Source-linked AI summary
Simultaneous Scheduling of Multiple Frequency Services in Stochastic Unit Commitment
Luis Badesa, Fei Teng, Goran Strbac
TL;DR
Low-carbon grids have less inertia, making alternative frequency services harder to schedule alongside uncertain energy production and dynamic frequency requirements. The paper develops a frequency-constrained SUC that co-optimises these services and demonstrates economic and environmental benefits in Great Britain’s 2030 system.
Problem
Reduced inertia increases the need for alternative frequency services, but jointly scheduling them under uncertainty is difficult because steady-state optimisation must be linked with frequency dynamics.
Method
The paper formulates an SUC model that co-optimises energy, synchronised and synthetic inertia, EFR, PFR, and a dynamically reduced largest power infeed using linearised frequency constraints.
Results
The GB 2030 case studies show that simultaneously optimising diverse frequency services produces significant economic savings and carbon reduction while revealing service synergies and conflicts.
Takeaways & Limitations
Co-optimising alternative frequency services is important for accurately capturing their value in frequency-secured stochastic scheduling.
Abstract
from arXiv · showhide
The reduced level of system inertia in low-carbon power grids increases the need for alternative frequency services. However, simultaneously optimising the provision of these services in the scheduling process, subject to significant uncertainty, is a complex task given the challenge of linking the steady-state optimisation with frequency dynamics. This paper proposes a novel frequency-constrained Stochastic Unit Commitment (SUC) model which, for the first time, co-optimises energy production along with the provision of synchronised and synthetic inertia, Enhanced Frequency Response (EFR), Primary Frequency Response (PFR) and a dynamically-reduced largest power infeed. The contribution of load damping is modelled through a linear inner approximation. The effectiveness of the proposed model is demonstrated through several case studies for Great Britain's 2030 power system, which highlight the synergies and conflicts among alternative frequency services, as well as the significant economic savings and carbon reduction achieved by simultaneously optimising all these services.
NOMENCLATURE
This nomenclature defines the paper’s indices, sets, constants, and frequency-related parameters for the scheduling model.
- Indices and Sets: The model indexes generators, binary-expansion bits, scenario-tree nodes, overestimating planes, and storage units.
- Constants and Variables: Frequency-dynamic parameters include the nadir time, admissible frequency deviation, quasi-steady-state deviation, and node probabilities.
- Constants and Variables: The nomenclature also records marginal, no-load, and startup costs for generating units.
- Constants and Variables: System parameters include load damping, nominal frequency, inertia constants, demand, must-run units, and maximum unit outputs.
- Constants and Variables: Operational limits specify the largest power-infeed bound, admissible RoCoF, total EFR bound, and EFR and PFR delivery times.
Decision Variables (continuous unless otherwise indicated)
The paper introduces decision variables and motivates frequency-aware stochastic scheduling for systems with high renewable penetration and reduced inertia.
- Decision Variables (continuous unless otherwise indicated): Decision variables represent auxiliary linearisation terms, unit starts, online units, generation, frequency-response provision, and binary PFR expansions.
- Decision Variables (continuous unless otherwise indicated): The model tracks operating cost, post-loss system inertia, total PFR, and total EFR.
- I. INTRODUCTION: Nonsynchronous renewable sources displace thermal units without currently providing inertia, reducing system inertia and increasing instability risk.
- I. INTRODUCTION: High renewable penetration makes transient frequency-nadir requirements more important than relying only on predefined total-FR thresholds.
- I. INTRODUCTION: The paper addresses the mathematical challenge of embedding differential-equation-driven frequency evolution into algebraic UC optimisation.
- I. INTRODUCTION: Alternative services include faster EFR, dynamically reduced largest power infeed, and synthetic inertia, whose interactions remain important scheduling questions.
- I. INTRODUCTION: The proposed SUC co-optimises synchronised and synthetic inertia, PFR, EFR, and a dynamically reduced largest power infeed using linearised frequency constraints.
II. STOCHASTIC UNIT COMMITMENT
The SUC minimises expected operating cost under renewable uncertainty using a scenario tree and receding-horizon decisions, while enforcing operational and frequency-security requirements.
- II. STOCHASTIC UNIT COMMITMENT: Renewable uncertainty is represented by quantile-based net-demand scenarios in a scenario tree.
- II. STOCHASTIC UNIT COMMITMENT: The tree branches only at the current-time node to retain similar results while reducing computational time.
