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Decentralized and Equilibrium-Set-Oriented Stability Analysis and Control for Power Systems
Peng Yang, Liaoyuan Yang, Feng Liu
TL;DR
Fluctuating renewable generation challenges centralized stability analysis tied to a single known equilibrium. The paper develops an IODP-based decentralized framework that certifies equilibrium-set stability through local device conditions and adds passivation control for devices with insufficient IODP. Case studies on a modified IEEE 39-bus system demonstrate stability certification under varying operating points with reduced computational burden.
Problem
Conventional stability methods are centralized and equilibrium-point-oriented, making fixed equilibrium information difficult to use under highly variable renewable operating conditions.
Method
The framework uses input–output differential passivity, local IODP indices and certified regions, plus I/O-transformation-based passivation control for devices with insufficient IODP.
Results
Case studies on a modified IEEE 39-bus system show consistent stability certification across varying operating points while substantially reducing computational burden through decentralized local verification.
Takeaways & Limitations
The framework provides a scalable equilibrium-set-oriented approach for stability certification and control in renewable-rich grids with heterogeneous devices and changing operating conditions.
Abstract
from arXiv · showhide
Conventional power-system stability analysis is largely centralized and centered on a single equilibrium point, which becomes increasingly restrictive in the presence of large-scale fluctuating renewable generation. This paper develops a decentralized framework for stability analysis and control that certifies the asymptotic stability of an equilibrium set rather than that of a given single operating point. To this end, we introduce a new notion termed input--output differential passivity (IODP), which decomposes equilibrium-set stability of the interconnected system into local requirements imposed on individual devices. These requirements are formulated without embedding a particular operating equilibrium into the local conditions; once the certified regions are constructed, stability verification for a given operating scenario reduces to checking whether its equilibrium lies in the certified set. The proposed conditions require each bus to possess a sufficient level of IODP, quantified by an IODP index. To compensate for an IODP shortage, we further develop an I/O-transformation-based passivation controller that reshapes the local input--output behavior of the corresponding device. In this way, all grid-connected components can be made to satisfy the decentralized conditions for system-wide stability. The proposed framework is validated on a modified IEEE 39-bus system. Simulation results demonstrate that it provides a scalable and equilibrium-set-oriented solution for stability certification and control under highly variable operating conditions.
I. INTRODUCTION
Fluctuating renewable generation and heterogeneous inverter-based devices expose limits in centralized, single-equilibrium stability analysis. The paper introduces a decentralized framework that uses local device information to certify and control stability across equilibrium sets.
- Motivation: Renewable-rich grids increase operational complexity while conventional analysis requires a fixed, explicitly known equilibrium point.Centralized methods can also impose computation and communication burdens and raise model-privacy concerns.
- Related work: Existing decentralized methods remain largely equilibrium-point-oriented and often rely on linearization, explicit equilibrium information, or cross-subsystem data.Prior approaches include linear and nonlinear criteria based on positive realness, Nyquist analysis, eigenvalue approximation, passivity, dissipativity, input-to-state stability, and sum-of-squares techniques.
- Proposed framework: IODP formulates passivity through input and output differentials, removing explicit dependence on a particular equilibrium point.This supports local conditions suitable for decentralized, equilibrium-set-oriented analysis.
- Proposed framework: The framework introduces device-level IODP indices and certified regions, reducing scenario certification to checking local equilibrium components against those regions.The required conditions use local device models without centralized computation.
- Control design: An I/O-transformation-based passivation controller reshapes devices with insufficient IODP so they can satisfy conditions required for system-wide stability.The control layer quantifies and regulates each device’s passivity shortage or excess and remains applicable under changing operating equilibria.
- Power-system model: The power system is modeled as a graph-coupled dynamic–static DAE with dynamic and static buses connected through network algebraic constraints.Dynamic buses use input-state-output models, while static buses use input-output relations and share voltage-current port conventions.
B. Problem Statement
The paper targets decentralized, equilibrium-set-oriented stability certification and local passivation control for interconnected power systems. It introduces IODP as a differential, equilibrium-independent bus-level property whose indices quantify passivity margins.
