Statistical Analysis of Electricity Prices in Germany Using Benford’s Law

Abstract

The year 2022 was marked by a significant increase in electricity prices in Germany, with prices reaching extreme levels due to various geopolitical and climatic factors. This research analyzes the evolution of electricity prices in Germany from 2015 to 2024 and applies Benford’s Law to examine the distribution of the first digits of these prices. Historical electricity price data from Germany, obtained from publicly available sources, were used for the analysis. We applied Benford’s Law to determine the frequency of occurrence of the first digits of electricity prices and compared the results with the expected distribution according to Benford’s Law. We also considered the impact of negative electricity prices. The results suggest that external factors, such as geopolitical events and climatic conditions, have a significant impact on the volatility of electricity prices. Benford’s Law can be a useful tool for analyzing electricity prices, although its application to this market shows certain deviations.

1. Introduction

The year 2022 was marked by a significant increase in electricity prices. Electricity traders observed price increases at the level of several hundred EUR/MWh. The year 2021 also saw a rise in prices due to the COVID-19 pandemic. When we examine electricity prices in Germany, the number of instances where prices exceeded 100 EUR/MWh is shown in

Table 1. Frequency of electricity prices higher than 100 EUR/MWh in Germany.

Year	Frequency
2015	0
2016	1
2017	61
2018	13
2019	7
2020	25
2021	2685
2022	7376
2023	4129
2024	357

. Over the entire observed period (2015 to 2024), the number of instances where the electricity price exceeded 100 EUR/MWh was recorded 14,654 times which is approximately 18% from history data, specifically:

Number price between 100–199.9 EUR/MWh 9052 times

Number price between 200–299.9 EUR/MWh 3020 times

Number price between 300–399.9 EUR/MWh 1304 times

Number price between 400–499.9 EUR/MWh 761 times

Number price between 500–599.9 EUR/MWh 356 times

Number price between 600–699.9 EUR/MWh 106 times

Number price between 700–799.9 EUR/MWh 48 times

Number price between 700–799.9 EUR/MWh 7 times

Number price between 800–899.9 EUR/MWh 0 times

All analyses in Chapters 1 and 2 were performed in Excel by Microsoft VBA.

It was not until 2021 that electricity prices began to rise significantly in Germany, starting around September 2021. Since 2021, the frequency of prices exceeding 100 EUR/MWh has not returned to pre-2021 levels. This increase is mainly due to various fundamental factors, including the COVID-19 pandemic, the war in Ukraine, and various political decisions. A detailed analysis of electricity price trends in 2022 indicates that one of the most significant factors influencing electricity prices in that year was the war in Ukraine. This conflict led to a substantial disruption of natural gas and oil supplies from Russia, a major supplier of energy resources to Europe. Sanctions imposed on Russia and restrictions on gas supplies caused a sharp rise in energy prices on global markets. Since natural gas is often used for electricity generation, the increase in its prices had a direct impact on the cost of electricity production.

Following the lifting of pandemic restrictions in 2021, there was a significant revival in economic activity, which increased energy demand. Industrial sectors sought to make up for lost time by ramping up production, resulting in higher electricity demand. This increased demand contributed to higher prices as supply could not keep pace with the growing demand. The COVID-19 pandemic caused significant disruptions in global supply chains, leading to shortages of critical components and materials needed for electricity production and distribution. The shortage of components, such as semiconductors and other technologies, affected the production and maintenance of energy equipment, subsequently impacting the stability and reliability of electricity supply.

Extreme climatic conditions also contributed to higher electricity prices. In 2022, several extreme heatwaves and droughts were recorded, affecting electricity production from renewable sources such as hydroelectric plants.

2. Analysis of Electricity Price Trends from 2015 to 2024

Recent electricity price data indicate increasing volatility in electricity prices. For instance, 2022 was recorded as the most volatile year in recent years. In Germany (Figure 1), the highest recorded electricity price was EUR871/MWh, while the lowest was −EUR19.04/MWh, resulting in a price range of EUR890.04. The primary causes of commodity price fluctuations are various fundamental factors. In 2022, several such factors were identified [1,2,3]:

• Energy Crisis Due to the Russia–Ukraine Conflict: The war between Russia and Ukraine led to significant restrictions on the supply of natural gas and oil from Russia to Europe, causing an increase in the prices of these commodities and subsequently electricity, which is also generated from gas.
• Increase in Gas Prices: Gas is a key input for electricity generation in gas-fired power plants. The rise in its price directly impacted the cost of electricity production, thereby affecting its market price.
• Increased Demand for Energy: Following the COVID-19 pandemic, economic activities resumed, leading to an increased demand for energy, including electricity.
• Supply Constraints of Renewable Sources: Unfavorable weather conditions during certain periods of 2022 limited electricity production from renewable sources such as wind and solar power plants.
• Insufficient Capacity of Nuclear Power Plants: Countries like France had to reduce electricity production from nuclear power plants due to technical issues and maintenance, reducing the overall availability of electricity.
• Regulations and Emission Allowances: The European Union’s policies on reducing CO₂ emissions led to higher prices for emission allowances, increasing the cost of electricity production from fossil fuels.

