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Towards Optimal System Scheduling with Synthetic Inertia Provision from Wind Turbines

Zhongda Chu, Uros Markovic, Gabriela Hug, Fei Teng

arXiv:1911.02857v3eess.SYcs.SY

TL;DR

Converter-interfaced wind reduces system inertia, motivating system-level treatment of turbine SI dynamics. The paper integrates a WT control scheme and frequency constraints into stochastic scheduling; its cases show that underproduction affects costs and optimized SI outperforms fixed settings.

  • Problem

    As wind replaces conventional generation, converter interfaces reduce system inertia, while existing system-level WT SI studies do not capture detailed turbine dynamics for scheduling.

    Source: §I · p. 1

  • Method

    The paper combines WT SI control that accounts for underproduction with analytical frequency constraints in a stochastic unit-commitment model that optimizes SI provision.

    Source: §I · p. 2

  • Results

    Up to 170 M£, system operation costs are underestimated when WT underproduction is neglected; fixed SI's maximum cost saving is roughly 25% lower than optimal provision.

    Source: C. Impact of SI Penetration Level and… · p. 8, D. Value of Dynamically Optimizing SI… · p. 8

  • Takeaways & Limitations

    Cost savings increase considerably through 40% SI penetration, suggesting that not all WTs need SI capability.

    Source: C. Impact of SI Penetration Level and… · p. 8

Abstract

from arXiv · show

The undergoing transition from conventional to converter-interfaced renewable generation leads to significant challenges in maintaining frequency stability due to declining system inertia. In this paper, a novel control framework for Synthetic Inertia (SI) provision from Wind Turbines (WTs) is proposed, which eliminates the secondary frequency dip and allows the dynamics of SI from WTs to be analytically integrated into the system frequency dynamics. Furthermore, analytical system frequency constraints with SI provision from WTs are developed and incorporated into a stochastic system scheduling model, which enables the provision of SI from WTs to be dynamically optimized on a system level. Several case studies are carried out on a Great Britain 2030 power system with different penetration levels of wind generation and inclusion of frequency response requirements in order to assess the performance of the proposed model and analyze the influence of the improved SI control scheme on the potential secondary frequency dip. The results demonstrate that the inclusion of SI provision from WTs into UC can drastically impact the overall system costs.

I. INTRODUCTION

Converter-interfaced wind reduces system inertia, while kinetic-energy extraction for synthetic inertia creates post-support underproduction. The paper proposes control and scheduling methods that account for turbine dynamics while maintaining frequency limits and mechanical stability.

I. INTRODUCTION

  • Converter-interfaced WTs decouple rotor kinetic energy from the grid, while kinetic-energy-extraction SI causes post-support underproduction as rotors recover.Source: §I · p. 1
  • Device-level optimal SI need not be system-optimal, and existing system-perspective studies either rely on simulations or omit detailed WT dynamics.Source: §I · pp. 1–2
  • The proposed stochastic scheduling model optimizes WT synthetic inertia to minimize operating cost while enforcing frequency limits and turbine mechanical stability.Source: §I · p. 2
  • The proposed SI control accounts for WT underproduction, eliminates the associated secondary frequency dip, and approximates underproduction as negative system damping with proven conservativeness.Source: §I · p. 2
  • The highly nonlinear n-dimensional frequency-nadir constraint is efficiently linearized with guaranteed conservativeness and quantified accuracy.Source: §I · p. 2

A. VSWT Modeling

  • The VSWT model combines mechanical, aerodynamic, and MPPT components, neglects faster electrical dynamics, and assumes MPPT rather than deloading operation.Source: A. VSWT Modeling · p. 2

B. SI Control Scheme

The proposed wind-turbine synthetic-inertia control uses mechanical-power feedback to make the post-support power reduction gradual, eliminating the associated secondary frequency dip. It also expresses generation loss in terms of system-frequency deviation for integration into frequency dynamics.

B. SI Control Scheme

  • The improved operating trajectory removes the step power change and enables recovery at constant electrical power until MPPT operation is restored.The paper notes that further balancing of additional power supply and recovery speed is not addressed.Source: B. SI Control Scheme · p. 3
  • Without mechanical-power estimation, sudden power loss after synthetic-inertia provision produces a significant secondary dip and a worse frequency nadir.With the estimator, frequency deviation is slightly larger because less power is injected.Source: B. SI Control Scheme · p. 3
  • The generation loss from deviation from the maximum power point is expressed as a function of system-frequency deviation rather than rotor speed.The derivation assumes negligible mechanical-power-estimator error.Source: B. SI Control Scheme · p. 3
  • The described kinetic-energy extraction control assumes below-rated wind speed; pitch control at higher wind speeds is not detailed.The passage states pitch control can provide specified virtual inertia without other impact on system frequency response.Source: B. SI Control Scheme · p. 3
  • During support, turbine power combines RoCoF control with mechanical-power-estimator feedback until the frequency nadir, when mechanical and electrical power become equal.The estimator captures mechanical-power variation from measured rotor speed and adjusts SI feedback and the electrical-power reference.Source: B. SI Control Scheme · pp. 2–3

C. MPE Approximation

The section approximates the wind-turbine mechanical-power effect as a damping term so it can be incorporated into system frequency dynamics. The aerodynamic-power approximation is conservative, though less accurate than the damping fit.

