[Paper Review] Integrated System Models for Networks with Generators & Inverters

D. Venkatramanan, Manish K. Singh+3 more|arXiv (Cornell University)|Mar 15, 2022

Hermeneutics and Narrative Identity6 citations

TL;DR

This paper presents a unified, circuit-theoretic modeling framework for power systems integrating synchronous generators and both grid-following (GFL) and grid-forming (GFM) inverters across multiple timescales and reference frames. By leveraging dependent voltage-source models and consistent signal representations, it derives differential-algebraic-equation (DAE) and power-flow models from first principles, clarifying the distinction between physics-based dynamics and modeling approximations.

ABSTRACT

Synchronous generators and inverter-based resources are complex systems with dynamics that cut across multiple intertwined physical domains and control loops. Modeling individual generators and inverters is, in itself, a very involved activity and has attracted dedicated attention from power engineers and control theorists over the years. Control and stability challenges associated with increasing penetration of grid-following inverters have generated tremendous interest in grid-forming inverter technology. The envisioned coexistence of inverter technologies alongside rotating machines call for modeling frameworks that can accurately describe networked dynamics of interconnected generators and inverters across timescales. We put forth a comprehensive integrated system model for such a setting by: i) adopting a combination of circuit- and system-theoretic constructs, ii) unifying representations of three-phase signals across reference-frame transformations and phasor types, and iii) leveraging domain-level knowledge, engineering insights, and reasonable approximations. A running theme through our effort is to offer a clear distinction between physics-based models and the task of modeling. Among several insights spanning the spectrum from analytical to practical, we highlight how differential-algebraic-equation models and algebraic power-flow phasor models fall out of the detailed originating electromagnetic transient models.

Motivation & Objective

• Address the lack of a unified modeling framework for heterogeneous power systems with synchronous generators and both GFL and GFM inverters.
• Resolve ambiguities in representing three-phase signals, voltages, frequencies, and phase angles across different reference frames (abc, dq, phasor) and timescales.
• Clarify the distinction between physics-based electromagnetic and electromechanical dynamics and modeling approximations in control and simulation.
• Provide a rigorous foundation for power-flow formulations that explicitly include frequency dynamics and droop control effects in steady-state models.
• Enable consistent simulation, control design, and stability analysis for inverter-dominant grids by unifying representations across resources and timescales.

Proposed method

• Formulate generators and inverters using dependent voltage-source models with network-facing electrical quantities (voltage magnitude, phase angle, frequency).
• Apply reference-frame transformations (abc → dq → phasor) with minimal assumptions, preserving signal clarity and consistency across domains.
• Derive differential-algebraic-equation (DAE) models from electromagnetic transient dynamics using first-principles electromechanical and control equations.
• Construct steady-state power-flow models by projecting DAE models onto sinusoidal steady-state phasor domain, incorporating droop and VSC control laws.
• Use frequency-dependent admittance matrices (G(ωss) + jB(ωss)) to model network dynamics in the steady-state regime.
• Integrate resource-specific models (SG, GFL, GFM) into a unified system by defining consistent power injection equations with frequency and voltage dependencies.

Experimental results

Research questions

• RQ1 Are GFL inverters current sources and GFM inverters voltage sources in precise circuit-theoretic terms, and what are the implications for system modeling?
• RQ2 What is the exact mathematical and physical relationship between dynamic models in the abc frame, dq frame, and sinusoidal steady-state phasor domain?
• RQ3 Can steady-state power-flow models for SGs, GFL IBRs, and GFM IBRs be rigorously derived from dynamic models without ad hoc assumptions?
• RQ4 How do the differences between electrical-radian synchronous frequency and actual network operating frequency manifest in models across timescales?
• RQ5 What simplifications of true-to-form dynamic models yield DAE and power-flow models, and what assumptions underlie these approximations?

Key findings

• The steady-state frequency ωss is explicitly solved as a system variable, with ωss = ωs + (ΣP⋆ₙ) / (Σ(1/Mₙᴾ) + Σ(1/Rₙ𝒹)) under lossless network assumptions.
• The power-flow formulation is well-posed with 4N + 1 unknowns (P, Q, |E|, δ, ωss) and 4N equations from network and resource models, with a reference bus angle δr = 0 ensuring uniqueness.
• GFL and GFM inverters are shown to behave as PQ and PV buses respectively in steady-state, but only when droop terms are included; otherwise, they reduce to idealized PQ/PV models.
• The DAE model emerges from dynamic electromagnetic and control equations by retaining time derivatives of voltage and frequency, while power-flow models result from steady-state projection of these equations.
• The voltage-behind-reactance model for steady-state operation is rigorously derived from the voltage-behind-RL model under dynamic conditions, validating common power-flow assumptions.
• The paper establishes that the slack bus and angle reference are not arbitrary but emerge from the system's dynamic structure and frequency dynamics, resolving long-standing ambiguities in classical power-flow formulations.