When examining electricity prices, frequency charts were created for each year from 2015 to 2024 in Germany (Figure 2). The x-axis represents price ranges with a bin width of EUR10/MWh. The highest and lowest prices in a given year were identified to determine the number of bins, thus establishing the number of columns on the x-axis. From the data analysis, the following conclusions can be drawn:

• The results of this research indicate that the number of columns, and thus the number of price ranges, was significantly smaller from 2015 to 2020 compared to 2021 to 2024. In 2022, the highest number of price ranges was recorded in the observed period. As previously mentioned and shown in Figure 1, this is due to the high volatility of electricity prices in 2022. The results also show that the number of price ranges from 2015 to 2021 was approximately 4. This means that electricity prices varied by up to 4 * EUR10/MWh = EUR40/MWh during this period. In 2022, this number was significantly higher. However, after 2022, the number of price ranges did not return to 4 but increased to approximately 16 for Germany. This suggests that the trend in electricity price evolution did not return to pre-2022 levels after 2022. As of 2024, the number of price ranges in Germany is approximately 8, although this considers data only up to the end of May.
• Table 2. Maximum range for electricity price in years 2015 to 2024 in Czech Republic, Germany, Hungary, Poland and Slovakia.

  Year	CZ [€/MWh]	GE [€/MWh]	HU [€/MWh]	PL [€/MWh]	SK [€/MWh]
  2015	−20 to −10	−80 to −70	0 to 10	10 to 20	−20 to −10
  2016	−20 to −10	−140 to −130	−10 to 0	10 to 20	−20 to −10
  2017	−50 to −40	−90 to −80	0 to 10	10 to 20	−50 to −40
  2018	−40 to −30	−80 to −70	−30 to −20	10 to 20	−40 to −30
  2019	−50 to −40	−100 to −90	0 to 10	0 to 10	−30 to −20
  2020	−70 to −60	−90 to −80	−10 to 0	10 to 20	−70 to −60
  2021	−40 to −30	−70 to −60	−40 to −30	10 to 20	−40 to −30
  2022	−30 to −20	−20 to −10	0 to 10	10 to 20	−10 to 0
  2023	−70 to −60	−500 to −490	−50 to −490	−20 to −10	−30 to −20
  2024	−140 to −130	−140 to −130	−150 to −140	−40 to −50	−60 to −50

  displays the minimum range of electricity prices for each year from 2015 to 2024 for the countries Czech Republic, Germany, Hungary, Poland, and Slovakia. It is important to note that the frequency within these minimum ranges is low. For instance, in 2018, the minimum range of electricity prices for Germany was −80 to −70 EUR/MWh. However, the price appeared in this range only three times. The analysis was conducted on hourly data, and thus, in Germany, the price appeared in the range of −80 to −70 EUR/MWh for only three hours. Similarly, the same applies to the maximum ranges of electricity prices; their frequency is low.
• Based on Figure 2, it is possible to observe that from 2015 to 2020, the distribution of electricity prices follows a Gaussian curve. However, 2021 and 2022 were different. The peak (the range with the highest values) was approximately in the center before 2021, and the distribution was relatively even. In 2021 and 2022, however, the peak shifted to the left towards lower prices. In these years, there are extremes towards higher ranges, indicating extreme electricity prices. These extremes are not very frequent, though. The main distribution of electricity prices in these years appears to the left towards lower prices. This is also due to the frequency (although low) of extreme electricity prices towards the right.
• Figure 3 shows the graph of maximum and minimum electricity prices for the years 2015 to 2024 for the countries Czech Republic, Germany, Hungary, Poland, and Slovakia. For the reasons mentioned above, the greatest range between the maximum and minimum electricity prices was in 2022. Similarly, in 2021, the range was higher than in the years 2015 to 2020. In 2023 and 2024, the range returned to normal values as before 2021. From 2015 to 2020, electricity prices remained relatively stable in most of these countries, with a slight increase in some cases. However, around 2020, there was a significant surge in prices, culminating in a peak in 2022. This sharp increase likely reflects market disruptions caused by external factors such as geopolitical events, energy crises and so on. After 2022, electricity prices began to decline again in these countries, with the rate of decline varying across nations. In Germany, the prices experienced a dramatic drop after 2022, even falling below zero in 2023, indicating a period of negative electricity prices, possibly due to oversupply or specific market conditions. Poland recorded the highest price peak in 2022, with prices exceeding 1000 EUR/MWh, followed by a sharp decline in the subsequent years.
• Table 3. Frequency of negative electricity prices in years 2015 to 2024 in Czech Republic, Germany, Hungary, Poland and Slovakia.