C. MPE Approximation

  • The convoluted mechanical-power-loss expression is approximated as an additional damping term, with a quadratic fit used for its relationship to frequency deviation.The figure distinguishes the aerodynamic-power approximation from the damping term in per unit.Source: C. MPE Approximation · p. 3, Figure 4 · p. 4
  • The damping fit is accurate, while the aerodynamic-power approximation has a noticeable mismatch but remains conservative.The proposition states that the approximation is no greater than the original over the specified frequency interval, with equality at the endpoints.Source: C. MPE Approximation · p. 4
  • The aerodynamic-power approximation is bounded by the original mechanical-power effect, with equality only at the interval endpoints.Source: C. MPE Approximation · p. 4

D. Aggregated SI from Wind Farm

The proposed wind-farm model aggregates available synthetic inertia and its associated mechanical-power loss using the wind-speed distribution. It represents turbines with an identical dynamical model and assumes the distribution is available to the system operator.

D. Aggregated SI from Wind Farm

  • The aggregated wind-farm model represents total available synthetic inertia and its corresponding mechanical-power loss using the wind-speed distribution.Source: D. Aggregated SI from Wind Farm · p. 4
  • The aggregation addresses the difficulty of representing every wind turbine individually during system scheduling.Source: D. Aggregated SI from Wind Farm · p. 4
  • The model assumes identical turbine dynamics and wind-speed distribution information available to the system operator.With individual turbines and system-wide wind-speed information, the discussion applies to aggregated system SI.Source: D. Aggregated SI from Wind Farm · p. 4

1) SI Capacity Estimation:

The section estimates available synthetic-inertia capacity from turbine physical limits and the initial operating point. For kinetic-energy extraction, capacity first rises with wind speed, then declines near rated speed and becomes constant under the turbine power limit.

1) SI Capacity Estimation:

  • A turbine’s SI capacity must respect its minimum permissible rotor speed and available kinetic energy, while converter capacity also limits provision near rated operation.Source: 1) SI Capacity Estimation · p. 4
  • For kinetic-energy extraction, maximum SI capacity rises with wind speed, then declines near rated speed and remains constant under the maximum turbine power limit.Source: 1) SI Capacity Estimation · p. 4
  • The wind farm’s total SI capacity is computed from the capacities of its turbines and the wind-speed distribution within the farm.Source: 1) SI Capacity Estimation · p. 4

2) Mechanical Power Loss:

The section aggregates wind-farm mechanical-power loss as a system-damping term and integrates the resulting SI effect into frequency modeling. The resulting frequency metrics depend on aggregate inertia and are used to express stability limits.

2) Mechanical Power Loss:

  • Aggregated wind-farm mechanical-power loss can be rewritten as a system-damping term, and a feasible turbine allocation has actual loss no greater than the estimate.Source: 2) Mechanical Power Loss · p. 5

A. System Frequency Modeling with WT SI Control

B. Approximation of Nadir Constraint

The frequency-nadir constraint is reformulated for UC by approximating its nonlinear feasible region with linear constraints. The inner approximation is conservative, with its accuracy traded against computational burden.

B. Approximation of Nadir Constraint

  • The nadir constraint is nonconvex because it contains a bilinear term and a quadratic term, so the paper approximates it with linear constraints for MILP-based UC.The approach avoids integer variables from binary expansions.Source: B. Approximation of Nadir Constraint · p. 5
  • The feasible region is transformed into a multidimensional hyperboloid and inner-approximated by n·m hyperplanes, yielding a conservative approximation when H_sj ≥ 0.The number of layers and planes controls the approximation.Source: B. Approximation of Nadir Constraint · pp. 5–6, Figure 6 · p. 6
  • Increasing the number of planes can improve accuracy but raises computational burden, so n and m should balance conservativeness against tractability.A sample uses n = 2 layers and m = 12 planes; layers become denser near the hyperboloid vertex, where curvature is greater.Source: B. Approximation of Nadir Constraint · p. 6

C. Stochastic Unit Commitment

The paper adopts a stochastic unit commitment model that minimizes expected cost across a scenario tree. It includes system frequency security alongside unit-operation constraints.