- Objectives: The framework seeks to certify asymptotic stability of an equilibrium set without specifying a particular equilibrium or using more than local bus-level models.It also aims to passivate buses that fail the decentralized conditions so the overall system stabilizes toward the equilibrium set.
- IODP framework: IODP formulates differential passivity at both dynamic and static buses, providing the local properties used in decentralized equilibrium-set-oriented analysis.Dynamic and static definitions are assigned IODP indices (σ_i, ρ_i) over specified domains.
- IODP framework: IODP uses differential variables at both ports and does not anchor its dissipation conditions to a specific equilibrium.This formulation supports stability analysis of equilibrium sets rather than a single operating point.
- IODP indices: The IODP index pair (σ_i, ρ_i) measures differential passivity margins for the input and output channels, although multiple feasible pairs may exist for one subsystem and domain.Positive, zero, and negative values correspond to IODP excess, critical IODP, and shortage, respectively.
- Scope of the definitions: The definitions can hold on a specified domain, distinguishing the proposed formulation from related differentiated passivity concepts that require validity over the full state-input space.The domain-based formulation is contrasted with delta dissipativity and Krasovskii passivity.
B. Krasovskii-type Storage Function for IODP
The paper uses a Krasovskii-type storage function to reformulate IODP as a local matrix inequality and numerically compute certified IODP regions. A synchronous-generator example yields a five-dimensional certified region and an illustrative two-dimensional cross-section.
- Krasovskii-type storage function: The Krasovskii-type storage function reformulates IODP through a matrix-inequality characterization based on local differential information.The storage function is constructed from the system dynamics and a positive-definite matrix.
- IODP condition: Proposition 1 states that a positive-definite matrix and positive scalar satisfying the inequality for every point in D_i certify IODP(σ_i, ρ_i; D_i).The condition is imposed over the local domain rather than at one operating equilibrium.
- Numerical computation: The IODP index can be computed by fixing sampled points in D_i and solving the resulting LMI in P_i, σ_i, and ρ_i with semidefinite programming.Alternatively, fixing the decision variables allows scanning over (x_i, u_i) to identify a certified region D_i.
- Synchronous-generator example: The synchronous-generator example applies Proposition 1 to compute the generator's IODP region under the listed parameters.The example uses a flux-decay third-order synchronous-generator model with an algebraic equation and an optional frequency-integral coefficient K_I.
- Synchronous-generator example: IODP(-8.5098, 0; D_i) is obtained for the synchronous generator, where D_i is a five-dimensional region whose V^-(δ−θ) cross-section is plotted in Fig. 3.The plotted cross-section is a subset of the certified domain.
D. IODP Properties of Static Subsystems
The paper characterizes IODP for representative static subsystems, including the network, constant-impedance loads, and ZIP loads. For ZIP loads, the result holds on a domain where the symmetric Jacobian part is positive semidefinite.
- Static subsystem properties: The power network modeled as a static subsystem is IODP(0, 0; R^m).The paper presents this as a previously established global property of the network subsystem.
- ZIP load model: A ZIP load models voltage-dependent active and reactive power consumption through constant-impedance, constant-current, and constant-power components.It is represented as a static input-output mapping from DQ voltage components to negative DQ current components.
- ZIP load model: The ZIP load satisfies IODP(0, 0; D_i) on a domain D_i defined by the model's Jacobian condition.The proof uses the fact that the symmetric part of the Jacobian is positive semidefinite throughout D_i.
- Static subsystem properties: A constant-impedance load is globally IODP(0, 0; R^2).
IV. IODP-Based Equilibrium-Set Stability Theory
The stability theory converts local device-level IODP conditions into a decentralized certificate for asymptotic stability of the interconnected system's equilibrium set. Verification for a particular operating scenario reduces to checking local equilibrium membership in the certified domains.
- Equilibrium-set stability theory: The theory establishes that device-level IODP properties can guarantee equilibrium-set stability for the interconnected differential-algebraic power-system model.The proof is built from local bus-level requirements and their interconnection structure.