  Year	CZ [€/MWh]	GE [€/MWh]	HU [€/MWh]	PL [€/MWh]	SK [€/MWh]
  2015	72	110	0	0	72
  2016	58	97	6	0	58
  2017	112	146	113	0	102
  2018	42	134	2	0	42
  2019	58	211	0	0	51
  2020	119	298	4	0	115
  2021	33	139	8	0	24
  2022	80	69	0	0	2
  2023	134	301	74	43	90
  2024	119	152	125	43	38

  shows the number of negative electricity prices for the individual years from 2015 to 2024 for the countries Czech Republic, Germany, Hungary, Poland, and Slovakia. Naturally, the lowest number of negative prices in recent years was in 2022, when high electricity prices were a problem in these and other countries. However, the table shows a trend of negative prices in the recent period. As other publications describe, the main reason for negative electricity prices is excessive production from photovoltaics and wind turbines. In the case of the Czech Republic, Germany, and Slovakia, a significant number of negative prices can be seen over the entire observed period. Moreover, the analyses were conducted on data up to the end of April 2024, when photovoltaic plants had not yet produced their highest amount of energy. Interestingly, Poland recorded its first negative prices in 2023 during the observed period. The reason might be that the amount of electricity produced from photovoltaics increased by 7.5 times from 2020 to 2023.

-

Due to the volatility of electricity prices, there has been an increased emphasis on the possibilities of electricity price prediction and deeper analysis of electricity price data. In 1935, Benford’s Law was discovered, which describes the frequency with which the digits 1 through 9 appear as the leading significant digits in a wide range of data sets. While digit one appears in about one-third of cases, the probability for digit two is 17.6%, for digit three is 12.5%, and this proportion gradually decreases to the digit nine, which appears as the leading digit with a probability of 4.6%. Scientists discovered that this relationship was actually identified by astronomer Newcomb in 1881, but at that time, he could not convince many people of its validity. Both discoverers noticed while reading logarithmic tables in the library that the first pages were much dirtier than the others, indicating that their colleagues from various fields were more frequently looking up logarithmic values for numbers starting with one. Benford found that the law applies to various sets of data—population numbers, death rates, chemical and physical constants, baseball statistics, half-lives of radioactive isotopes, prime numbers, elements of the Fibonacci sequence, and many others. On the other hand, Benford’s Law does not apply to random sets with certain restrictions, such as lottery numbers, telephone numbers, dates, heights, and weights of groups of people. According to scientists, the law also applies to the second digit, but with less reliability; for further digits, the numbers have a random distribution with equal probability of occurrence. Scientists have not yet found a rule that can determine whether Benford’s Law applies to a given set of numbers or not [4]. Benford’s Law has been applied to many different data sets [5,6,7,8,9,10,11,12]:

-

To determine whether the lists of COVID-19 infection count, which claim to measure actual events, have been manipulated,

-

Analysis of lightning data and evaluation of negative effects with precise lightning parameters in kA units,

-

Detecting image tampering during resizing and compression,

-

Identifying anomalies in the number of publications per researcher and the number of researchers per publication,

-

Analyzing the distribution of initial letters in novels and similar studies,

-

Evaluating the quality of economic data in enterprises,

-

Application to Chinese texts,

-

Detecting whether a given text is typed by a human or a machine (robot).

However, to date, there is no research that focuses on the application of Benford’s Law to electricity prices or other commodities. As previously mentioned, Benford’s Law has been applied to various types of data, not just in technical environments. This paper aims to investigate the following questions through research:

(a)

Is a one-year sample of electricity price data sufficient?

(b)

Considering that electricity prices can be negative, how should negative numbers be handled? In practice, the situation may arise where the electricity price is 41 EUR/MWh as well as −41 EUR/MWh. Will the research results utilizing Benford’s Law differ when considering negative numbers versus not considering negative numbers?

(c)

Can Benford’s Law be used for potential electricity price prediction?

(d)

Can the statistical results be utilized in other ways?

3. Methodology and Data

The aim of the research was to apply Benford’s Law to electricity prices and determine whether it can be effectively used for electricity price analysis. Since this area has not been previously explored, we established the following objectives:

• To verify whether Benford’s Law can be applied to electricity prices.
• To compare the results of the analysis for the years before 2020 and from 2020 onward, due to the differing conditions in the electricity market.
• To determine the future direction of research based on the results obtained.

It may seem surprising that the first person to encounter this phenomenon was the American astronomer Simon Newcomb in 1881. He wrote a two-page article about it [7], in which he noticed that the pages at the beginning of logarithmic tables were more worn and more frequently used than those towards the end. At that time, no one paid much attention to this article.

In the 1930s, Frank Benford observed the same phenomenon regarding the wear of logarithmic tables. Apparently unaware of Newcomb’s article, Benford published his own more extensive article in 1938 [8], in which he collected 20,229 observations from various fields (human populations, values of constants, numbers in newspapers, lengths of rivers, addresses, etc.).

Both concluded the same: people using logarithmic tables encounter numbers starting with the digits 1 and 2 much more frequently than those starting with the digits 8 and 9.

According to the rules of the law, the first significant digit probability can be expressed. Benford found various datasets and stated a few dataset properties, which made the dataset best suited according to BL.

