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Synchronization in Electric Power Networks with Inherent Heterogeneity Up to 100% Inverter-Based Renewable Generation

Amirhossein Sajadi, Rick Wallace Kenyon, Bri-Mathias Hodge

arXiv:2104.04066v2eess.SYcs.SY

TL;DR

The paper studies synchronization in heterogeneous power networks as renewable generation grows. It derives conditions and analyzes generator parameters, finding damping critical and reporting stable synchronization feasibility in studied networks with up to 100% renewable generation.

  • Problem

    The paper addresses synchronization conditions for heterogeneous networks as conventional generators are replaced by renewable technologies, where prior closed-form work assumed homogeneous coupling damping factors.

    Source: §I · p. 3, §III · p. 11

  • Method

    The authors model generator frequency response with second-order dynamics, derive stability conditions for heterogeneous coupling factors, and conduct parametric and numerical analyses.

    Source: §I · p. 3, B. Dynamic Model of Power Network · pp. 5, 7

  • Results

    100% of studied cases with a stable power-flow equilibrium were dynamically stable; GFM damping was identified as a primary synchronization-enhancement mechanism under low-inertia conditions.

    Source: E. Application in Modern Power Networks · p. 10, §III · p. 11

  • Takeaways & Limitations

    The results provide a scientific basis for studying stability, dynamics, control, and operation of networks with 100% renewable generation.

    Source: §III · p. 11

    Networks with 100% renewable generation may achieve small-signal stable synchronization yet violate industrial control and operation standards.

    Source: §III · p. 11

Abstract

from arXiv · show

The synchronized operation of power generators is the foundation of electric power network stability and a key to the prevention of undesired power outages and blackouts. Here, we derive the conditions that guarantee synchronization in power networks with inherent generator heterogeneity when subjected to small perturbations, and perform a parametric sensitivity analysis to understand synchronization with varied types of generators. As inverter-based resources, which are the primary interfacing technology for many renewable sources of energy, have supplanted synchronous generators in ever growing numbers, the center of attention on associated integration challenges have resided primarily on the role of declining system inertia. Our results instead highlight the critical role of generator damping in achieving a stable state of synchronization. Additionally, we report the feasibility of operating interconnected electric grids with up to 100% power contribution from inverter-based renewable generation technologies. Our study has important implications as it sets the basis for the development of advanced control architectures and grid optimization methods that ensure synchronization and further pave the path towards the decarbonization of the electric power sector.

I. Introduction

The paper addresses synchronization in power networks with increasingly heterogeneous generators and develops conditions and analyses for stable operation, including networks with up to 100% inverter-based generation.

I. Introduction

  • The study targets decarbonizing networks as renewable deployment grows and power systems transition toward up to 100% renewable-based generation.The passage reports that solar and wind accounted for more than 72% of new global electricity generation capacity in 2020.Source: §I · pp. 1–2
  • Prior synchronization analyses include oscillator approximations and closed-form studies, but the cited closed-form work assumes homogeneous coupling damping factors.The authors argue that this assumption falls short of representing intrinsic heterogeneity in present and potential future networks.Source: §I · pp. 2–3
  • Heterogeneous coupling factors are used to derive necessary and sufficient synchronization conditions: at least one generator must have a frequency-forming relationship, and all must have positive damping.The authors contrast this with prior work assuming homogeneous coupling factors.Source: §I · p. 3
  • The paper uses second-order system dynamics to analyze generator heterogeneity and reports that damping is critical to synchronization, while reduced inertia offers benefits.It also reports feasibility up to 100% inverter-based generation, provided sufficient damping is available as fast-responding headroom reserve.Source: §I · p. 3

II. Results

The section defines synchronized operation and uses a mechanical analogy to motivate how inertia, damping, and heterogeneity shape synchronization after small disturbances.