- Decentralized stability conditions: Dynamic buses must satisfy IODP(0, 0; D_i) with a storage function, while static buses must satisfy the same condition on a domain D_i.These are the decentralized bus-level conditions.
- Certified domain construction: The global domain is formed as the Cartesian product of local IODP domains and intersected with the algebraic manifold.The resulting set D_G is used to define the certified operating region for the interconnected model.
- Scenario verification: For a particular operating scenario, certification requires computing its operating equilibrium and verifying that each local equilibrium component lies in its corresponding D_i.Equivalently, the equilibrium must belong to D_G.
B. From Local IODP to Equilibrium-Set Stability
The proof aggregates local IODP storage functions into a system-level function whose decay establishes asymptotic stability of the equilibrium set. Under the stated regularity assumptions, isolated equilibria in that set are also asymptotically stable.
- Lyapunov construction: The sum of local IODP storage functions serves as a W-function candidate for proving equilibrium-set stability.The aggregated function is bounded above and below by class-K functions of the system dynamics.
- Lyapunov construction: The aggregate storage function is constructed from local class-K bounds by minimizing sums over decompositions of the system differential norm.The resulting functions α, β, and γ remain class-K functions.
- Dissipation: Interconnection passivity gives a nonpositive network contribution, yielding storage decay of the form Ṡ ≤ −γ(∥f(x, u)∥).This decay is the key dissipation step in the equilibrium-set proof.
- Assumptions: The proof assumes that the equilibrium set is nonempty and bounded and that the power-system DAE is algebraically nonsingular within the certified domain.Algebraic nonsingularity ensures local solvability, index-one DAE behavior, and locally unique solution trajectories.
- Stability result: Under Assumptions 1–2 and the decentralized conditions, Theorem 5 establishes asymptotic stability of the equilibrium set.If an isolated equilibrium exists within the set, every isolated equilibrium in the set is also asymptotically stable.
- Stability proof: The convergence argument concludes that dist((x(t), u(t)), E) tends to zero, proving asymptotic stability of E.The same reasoning establishes asymptotic stability for an isolated equilibrium contained in E.
V. IODP-Based Passivation Control Design
The passivation controller uses an affine I/O transformation to reshape a device’s differential input–output behavior and achieve prescribed IODP levels. Under an invertibility assumption, the controlled subsystem satisfies the target IODP condition on an induced domain while preserving the desired steady-state relationship.
- Controller objective: IODP-based passivation reshapes a subsystem’s input–output behavior when its original IODP level is insufficient for decentralized stability certification.The controller is not restricted to two-dimensional ports.
- Passivity shaping: Theorem 6 shows that a suitable invertible transformation matrix reshapes IODP(σ_i, ρ_i; D_i) into prescribed IODP(ϛ_i, ϱ_i; eD_i).The controlled system uses transformed input and output variables and satisfies the target condition on the induced domain.
- I/O transformation: The controller applies an invertible affine transformation between original and external port variables, with bias terms selected for the desired steady-state relationship.The external variables are seen by the network, while the original variables remain internal controller signals.
- Well-posedness: Under Assumption 3, the transformation defines the original input uniquely from the state and transformed input, producing well-defined controlled dynamics and an induced IODP domain.For affine output relationships, Assumption 3 reduces to checking matrix invertibility.
- Steady-state preservation: The transformation preserves the desired steady-state operating point by using input and output bias terms to compensate constant offsets.The physical meaning of the external port variables is unchanged, while the closed-loop voltage–current relationship is reshaped.
B. Constructive Solution of the I/O Transformation Matrix
The I/O transformation matrix is computed constructively from polynomial equations in its block coefficients. Although real-root existence is generally difficult to verify, a sufficient condition guarantees a feasible transformation for zero target IODP indices.
- General construction: The full transformation design yields 12 scalar equations in 16 unknowns, so fixing four variables produces a determined polynomial system.A real solution is selected and its matrix invertibility must then be checked.