The definition of a significant number is as follows [9]: every real number excluding zero and denoting it as x, the first significant decimal number of the real number x, denoted as D₁(x), is the unique integer j ∈ {1, 2, …, 9} satisfying conditions [10]

10^k j ≤ |x| < 10^k (j + 1)

(1)

for some (unique) k ∈ Z. Additionally, for every number m ≥ 2, m ∈ N, the m-th significant decimal digit of x is denoted as D_m(x). This is defined inductively as a unique integer number j ∈ {0, 1, …, 9} in the following formula [7]:

$${10}^{\mathrm{k}}\left({\sum }_{\mathrm{i}=1}^{\mathrm{m}-1}{\mathrm{D}}_{\mathrm{i}}\left(\mathrm{x}\right){10}^{\mathrm{m}-1}+\mathrm{j}\right)\le \left|\mathrm{x}\right|<{10}^{\mathrm{k}}\left({\sum }_{\mathrm{i}=1}^{\mathrm{m}-1}{\mathrm{D}}_{\mathrm{i}}\left(\mathrm{x}\right){10}^{\mathrm{m}-1}+\mathrm{j}+1\right)$$

(2)

for some unique number k ∈ Z; for convenience, D_m(0): = 0 for all m ∈ N. By definition, the first significant digit D₁(x) of x! = 0 is never zero. However, when considering the second, third, fourth and so on, these significant digits may be any integers including zeros, i.e., they are defined by the set of {0, 1, …, 9} decimal numbers.

Thus, Benford’s law is a statistical law describing the probability that a specific digit becomes the first significant digit of the number from a dataset. However, the dataset that follows BL must fulfill the following basic requirements:

1. The dataset must not be radically restricted in value range (e.g., the dataset of people’s height or IQ is a radical restriction because of a very small range of possible values) [10].

2. The dataset must not be influenced by any kind of artificial effects caused by human actions aiming to change the values intentionally [10].

3. The value range in the dataset must be large enough [11]:

Fdiff > 3, according to Fdiff = log(max) − log(min)

(3)

where max and min stand for maximal and minimal dataset values, respectively.

4. The dataset should be large enough.

The Formula (2) describes the calculation of the first significant digit probability according to BL [12]:

F_d = log₁₀ (d + 1/d)

(4)

where F_d is the probability value. (In the case of decimal numbers, we have to use decimal logarithm), and

d ϵ D = (1, 2… 9)

(5)

Then, for digit 1:

F₁ = log₁₀ (1 + 1/1) = 0.30103

(6)

i.e., the probability that the decimal digit 1 is the first significant digit in approximately 30,103% of cases. The sum of all probabilities is 1, i.e., 100% probability for all decimal digits d ϵ D = (1, 2… 9), the digit 0 cannot be the first significant digit, and dataset samples with the value 0 (e.g., 0 or 0.000) are excluded from the dataset because they do not contain a significant digit.

From the equations, we can observe a descending trend of the probability. The probability distribution mentioned above is valid for decimal number systems; however, BL is not restricted to this system, and it is also applicable for other numeric systems, e.g., octal or hexadecimal numeric systems. Then, we can write the first significant digit probability equation in a more general form, assuming that the difference between 2 logarithms equals the logarithm of the ratio:

F_d = log_n (d + 1) − log_n (d)

(7)

where d denotes a particular digit from a particular numeric system, and n denotes the particular numeric system [10].

Analogously to the first significant digit, it is possible to define the second, third, fourth, and potentially further significant digits. For instance, the second significant digit for any non-zero real number is defined as the digit to the right of the first significant digit. The third significant digit then directly follows the second significant digit to the right, and the fourth significant digit follows the third significant digit to the right. Therefore, the second and subsequent significant digits can take on 10 values (0, 1, 2, 3, …, 9). For example, the number 245, understood as the value of a ratio variable (thus, 245.0), has its first significant digit as 2, the second as 4, the third as 5, and the fourth as 0. The number 8.556 has 8 as its first significant digit, 5 as both its second and third significant digits, and 6 as the fourth significant digit. In

Table 4. Probability estimates for BL.

Digit	BL
0	-
1	30.10%
2	17.61%
3	12.49%
4	9.69%
5	7.92%
6	6.69%
7	5.80%
8	5.12%
9	4.58%

we can see probability estimates for BL.

For data analysis, available data from the website energy-charts.com were used. To statistically process data using Benford’s Law, a sufficient amount of data is required. The analyses described in the next chapter were applied to data from 2015 to the present, specifically up to the end of April 2024. Data for Germany were selected for the analysis.

4. Results and Discussion

In the first step of the research, it was necessary to address the issue of the leading digit 0. Since Benford’s Law does not consider this digit, this problem needs to be resolved. Therefore, an initial experimental study was conducted for one market; specifically, the German market, which is one of the largest in Europe.

Table 5. Application BL on the German market in 2015 without considering the number 0.

