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Optimal Placement of Virtual Inertia in Power Grids
Bala Kameshwar Poolla, Saverio Bolognani, Florian Dorfler
TL;DR
The paper asks where virtual inertia should be placed to address stability risks from low and heterogeneous inertia in converter-dominated grids. It combines a linear network-reduced model with an H2 coherency metric, analytical allocation results, and gradient-based computation. The resulting methods provide closed-form solutions for selected cases and locally optimal allocations validated against heuristics in a three-region case study.
Problem
The paper addresses the limited attention given to where virtual inertia should be placed, despite evidence that placement affects grid performance.
Method
It uses a linear network-reduced power-system model with an H2 coherency metric, analytical results for selected cases, and a gradient-based computational approach for general allocation.
Results
The paper obtains closed-form global optimality results for particular instances and locally optimal solutions for general cases, including a three-region case study.
Takeaways & Limitations
The results suggest that disturbance location and inertia placement, rather than total inertia alone, dictate grid resilience within the studied formulation.
Abstract
from arXiv · showhide
A major transition in the operation of electric power grids is the replacement of synchronous machines by distributed generation connected via power electronic converters. The accompanying "loss of rotational inertia" and the fluctuations by renewable sources jeopardize the system stability, as testified by the ever-growing number of frequency incidents. As a remedy, numerous studies demonstrate how virtual inertia can be emulated through various devices, but few of them address the question of "where" to place this inertia. It is however strongly believed that the placement of virtual inertia hugely impacts system efficiency, as demonstrated by recent case studies. In this article, we carry out a comprehensive analysis in an attempt to address the optimal inertia placement problem. We consider a linear network-reduced power system model along with an H2 performance metric accounting for the network coherency. The optimal inertia placement problem turns out to be non-convex, yet we provide a set of closed-form global optimality results for particular problem instances as well as a computational approach resulting in locally optimal solutions. Further, we also consider the robust inertia allocation problem, wherein the optimization is carried out accounting for the worst-case disturbance location. We illustrate our results with a three-region power grid case study and compare our locally optimal solution with different placement heuristics in terms of different performance metrics.
I. INTRODUCTION
The paper studies where to place virtual inertia as low and heterogeneous inertia profiles threaten grid stability. It develops an H2-based framework, analytical results, and computational methods for inertia allocation.
- Low inertia and variable renewable generation can produce large frequency swings in transmission grids and microgrids.
- The paper asks where virtual inertia should be placed to alleviate destabilizing effects from spatially heterogeneous inertia profiles.
- The proposed H2 coherency metric penalizes angular differences and frequency excursions while measuring disturbance amplification.
- The framework optimizes inertia placement subject to capacity and budget constraints despite the problem's non-convexity.
- Closed-form results cover selected cases, while a gradient-based computational approach finds locally optimal solutions for large networks and arbitrary parameters.
- The model represents synchronous-machine, inverter-based, and frequency-responsive-load dynamics through aggregate inertia and damping parameters.
B. Coherency performance metric
The coherency metric measures the energy associated with angle differences and frequency deviations after disturbances. Its H2 interpretation covers both impulsive events and stochastic renewable or load fluctuations.
- The performance objective is the energy expended in returning to steady state, expressed through angle differences and frequency displacements.
- The angle-error weights define a connected penalty graph that may differ from the physical power grid.
- Choosing N = L penalizes local angle differences, whereas N = I_n − 1_n1_n^T/n penalizes global errors.
- The metric represents generalized energy in swing-mode oscillations when a_ij = b_ij and s_i = m_i.
- Disturbance localization is modeled through input scaling factors whose probabilities are assumed known from historical data and other sources.
- The squared H2 norm measures output energy for unit impulses and steady-state output variance under unit-variance white-noise inputs.
III. OPTIMAL INERTIA ALLOCATION
The optimal allocation problem adds virtual inertia to minimize the H2 norm under bus capacity and total-budget constraints. Its Lyapunov formulation is generally large-scale and non-convex.
- Each bus has positive inertia, and the optimization allocates additional virtual inertia to minimize the H2 norm.
- Bus-specific upper bounds represent available capacity or installation space, while the total budget represents aggregate device cost.