- Diagonal construction: Restricting the transformation blocks to scalar multiples of the identity reduces the design to four unknowns and three equations.Fixing one coefficient gives a determined quadratic system with at most eight complex solutions counting multiplicities.
- Feasibility limitation: Real-root existence is generally nontrivial, so the constructive procedure provides a sufficient special-case condition rather than a general guarantee.This limitation concerns verifying feasible real solutions before selecting an invertible matrix.
- Practical synthesis: Setting one original IODP index to zero provides a practical synthesis strategy that guarantees σ_iρ_i = 0 and enables a real, invertible transformation matrix.The strategy absorbs the remaining passivity shortage through the other channel.
- Example: In the SG example, an original IODP index of (−8.5098, 0) is transformed so the controlled bus satisfies IODP(0, 0; eD_1).The example uses Proposition 8 and verifies Assumption 3 because the output is affine in the input.
VI. Case Study
The case study validates the framework on a modified IEEE 39-bus system with heterogeneous devices. Local IODP conditions are derived independently of a nominal equilibrium, while each operating scenario is certified by testing its computed equilibrium against the certified regions.
- Certification procedure: The modified IEEE 39-bus case study derives local IODP conditions without fixing one nominal equilibrium.Scenario certification instead checks whether each computed equilibrium lies in the corresponding certified region.
A. System Description and Setup
The test system combines synchronous generators, droop-controlled and virtual-synchronous-generator inverter sources, constant-impedance loads, and intermediate buses. After local control design, all buses satisfy the decentralized condition, and a 5% load disturbance produces bounded trajectories that settle to new steady states.
- Device composition: The modified IEEE 39-bus system replaces two synchronous generators with droop-controlled inverter sources and adds inverter-based sources using droop or virtual synchronous-generator control.The inverter sources are distributed across buses 1, 7, 16, 18, 21, and 27.
- Network composition: The system contains 16 dynamic buses and 23 static buses, including 12 constant-impedance load buses and 11 intermediate connection buses.This setup represents heterogeneous dynamic and static components.
- IODP analysis: The dynamic-bus IODP indices are computed using Proposition 1, with recomputed pairs imposing ρ_i = 0 when the initial pair fails σ_iρ_i ≤ 1/4.The recomputed indices are then used for control design.
- Control design: The I/O transformation control is applied to every dynamic device, and Theorem 6 establishes the decentralized stabilization condition for each controlled dynamic bus.Static buses naturally satisfy IODP(0, 0; R^2).
- Time-domain validation: A 5% simultaneous increase in active and reactive loads at all load buses drives transient responses that remain bounded and settle to new steady-state values.The responses include generator-bus frequencies and representative-bus voltage magnitudes.
C. Random Operating Scenarios
Across random operating scenarios, the decentralized IODP membership test certifies stability consistently with centralized Jacobian analysis while allowing parallel local verification. The approach reduces computation time substantially, although its sufficient conditions can be conservative under severe fluctuations.
- Stability certification: Each operating scenario is certified by checking whether its local equilibrium components lie inside the corresponding IODP regions.The study compares this membership test with centralized Jacobian-eigenvalue analysis across sampled scenarios.
- Stability certification: Stable operating points certified by the proposed method are fully consistent with Jacobian-eigenvalue analysis.Operating points outside the IODP regions may still be Jacobian-stable because the certification condition is sufficient rather than necessary.
- Computational performance: 10.12–10.76 times: centralized computation exceeds decentralized computation in total runtime across tested scenarios.The decentralized runtime uses the longest local computation time because local checks can run in parallel.
- Computational performance: O(n^3) centralized eigenvalue computation is applied to a 44-dimensional system Jacobian, whereas decentralized verification uses 4- or 5-dimensional single-device matrices.The smaller local matrices explain the observed computational advantage of decentralized verification.
- Framework validation: The framework combines equilibrium-set-oriented IODP certification with an I/O-transformation-based passivation controller for devices lacking sufficient IODP.The modified IEEE 39-bus case studies report stability certification under varying operating points and reduced computational burden through local verification.