Digit	Number of Values	BL %	Number of Digits in %	Delta
1	960	30.10%	11.08%	19.02%
2	2846	17.60%	32.85%	−15.25%
3	2505	12.50%	28.91%	−16.41%
4	1514	9.70%	17.47%	−7.77%
5	505	7.90%	5.83%	2.07%
6	112	6.70%	1.29%	5.41%
7	22	5.80%	0.25%	5.55%
8	18	5.10%	0.21%	4.89%
9	41	4.60%	0.47%	4.13%

shows the results of applying Benford’s Law to electricity prices in the German market. During the analysis of prices, it was found that in 2015, 2.7% of the values began with the digit 0. This can also be considered a statistical deviation. Therefore, electricity prices starting with the digit 0 were not considered further. Negative values are also not considered in Table 5. Negative prices in 2015 accounted for approximately 1.3%, which is also a low value. Comparing the entire history of data from 2015 to May 2024, about 1.4% of numbers start with the digit 0. Negative prices in 2015 accounted for approximately 1.9%, which is also a low value. Combined, this is approximately 3.3%, which is still a low value. Considering only the digits 1 to 9, the results for the German market indicate that:

• The digit 1 does not appear most frequently in 2015. It appears in exactly 11.08% of cases.
• As mentioned above, the digit 0 occurred in 2.7% of cases in 2015. Prices starting with the digit 0 range from 0.00 to 0.99, which are prices close to 1 EUR/MWh. Therefore, the 2.7% occurrence of the digit 0 can be added to the percentage occurrence of the digit 1. This increases the occurrence of the digit 1 from 11.08% to 13.78%.
• In Table 5, the “Delta” value indicates the difference between the percentage occurrence of a digit according to Benford’s Law and the actual values applied to electricity prices. The analysis results show that for the digits 2, 3, and 4, the Delta value is negative. This means that the occurrence of these digits is higher than defined by Benford’s Law. A low Delta value means that the result is closest to the value from Benford’s Law.
• The digit 5 has the lowest Delta value. Conversely, the digit 1 has the highest Delta value. As mentioned above, 2.7% of electricity prices start with the digit 0. If this number is added based on the explanation in point 2, the Delta value for the digit 1 is 16.32%.

Now, consider the scenario where we include negative electricity prices in the analysis in the following way: if the electricity price is −45 EUR/MWh, this price is added to the frequency count of the digit 4; if the price is −14.8 EUR/MWh, it is added to the frequency count of the digit 1, and so on. The results (

Table 6. Application BL on the German market in 2015 without considering the number 0 and with considering negative number.

Digit	Number of Values	BL %	Number of Digits in %	Delta
1	993	30.10%	11.54%	18.56%
2	2860	17.60%	33.25%	−15.65%
3	2505	12.50%	29.12%	−16.62%
4	1522	9.70%	17.69%	−7.99%
5	513	7.90%	5.96%	1.94%
6	118	6.70%	1.37%	5.33%
7	27	5.80%	0.31%	5.49%
8	21	5.10%	0.24%	4.86%
9	43	4.60%	0.50%	4.10%

) show the following:

• The most frequent digit is again digit 2, which appears in 33.25% of cases.
• The digit 1 has the highest Delta value. Given the small number of negative prices in 2015, the results did not change significantly.

Similarly to the previous analysis, the frequency of the digits does not match the frequency predicted by Benford’s Law. The results could have been influenced by several factors. Firstly, in 2015, negative numbers did not occur as frequently as in other years, as shown in Table 3.

In the next step, the analysis was applied to subsequent years, but without considering negative electricity prices.

Table 7. Application BL on the German market in 2016 without considering the number 0.

Digit	Number of Values	BL %	Number of Digits in %	Delta
1	1038	30.1%	11.82%	18.28%
2	3565	17.6%	40.59%	−22.99%
3	2693	12.5%	30.66%	−18.16%
4	794	9.7%	9.04%	0.66%
5	282	7.9%	3.21%	4.69%
6	98	6.7%	1.12%	5.58%
7	66	5.8%	0.75%	5.05%
8	70	5.1%	0.80%	4.30%
9	51	4.6%	0.58%	4.02%

,

Table 8. Application BL on the German market in 2017 without considering the number 0.

Digit	Number of Values	BL %	Number of Digits in %	Delta
1	743	30.1%	8.48%	21.62%
2	1842	17.6%	21.03%	−3.43%
3	3478	12.5%	39.70%	−27.20%
4	1486	9.7%	16.96%	−7.26%
5	437	7.9%	4.99%	2.91%
6	180	6.7%	2.05%	4.65%
7	135	5.8%	1.54%	4.26%
8	124	5.1%	1.42%	3.68%
9	149	4.6%	1.70%	2.90%

,

Table 9. Application BL on the German market in 2018 without considering the number 0.

Digit	Number of Values	BL %	Number of Digits in %	Delta
1	399	30.1%	4.55%	25.55%
2	812	17.6%	9.27%	8.33%
3	1726	12.5%	19.70%	−7.20%
4	2447	9.7%	27.93%	−18.23%
5	1740	7.9%	19.86%	−11.96%
6	872	6.7%	9.95%	−3.25%
7	410	5.8%	4.68%	1.12%
8	121	5.1%	1.38%	3.72%
9	56	4.6%	0.64%	3.96%

,

Table 10. Application BL on the German market in 2019 without considering the number 0.