- II. STOCHASTIC UNIT COMMITMENT: A 24-hour hourly SUC is repeatedly recomputed, applying current decisions while discarding future decisions as forecasts update.
- II. STOCHASTIC UNIT COMMITMENT: The objective minimises expected operational cost across scenario-tree nodes and includes generating-unit operating costs.
- II. STOCHASTIC UNIT COMMITMENT: Equivalent generating units are clustered to reduce computational burden.
- II. STOCHASTIC UNIT COMMITMENT: The formulation includes power-system behaviour constraints, while net-demand uncertainty is detailed in the referenced SUC formulation.
- II. STOCHASTIC UNIT COMMITMENT: SUC is computationally demanding but is tested because it can provide more cost-effective operation at high renewable penetration.
- II. STOCHASTIC UNIT COMMITMENT: Frequency security requires sufficient inertia and frequency response at all times, including compliance with RoCoF, nadir, and quasi-steady-state limits.
A. Frequency-Security Constraints
The frequency-security formulation derives RoCoF, quasi-steady-state, and nadir constraints from post-outage dynamics, then approximates damping and linearises the resulting optimisation constraints.
- A. Frequency-Security Constraints: Dynamic-frequency constraints are derived by solving the swing equation for frequency deviation after a generation outage.
- A. Frequency-Security Constraints: EFR and PFR are modelled as ramped responses reaching their service levels after delivery times Ts and Tg.
- A. Frequency-Security Constraints: The RoCoF constraint follows from the initial post-outage frequency derivative, whose maximum occurs at t = 0.
- A. Frequency-Security Constraints: The post-loss system inertia is explicitly calculated before imposing the dynamic-frequency requirements.
- A. Frequency-Security Constraints: The quasi-steady-state constraint assumes RoCoF is effectively zero after the transient response.
- A. Frequency-Security Constraints: Total EFR and PFR are aggregated as RS and RG across storage units and generators, respectively.
- A. Frequency-Security Constraints: The nadir requirement bounds the maximum frequency deviation at the time t∗ when the nadir is reached.
- A. Frequency-Security Constraints: The nadir time is found by setting the derivative of frequency deviation to zero, within the interval [Ts, Tg).
B. Contribution of Load Damping to Supporting the Nadir
The paper incorporates load damping into the frequency-nadir constraint using a linear inner approximation. The approximation captures part of damping’s support while remaining more conservative than neglecting damping entirely.
- Linear approximation: A linear term is proposed for the nadir constraint to account for load damping.The frequency deviation at the nadir is obtained while considering damping in the system-frequency equation.
- Linear approximation: The relation between total PFR and demand-side damping is represented as a convex, monotonically decreasing function of demand.This function can therefore be inner-approximated by a line and incorporated into the nadir constraint.
- Accuracy and conservatism: The linear inner approximation underestimates load damping’s actual contribution, producing a tighter feasible region than the true constraint.It is nevertheless less conservative than simply omitting damping.
C. Linearisation of the Frequency-Nadir Constraint
The nonconvex frequency-nadir constraint is converted into a mixed-integer linear form using inner approximations and binary expansion. Accuracy can be increased systematically, at the cost of greater computational burden.
- Constraint linearisation: The proposed linearisation enables the nonconvex nadir constraint to be implemented in an MILP while guaranteeing frequency security.RoCoF and quasi-steady-state constraints are already linear, whereas the nadir constraint requires additional treatment.
- Nonlinear terms: The constraint’s nonlinear terms comprise two continuous-variable products, H · RG and RS · RG, plus the quadratic term (PL − RS)^2.These terms arise on opposite sides of the nadir constraint.
- Approximation method: The quadratic right-hand side is inner-approximated with overestimating planes, while RG is represented by binary expansion to linearise both left-hand-side products.Choosing RG for binary expansion allows the same representation to support both products.
- Approximation method: The products involving H and RS are linearised exactly after binary expansion using standard big-M constraints.The resulting auxiliary variables are m_l = H · z_l and k_l = RS · z_l.
- Accuracy and cost: Increasing the number of planes and binary variables improves approximation accuracy but increases the optimisation model’s computational burden.The paper quantifies this accuracy–computational-cost tradeoff in Section IV-B.
IV. CASE STUDIES
Case studies evaluate the frequency-secured scheduling framework on Great Britain’s 2030 power system. The studies examine economic and environmental benefits under specified generation, storage, demand, uncertainty, and frequency-security settings.
- Study objectives: The framework targets cost-effective low-carbon operation and can identify beneficial practices such as BESS-provided EFR and part-loading large generating units.The case studies assess both economic and environmental benefits.
