[Paper Review] Almost global asymptotic stability of a grid-connected synchronous generator

Vivek Natarajan, George H. Weiss|arXiv (Cornell University)|Oct 16, 2016

Microgrid Control and Optimization4 citations

TL;DR

This paper establishes sufficient conditions for almost global asymptotic stability (aGAS) of a grid-connected synchronous generator with constant field current and frequency droop control, using a fourth-order model derived via Park transformation. The key result is that under these conditions, all initial states converge to a stable periodic trajectory—matching grid frequency—enabling robust synchronverter design with guaranteed stability.

ABSTRACT

We study the global asymptotic behavior of a grid-connected constant field current synchronous generator (SG). The grid is regarded as an "infinite bus", i.e. a three-phase AC voltage source. The generator does not include any controller other than the frequency droop loop. This means that the mechanical torque applied to this generator is an affine function of its angular velocity. The negative slope of this function is the frequency droop constant. We derive sufficient conditions on the SG parameters under which there exist exactly two periodic state trajectories for the SG, one stable and another unstable, and for almost all initial states, the state trajectory of the SG converges to the stable periodic trajectory (all the angles are measured modulo $2\\pi$). Along both periodic state trajectories, the angular velocity of the SG is equal to the grid frequency. Our sufficient conditions are easy to check computationally. An important tool in our analysis is an integro-differential equation called the {\\em exact swing equation}, which resembles a forced pendulum equation and is equivalent to our fourth order model of the grid-connected SG. Apart from our objective of providing an analytical proof for a global asymptotic behavior observed in a classical dynamical system, a key motivation for this work is the development of synchronverters which are inverters that mimic the behavior of SGs. Understanding the global dynamics of SGs can guide the choice of synchronverter parameters and operation. As an application we find a set of stable nominal parameters for a 500 kW synchronverter.

Motivation & Objective

• To analytically prove almost global asymptotic stability (aGAS) of a grid-connected synchronous generator without controller feedback beyond frequency droop.
• To identify computationally checkable sufficient conditions on generator parameters ensuring one stable and one unstable periodic trajectory.
• To provide a foundation for designing stable synchronverters by modeling them after the dynamics of real synchronous generators.
• To avoid common simplifications like static stator flux approximations, ensuring the analysis reflects the true nonlinear dynamics.
• To demonstrate that increasing effective inductance and resistance via virtual inductors can stabilize otherwise unstable parameter sets.

Proposed method

• Derives a fourth-order nonlinear time-invariant model using Park transformation to transform the generator dynamics into a coordinate system where periodic trajectories become equilibrium points.
• Introduces the exact swing equation, an integro-differential equation resembling a forced pendulum, equivalent to the full fourth-order model.
• Applies dynamical systems theory to analyze stability, focusing on hyperbolic equilibrium points and their basins of attraction.
• Uses a modified inverter structure with a virtual inductor (nLs, nRs) to increase effective stator inductance and resistance by factor n, enabling stability under otherwise unstable parameters.
• Employs numerical validation and plots of key functions (e.g., 𝒩, ω_max^d, ω_min^d) to verify satisfaction of sufficient stability conditions in Theorem 6.3.
• Validates results via numerical simulations showing convergence to stable periodic orbits under the derived conditions.

Experimental results

Research questions

• RQ1 Under what conditions on the generator parameters does the grid-connected synchronous generator exhibit almost global asymptotic stability?
• RQ2 How does the inclusion of a virtual inductor (via n-fold scaling of inductance and resistance) affect the stability of the system?
• RQ3 Can the exact swing equation model capture the true nonlinear dynamics without approximating stator flux dynamics?
• RQ4 What happens to the system's stability when resistance is slightly increased beyond a critical threshold, and why does this break the aGAS property?
• RQ5 Why do standard reduced-order models fail to predict the correct global stability behavior compared to the full model?

Key findings

• For the nominal 500 kW synchronverter with n=30, the modified parameters (Ls=825.06 mH, Rs=32.4 Ω, mi_f=51.67 V·s) satisfy the sufficient conditions in Theorem 6.3, ensuring almost global asymptotic stability.
• The function 𝒩 plotted in Figure 7 confirms the existence of a unique stable equilibrium in the transformed system, validating the stability conditions.
• The plots of ω_max^d, ω_min^d, and 2ω_min^d in Figure 8 show that the conditions in Theorem 6.3 are satisfied, confirming aGAS for the system with virtual inductor.
• Even with n=1 (no virtual inductor), numerical simulations suggest aGAS may still hold, but this cannot be proven analytically due to sensitivity to small parameter changes.
• A small increase in Rs (to 1% of V_rms) causes the system to lose aGAS, resulting in multiple unstable equilibria and periodic solutions, indicating a delicate stability boundary.
• When D_p is reduced to 15 N·m/(rad/sec), the system exhibits two sequences of unstable equilibria and periodic solutions, demonstrating complex dynamics not seen in standard forced pendulum systems.