The focus of developing economies in recent years has shifted from the growth of per capita national income to sustainability. A sustainable economy is now possible with a sustainable environment. Therefore, energy utilization rates of countries have become a very important issue with the sustainability framework. In this study, the energy consumption convergence of the 10 European countries with the highest per capita energy consumption is analyzed using the data set covering the period between 1965 and 2022. Conventional unit root tests and many recent unit root tests with extended residuals are used to detect energy convergence. Conventional unit root tests did not provide sufficient evidence for convergence, and the classical unit root tests were found to be very weak in rejecting the null hypothesis. When the residuals obtained from the classical unit root tests are analyzed, it is found that the residuals of many countries are not normally distributed. Therefore, RALS-type unit root tests with extended residuals were applied to the data sets. Although no convergence was detected for any country because of classical unit root tests, it was found that some European countries showed convergence behavior in the results of RALS-type unit root tests. Since the data set covers 58 years, it is known that many economic fluctuations have been experienced in this long period. For this reason, the most recent RALS type unit root test, which considers structural breaks, was applied to the data sets. No convergence was detected in the classical unit root tests, and because of the unbroken RALS-type unit root tests, it was found that three countries converged to the group average. As a result of the latest RALS-type unit root test with breaks found that 7 out of 10 selected European countries (Germany, Belgium, France, Netherlands, Italy, Romania, and Greece) converged to the group average. [ABSTRACT FROM AUTHOR]