- The Lyapunov constraint is bilinear, and inertia variables also enter the system matrix through inverse inertia terms.
- Consequently, the allocation problem is non-convex and typically large-scale.
- The paper develops bounds, simplified formulations, two-area analysis, and a numerical method for locally optimal solutions.
- Zero inertia yields an ill-posed model whose algebraic and dynamic state counts depend on system parameters.
A. Performance bounds
The performance bounds separate network topology from the inertia decision variables and clarify how disturbances, damping, frequency penalties, and inertia interact. For N = L, the topology-dependent offset reduces to n − 1.
- Theorem 2 establishes upper and lower bounds for the squared H2 objective of the optimal inertia allocation problem.
- The proof derives the bounds by expressing the observability Gramian as a block matrix and expanding the Lyapunov constraint.
- The network topology enters the bounds only through a constant factor decoupled from the inertia decision variables.
- For N = L, the topology-dependent term is Trace(NL†) = n − 1.
- The bounds depend on extrema of disturbance strengths and damping coefficients, including their ratios, together with frequency penalty weights and inertia variables.
B. Noteworthy cases
The paper identifies tractable inertia-allocation cases under specialized performance metrics and disturbance assumptions. These cases yield convex reformulations, topology-independent solutions, or allocation-independent costs.
- Primary control effort: Primary-control effort minimization can be equivalently formulated as a convex optimization problem.The formulation uses S = D and N = 0.
- Primary control effort: The unique primary-control optimum is topology-independent and depends on disturbance locations and strengths encoded by v_i.With a concentrated disturbance, maximal inertia is allocated at the disturbed node; without capacity constraints, allocation is proportional to v_i^1/2.
- Uniform disturbance-damping ratio: Under a uniform disturbance-damping ratio λ = v_i/d_i, the general allocation problem becomes convex.This reformulation is independent of network topology because disturbances are dissipated at every node in the same proportion.
- Kinetic energy penalization: When S = cM under the uniform-ratio assumption, the squared H2 performance metric is independent of inertia allocation.The result reflects penalization of kinetic-energy variation as it decays to zero.
C. Explicit results for a two-area network
The two-area case admits an analytical Lyapunov solution and reveals how disturbance-damping ratios shape optimal placement and the geometry of the cost function. Disturbance location strongly influences where inertia is allocated.
- Analytical solution: The two-area Lyapunov equation has a closed-form solution, reducing the cost to a rational function of fourth-order polynomials in the inertia coefficients.The resulting optimization problem admits a unique minimizer.
- Allocation structure: Identical v_i/d_i ratios and frequency penalties produce identical optimal allocations when capacity constraints are absent.The budget constraint becomes active for sufficiently large inertia bounds.
- Allocation structure: When v_i/d_i > v_j/d_j, the optimal allocation favors node i, while sufficiently uniform ratios make the cost strongly convex.For a disturbance affecting only one node, strong convexity is lost.
- Disturbance location: As disturbances vary from node 2 to node 1, optimal inertia is allocated dominantly at the disturbance location.Figure 2 uses d1 = 1 < d2 = 2, m_bdg = 25, and a12 = 1.
D. A computational method for the general case
For the general nonconvex allocation problem, the paper derives an explicit gradient and uses it in a computational method for locally optimal solutions. The method also supports sparsity-promoting allocation of limited inertia units.
- General formulation: The full allocation problem is expressed as a cost f(m) through the Lyapunov-equation solution P(m).This extends the tractable special-case analyses to arbitrary parameters.
- Gradient computation: An efficient algorithm computes the explicit gradient ∇f(m) of the allocation cost.The gradient formulation incorporates the nonlinear Lyapunov equations into gradient information, avoiding their treatment as a large-scale optimization block.
- Gradient computation: The numerical procedure takes the current inertia vector m and outputs a numerical evaluation g of ∇f(m).This is the stated input-output role of Algorithm 1.
A(0), CTC
The gradient derivation uses perturbation analysis of the Lyapunov equation and Taylor or power-series expansions. First- and second-order Lyapunov equations determine the expansion terms needed for differentiation.
- Perturbation analysis: The objective is differentiated by perturbing m along a direction μ and expanding A(m + δμ), B(m + δμ), and P in series around δ = 0.The derivation relies on Taylor and power-series expansions.