A. Synchronization in Electric Power Networks

  • Synchronization means all generators’ electric-angle evolution rates match; at steady state, the network has a common frequency.The section links synchronized operation to stable power flow and notes historical frequency-synchronization failures that interrupted delivery.Source: A. Synchronization in Electric Power… · p. 4
  • The mechanical network’s spring force corresponds to transmission-line power-flow changes, and synchronization corresponds to damping transients and regaining a steady distance with identical speeds.The analogy motivates applying mechanical-network synchronization intuition to electric networks.Source: A. Synchronization in Electric Power… · pp. 4–5
  • Smaller inertial mass and greater damping yield more robust natural synchronization in the mechanical analogy by reducing induced kinetic energy and dissipating it faster.The induced kinetic energy is proportional to inertial mass, while dissipation capability is proportional to damping.Source: A. Synchronization in Electric Power… · p. 5
  • Homogeneous inertial and damping coefficients limit the range of observed oscillatory modes relative to heterogeneous coefficients, potentially distorting network-dynamics analyses.The section argues that homogeneity is rarely valid for real-world large-scale networks.Source: A. Synchronization in Electric Power… · p. 5

B. Dynamic Model of Power Network

The model represents generator frequency response with second-order dynamics, reduces the network to dynamic elements, and linearizes it to assess synchronization after small perturbations.

B. Dynamic Model of Power Network

  • Synchronous generators and grid-forming inverters use second-order frequency-response models, while grid-following inverters follow measured grid frequency and are eliminated as algebraic elements in Kron reduction.The model’s inertia and damping coefficients represent effective contributions to frequency deviation and transient-oscillation dissipation; grid-forming inverter parameters can be adjusted adaptively.Source: B. Dynamic Model of Power Network · pp. 5–6
  • The network is formed from line admittances and power-flow equilibrium, then Kron-reduced to retain dynamic nodes while eliminating algebraic elements.Loads are treated as constant over the cycles-to-seconds timescale considered.Source: B. Dynamic Model of Power Network · p. 7
  • For small perturbations, the model is linearized and transformed to relative angles against a reference generator, with network coupling, damping, and inertia represented in the resulting dynamics.The transformed state uses n − 1 relative electric angles and absolute electric speeds.Source: B. Dynamic Model of Power Network · p. 7

C. Conditions of Stability

The section defines stability through the characteristic matrix’s eigenvalues and gives the conditions required for stable synchronization.

C. Conditions of Stability

  • Stable synchronization requires at least one non-GFL generator and positive damping coefficients for every generator.The stated condition is necessary and sufficient.Source: C. Conditions of Stability · p. 8
  • The equilibrium is stable if and only if every root of the characteristic polynomial has a non-positive real part.The roots are the eigenvalues of the characteristic matrix.Source: C. Conditions of Stability · p. 8
  • For a network with n generators, the characteristic-matrix condition yields 2n − 1 eigenvalues whose eigenvectors are linearly independent; stability holds iff all roots of p(λ) have non-positive real parts.The characteristic matrix is defined for the network described in (4).Source: C. Conditions of Stability · p. 8

D. Parametric Analysis of Synchronization

The analysis separates generator internal modes from a real network coupling mode and examines how damping and inertia shape synchronization. It finds damping more significant than inertia for stability, with the 3-generator benchmark corroborating the parameter relationships.

D. Parametric Analysis of Synchronization

  • In the homogeneous special case, two internal modes per generator are complex conjugates; increased damping and reduced inertia move stable-network eigenvalues left.The real parts of the internal modes move left with increased damping, while reduced inertia shifts the eigenvalues further left.Source: D. Parametric Analysis of… · pp. 8–9
  • With heterogeneous damping, an additional real coupling mode depends on generator damping and inverse inertia, alongside each generator’s complex-conjugate internal-mode pair.The coupling mode distinguishes this formalism from prior work assuming homogeneous damping.Source: D. Parametric Analysis of… · pp. 8–9
  • Natural oscillation frequency is independent of damping and varies inversely with the square root of inertia, while damping ratio rises with damping and falls with the square root of inertia.Higher damping and lower inertia correspond to less deviant oscillations, shorter settling time, and a reduced transient envelope.Source: D. Parametric Analysis of… · p. 9
  • The 3-generator benchmark corroborates the parameter relationships, and adjusting damping and inertia can enhance network stability.The benchmark results are presented in Figs. 3 and 4 and Table I.Source: D. Parametric Analysis of… · p. 9
  • Generator damping is more significant than inertia for stable synchronization, and reduced inertia alone need not worsen small-perturbation performance.Multi-loop droop grid-forming inverters that provide damping independently of inertia are identified as promising for grid stability.Source: D. Parametric Analysis of… · p. 10