Digit	Number of Values	BL %	Number of Digits in %	Delta
1	266	30.1%	3.04%	27.06%
2	1255	17.6%	14.33%	3.27%
3	3118	12.5%	35.59%	−23.09%
4	2356	9.7%	26.89%	−17.19%
5	1088	7.9%	12.42%	−4.52%
6	249	6.7%	2.84%	3.86%
7	81	5.8%	0.92%	4.88%
8	41	5.1%	0.47%	4.63%
9	34	4.6%	0.39%	4.21%

,

Table 11. Application BL on the German market in 2020 without considering the number 0.

Digit	Number of Values	BL %	Number of Digits in %	Delta
1	1108	30.1%	12.61%	17.49%
2	2292	17.6%	26.09%	−8.49%
3	2411	12.5%	27.45%	−14.95%
4	1505	9.7%	17.13%	−7.43%
5	604	7.9%	6.88%	1.02%
6	168	6.7%	1.91%	4.79%
7	101	5.8%	1.15%	4.65%
8	86	5.1%	0.98%	4.12%
9	74	4.6%	0.84%	3.76%

,

Table 12. Application BL on the German market in 2021 without considering the number 0.

Digit	Number of Values	BL %	Number of Digits in %	Delta
1	1977	30.1%	22.57%	7.53%
2	763	17.6%	8.71%	8.89%
3	548	12.5%	6.26%	6.24%
4	997	9.7%	11.38%	−1.68%
5	1152	7.9%	13.15%	−5.25%
6	968	6.7%	11.05%	−4.35%
7	847	5.8%	9.67%	−3.87%
8	726	5.1%	8.29%	−3.19%
9	578	4.6%	6.60%	−2.00%

,

Table 13. Application BL on the German market in 2022 without considering the number 0.

Digit	Number of Values	BL %	Number of Digits in %	Delta
1	2826	30.1%	32.26%	−2.16%
2	2384	17.6%	27.21%	−9.61%
3	1211	12.5%	13.82%	−1.32%
4	792	9.7%	9.04%	0.66%
5	437	7.9%	4.99%	2.91%
6	220	6.7%	2.51%	4.19%
7	257	5.8%	2.93%	2.87%
8	261	5.1%	2.98%	2.12%
9	248	4.6%	2.83%	1.77%

,

Table 14. Application BL on the German market in 2023 without considering the number 0.

Digit	Number of Values	BL %	Number of Digits in %	Delta
1	4229	30.1%	48.28%	−18.18%
2	288	17.6%	3.29%	14.31%
3	155	12.5%	1.77%	10.73%
4	221	9.7%	2.52%	7.18%
5	275	7.9%	3.14%	4.76%
6	342	6.7%	3.90%	2.80%
7	630	5.8%	7.19%	−1.39%
8	950	5.1%	10.84%	−5.74%
9	1245	4.6%	14.21%	−9.61%

and

Table 15. Application BL on the German market in 2024 without considering the number 0.

Digit	Number of Values	BL %	Number of Digits in %	Delta
1	448	30.10%	13.34%	16.76%
2	71	17.60%	2.11%	15.49%
3	112	12.50%	3.33%	9.17%
4	228	9.70%	6.79%	2.91%
5	445	7.90%	13.25%	−5.35%
6	577	6.70%	17.18%	−10.48%
7	567	5.80%	16.88%	−11.08%
8	448	5.10%	13.34%	−8.24%
9	234	4.60%	6.97%	−2.37%

display the results of applying Benford’s Law to electricity prices for the years 2016 to 2024 in Germany.

In Figure 4, the comparison of the frequency of the digit 1 for the years 2015 (blue curve) to 2024 with the frequency of 30.1% according to Benford’s Law (red curve) is shown. From the results displayed in Table 5, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15 and

Table 16. Application BL on the German market in 2015–2024 without considering the number 0.

Digit	Number of Values	BL %	Number of Digits in %	Delta
1	13,994	30.10%	17.03%	13.07%
2	16,118	17.60%	19.62%	−2.02%
3	17,957	12.50%	21.86%	−9.36%
4	12,340	9.70%	15.02%	−5.32%
5	6965	7.90%	8.48%	−0.58%
6	3786	6.70%	4.61%	2.09%
7	3116	5.80%	3.79%	2.01%
8	2845	5.10%	3.46%	1.64%
9	2710	4.60%	3.30%	1.30%

, the following conclusions can be drawn:

