecasting method reached 6.30. It is much larger than the root mean square error (RMSE) of 5.74 for the hourly total load data used alone, so the input in this model retains only the total load for each period.

(2) Model B: Time-series model considering hourly total load and wind–load ratio. The input variables selected in this model are the hourly total load of each period $L_1$–$L_{24}$ and the wind–load ratio of each period $W{S}_1$–$W{S}_{24}$. This model is constructed to verify the vital influence of the wind–load ratio in the forecasting model.

(3) Model C: Time-series model considering hourly total load, wind and solar–load ratio.

The input variables selected in this model are the hourly total load of each period $L_1$–$L_{24}$ and the wind-and-solar–load ratio $W{S}_1$–$W{S}_{24}$. In the course of the previous discussion, the Pearson correlation coefficients of the wind–load ratio and wind and solar–load ratio are very close to that of the electricity price. From the observation of data in Table 1, it is found that they have different effects on the coefficient of variation of electricity price. Therefore, model C can verify whether the input of the solar–load ratio can further improve the prediction effect.

4.2. Analysis of the Experiment

Based on the data of the hourly electricity price, total load, wind load and solar power load in the same period of 2020 and 2021 (20 January–20 March) in the Nordic electricity market DK1 region, three models with different input variables are trained to predict the hourly electricity price in last ten days. Compared with the test set, the mean absolute error (MAE) and root mean square error (RMSE) are used to evaluate the model.

Table 2. Comparison of model prediction errors.

Time	Mean Price/ Euro ${(\mathrm{MW}·\mathrm{h})}^{-1}$	Model A/ Euro ${(\mathrm{MW}·\mathrm{h})}^{-1}$		Model B/ Euro ${(\mathrm{MW}·\mathrm{h})}^{-1}$		Model C/ Euro ${(\mathrm{MW}·\mathrm{h})}^{-1}$
		MAE	RMSE	MAE	RMSE	MAE	RMSE
00:00	12.08	6.7	7.8	5.9	8.1	7.3	8.3
01:00	11.68	8.7	10.3	6.2	6.9	6.1	7.7
02:00	11.39	8.5	9.3	6.2	7.6	6.8	8.4
03:00	12.26	5.6	7.4	6.8	9.0	6.9	8.0
04:00	13.21	5.8	7.7	6.8	8.2	7.4	8.7
05:00	17.86	13.1	15.2	8.1	11.8	9.1	12.2
06:00	22.45	10.0	12.3	5.3	7.1	5.7	7.5
07:00	25.44	3.5	3.9	9.3	11.0	7.7	10.0
08:00	24.70	9.7	12.1	13.2	15.6	11.3	15.1
09:00	22.85	13.0	15.1	13.4	17.6	12.3	18.2
10:00	21.62	12.6	17.1	17.1	21.5	16.2	21.5
11:00	20.32	16.7	24.9	15.7	22.7	17.3	21.2
12:00	19.38	14.7	24.3	15.7	22.5	17.0	22.7
13:00	19.21	16.4	23.8	16.7	23.1	15.6	22.3
14:00	19.38	13.9	16.6	12.4	16.8	14.0	17.6
15:00	21.06	8.8	11.7	9.6	12.5	10.9	13.6
16:00	24.83	6.8	8.3	8.4	9.7	7.3	9.9
17:00	26.24	11.3	13.7	9.6	11.9	9.3	11.0
18:00	22.30	7.7	10.2	7.0	8.7	7.6	9.5
19:00	18.91	6.1	8.9	5.5	6.4	5.7	6.9
20:00	17.69	6.6	8.7	6.6	7.6	5.1	6.4
21:00	16.97	6.5	9.4	6.1	8.0	6.2	7.9
22:00	14.79	5.4	7.0	3.4	4.8	5.3	5.8
23:00	12.37	5.5	6.9	5.2	6.1	5.1	6.0
Mean	18.71	9.3	12.2	9.2	11.9	9.3	11.9

shows the performance of different models under the SSA-DELM algorithm in each period, and

Table 3. Test set error performance on Model C in 2020 and 2021.

Algorithms	2020		2021
	MAE	RMSE	MAE	RMSE
BP	11.3	13.5	14.9	19.0
LSTM	11.8	13.8	11.1	14.4
DELM	7.9	9.4	16.3	20.4
SSA-DELM	3.8	4.7	9.3	11.9

shows the average prediction error of each prediction method under the three models. Through Table 3, it can be found that the prediction effect of model A is far inferior to that of model B and model C, and the performance of model B is similar to that of model C. The average effect of model B with the wind–load ratio is slightly better than that of model C with the wind-and-solar–load ratio. However, combined with Table 2, it is not difficult to find that the prediction effect of model C with the wind–load and solar–load ratios is significantly better than that of model B in the four periods of 11, 12, 13 and 14. These four periods are the periods with the largest solar output in the whole day, and the influence of the solar–load ratio on the electricity price prediction cannot be ignored. In addition, by comparing the performance of models A and B, combined with the data in Table 1, it can be observed that the prediction effect is significantly improved in the periods of 0–5 and 21–23 with a high wind load, which shows that the wind–load ratio has a great influence on the price prediction effect in almost each period.

In addition, on the basis of testing models with different input variables, this paper also tests the superiority of the SSA-DELM algorithm in the application scenario of electricity price time series with a high coefficient of variation. By comparing the prediction effects of BP, LSTM and DELM on model C, as shown in Figure 5, the evaluation criterion is MAE, and the effect of the DELM model is better than that of BP and LSTM models. The SSA optimization algorithm further improves the prediction accuracy of DELM. As shown in Table 3, the SSA-DELM algorithm shows excellent test results on the 2020 data set, which shows that the SSA-DELM algorithm has significant advantages in the application scenarios with a high COV.

5. Conclusions

In this paper, an electricity price forecasting model based on SSA-DELM was constructed for the electricity market that contains a high proportion of wind and solar power. Based on the data set of DK1 in the Nordic market, conclusions can be summarized as follows. First, the wind–load ratio and the solar–load ratio play essential roles in electricity price forecasting. Applying them as input variables can significantly improve the predictive accuracy of tariff models, as shown in Section 4, the performance was improved by 27.9% compared to the model with the load as an input variable. Second, the SSA-DELM possesses a better electricity price forecasting performance in the scenario with a high COV and is more suitable for disordered time series models, which can be confirmed in comparison with LSTM. In terms of index RMSE, SSA-DELM has a 65.9% improvement over the commonly used LSTM. Moreover, the developed SSA-DELM performs well on a data set with a high COV, which is a statistical characteristic of the data. It should also perform well on data sets with the same characteristic, which is worth further investigation.