E. Application in Modern Power Networks

Across six network benchmarks, stable power-flow equilibria were dynamically stable in every tested case. The results support small-signal feasibility at up to 100% renewable generation, while showing damping’s importance and retaining a non-GFL generator requirement.

E. Application in Modern Power Networks

  • 100% of cases with a stable power-flow equilibrium were dynamically stable across the six studied networks and tested inertia and damping ranges.The simulations used 1,000 random parameter and network-condition cases per benchmark.Source: E. Application in Modern Power Networks · p. 10
  • Stable synchronized renewable operation requires a non-GFL generator to remain interconnected, a condition GFM inverters or hydro plants can satisfy.The discussion attributes GFM capability to fast digital controllers that can provide desired inertia and damping.Source: E. Application in Modern Power Networks · p. 10
  • Up to 100% renewable-based generation is feasible from a small-signal stability perspective, but this does not establish compliance with practical operating standards.A stable case is defined as bounded response; practical standards may introduce nonlinear elements such as under-frequency load shedding and frequency ride-through.Source: E. Application in Modern Power Networks · pp. 10–11
  • Across all six grids, low-inertia, high-damping conditions had lower hertz-sec scores, while high-inertia, low-damping conditions had higher scores.Hertz-sec variation tracked damping more closely than inertia.Source: E. Application in Modern Power Networks · p. 11

III. Discussion

The discussion argues that damping, particularly adjustable GFM-inverter damping, is central to synchronization in emerging power grids. It frames the findings as a basis for control development while noting operational and timescale boundaries.

III. Discussion

  • Adjusting GFM-inverter damping is identified as the primary mechanism for enhancing synchronization, particularly under low-inertia operating conditions.The paper contrasts this emphasis with contemporary research dialogue focused primarily on mechanical inertia.Source: §III · p. 11
  • Even with stable synchronization under small-signal perturbations, renewable grids may violate industrial requirements for UFLS, ROCOF relays, or balancing-authority frequency response.The authors state that coordinated advanced control may manage these response issues using the identified stability-enhancement mechanisms.Source: §III · p. 11
  • Conventional grid controls may be inadequate for 100% renewable grids, while adjustable inverter resources are often not obligated to provide frequency support.The paper cites limited headroom and current market structures as reasons support may not be provided.Source: §III · p. 12
  • The authors propose reconsidering control and automation structures and developing adaptive approaches for grids with high shares of grid-forming inverters.Examples include protection that tracks inertia and real-time adjustment of droop using available headroom reserve.Source: §III · p. 12
  • The work addresses frequency synchronization over cycles-to-seconds timescales, not the broader decarbonization challenges spanning minutes to years or milliseconds to cycles.The discussion identifies resource adequacy and nonlinear transients as separate challenges.Source: §III · p. 12
  • The findings may provide a scientific basis for studying stability, dynamics, control, and operation of bulk networks with 100% renewable generation.The authors also point to grid planning and optimization frameworks that use power-electronic devices’ functionalities and responsiveness.Source: §III · pp. 11, 13

Methods

The methods establish an electrically equivalent network, solve its operating point, and analyze small-perturbation dynamics. They also define a frequency-response metric and identify where results data and code are available.

