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Voltage source inverter-fed permanent magnet synchronous motors (VSI-PMSMs) are widely used in industrial actuation and mechatronic systems in water pumping stations, as well as in the traction of transportation systems (such as electric vehicles and electric trains or ships with electric propulsion). The dynamic model of VSI-PMSMs is multivariable and exhibits complicated nonlinear dynamics. The inverters’ currents, which are generated through a pulsewidth modulation process, are used to control the stator currents of the PMSM, which in turn control the rotational speed of this electric machine. So far, several nonlinear control schemes for VSI-PMSMs have been developed, having as primary objectives the precise tracking of setpoints by the system’s state variables and robustness to parametric changes or external perturbations. However, little has been done for the solution of the associated nonlinear optimal control problem. The purpose of this study/paper is to provide a novel nonlinear optimal control method for VSI-fed three-phase PMSMs.

The present article proposes a nonlinear optimal control approach for VSI-PMSMs. The nonlinear dynamic model of VSI-PMSMs undergoes approximate linearization around a temporary operating point, which is recomputed at each iteration of the control method. This temporary operating point is defined by the present value of the voltage source inverter-fed PMSM state vector and by the last sampled value of the motor’s control input vector. The linearization relies on Taylor series expansion and the calculation of the system’s Jacobian matrices. For the approximately linearized model of the voltage source inverter-fed PMSM, an H-infinity feedback controller is designed. For the computation of the controller’s feedback gains, an algebraic Riccati equation is iteratively solved at each time-step of the control method. The global asymptotic stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimation-based control for this system, the H-infinity Kalman filter is proposed as a state observer. The proposed control method achieves fast and accurate tracking of the reference setpoints of the VSI-fed PMSM under moderate variations of the control inputs.

The proposed H-infinity controller provides the solution to the optimal control problem for the VSI-PMSM system under model uncertainty and external perturbations. Actually, this controller represents a min–max differential game taking place between the control inputs, which try to minimize a cost function that contains a quadratic term of the state vector’s tracking error, the model uncertainty, and exogenous disturbance terms, which try to maximize this cost function. To select the feedback gains of the stabilizing feedback controller, an algebraic Riccati equation is repetitively solved at each time-step of the control algorithm. To analyze the stability properties of the control scheme, the Lyapunov method is used. It is proven that the VSI-PMSM loop has the H-infinity tracking performance property, which signifies robustness against model uncertainty and disturbances. Moreover, under moderate conditions, the global asymptotic stability properties of this control scheme are proven. The proposed control method achieves fast tracking of reference setpoints by the VSI-PMSM state variables, while keeping also moderate the variations of the control inputs. The latter property indicates that energy consumption by the VSI-PMSM control loop can be minimized.

The proposed nonlinear optimal control method for the VSI-PMSM system exhibits several advantages: Comparing to global linearization-based control methods, such as Lie algebra-based control or differential flatness theory-based control, the nonlinear optimal control scheme avoids complicated state variable transformations (diffeomorphisms). Besides, its control inputs are applied directly to the initial nonlinear model of the VSI-PMSM system, and thus inverse transformations and the related singularity problems are also avoided. Compared with backstepping control, the nonlinear optimal control scheme does not require the state-space description of the controlled system to be found in the triangular (backstepping integral) form. Compared with sliding-mode control, there is no need to define in an often intuitive manner the sliding surfaces of the controlled system. Finally, compared with local model-based control, the article’s nonlinear optimal control method avoids linearization around multiple operating points and does not need the solution of multiple Riccati equations or LMIs. As a result of this, the nonlinear optimal control method requires less computational effort.

Voltage source inverter-fed permanent magnet synchronous motors (VSI-PMSMs) are widely used in industrial actuation and mechatronic systems in water pumping stations, as well as in the traction of transportation systems (such as electric vehicles and electric trains or ships with electric propulsion), The solution of the associated nonlinear control problem enables reliable and precise functioning of VSI-fd PMSMs. This in turn has a positive impact in all related industrial applications and in tasks of electric traction and propulsion where VSI-fed PMSMs are used. It is particularly important for electric transportation systems and for the wide use of electric vehicles as expected by green policies which aim at deploying electromotion and at achieving the Net Zero objective.

