## 1 Introduction  
The operation of transformers is crucial in the context of power system management. Unplanned long-term shutdown of units is a significant expense for transmission network operators. For this reason, the demand for ensuring low failure frequency of transformers is natural. The failure rate is also a subject of regular worldwide research [1]. The operational risk of the transformer is closely related to its technical condition. For its assessment, several methods can be used, which have been explained in a detailed study [2] and in the American Transformer Diagnostics Guide [3]. These techniques can be performed as periodic maintenance or advanced diagnostic procedures or can be implemented in the form of online monitoring systems. Most common techniques, such as oil diagnostics, insulation resistance tests, turns ratio measurements, and inspection of accessories, are performed relatively often and in the state of normal trouble-free operation, with the general technical condition of the transformer being determined on their basis. In the case of transformers with developing defects or aged units, some advanced methods are used to assess their technical condition, such as frequency domain spectroscopy (FDS) and frequency response analysis (FRA) measurements, advanced bushings and on-load tap changer (OLTC) diagnostics, as well as measurement of partial discharges. The use of diagnostic procedures is mostly limited to framework operation instructions, which are specific for every country or power system operator. One of these is the American Transformer Maintenance Instruction [4], which indicates time intervals between inspections of various transformer parts and its external equipment. Due to organizational and financial factors, however, standard assessment of the technical condition is based on periodic tests that are easy to perform. In this case, insulation oil testing is of particular importance and is useful in transformer fault diagnosis [5]. The use of advanced methods usually occurs only in a pre-emergency situation or when making decisions regarding the scope of repair works, which is also connected with predicting units' further operation. The use of online monitoring of various operation parameters has created a new quality in the current assessment of transformers' technical conditions. However, in line with exploitation experiences, it is recommended that the data obtained from monitoring are generalized to the form of one indicator, usually referred to as the health index. This allows one to synthetically present the technical condition of the transformer in the form of a single numerical indicator, which is more understandable at the level of company management. It also allows a comparison between many transformers with different parameters. The health index (HI) can be determined based on both a large and relatively small number of specific parameters. This allows for the creation of indicators of varying complexity. At the same time, expert knowledge is required which takes into account such elements as the construction and type of the transformer, principles of operation and service, and the involvement of advanced diagnostic technologies. Health index algorithms take different forms, including the expert systems based on Canadian [6] and Italian [7] experiences, intelligent algorithms using fuzzy logic [8], artificial neural networks [9], and systems based on the mathematical Markov model [10]. In this paper, the authors propose a transformer health index based on the results of advanced oil diagnostics. It is a common and most frequently performed transformer monitoring procedure under the conditions of the Polish power system. The non-complex acquisition of oil samples and the relatively low cost of their physicochemical properties analysis, as well as the increasing popularity of oil online monitoring systems, creates the conditions for ongoing assessment of technical conditions based on the value of the health index.