- System setup: The GB 2030 test system spans 20GW to 60GW demand and includes pumped storage plus a 1GWh BESS rated at 200MW with EFR capability.The pumped-storage unit has 10GWh capacity, 2.6GW rating, and 75% round-trip efficiency; the BESS has 90% efficiency.
- Frequency-security settings: The simulations impose Δf_max = 0.8Hz, Δf_ss_max = 0.5Hz, and RoCoF_max = 0.5Hz/s as dynamic-frequency requirements.Synthetic inertia from wind turbines is considered only in Section IV-F.
- Stochastic solution setup: The SUC uses a scenario tree branching only at the current-time node with seven net-demand quantiles ranging from 0.005 to 0.995.MILP solutions use a 0.1% duality gap and are computed with FICO Xpress 8.0.
A. Validation of the Frequency-Security Constraints
The proposed frequency-security constraints are validated by feeding an SUC solution into MATLAB/Simulink dynamic simulations. The simulated frequency nadir, RoCoF, and quasi-steady-state deviation remain within their optimisation limits.
- A. Validation of the Frequency-Security Constraints: The validation uses MATLAB/Simulink models of generator and BESS dynamics based on first-order control representations.Generator dynamics include droop control, governor dynamics, and saturation; BESS dynamics use a first-order block.
- A. Validation of the Frequency-Security Constraints: 0.72Hz simulated nadir respects the 0.8Hz optimisation limit for an SUC solution with a binding nadir constraint.The solution schedules H = 132GWs, RS = 0.22GW, RG = 2.24GW, PL = 1.66GW, with PD = 38.3GW.
- A. Validation of the Frequency-Security Constraints: 0.31Hz/s simulated RoCoF and 0.35Hz quasi-steady-state deviation are also within their specified limits.The example uses generator and BESS time constants of τg = 5s and τb = 0.1s.
- A. Validation of the Frequency-Security Constraints: The conservative nadir result reflects both the analytical approximation and the increasing-ramp representation of frequency-response power injection.The increasing-ramp assumption conservatively approximates generic droop control.
B. Assessment of the Proposed Analytical Approximations
The paper evaluates three conservative linearisation approximations for the frequency-nadir constraint, quantifying their accuracy and computational tradeoffs. Greater approximation precision generally reduces conservativeness but increases optimisation burden.
- B. Assessment of the Proposed Analytical Approximations: Three approximations linearise the nadir constraint: a load-damping term, overestimating planes for a squared term, and binary expansion of RG.Their accuracy and computational performance are assessed at different precision levels.
- B. Assessment of the Proposed Analytical Approximations: 6% average conservativeness (0.05Hz) results from the proposed load-damping approximation, versus 25% (0.19Hz) when damping is ignored.In the worst case, the approximation increases the requirement by 9% (0.07Hz), compared with 39% (0.31Hz) without damping.
- B. Assessment of the Proposed Analytical Approximations: More overestimating planes reduce conservativeness but increase SUC computation time.Increasing planes from 2 to 4 and 8 reduces frequency-service cost by 0.4% and 0.7%, while increasing computation time by 7% and 60%, respectively.
- B. Assessment of the Proposed Analytical Approximations: Using fewer binary-expansion bits for RG significantly reduces computation time at the cost of increased conservativeness.The base case uses 12 bits, with 4095MW as the highest representable value.
- B. Assessment of the Proposed Analytical Approximations: The subsequent SUC simulations use 2 planes and 7 bits for RG to balance objective accuracy and computational efficiency.This configuration removes 5 least significant bits from the 12-bit base case.
C. Value of Defining and Optimising EFR as a distinct service
The study compares fixed and optimised EFR provision against a PFR-only baseline in Great Britain’s 2030 system. Optimising EFR yields increasing value with wind penetration and enables synergies between frequency response, energy, and reserve services.
- C. Value of Defining and Optimising EFR as a distinct service: The three strategies are “Just PFR,” “Fixed EFR,” and “Optimised EFR,” with the latter co-optimising EFR, PFR, and inertia.The largest power infeed is held constant at PL = PmaxL in these comparisons.
- C. Value of Defining and Optimising EFR as a distinct service: EFR value increases with wind penetration as declining system inertia makes EFR more valuable.Savings are evaluated for 0, 10, 20, 30, and 40GW wind capacities using a 200MW-rated, 5h BESS.
