## 1 Introduction  
The growing energy demand and environmental problems have led to the integration of renewable energy technologies to the power grid [1]. These technologies are integrated into the power system via inverters, requiring the HVDC system for long-distance transmission [2; 3]. Their development over the past few decades, and nonlinear loads such as inverters, arc furnaces, personal computers, AC and DC drives, etc. have brought about a significant increase in harmonics [4; 5; 6]. Nonlinear loads are notorious for generating harmonics and require transformers to connect them safely to the network. Transformers are usually designed under the assumptions of sinusoidal conditions [7; 8]. It is known [7] that the harmonic content in the load current increases losses in the transformer and eventually the temperature, resulting in its premature damage. Bushings are perhaps the weakest parts of transformers, electrically and physically, and their failure is one of the major causes of transformer malfunctions [9]. The contribution of bushings and tank walls to transformer failures are about 20% and 21%, respectively, which is a significant percentage among all the damages caused by other parts of the transformer [10]. Therefore, with the increasing trend of nonlinear loads in the distribution network, the transformer is even more prone to fail. Consequently, to reduce power losses and heating in bushing regions, low permeability materials [11], and magnetic shunts [12] have been proposed in the literature [7]. At the industry design stage, it is very important to compute transformers stray losses caused by the alternating electromagnetic field (EMF). This way, the derivation of new formulas is desirable and is sought to improve calculation and estimation methods. Several studies have been carried out recently with these objectives in mind [13; 14; 15; 16; 17; 18; 19; 20]. Thus, analytical approaches have proven to be useful in calculating electromagnetic losses [13]. Generally speaking, there are two analytical methods to estimate eddy current (EC) losses in transformer tanks: (i) Application of [PERSON]'s theorem [15] and (ii) direct calculation of the EMF by solving [PERSON]'s equations [13; 20]. In the first method (see [PERSON]'s analytical method [15]), [PERSON]'s theorem is used to estimate the power dissipation. However, the power dissipation formula contains semi-empirical parameters selected by nonrigorous methods. The precision of the second method is limited only by the difficulty of reproducing real geometries. In [19], the axial component of the electric field (EF) in the tank wall was ignored while solving [PERSON]'s equations, which negatively impacted the accuracy of the obtained results [20]. In [20], analytical formulas for calculating EMF and EC losses in transformer tank walls were obtained by solving [PERSON]'s equations in the case of the axial geometry of the tank wall. The results demonstrated a good accuracy compared with the previous methods. However, a load current with only the fundamental frequency was considered. Moreover, the obtained formulae involve Bessel functions, which might not be easily implemented in an industrial setting. None of the previously published works [13; 14; 15; 16; 17; 18; 19; 20] consider harmonics in the load current while deriving analytical models for computing losses in transformer tank walls. Due to the increasing applications of nonlinear loads, the computation of stray losses in transformer tanks requires the consideration of the harmonic content in the load current, with proper computation of the EMF. Therefore, it is of great importance to study the influence of the load current spectrum on the stray losses and heating of transformers. In our work, we propose simplified analytical formulas to calculate the EMF and EC losses in the presence of harmonics, which are derived for the bushing region of transformer tanks. The obtained formulas enable the analysis of the contribution of each harmonic to the power losses.