The results of numerous studies show that the control of power grid modes is carried out mainly using a technical criterion. The economic criterion is taken into account through the use of complex and inaccurate models that do not accurately predict the result. The emergence of market relations in the energy sector makes power systems economic entities in terms of production and satisfaction of demand for electricity by various economic entities (industry, households, businesses, etc.). Under these conditions, electricity is a commodity with a corresponding price and quality indicators. This requires the application of the risk assessment methodology as an economic category in the activities of power systems as a business entity. The methodology of risk assessment in market conditions requires business entities to search for methods to minimize risk as a possibility of adverse events. Under these conditions, it becomes possible to make the best management decisions regarding the most important criterion that reflects the interests of business entities at a given time. However, the imperfection of the relevant methodology for risk assessment in the energy sector delays their application in the industry. At the same time, when making management decisions, three possible levels can be distinguished: decision-making in conditions of certainty, when the result is presented in a deterministic form and can be determined in advance; decision-making under conditions of risk, when the outcome cannot be determined in advance, but there is information on the probability of distribution of possible consequences; decision-making in conditions where the outcome is random and there is no information about the consequences of the decision. An analysis of scientific publications shows that some authors’ works are devoted to solving the issues of applying the theory and principles of risks in the energy sector, in which the problem is solved only at the first two levels. At the same time, the operation of energy facilities is characterized by a high level of uncertainty and incomplete information about the consequences of such decisions. Therefore, the development of a methodology for making management decisions in the energy sector based on the theory and practice of risks, taking into account the high level of uncertainty and incomplete information, is an urgent scientific task. Implementation of algorithms and programs for controlling the modes of power grids based on them can meet the requirements for reliable and high-quality energy supply to the most demanding consumers and create favorable conditions for their business. This work is devoted to the development of scientific and methodological foundations for determining the voltage risk in power system networks, taking into account the uncertain nature of the loads and its impact on consumers. Based on the results of the study, a mathematical model of the risk of voltage collapses in networks, an algorithm and a methodology for its calculation were proposed.

Work-based learning models have been applied to varying extent across countries. This raises questions about the factors that determine the feasibility of their implementation in small and medium-sized enterprises (SMEs). The multifaceted impact of these factors and the indicators reflecting them varies significantly, which makes it necessary to rank them appropriately. This study uses a pairwise comparison of the indicators’ importance based on their transitivity to determine such ranking. This new methodology is superior to existing approaches in that it allows the researcher to determine the importance rankings of a virtually unlimited number of indicators without compromising the adequacy of the assessment. It also significantly reduces the scope of expert judgement. The methodology also accounts for interactions between indicators, further improving the relevance of the rankings. Finally, the proposed methodology is versatile as it can be applied to rank the importance of any type of indicators reflecting the state of socio-economic systems.

Many years of experience show that in most cases the company’s development strategies, which are an essential condition for their competitiveness, have been unsuccessful. The main reason for this situation is that the possibilities of implementing such strategies have not been properly assessed. This is a difficult task, as it includes both internal and external situations of the company, which are reflected by many indicators. The company’s ability to properly evaluate them can be described by the concept of its strategic potential. By its very nature, it is a complex phenomenon, so multi-criteria methods are suitable for quantitative assessment of its condition. Today, they are all based on linear estimation, i.e. the importance of the indicator does not depend on its value. In real life, everything happens non-linearly, so it is appropriate to integrate such an approach into the assessment of the state of socioeconomic phenomena or processes. In this case, the transformed value of the indicator follows a double exponential distribution, which is one of the three limiting cases of the distribution of statistics, corresponding to the curve. The assessment of the strategic potential of four construction companies was carried out based on the known four linear multi-criteria evaluation methods, both those that do not require and those that do require knowledge of the weights of the indicators ‒ sum of places, geometric mean, SAW, TOPSIS and PROETHEE. Non-linear evaluation was carried out in a geometric-analytical way. The obtained results, expressed in ratings, are the same when evaluated by PROMETHEE II and the geometric-analytical method and are equally different compared to the SAW and TOPSIS methods. On the other hand, compared to the PROMETHEE II method, the geometric-analytical method is much simpler and therefore easily automated. In addition, it can be applied in cases where some of the indicator values are negative.

Nuclear energy is a crucial component in decarbonizing the economy, serving as a low carbon dioxide (CO₂) emission energy source. In this context, nuclear safety remains a critical issue, especially in relation to the operation of nuclear power plants. The article presents a new fractal-cluster method for controlling the tightness of the fuel element shell, which is important for nuclear safety. The method provides high-quality control of defects on the inner and outer surfaces of the shell without removing elements from the fuel assembly and from the nuclear reactor. This makes it possible to promptly detect damage that could threaten the containment integrity and safety of nuclear power plants. Studies have shown that the fractal dimension varies from 2.1 to 2.5, which indicates significant surface heterogeneity caused by mechanical damage or corrosion processes. The proposed method allows not only to detect defects but also to assess their impact on the integrity of the vessel. Real-time data integration with leakage control systems ensures timely threat responses. This approach enhances fuel shell leakage control, contributing to improved nuclear safety and reduced operational risks.