• From 2015 to 2020, the frequency of the digit 1 was relatively low and did not approach the expected frequency of 30.1%.
• In 2021, the frequency of the digit 1 exceeded 20% and gradually increased. In 2024, it is currently below 20%, but the data only extend to the end of May.
• In 2022 and 2023, the frequency of the digit 1 reached its highest values. This can be attributed to the fact that in these years, there was a high number of electricity prices above 100 EUR/MWh—see Table 1. All prices ranging from 100 to 199.99 were counted under the digit 1.
• In Table 16, data for the entire period from 2015 to 2024 are displayed. The results show that the frequency of the digit 1 is 17.03%. This still does not approach the value of 30.1%. Among all digits, the highest Delta value is for the digit 1.
• Figure 5 shows the comparison of the frequency of the digit 2 for the years 2015 (blue curve) to 2024 with a frequency of 17.6% according to Benford’s Law (red curve). The frequency of the digit 2 for the years 2015 to 2024 hovers around 17.6%. Considering all available data, the frequency of the digit 2 is 19.62%. The Delta value is −2.02%, which is relatively low. Therefore, it can be said that Benford’s Law can be applied to the digit 2.
• The comparison of the frequency of the digit 3 is shown in Figure 6. The results indicate that in recent years, low Delta values can be observed. For the entire data history from 2015 to 2024, the Delta value for the digit 3 is −9.36%, which is relatively high. Therefore, Benford’s Law cannot be applied to the digit 3.
• Figure 7 represents the trend of the frequency of the digit 4 during the years 2015 to 2024. Values range from 2.52% to 27.93%, which are high values. The frequency value for the entire data set is 15.02%, which is also high. Similar to the case of digit 3, Benford’s Law cannot be applied to digit 4.
• The frequency of the digit 5 ranges from 3.14% to 19.86% between 2015 and 2024 (Figure 8). Considering all data from 2015, the frequency of the digit 5 is 8.48%. The values in individual years, as well as for the entire period, are high, and thus Benford’s Law cannot be applied to the digit 5.
• The frequency of the digit 6 (Figure 9) ranges from 1.12% to 17.18%, while according to Benford’s Law, the frequency of the digit 6 should be approximately 6.7%. The value of 17.18% is from 2024, where the data sample for the whole year is not yet available. If 2024 is excluded, the highest frequency of the digit 6 is 11.05%. The frequency of the digit 6 fluctuates significantly during this period. When applied to the entire data set, the frequency of the digit 6 is 4.61%, and the Delta value is 2.09%, which can be considered relatively low. Therefore, Benford’s Law could be applied to the digit 6 for the entire data set, but not for each year individually.
• The trend of the frequency of the digit 7 (Figure 10) is similar to that of the digit 6. The lowest frequency is 0.25%, and the highest is 16.88%. Considering that 2024 (where the frequency is 16.88%) is not yet complete, the highest value is 9.67%. Similarly, for the digit 7, when applied to the entire data set, the frequency is 3.79%, and the Delta value is 2.01%, which is relatively low. Therefore, Benford’s Law could be applied to the digit 7 for the entire data set, but not for each year individually.
• The frequency of the digit 8 ranges from 0.21% to 13.34% (Figure 11). Excluding 2024, the maximum value is 10.84%. For the entire data history, the frequency of the digit 8 is 3.46%, while according to Benford’s Law, it should be 5.1%. Therefore, as in the previous cases, Benford’s Law could be applied to the entire data set, but not to each year individually.
• The lowest frequency of the digit 9 is 0.39%, while the highest is 14.21% (Figure 12). For the entire data history, the frequency of the digit 9 is 3.3%, while according to Benford’s Law, it should be 4.6%. Therefore, as in the previous cases, Benford’s Law could be applied to the entire data set, but not to each year individually.

During this period, electricity prices were characterized by a relatively stable trend, influenced mainly by long-term factors such as the increasing share of renewable energy sources, low gas prices, and stable geopolitical conditions. Electricity prices followed a Gaussian distribution, with the main peak of the distribution being symmetric and located in the center. This indicates that there were no significant price fluctuations during this period.

In contrast, the period 2020–2022 was marked by significant changes in the electricity market, reflected in the price developments. The main factors were the COVID-19 pandemic, which caused a sharp drop in demand, the subsequent economic recovery leading to rising prices, and particularly the energy crisis due to the Russia–Ukraine conflict, which resulted in extremely volatile electricity prices. During this period, there was a notable increase in the number of price extremes, either very high or even negative. These changes caused the price distribution to become asymmetric, with the main peak of the distribution shifting towards lower prices, but with distinct extremes on both sides of the spectrum.

We are increasingly observing negative electricity prices on exchanges. Several analysts agree that the reason for this lies in photovoltaic power plants and wind turbines. Whether this is the case might be revealed by further research, which could determine a possible correlation between production from photovoltaics and wind turbines and electricity prices. Looking at current electricity prices, it is apparent that in countries where electricity is generated from conventional sources (nuclear, coal, etc.), negative prices do not appear as frequently. This is also due to the fact that the share of photovoltaic and wind power plants is not as significant. The issue of applying Benford’s law to negative prices would require further research. The range of negative electricity prices is not significant. Therefore, similar to the case of positive prices, the research would focus on analyzing the second digit in negative prices.

5. Conclusions

The application of Benford’s Law to electricity prices has not been previously studied. The results presented in this article indicate that Benford’s Law can be an effective tool for analyzing electricity prices, but with certain limitations and specifics. The analysis showed that for electricity prices in the German market from 2015 to May 2024, not all digits occurred as predicted by Benford’s Law. The most significant deviations were observed for the digit 1, whose frequency was significantly lower than Benford’s Law predicts.

