Placement and Implementation of Grid-Forming and Grid-Following Virtual Inertia and Fast Frequency Response

Bala Kameshwar Poolla; Dominic Groß; Florian Dörfler

Source-linked AI summary

Placement and Implementation of Grid-Forming and Grid-Following Virtual Inertia and Fast Frequency Response

Bala Kameshwar Poolla, Dominic Groß, Florian Dörfler

arXiv:1807.01942v4Jul 5, 2018math.OC

TL;DR

Low-inertia power systems lose the synchronous-machine inertia and damping needed for frequency stability. This paper develops explicit grid-following and grid-forming virtual-inertia models, optimizes their tuning and placement using an H2-norm approach, and evaluates them in a high-fidelity South-East Australian system. Simulations show improved resilience for both implementations across a range of disturbances, with differences mainly associated with maximum power injection.

Problem
Power-electronics-interfaced renewable generation lacks synchronous machines' natural inertia and damping, creating frequency-stability concerns in low-inertia systems.
Method
The paper models grid-following and grid-forming virtual inertia as dynamic feedback controllers and uses system-norm optimization to tune their parameters and placement.
Results
For a range of disturbances, both virtual-inertia implementations improved system resilience relative to the system without virtual inertia.
Takeaways & Limitations
System robustness depends on both the amount and spatial distribution of virtual inertia, while implementation differences are mainly related to maximum power injection.

Abstract

from arXiv · show

The electric power system is witnessing a shift in the technology of generation. Conventional thermal generation based on synchronous machines is gradually being replaced by power electronics interfaced renewable generation. This new mode of generation, however, lacks the natural inertia and governor damping which are quintessential features of synchronous machines. The loss of these features results in increasing frequency excursions and, ultimately, system instability. Among the numerous studies on mitigating these undesirable effects, the main approach involves virtual inertia emulation to mimic the behavior of synchronous machines. In this work, explicit models of grid-following and grid-forming virtual inertia (VI) devices are developed for inertia emulation in low-inertia systems. An optimization problem is formulated to optimize the parameters and location of these devices in a power system to increase its resilience. Finally, a case study based on a high-fidelity model of the South-East Australian system is used to illustrate the effectiveness of such devices.

I. INTRODUCTION

The paper addresses frequency-stability concerns in low-inertia systems by modeling and optimally placing grid-following and grid-forming virtual inertia devices. It evaluates the approach using detailed power-system models and a high-fidelity South-East Australian case study.

Power-electronics-interfaced renewable generation replaces synchronous machines but lacks their inherent rotational inertia and damping, raising frequency-stability concerns.
Virtual inertia and fast frequency response are studied as short-term substitutes for machine inertia in systems with reduced synchronous generation.
The paper develops explicit converter-based models for grid-following and grid-forming virtual inertia, including PLL and grid-forming control dynamics.
A computationally efficient H2-norm algorithm tunes virtual-inertia parameters and placement by treating the devices as feedback controllers.
A modified high-fidelity South-East Australian power-system model supports extensive simulations of device impacts, optimal tuning, and validation of linearized models.

A. Modelling of virtual inertia devices

The paper models grid-following and grid-forming virtual inertia as local dynamic feedback controllers with distinct converter interfaces and frequency-response mechanisms.

Both implementations are modeled as local dynamic feedback controllers, and the framework can accommodate arbitrary controller transfer functions.
Grid-following: Grid-following virtual inertia injects active power according to frequency deviation and RoCoF estimated by a phase-locked loop.
Grid-following: Filtering within the PLL enables an explicit RoCoF estimate, unlike the standard SRF-PLL with a PI loop filter.
Grid-following: The grid-following implementation uses a current source that injects three-phase current and tracks active-power references.
Grid-forming: Grid-forming virtual inertia uses a voltage source whose generated voltage angle depends on device power in-feed.
Grid-forming: The grid-forming model uses virtual inertia and damping constants, while regulating voltage amplitude at the connected bus's nominal operating voltage.

B. Disturbance model

The disturbance model represents faults such as load changes, renewable fluctuations, and generator outages through bus current injections. Performance is assessed using conventional transient metrics and H2-based measures of frequency imbalance and control effort.

Faults are modeled as disturbances acting at voltage buses through current injections, covering load steps, renewable fluctuations, and generator outages.The disturbance vector represents changes in load or generation at each bus.
Frequency nadir and maximum RoCoF are defined from the post-disturbance evolution of bus-frequency deviations.The generator RoCoF is computed by filtering the frequency derivative through a low-pass filter.
The evaluated outputs include generator frequencies, RoCoF, mechanical power injections, and active-power injections from VI devices.Peak VI and governor power injections are also defined for step disturbances.
The metric set quantifies frequency and RoCoF imbalance together with converter virtual-inertia, damping, and generator mechanical efforts over a time horizon.Nonnegative weights trade off the relative contributions of these efforts.
The H2 norm is used for control design because it measures short-term energy imbalance and yields tractable optimization problems beyond classical step-based metrics.Conventional nadir, RoCoF, and power-injection metrics remain part of the evaluation.