Unlike past approaches, in the new nonlinear optimal control method, linearization is performed around a temporary operating point, which is defined by the present value of the system’s state vector and by the last sampled value of the control input vector and not at points that belong to the desirable trajectory (setpoints). Besides, the Riccati equation, which is used for computing the feedback gains of the controller, is new, as is the global stability proof for this control method. Comparing with nonlinear model predictive control, which is a popular approach for treating the optimal control problem in industry, the new nonlinear optimal (H-infinity) control scheme is of proven global stability, and the convergence of its iterative search for the optimum does not depend on initial conditions and trials with multiple sets of controller parameters. It is also noteworthy that the nonlinear optimal control method is applicable to a wider class of dynamical systems than approaches based on the solution of state-dependent Riccati equations (SDRE). The SDRE approaches can be applied only to dynamical systems that can be transformed to the linear parameter varying form. Besides, the nonlinear optimal control method performs better than nonlinear optimal control schemes which use approximation of the solution of the Hamilton–Jacobi–Bellman equation by Galerkin series expansions.

To provide a general framework for the electromagnetic analysis of axial flux motors and generators.

The procedure is based on the solution of Maxwell's equation in a cylindrical frame. All field sources (permanent magnets, windings) are subdivided into filamentary windings. The expansion of the 2D air‐gap magnetic field into a Fourier series is computed at every radius. The contributions of the harmonics are then added to achieve the expressions of the stator and rotor flux densities, back emf and developed torque. Slotting and skewing are taken into account also.

The model can be written in a compact form by introducing a generalisation of the space vectors theory. The analysis is proved to be in accordance both with the finite element analysis and with experimental data.

The model does not take into account eddy‐currents and non‐linearities. It does not take into account also specifically 3D phenomena, as the radial components of the flux densities.

The analysis is of practical interest from the standpoint both of control and of machine design. In this latter occurrence, it represents a valid alternative over computationally heavier 3D finite elements models.

Although the procedure is partly based on previous analyses, it is original in the way it exploits the basic theory in order to introduce skewing, slotting, and finite length of the iron. With respect to other models introduced so far, the present one is more compact in the end, its parameters can be easily computed and their physical meaning is easily understood.

The paper aims to illustrate a numerical technique to calculate fields and inductances of rotating electrical machines.

The technique is based on an integral formulation of the nonlinear magnetostatic model in terms of the unknown magnetization. The solution is obtained by means of a Picard-Banach iteration whose convergence can be theoretically proved.

The proposed method has been used to build a model of a large turbine generator. In particular, the influence of end effects on flux linkages has been computed. It has been demonstrated that the 2D solution underestimates the flux linkages as well as the no load voltage of 2 per cent, while the leakage fluxes are computed by the 2D solution with errors as high as 20 per cent.

The method is advantageous in comparison to standard methods.

The purpose of this paper is to present a numerical approach for the computation of 3D magnetic fields in rotating electrical machines. The technique is suitable for the computation of flux densities and forces in the end windings of large synchronous turbo generators (TG).

The magnetostatic FEM model of the generator end windings is carried out for different displacements of the rotor axis to the stator magnetomotive force (MMF) axis. The method is based on a parallel integral formulation allowing to substantially reduce the computational effort.

The computational model requires only the discretization of magnetic materials and conductors and is fast enough for carrying out 3D analyses on a time scale fast enough for the needs of the designer. As far as the present application is concerned, the analysis of a synchronous generator in the class of 300‐400 MVA has shown that the most stressed elements of the armature conductors are those closer to the stator ends. The study demonstrates that the maximum stress component on the end windings is axial and is achieved when the MMF is aligned to the direct axis.

The present approach combining an efficient integral formulation, the sparsification of the relevant matrices and the parallel implementation of the related algorithms gives rise to an original computational tool that allows a more accurate description of the machine in comparison to other numerical simulations that can be found in the literature.