C. Stochastic Unit Commitment

  • The stochastic UC model minimizes expected cost over scenario-tree nodes representing renewable-energy uncertainty and generation outages.The model is adopted from prior work.Source: C. Stochastic Unit Commitment · p. 6
  • The model includes power balance, thermal and storage operation, and frequency-security constraints, using maximum instantaneous RoCoF for the RoCoF constraint.The section also depicts system frequency dynamics with SI from WTs.Source: C. Stochastic Unit Commitment · p. 6, Figure 7 · p. 6

IV. RESULTS

The case studies use a Great Britain 2030 power system with specified demand, disturbance, and frequency-response settings. Annual system operation is simulated using SUC with frequency constraints.

IV. RESULTS

  • The GB 2030 system case studies use load demand of 20–60 GW, damping of 0.5% of demand per 1 Hz, a 10 s FR delivery time, and a maximum power loss of 1800 MW.Thermal-plant system details are referenced to.Source: §IV · p. 6
  • The frequency limits are 0.8 Hz for nadir deviation, 0.5 Hz for steady-state deviation, and 0.5 Hz/s for RoCoF.Source: §IV · p. 6
  • Annual system operation is simulated under SUC with frequency constraints, solved using FICO Xpress through a C++ application via BCL.Source: §IV · p. 6

A. Validation of Proposed Frequency Dynamics and Constraints

In the GB 2030 studies, adding SI provision from WTs to frequency-constrained scheduling yields substantial system-cost savings, especially at high wind penetration. Validation indicates that the analytical dynamics and nadir constraints track simulated frequency behavior within stated bounds, with approximation accuracy balanced against computation.

A. Validation of Proposed Frequency Dynamics and Constraints

  • Analytical expressions closely approximate simulated maximum RoCoF and frequency nadir, but their frequency-response approximation is valid only until nadir.A noticeable mismatch appears thereafter.Source: Figure 8 · p. 6, A. Validation of Proposed Frequency… · p. 7
  • A sample SUC solution yields a 0.758 Hz frequency nadir and −0.17 Hz/s maximum RoCoF within prescribed limits; SI decay causes a discontinuity at nadir.WT power reduction is negligible in the first few seconds after the disturbance, when SI is most valuable; the modeled recovery period is not considered.Source: A. Validation of Proposed Frequency… · p. 7, Figure 9 · p. 7
  • With n = 4 cones and m = 12 planes, the nadir approximation has average conservativeness of 0.036 Hz (4.5%) and reasonable computation time.More cones and planes reduce conservativeness, but accuracy gains diminish as computational time increases.Source: A. Validation of Proposed Frequency… · p. 7, Table I · p. 7

B. Impact of SI Provision on System Operation

  • Adding WT SI to frequency-constrained scheduling produces significant cost savings, particularly at high wind penetration, compared with frequency constraints without SI.Without frequency constraints, costs fall approximately linearly as wind capacity increases; with constraints but no SI, savings tend to saturate above 40 GW wind capacity.Source: B. Impact of SI Provision on System… · pp. 7–8, Figure 10 · p. 7
  • At 60 GW installed wind capacity, WT SI coincides with 9 GW less online synchronous generation and about 4 GW more wind utilization during abundant-wind periods.Scheduled WT SI time constant ranges from 1.1 to 2.2 s and varies inversely with synchronous-generation capacity.Source: B. Impact of SI Provision on System… · p. 8, Figure 11 · p. 8

C. Impact of SI Penetration Level and WT Underproduction

SI cost savings rise with penetration up to 40%, while omitting WT underproduction can substantially understate costs and frequency-security risks. Dynamically optimized SI provision yields greater savings than fixed-SI control, and the paper identifies further model extensions for future work.

C. Impact of SI Penetration Level and WT Underproduction

D. Value of Dynamically Optimizing SI Provision

  • At 50% SI penetration, maximum system cost saving under fixed-SI control is roughly 25% lower than under optimal SI provision, whose equivalent WT inertia ranges from 0 to 4.16 s.Fixed-SI savings increase with SI time constants up to 3.2 s, after which WT-underproduction management diminishes added benefits. When maximum system SI falls below the fixed value, the maximum SI is applied to avoid SUC infeasibility.Source: D. Value of Dynamically Optimizing SI… · p. 8

V. CONCLUSION AND FUTURE WORK

  • The conclusion identifies WT underproduction as important to both frequency security and turbine mechanical stability, and therefore relevant to SI design and scheduling.Future work includes optimal SI allocation by wind farm, potential real-time model predictive control, minimizing wind-energy loss and mechanical stress, and reconsidering the center-of-inertia frequency model using local generator frequencies.Source: §V · p. 9
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