B. Design constraints

VI design is constrained by aggregate damping limits, individual gain limits, and converter power ratings. The power-injection approximation uses separate a priori bounds for RoCoF and frequency deviation because their peaks need not coincide.

The sum of VI damping gains is upper-bounded to respect grid-code and primary-control-reserve constraints.The bound is written as Σ_k d̃_k ≤ d_sum.
Individual inertia and damping gains are constrained to account for each converter’s maximum power rating.The gains are also restricted to be non-negative.
The approximation reflects that inertia and damping responses do not necessarily reach their peak values simultaneously.Empirical observations indicate that maximum frequency deviation and maximum RoCoF can occur at different times.
The maximum-power constraint approximates VI injection using separate limits for maximum RoCoF and maximum frequency deviation.The approximation relies on a priori estimates of |ω̇|max and |ω|max.

IV. CLOSED-LOOP SYSTEM AND H2 OPTIMIZATION

The paper formulates VI placement and tuning as system-norm minimization for linearized systems containing grid-forming or grid-following devices. Separate closed-loop representations define the feedback variables and gains for each implementation.

VI placement and tuning optimize grid-following and grid-forming gain matrices Kfoll and Kform to improve the post-fault response of a low-inertia system.
The optimization combines the power-system and disturbance models with either grid-following or grid-forming VI models and minimizes an input-output system norm.The decision variables include virtual inertia and damping gains.
Grid-forming devices expose internal frequency and active-power injection as outputs, with the derivative of internal frequency as the control input.Their closed-loop interconnection uses the gain matrix Kform.
Grid-following devices expose PLL frequency and RoCoF estimates as outputs, with the active-power set-point as the control input.Their closed-loop interconnection uses the gain matrix Kfoll.
The overall model defines system states, control inputs, outputs, and performance outputs before linearization around a nominal operating point.The algebraic network equation and the unobservable absolute-angle mode are removed under stated regularity conditions.

B. Virtual inertia as output feedback

VI devices are represented as static output-feedback controllers whose gains determine grid-following and grid-forming control inputs. The performance output and closed-loop dynamics then support H2-based evaluation and tuning.

The control inputs for VI devices are given by static output feedback.
Grid-following and grid-forming implementations use feedback matrices Kfoll and Kform, respectively.Their feedback gains are defined from the corresponding inertia and damping parameters.
The feedback gains satisfy α̃_k = −d̃_k m̃_k^-1 for the indexed VI devices.The model includes nc virtual inertia devices.
The performance output penalizes frequency deviations, RoCoF, and power injections from VI devices and generators over an infinite horizon for impulse disturbances.
With Acl = A + BKC, the interconnected VI model provides the closed-loop dynamics used to form the system for analysis.

C. H2 norm optimization

The paper formulates H2-norm optimization for tuning virtual inertia device gains under local-feedback and control-gain constraints. The resulting problem is generally non-convex, but its gradient can be computed efficiently for scalable optimization.

The H2 norm is computed from the positive definite observability Gramian solving a Lyapunov equation.The Gramian is parameterized by the feedback gain K for the system matrices A, B, C, and C_p.
The optimization encodes purely local virtual inertia feedback through structural constraints on K.A separate constraint set limits the admissible control gains.
The problem can tune the gain of any virtual inertia device and can promote sparse allocations with an ℓ1 penalty.
Evaluating the Lyapunov-based cost is generally non-convex and potentially very large-scale.The cost is nonlinear in both the Gramian P and gain matrix K.
The gain gradient can be used directly with scalable first-order methods or to accelerate higher-order methods.Projected gradient methods are given as an example.

D. Complexity of the gradient computation

The proposed gradient computation reduces the cost of H2 optimization relative to earlier approaches while retaining detailed virtual inertia device models. It requires only two Lyapunov equations and has O(n^3) complexity.

O((n + 1)n^3) complexity in prior work comes from solving n −1 Lyapunov equations of dimension 2n.
The proposed method avoids the higher computation burden of sequential linear programming that combines eigenvalue calculations, time-domain simulations, and linear-program solutions.That comparison concerns prior optimization of grid-following virtual inertia and damping allocation.

V. TEST CASE DESCRIPTION

The case study uses a detailed 14-generator, 59-bus South-East Australian system modified to represent low inertia and augmented with virtual inertia devices. Linearized predictions are compared with nonlinear responses for both converter implementations.

