**3. Material and Methods**

The article presents variables (indicators) that describe the level of use of renewable energy sources (RES) for electricity generation. Twenty-eight countries were surveyed, including 27 countries of the current European Union and the United Kingdom, which was still formally a member of the EU until the end of 2020. The following years were selected for the study: 2004, 2009, 2014, and 2019. The years 2004, 2009, 2014, and 2019 were selected to show the changes in the studied quantities at equal time intervals of five years, counting backwards to 2019, from which the most recent complete data are derived. At the same time, 2004 is the year of enlargement of the European Union by 10 new member states. According to the authors, data from a greater number of years would limit the transparency of the presentation of the problem and not increase the accuracy of the WAP method used. The data from all the years 2004–2019 were used in the panel model. The data come from the Eurostat database [44].

In this study, one of the methods of multidimensional comparative analysis (WAP)— Hellwig's taxonomic measure of development—was used to assess the level of development of electricity production from renewable sources. It is one of the methods of linear ordering, which allows for the ranking of objects in order from the best to the worst according to the level of a complex phenomenon. Multidimensional comparative analyses are a willingly used research method, as evidenced by the works of Cheba and Szopik-Depczy ´nska [45], Rollnik-Sadowska and D ˛abrowska [46], and Gineviˇcius [47].

The concept of a complex phenomenon is closely related to the concept of a diagnostic variable and a synthetic (aggregate) variable. Diagnostic variables are the variables describing the examined complex phenomenon, whereas a synthetic variable is 'a variable which, based on a set of normalized diagnostic variables, determines quantitatively the level (degree of development) of the considered phenomenon in the studied objects' [48]. The synthetic measure is unitless.

The selection of diagnostic variables is based on substantive and formal criteria [48,49]. The basic substantive criterion is the importance of a given variable in the description of a complex phenomenon under study (e.g., according to expert opinion). Formal criteria include a high degree of variability and weak correlation of diagnostic features. The variables that qualify to the set of diagnostic variables, apart from having a significant impact on the studied complex phenomenon, should also be characterized by an appropriate degree of variation and should be weakly correlated among themselves (then, they do not duplicate the information transmitted by other variables).

In the next step, the diagnostic variables are identified, i.e., the nature of the impact of particular variables on a complex phenomenon is determined. In practice, it means a

division of the set of diagnostic variables into two subsets: variables—stimulants (S) and variables—destimulants (D).

Next, the diagnostic variables are normalised. Normalisation aims to bring the values of individual variables to comparability (by being rid of denominators and standardising the ranges of values taken by diagnostic characteristics) [50]. In this article, the standardisation method was used to normalise the values of individual diagnostic variables. Variables normalized by this method are characterized by an arithmetic mean equal to zero and a standard deviation equal to unity.

In the next step, the so-called development pattern is determined, i.e., an 'ideal' object having the most favourable values of diagnostic variables (i.e., in the case of stimulants— the highest values, while in the case of destimulants—the lowest values).

Then Euclidean distances *di* of particular objects from the so-called development pattern were calculated according to the following formula:

$$\begin{aligned} d\_i &= \sqrt{\sum\_{j=1}^k \left(z\_{ij} - z\_{0j}\right)^2}, \\ (i &= 1, 2, \dots, n), (j = 1, 2, \dots, k), \end{aligned} \tag{1}$$

where:

*k*—number of diagnostic variables;

*n*—number of objects (here: countries).

Subsequently, a synthetic measure was constructed, describing the level of use of renewable sources for electricity generation in each of the countries included in the study. The paper uses the following formula aggregating the normalized diagnostic variables:

$$\begin{aligned} z\_i &= 1 - \frac{d\_i}{d\_0}, \\ (i &= 1, 2, \dots, n), \end{aligned} \tag{2}$$

where:

$$d\_0 = 
max d\_i.$$

The above formula is counted among model aggregating functions [48,49]. Formula (2) does not take into account weights, i.e., it assumes equal importance of all diagnostic features that describe the examined complex phenomenon. The synthetic measure takes values in the range [0; 1]. The level of development of renewable sources for electricity generation is the higher, the closer the synthetic measure is to one.