- C. Value of Defining and Optimising EFR as a distinct service: More than 33% higher savings result from optimising EFR provision compared with providing a fixed amount, across all wind-penetration levels.A 200MW BESS can provide up to 400MW of EFR at times and can also provide reserve.
- C. Value of Defining and Optimising EFR as a distinct service: A BESS can provide up to 400MW of EFR from a 200MW rating by shifting rapidly between fully charging and fully discharging.Figure 7 tracks net demand, BESS state of charge, and scheduled EFR over a two-day period.
- C. Value of Defining and Optimising EFR as a distinct service: Charging during low-net-demand periods supports both EFR provision and lower energy costs, while the BESS can provide reserve when needed.This creates a synergy unavailable when fixed EFR forces the BESS to remain idle.
- C. Value of Defining and Optimising EFR as a distinct service: Optimised-EFR savings are largest with small BESS volumes and become very limited above 800MW of available BESS.At larger volumes, EFR is sufficient even during low-net-demand periods.
D. Value of Dynamically-Reduced Largest Power Infeed
Dynamically reducing the largest power infeed can lower frequency-service needs and operating costs, but its value depends on nuclear-fleet characteristics and wind conditions. Co-optimising EFR and deloading reveals both complementary value and competition when availability is high.
- D. Value of Dynamically-Reduced Largest Power Infeed: Deloading large nuclear units can reduce frequency-service requirements because the largest single-unit infeed drives Great Britain's frequency-response need.The strategy may be cost-effective despite nuclear's low-cost, zero-emissions energy.
- D. Value of Dynamically-Reduced Largest Power Infeed: Considerable operating-cost savings arise from dynamically reducing the largest power infeed, but savings decline with more large units and slower ramp rates.The cases vary deloading capability, nuclear-fleet size, and ramp rate under 40GW of wind.
- D. Value of Dynamically-Reduced Largest Power Infeed: Deloading can reduce carbon emissions by lowering frequency-service needs, keeping fewer part-loaded conventional generators online, and accommodating more renewable energy.This benefit occurs despite reduced output from carbon-free nuclear plants.
- D. Value of Dynamically-Reduced Largest Power Infeed: Adopting nuclear deloading reduces nuclear load factor, and subsidies could reduce the strategy's savings.The paper identifies compensation for reduced nuclear production as a practical economic consideration.
- E. Full Co-Optimisation of Frequency Services: EFR is more beneficial than nuclear deloading at low wind, whereas deloading becomes more valuable at high wind and delivers post-fault frequency response virtually immediately.EFR is cost-free in this framework, while deloading's energy-cost impact depends on whether nuclear output is replaced by thermal generation or wind.
- E. Full Co-Optimisation of Frequency Services: Full Optimisation savings are significantly below the sum of separate EFR and deloading savings in high-availability cases, indicating competition between the services.The comparison uses the “Just PFR” strategy as the reference.
F. Impact of Damping and SI in the Value of Frequency Services
Higher load damping and synthetic inertia reduce the value and required volume of alternative frequency services, while stochastic scheduling yields lower operating cost and wind curtailment than deterministic scheduling. The model also identifies remaining scope for richer frequency products, locational frequency modelling, and pricing analysis.
- F. Impact of Damping and SI in the Value of Frequency Services: Higher load damping leaves EFR and deloading savings largely unchanged when availability is limited but makes them more sensitive when availability increases.The results suggest damping reduces the required service volume without changing the services' competitiveness.
- F. Impact of Damping and SI in the Value of Frequency Services: When wind turbines provide synthetic inertia with a conventional-like inertia time-constant, the benefits and need for alternative frequency services become very limited.With limited synthetic-inertia capability, EFR and nuclear deloading retain clear benefits.
- F. Impact of Damping and SI in the Value of Frequency Services: Deterministic UC produces higher operating cost and wind curtailment than SUC, while yielding lower savings from EFR and part-loading nuclear.The paper attributes this to deterministic UC scheduling more slow-start CCGTs and therefore more inertia.
- F. Impact of Damping and SI in the Value of Frequency Services: The proposed SUC co-optimises uncertain energy production with synchronised and synthetic inertia, PFR, EFR, and dynamically reduced largest power infeed using linear frequency-security constraints.The framework is designed for implementation in computationally demanding stochastic unit commitment.
- V. CONCLUSION AND FUTURE WORK: Future work should address additional delivery-time-distinct frequency products, locational frequency evolution, and pricing schemes for inertia and frequency response.These are identified as three main enhancement areas for the proposed model.