One of the main reasons for these deviations is the fact that Benford’s Law does not account for prices starting with the digit 0, as well as negative prices that occur in the electricity market. These anomalies were removed in our analysis, which may have impacted the overall results. Prices ranging from 0 to 0.99 could be considered as starting with the digit 1 since the difference in price between 0 and 1 is minimal. On the other hand, prices starting with the digit 1 can range from 0 EUR/MWh to 19.99 EUR/MWh and 100 EUR/MWh to 199.99 EUR/MWh, which is not as common. This is also the reason why the frequencies of the digits 2, 3, and 4 are the highest. This is due to the frequent occurrence of prices ranging from 20 EUR/MWh to 49.99 EUR/MWh.

A significant part of our analysis was also monitoring the frequency of individual digits in various price ranges. We found that electricity prices above 100 EUR/MWh and negative prices affect the frequency of individual digits and should be included in future analyses for more accurate results.

The direction for future research could be realized in:

• Extending the scope of analysis: It would be beneficial to extend the analysis to other electricity markets in Europe and other parts of the world to determine whether the observed deviations are specific to the German market or represent a global phenomenon.
• Detailed analysis of prices above 100 EUR/MWh: As mentioned in the conclusion, prices above 100 EUR/MWh significantly impact digit frequency. Future research should focus on a more detailed examination of this impact and identifying patterns within these high prices.
• Including negative prices: Negative electricity prices, although relatively rare (2–4%), can have a significant impact on the analysis. Future studies should include these prices and examine how they affect the overall frequency of digits according to Benford’s Law. This aspect needs further attention as negative prices are becoming increasingly common across almost all markets, not just in Germany.
• Using Benford’s Law for prediction: Research should also focus on the potential use of Benford’s Law for predicting electricity prices. Although this area is not yet sufficiently explored, the potential for predictive analysis could be significant. Even if the application of Benford’s Law to electricity prices did not yield the desired results, the statistical frequency results of individual digits could be useful for further research, such as determining the probability of price ranges starting with a specific digit. Further research could focus on the analysis of the second digit. The results indicate that the most frequent digits are 1 (17.03%), 2 (19.62%), and 3 (21.86%). Considering that in 58.51% of cases, the price ranges between 1 EUR/MWh and 39.99 EUR/MWh, analyzing the second digit could potentially increase the accuracy of electricity price predictions.
• Other statistical methods: In addition to Benford’s Law, other statistical methods should be explored for analyzing electricity prices. This could provide a more comprehensive view of the dynamics of the electricity market.

The results of the research and analysis of electricity prices in Germany suggest that a one-year data sample shows higher Delta values compared to analyses focused on a larger data history, specifically in this case, data from 2015 to 2024.

These directions for future research could contribute to a deeper understanding of electricity prices and the development of more effective tools for their analysis and prediction. Commodity prices are relatively specific, as it is unlikely (for example, in the case of electricity prices) that they would fall in the range of 0 EUR/MWh to 19.99 EUR/MWh and 100 EUR/MWh to 199.99 EUR/MWh, starting with the digit 1, in 30.1% of cases. According to Benford’s Law, the digit 1 should appear in 30.1% of cases.

In conclusion, despite the results, it can be stated that the application of Benford’s Law to electricity prices requires deeper analysis. As mentioned earlier, one possible approach is to exclude data from years that were atypical. However, since the onset of the pandemic, electricity prices have been unstable, and the years 2021 to 2024 are not representative of typical market conditions. These global factors have caused high volatility in the electricity market, which is likely to persist. Given that the application did not yield the expected results, future research should consider focusing on the second digit, which is not as clearly defined as the first digit. The second digit may appear more frequently in any form from 0 to 9. As previously noted, the first digit is significantly influenced by the electricity price itself.

Benford’s Law, which describes the distribution of leading digits in various data sets, can offer valuable insights when analyzing electricity prices. Although we observed some deviations from the expected values according to Benford’s Law, the statistical analysis of digit frequency can reveal interesting patterns that could form the basis for further research and potentially for price prediction. For instance, if we know that a certain price range has a high probability of starting with a specific leading digit, this information could be used in modeling or forecasting future prices.

Our results suggest that while Benford’s Law did not fully predict the distribution of electricity prices, the frequency of occurrence of individual digits provides fundamental data that could be utilized in predictive models. Specifically, we identified that certain digits, such as digit 1, occur at a frequency that differs from what is expected by Benford’s Law, particularly in years with extreme prices. This information can be used to identify likely price intervals in the future, which is a crucial step in developing predictive algorithms. In conclusion, it is important to emphasize that while this research focused on the analysis of electricity prices using Benford’s Law, similar analytical approaches could potentially be applied to other commodities, such as gas. Like electricity, gas is subject to significant market and geopolitical factors that influence its price. We anticipate that applying Benford’s Law to this commodity could reveal similar patterns and deviations as observed with electricity. However, this aspect requires further research to verify its suitability and effectiveness in predicting the prices of other commodities. By expanding this application, the results of our research could be applied more broadly and contribute to a better understanding of price dynamics across various markets.