Analytical Formulation and Equilibrium Structure of a 26-State Nonlinear Dynamical System for DFIG

Abstract

Abstract

We formulate and analyze a 26-dimensional nonlinear dynamical system governing a doubly-fed induction generator (DFIG) wind energy conversion system coupled to an infinite bus through a dynamic transmission line. Seven interacting subsystems—aerodynamics, a two-mass drivetrain, a fourth-order machine, rotor- and grid-side converter controllers, a phase-locked loop, and a pitch regulator—are assembled into a single vector field x˙=f(x,u) on R26, derived in dimensionless coordinates. Strict positivity of the determinant Δ=LsLr−Lm2=σLsLr for every physically admissible machine renders the flux–current map invertible, so the right-hand side is well defined; the nodal Kirchhoff constraint forms a semi-explicit differential-algebraic relation that we eliminate to obtain an explicit ordinary differential equation. The central contribution is a constructive scheme for the equilibria: the 26 stationarity conditions f(x★,u)=0 are solved by an iterative voltage-matching procedure converging to a residual below 10−11 per unit—essentially machine precision—which removes the spurious start-up transients common in reported simulations. Analytically chosen feedback gains induce a hierarchy of well-separated time scales, placing the closed loop in the multiple-time-scale class; the separation is made quantitative through explicit small parameters εi formed from the ratios of subsystem time constants. Numerical integration of a GE 3.6 MW configuration confirms the construction: under stationary forcing, the rotor speed stays within 1.32×10−5 pu of the equilibrium, and under a large-amplitude wind program (11→14→9 m/s) spanning the full operating envelope, it is regulated to within 0.065%, while the DC-link voltage deviation remains below 2.4×10−5 pu and the power balance closes with residual below 10−3 pu, the ≈2% mechanical–electrical gap being the modeled losses. Linearization about the computed equilibrium yields a Jacobian whose spectrum lies entirely in the open left half-plane, establishing local asymptotic stability and exposing the individual electromagnetic, torsional, and control modes. The model furnishes a rigorously initialized, analytically transparent basis for linearization, spectral stability analysis, and bifurcation study. Its practical value is that a consistent equilibrium and a certified spectrum remove the start-up transients and undocumented tuning that otherwise let initialization artifacts masquerade as genuine dynamics, so that the model can serve as a trustworthy building block for weak-grid and wind-farm stability studies.