V. TEST CASE DESCRIPTION: The test case is based on a 14-generator, 59-bus South-East Australian power system with higher-order turbine, governor, PSS, and AVR models.The system has a string topology and weak coupling between South Australia and the rest of the system.
V. TEST CASE DESCRIPTION: Four synchronous machines are replaced by constant power sources to create a low-inertia scenario, and 15 virtual inertia devices are distributed across the system.The replacement sources preserve the original generators’ active and reactive power injections regardless of frequency or voltage.
V. TEST CASE DESCRIPTION: The study compares the original system with closed-loop grid-following and grid-forming virtual inertia and damping implementations.The comparison focuses on the defined grid-stability performance metrics.
V. TEST CASE DESCRIPTION: Relative linearization errors are evaluated for step disturbances from −250 MW to +250 MW at six locations.The nonlinear and linearized models are compared for both converter configurations.
V. TEST CASE DESCRIPTION: The optimization uses identical weighted performance outputs for both implementations and imposes bounds on total damping, individual damping, and individual inertia.The constraints limit total damping and roughly limit converter output to 40 MW for normal-regime frequency deviations.

C. Contrasting allocations for different VI implementations

Optimized virtual inertia allocations are spatially nonuniform, with both implementations concentrating inertia in area 5. Both improve frequency nadirs, while grid-following devices trade lower mean nadir for greater variability and substantially larger injections.

C. Contrasting allocations for different VI implementations: Area 5 receives most optimized virtual inertia in both grid-forming and grid-following configurations, whereas uniform allocations are typically not optimal.Grid-following allocation is negligible at some nodes outside area 5, while grid-forming gains are distributed across all nodes.
D. Impact of VI devices on frequency stability: Virtual inertia devices reduce the mean and variance of frequency-nadir distributions relative to the system without virtual inertia.The distributions are evaluated over the disturbance cases shown for the South-East Australian system.
D. Impact of VI devices on frequency stability: Grid-following virtual inertia has a smaller mean frequency nadir but larger variance and a longer tail than grid-forming virtual inertia.This performance comes with larger peak power injections for some disturbances.
D. Impact of VI devices on frequency stability: Roughly three times larger maximum power injection occurs for grid-following than grid-forming virtual inertia over the same disturbances.
D. Impact of VI devices on frequency stability: Virtual inertia has a modest apparent effect on maximum generator RoCoF, but H2 norms capture post-first-swing RoCoF effects that this metric misses.The reported maximum generator RoCoF is measured at generator buses and is typically attained during the first swing.
D. Impact of VI devices on frequency stability: Overall, virtual inertia devices have a positive impact on frequency stability, with implementation differences mainly associated with maximum power injection.

E. Time-domain responses

A 200 MW load increase at node 508 was used to examine time-domain responses in the low-inertia system with grid-following and grid-forming VI. Both implementations improved frequency stability, with grid-forming VI generally achieving better absolute frequency metrics and lower control effort.

E. Time-domain responses: Both VI implementations improve frequency nadir and maximum RoCoF, with grid-forming VI performing better in absolute values.The comparison concerns the 200 MW load increase at node 508.
E. Time-domain responses: Grid-forming VI injects less maximum active power than grid-following VI for both individual-device and combined-device measures.The paper characterizes this as better performance with lesser control effort.
E. Time-domain responses: Active power injections from VI devices reduce the maximum governor response relative to the low-inertia system.The comparison is made for the maximum governor response of a single synchronous machine.
E. Time-domain responses: The H2 norm decreases for both VI implementations and serves as an effective proxy for time-domain power-system metrics.This links the optimization objective to the observed time-domain behavior.
E. Time-domain responses: The grid-forming VI optimization takes around 60s, compared with 160s for grid-following VI under identical penalties.The timings were obtained using MATLAB on a Core i7-6600U CPU.

APPENDIX

The appendix develops gradient-based computation for the H2 optimization under stability, sparsity, and parameter constraints. Projected gradients can support locally optimal solutions for very large systems when constraint projections are efficient.

APPENDIX: Computing the H2 norm for a given gain K requires solving Lyapunov equations, while gradient projection reduces optimization variables under sparsity constraints.The non-zero parameter vector differs between grid-following and grid-forming implementations.
APPENDIX: The H2 norm and its gradient are well defined only for a stable closed-loop system because the norm is infinite for unstable systems.The optimization therefore requires an initial stabilizing gain satisfying the sparsity and constraint sets.
APPENDIX: Projected gradient methods can find locally optimal solutions for very large systems when projections onto the constraint set are efficient.If projection is inefficient, the same gradient computation can accelerate higher-order methods.