This paper presents an analytical solution for the Maximum-Torque-per-Ampere (MTPA) operation of synchronous machines (SM) with anisotropy and magnetic cross-coupling for the application in wind turbine systems and airborne wind energy systems. For a given reference torque, the analytical MTPA solution provides the optimal stator current references which produce the desired torque while minimizing the stator copper losses. From an implementation point of view, the proposed analytical method is

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... ppealing in terms of its fast online computation (compared to classical numerical methods) and its efficiency enhancement of the electrical drive system. The efficiency of the analytical MTPA operation, with and without consideration of cross-coupling, is compared to the conventional method with zero direct current. Nomenclature N, R, C are the sets of natural, real, and complex numbers. x ∈ R or x ∈ C is a real or complex scalar. x ∈ R n (bold) is a real valued vector with n ∈ N. X ∈ R n×m (capital bold) is a real valued matrix with n ∈ N rows and m ∈ N columns. x k s = (x d s , x q s ) ∈ R n×m is a stator space vector expressed in the synchronously rotating k-coordinate system (with orthogonal axes known as the direct (d) and quadrature (q) axes) and may represent voltage, current or flux linkage.