Passivity-Based Stability in Power Systems

• Passivity-based power system stability analysis is a framework that uses energy storage functions and input–output inequalities to certify the stability of heterogeneous power networks.
• It underpins decentralized, sample-data, and model-free controller designs that maintain both small- and large-signal stability in systems with high inverter and converter penetration.
• The approach integrates mathematical foundations from port-Hamiltonian and Lyapunov theories, enabling scalable, robust stability certificates even in the presence of uncertainties.

Passivity-based power system stability analysis is a framework that leverages the dissipativity properties of components and controllers in electric power systems to guarantee small- and large-signal stability using energy-like storage functions and input–output inequalities. Passivity, in this context, formalizes the requirement that a subsystem (generator, inverter, converter, or network) neither generates net energy nor self-excites when interconnected, allowing scalable and modular stability certificates that are robust to heterogeneity and uncertainty. The approach has deep roots in port-Hamiltonian system theory and Lyapunov functions, and underpins the design of decentralized, sample-data, and even model-free controllers in state-of-the-art power system engineering.

1. Mathematical Foundations of Passivity in Power Systems

Passivity theory in power systems exploits storage functions $V(x)\ge 0$ and inequalities of the form

$\dot V(x)\le u^\top y$

to ensure that, for prescribed input–output pairs $(u,y)$, the subsystem is energy-dissipative. In the context of converters and networks, passivity is frequently characterized using incremental (shifted) storage functions about nonzero equilibria (shifted passivity): $$\dot H(\tilde x) = -\tilde x^\top Q R Q\,\tilde x + \tilde y^\top \tilde u,$$ where $H$ is the quadratic Hamiltonian, $Q$ the system matrix, $R$ the dissipation matrix, and the tilde denotes deviations from equilibrium (Moreschini et al., 26 Aug 2025). For standard port-Hamiltonian realizations of power electronic and electromechanical systems, this structure arises from the underlying physics.

The power system network itself can be cast as a static passive map in nodal coordinates: $$\begin{pmatrix}I_a \ I_b\end{pmatrix} = \begin{pmatrix}G_n & -B_n\ B_n & G_n\end{pmatrix} \begin{pmatrix}V_a\ V_b\end{pmatrix},$$ where $G_n$ and $B_n$ are nodal conductance and susceptance matrices (Spanias et al., 2018). Bus dynamics—generators, inverters, loads—are modeled as general non-linear multivariable dynamical systems, each stipulated to satisfy local input–output passivity with respect to prescribed ports.

Beyond classical passivity, passivity-short (where a device is not strictly passive but exhibits a bounded energy shortage quantified by indices) and output-feedback passivity-indexed conditions are central to modern distributed analyses (Xu et al., 2019, Yang et al., 2019, Yang et al., 2019). Output-feedback passivity with index $\sigma$ ensures robust dissipativity in the presence of non-passive network-induced couplings.

2. Passivity-Preserving Control Design and Discretization

Passivity-based control (PBC), particularly PID-PBC, is constructed by wrapping a passive controller (PID with positive gains) around a passive system output. In power converters, the continuous PID-PBC protocol is: $$\dot{\xi} = \tilde y, \quad u = -K_P\tilde y - K_I\xi - K_D\dot{\tilde y},$$ with Lyapunov certification through composite energy and dissipation storage (Zonetti et al., 2021).

Crucially, naive digitization (e.g., forward Euler) of such controllers destroys discrete-time passivity, even for small sampling steps, due to aliasing of the dissipativity property at the sampling boundary. A passivity-preserving discretization requires using implicit midpoint (symplectic) methods: $$x_{k+1}=x_k + h\,f(z_k) + h\,g(z_k)\,u_k, \quad z_k = \frac{x_k + x_{k+1}}{2},$$ ensuring that discrete incremental storage $\Delta H(x_k)$ satisfies a discrete dissipation inequality

$$\Delta H(x_{k+1}) - \Delta H(x_k) \le h\,\tilde y_k^\top \tilde u_k,$$

thus preserving strong stability claims for the sampled-data closed loop (Moreschini et al., 26 Aug 2025).

Global asymptotic stability is then achieved if controller gains and the system's damping satisfy the continuous-time “damping injection” condition $R + g(x^*)K_Pg(x^*)^\top \succ 0$; the implicit midpoint method avoids imposing arbitrary smallness constraints on the sampling time, modulo algebraic regularity conditions for the implicit update.

3. Distributed and Modular Passivity-Indexed Stability Criteria

Recent advances provide fully distributed, output-feedback passivity-indexed (OFP) stability certifications applicable to large heterogeneous and nonlinear networks (Yang et al., 2019, Yang et al., 2019). Each bus is required to satisfy a local OFP$(\sigma)$ inequality: $$\dot S_i \le w_i(u_i, y_i, \dot y_i) - \sigma_i(y_i - y^*_i)^\top \dot y_i,$$ where $w_i$ is a “differential supply rate” involving the bus's output variables and their derivatives. The network's aggregate susceptibility to non-passiveness is captured by the minimal eigenvalue $\lambda$ of the network’s energy-function Hessian. The key stability condition for all $i$ is: $\sigma_i > -\lambda,$ which decouples the local device passivity requirement from the detailed system structure. Such conditions can be checked and enforced via local measurements and controller tuning, and they enable novel scalable wide-area control design, including modular, data-driven synthesis via passivity-shortage indices and two-level Lyapunov matrix inequalities (Xu et al., 2019).