- Lyapunov expansion: The underlying Lyapunov equation is PA(m + δμ) + A(m + δμ)^TP + C^TC = 0.It defines the matrix P used in the perturbation-based gradient derivation.
- Lyapunov expansion: The perturbed Lyapunov equation yields equations for P^(0) and P^(1), including the first-order equation P^(1)A^(0) + A^(0)^TP^(1) + P^(0)A^(1) + A^(1)^TP^(0) = 0.The first equation is feasible with a positive semidefinite P^(0) satisfying P^(0)z0 = 0.
- Gradient construction: The gradient components are computed using directional derivatives with μ = e_i for each inertia coordinate.This establishes differentiability of f(m) for the stated domain.
E. The planning problem: Economic allocation of resources
The planning problem addresses economically limited deployment of virtual inertia by promoting sparse allocations rather than placing devices at every grid bus. An ℓ1 penalty preserves compatibility with the gradient-based optimization approach while exposing a performance–sparsity trade-off.
- ℓ1 regularization promotes sparse virtual-inertia allocations when economic constraints limit the number of deployed devices.The underlying allocation problem is generally combinatorial, so the modified objective adds a sparsity penalty.
- γ ≥ 0 trades off the sparsity penalty against the original inertia-allocation objective.
- The regularized objective remains suitable for the gradient algorithm because its added linear penalty is differentiable.The penalty can be incorporated within Algorithm 1, and the analytic results can likewise be re-derived for the regularized cost.
F. The min-max problem: optimal robust allocation
The robust allocation problem optimizes inertia placement against uncertainty in disturbance location, using a min–max formulation and gradient-enabled computation. In the case study, robust profiles reduce worst-case cost, while disturbance-specific and sparse allocations reveal strong placement effects.
- F. The min-max problem: optimal robust allocation: Rare, poorly predicted faults motivate an inertia profile that remains effective against the most detrimental disturbance location.
- F. The min-max problem: optimal robust allocation: The robust formulation uses a convex disturbance set and rewrites the inner maximization through an equivalent dual minimization.The objective is linear in the disturbance vector, enabling the dual reformulation by strong duality.
- F. The min-max problem: optimal robust allocation: Robust optimal profiles tend to make the cost indifferent to disturbance location and, for primary-control effort, equalize inertia allocations across buses.The primary-control special case induces a valley-filling strategy that uses the full budget and prioritizes buses with lowest inertia.
- IV. CASE STUDY: 12-BUS-THREE-REGION SYSTEM: Localized disturbances concentrate optimal inertia near the disturbed node, whereas adding inertia at undisturbed nodes can be detrimental.For a disturbance at node 4, inertia is assigned to buses 4 and 6 without sparsity regularization, and exclusively to bus 4 with γ > 2e−4 at negligible performance loss.
- IV. CASE STUDY: 12-BUS-THREE-REGION SYSTEM: γ = 6e−5 reduces the uniform-disturbance allocation from nine to seven buses with a 1.3% performance degradation.
- IV. CASE STUDY: 12-BUS-THREE-REGION SYSTEM: Time-domain simulations show the H2-optimal allocation is superior in frequency overshoot and angle differences, while also requiring the least control effort.The eigenvalue spectrum alone provides only partial information: another profile performs marginally better spectrally but worse in time-domain responses.
V. CONCLUSIONS
The paper studies virtual-inertia placement using an H2 network-coherency metric, derives analytical results for special cases, and develops a gradient-based method for general networks. A three-region validation supports locally optimal allocations over intuitive heuristics and emphasizes placement and disturbance location as central to resilience.
- The H2-based formulation produces a large-scale, non-convex optimization problem for virtual-inertia placement.
- Closed-form solutions are obtained for certain cost functions, problem instances, and the two-area case.
- An explicit gradient formulation provides a computational approach for locally optimal allocations and supports sparsity-promoting placement of finite inertia units.
- Three-region time-domain simulations demonstrate the efficacy of locally optimal allocations compared with intuitive heuristics.
- The results suggest that disturbance location and inertia placement, rather than total system inertia alone, dictate resilience within the studied framework.