Real-time optimization by indirect NMPC methods

Nonlinear Model Predictive Control (NMPC) applied to wind turbines is a flexible instrument for simultaneously optimizing total energy output and controlling turbines fatigue. NMPC requires the numerical solution of a sequence of optimal control problems on a finite prediction horizon. Commonly these optimal control problems are solved by direct methods: They need only a reduced amount of mathematical analysis, but the numerical effort increases very rapidly with increasing problem size. In real time computations, this limits system complexity, accuracy and horizon length. To overcome these problems, efficient indirect methods are employed in a new algorithm. In addition to the equations of motion, a set of differential equations for the adjoint variables and algebraic expressions for the controls are derived from the so-called Hamiltonian. The number of degrees of freedom of the optimization problem is small compared to direct methods, because controls (and states) have not to be discretized to form a finite-dimensional nonlinear optimization problem. Additional conditions for optimality allow the exclusion of suboptimal solutions. Constraints can be fulfilled on the whole prediction horizon, not just at the discretized time points. A main challenge for indirect methods in NMPC is the automatic detection of the time-varying solution structure, i.e. of all points within the respective finite time horizon at which constraints become active or inactive. A new algorithm has been developed for automatic structure analysis. As a benchmark a standard model for the design of wind turbine controllers is used; the chosen horizon length is 15 s and the length of the control interval 120 ms. Energy output is increased by almost four percent compared to an excellent reference controller.