Heterogeneous device models—synchronous generators, conventional droop inverters, quadratic droop inverters—are all encompassed, with passivity-index constraints mapped to explicit gain requirements on virtual inertia, governor, and excitation loops (Yang et al., 2019, Yang et al., 2019).

4. Nodal Admittance and Frequency-Domain Passivity-Sensitivity Analysis

Stability of interconnected systems of converter-interfaced generation can be diagnosed via the frequency-domain passivity of device and network admittances $Y(j\omega)$: $$\text{Device is passive at } \omega \iff Y(j\omega) + Y(j\omega)^\dagger \succeq 0.$$ Physically this means the system never injects net active power at that port and frequency (Lee et al., 10 Apr 2025).

Passivity-sensitivity methodology determines which device or network parameter modifications most effectively increase the frequency bands of guaranteed passivity (and thus damping). For scalar or $2\times2$ admittance models, the passivity index at frequency $\omega$ is the minimal eigenvalue of the admittance Hermitian part. Sensitivities $\partial S / \partial p_i$ (where $S$ is the passivity index) guide controller retuning: e.g., increasing filter inductance or decreasing PLL gain in weak grid–forming converter cases maximally widens the passive band.

The system-level extension collects all active and passive elements in the full nodal admittance; robustness can be certified by ensuring the minimum eigenvalue of the combined Hermitian admittance is positive over the frequency range of concern (Lee et al., 10 Apr 2025). This methodology circumvents expensive state-space eigenanalysis.

5. Application in Networked and Sampled-Data Power Systems

Passivity-based frameworks apply consistently to DC and AC microgrids, sampled-data converter control, and networks with high inverter/converter penetration. For DC microgrids, storage functions and IF–OFP indices are constructed for each bus, line, and controller stage, and aggregate Lyapunov functions are assembled for global certification (Malan et al., 2023, Anees et al., 2024). The parallel interconnection of passive (or index-specified) devices via the network remains Lyapunov and LaSalle stable if the per-device and per-line indices meet the passivity test.

For sampled-data (discrete-time) PID-PBC, the implicit midpoint discretization recovers discrete-time shifted passivity and carries over continuous-time stability guarantees into the fully digital controller setting, as demonstrated in simulations and experiments with robust voltage regulation under step changes, sampling period variations, and hardware-in-the-loop setup (Moreschini et al., 26 Aug 2025). This addresses a notorious challenge where direct discretization often results in loss of stability due to hidden energy increments.

Model-free, data-driven dissipativity-based control leverages the same operator inequalities—formulated in discrete time and parameterized by neural networks—to directly synthesize controllers enforcing Lyapunov decrease and strict dissipativity in black-box systems. This approach expands the domain of practical applicability beyond analytically tractable models (Wang et al., 25 Oct 2025).

6. Controller Design Trade-Offs and Passivity Index Region

Passivity certification is generally frequency-dependent, leading to trade-offs (“waterbed effect”): raising passivity indices in certain bands can shrink others, especially in the presence of physical limitations, non-minimum phase behavior, or non-passive devices. The Positive Damping (PD) region framework visualizes for SISO and MIMO systems the maximal achievable passivity index and the frequency bands over which passivization is possible via Nyquist or Nichols plots (Peng et al., 15 Jan 2026). The maximum index at a given frequency is determined by the ratio $\varsigma(\omega)=\Re G(j\omega)/|G(j\omega)|^2$, which yields the tightest trade-off for controller design.

Encoding the PD region requirement as an LMI enables direct integration in control design via the generalized KYP lemma, enabling systematic multiband optimization of passivity indices alongside classical performance metrics (Peng et al., 15 Jan 2026).

7. Practical Considerations and Impact

Passivity-based analysis has established itself as the leading method for scalable, modular, and robust stability certification in modern power systems. Its ability to accommodate heterogeneity, permit distributed or sample-data control, and avoid reliance on full-state knowledge or central eigenanalysis makes it uniquely suited to systems with high-dimension, high-inverter or converter penetration, and changing network topology. Recent work extends the theory to include passivity-short devices, admits dissipation inequalities for bus and line models, and uses adaptive or data-driven estimation of passivity indices to provide real-time or near-real-time control retuning (Xu et al., 2019, Wang et al., 25 Oct 2025).

Current trends include the integration of frequency- and time-domain criteria, interface-variable selection to decouple low/high-frequency behavior, and model-free enforcement of dissipativity via machine-learning constructs. The framework is adaptable to AC and DC grids, multi-terminal and meshed architectures, and grid-forming or grid-following control regimes (Moreschini et al., 26 Aug 2025, Malan et al., 2023, Anees et al., 2024). Open challenges remain in quantifying performance–robustness trade-offs, developing verifiable supply rates for broad classes of non-linear and stochastic loads, and further optimizing distributed adaptive passivity-index design.

In summary, passivity-based power system stability analysis achieves global or modular stability guarantees by leveraging the dissipativity properties of physical subsystems and controllers, supports rigorous control design in both continuous-time and sampled-data settings, and provides a critical toolset for scalable, robust, and adaptive stability analysis in the evolving landscape of power networks (Moreschini et al., 26 Aug 2025, Lee et al., 10 Apr 2025, Xu et al., 2019, Malan et al., 2023, Wang et al., 25 Oct 2025, Yang et al., 2019, Peng et al., 15 Jan 2026, Zonetti et al., 2021).