Methods

  • The network admittance matrix represents topology and links, with off-diagonal entries set by negative branch admittances and diagonal entries summing connected links.The matrix is described as square and symmetric.Source: Network admittance matrix · p. 13
  • Power-flow equations determine generator dispatch and bus voltage magnitudes and angles, providing an initial condition for linearization and dynamic analysis.The study used MATPOWER to obtain numerical power-flow solutions.Source: Network power flow · p. 14
  • Kron reduction retains dynamic boundary nodes and approximates the electrical connections between them after eliminating nodes without dynamic elements.The resulting equivalent network incorporates eliminated nodes and loads through adjusted admittance values between boundary nodes.Source: Kron reduction · p. 14
  • Linearization approximates nonlinear behavior around an equilibrium by retaining first-order Taylor terms, for dynamics under small perturbations.The system is linearized around an operating point with the input set to zero.Source: Model linearization · pp. 14–15
  • The second-order response description uses natural frequency to indicate the system’s natural oscillation mode and damping ratio to describe damping.The paper states these parameters implicitly represent contributing components to inertia and effective damping.Source: Second-order response characteristics · p. 15
  • Hertz-sec quantifies frequency response by integrating the absolute frequency deviation over its transient period.The metric is presented as a proxy for disturbance-induced changes in kinetic energy; f0 and f(t) denote pre- and post-disturbance frequency.Source: Hertz-sec metric · p. 15
  • Data for Figures 3–5 and Table 1, and code for Figures 2–5 and Table 1, are available through the cited GitHub repository.The raw data for Figure 1 are restricted; data for Figure 2 are embedded in its code, and Figure 4 PSCAD models are also available on GitHub.Source: Data Availability · p. 15, Code Availability · p. 15

Author Contributions Statement

A. Sajadi and R. W. Kenyon designed the research, and Sajadi carried out most analytical and numerical analyses. B. M. Hodge supervised the project, and all authors edited the manuscript.

Author Contributions Statement

Funding

The authors report that they had no funding to report.

Funding

  • The authors report no funding.The paper states, “Authors have no funding to report.”Source: Funding · p. 16

Figure Legends/Captions

The captions cover synchronization dynamics from a real network split and mechanical analogies to generator-parameter sensitivity, technology comparisons, and six-network benchmarks. They report that increased damping and reduced inertia improve frequency dynamics, with damping having the stronger impact across the benchmarks.

Table I

  • Table I presents dynamic performance for cases subjected to a small perturbation.The caption names the table's scope but supplies no cell values.Source: Table I · p. 19
  • Natural oscillation frequency is independent of generator damping and varies with the inverse square root of inertia, while natural damping ratio increases with damping and follows a square-root relationship with inertia.The text identifies damping as significant for achieving a stable synchronized frequency response.Source: Figure Legends/Captions · p. 19

Fig. 1

  • After the European network split on January 8, 2021, frequency traces showed two distinct frequencies and power delivery was interrupted for 1 hour and 3 minutes.The traces were recorded by Phasor Measurement Units across the synchronized continental network.Source: Figure 1 · p. 19

Fig. 2

  • In the mechanical analog, lower mass and greater damping yield faster speed settling and quicker kinetic-energy dissipation.The figure varies mass and damping under heterogeneous and homogeneous conditions.Source: Figure 2 · p. 20

Fig. 3

  • Reducing inertia and increasing damping improve the dynamic response, while increasing inertia and reducing damping have the opposite effect.The comparison also reports superior network-dynamics improvement for multi-loop droop GFM over GFL.Source: Figure 3 · p. 21

Fig. 4

  • With 100% multi-loop droop GFM, effective inertia approaches zero and the response is effectively first-order; 100% GFL is infeasible in the modeled network.All-SG, all-GFM-VSM, and two-thirds-GFL plus one-third-SG responses are described as very similar.Source: Figure 4 · p. 22

Fig. 5

  • Across six network benchmarks, higher damping and lower inertia correspond to improved frequency dynamics, with damping having the more pronounced impact.The figure summarizes 1,000 random operating conditions across six networks.Source: Figure 5 · p. 23
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