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A Counterexample to Two Representative Unit Aggregation Formulations for Unit Commitment
Zixuan Duan, Zhengshuo Li
TL;DR
Unit aggregation reduces symmetry among identical generators, but exactness requires every aggregate point to admit a feasible unit-level realization. The paper gives a counterexample for PCUC and TUA and tests their aggregate solutions on replication instances. Both formulations can produce optimal aggregate solutions without feasible unit-level realizations.
Problem
Exact aggregation requires every point admitted by an aggregate formulation to have a feasible unit-level realization, a requirement at issue for formulations targeting slow-ramping units.
Source: §I · p. 1, C. Feasible-Region Exactness of Unit… · p. 2Method
The paper constructs an aggregate-feasible counterexample for PCUC and TUA and tests optimal solutions from replication instances for unit-level realizability.
Source: A. A Common Counterexample to PCUC and… · p. 3, B. Further Computational Evidence · p. 3Results
PCUC had no feasible unit-level realization in 5 replication instances, while TUA had none in all 20; the counterexample also shows both formulations admit an unrealizable aggregate trajectory.
Source: B. Further Computational Evidence · p. 3, A. A Common Counterexample to PCUC and… · p. 3Takeaways & Limitations
PCUC and TUA can overestimate individual-unit ramping flexibility, and developing exact, efficiently solvable aggregation models remains a challenge.
Source: A. A Common Counterexample to PCUC and… · p. 3, §IV · p. 3The paper does not present an exact, efficiently solvable model for slow-ramping units; developing one remains a challenge.
Source: §IV · p. 3
Abstract
from arXiv · showhide
Unit aggregation removes symmetry among identical generators in unit commitment, but an aggregate formulation is feasible-region exact only if every aggregate trajectory it admits has a feasible unit-level realization. This letter shows that two commonly used aggregation formulations for slow-ramping units, i.e., p-clustered unit commitment (PCUC) and tight unit aggregation (TUA), cannot satisfy this requirement. We construct a counterexample that satisfies all aggregate constraints of both formulations yet admits no feasible disaggregation. This counterexample reveals a limitation common to both formulations: they do not guarantee intertemporal consistency of unit-level output allocations. Experiments on the replication cases reported in published literature further verify that such infeasibility of disaggregation can even occur in optimal solutions of aggregate models. This letter demonstrates that PCUC and TUA can still overestimate ramping flexibility. Developing exact aggregation models that can be solved efficiently for identical slow-ramping units remains a challenge.
B. Parameters
The paper sets out parameter and variable notation for unit commitment, then motivates and introduces its examination of PCUC and TUA for slow-ramping units.
B. Parameters
- The parameter notation uses P/P for minimum/maximum output, RU/RD for normal ramp limits, SU/SD for startup/shutdown capability, and TU/TD for minimum up/down time.The supplied notation gives paired labels without separate values.Source: B. Parameters · p. 1
C. Variables
- Unit variables u, v, w, and p denote on/off status, startup, shutdown, and above-minimum output; cluster variables U, V, W, and P denote corresponding counts or output.Cluster variables are indexed by cluster k and period t.Source: C. Variables · p. 1
I. INTRODUCTION
- PCUC and TUA are examined because exact aggregation for slow-ramping units remains a challenge, despite the symmetry-reduction goal of aggregation.The paper constructs an aggregate-feasible counterexample with no feasible unit-level realization.Source: §I · p. 1
A. Unit-Level Unit Commitment Formulation
The unit-level formulation minimizes generation and startup costs subject to system-wide constraints and unit operating limits.
A. Unit-Level Unit Commitment Formulation
- The UC objective minimizes total generation cost over scheduling periods, with operating and startup costs.The supplied objective summary names both cost components.Source: A. Unit-Level Unit Commitment… · p. 1
- System-wide constraints include power balance and branch capacity.These are given as examples of system constraints.Source: A. Unit-Level Unit Commitment… · p. 1
- Unit operating constraints govern commitment transitions, minimum up/down times, output bounds, and ramp-rate limits between consecutive intervals.Constraints (3)–(10) cover these operating requirements.Source: A. Unit-Level Unit Commitment… · pp. 1–2
B. CUC Formulation
The CUC formulation aggregates identical units within clusters, replacing unit-level binary decisions with counts and aggregate output variables.
B. CUC Formulation
- CUC replaces unit-level binary commitment variables with integer counts of online, starting, and shutting-down units and aggregate power variables.This eliminates symmetry and reduces the number of discrete variables.Source: B. CUC Formulation · p. 2
- Each cluster contains a set of identical units, and its online-unit count changes by the number starting minus the number shutting down.The variables U, V, and W are bounded by the cluster's unit count.Source: B. CUC Formulation · p. 2
- The CUC constraints are aggregate counterparts of the unit-level operating constraints.The listed aggregate inequalities include ramp-up and ramp-down limits.Source: B. CUC Formulation · p. 2
C. Feasible-Region Exactness of Unit Aggregation
Feasible-region exactness requires both representing every feasible unit schedule and realizing every aggregate point at unit level; the paper distinguishes fast- and slow-ramping clusters.
C. Feasible-Region Exactness of Unit Aggregation
- Feasible-region exactness requires completeness and unit-level realizability: every feasible unit schedule has an aggregate representation, and every admitted aggregate point has a feasible unit realization.The supplied definition introduces these as the two required conditions.Source: C. Feasible-Region Exactness of Unit… · p. 2
- A cluster is fast-ramping when both normal ramp limits cover the full operating range within one period; otherwise it is slow-ramping.This classification partitions the set of clusters.Source: C. Feasible-Region Exactness of Unit… · p. 2
- CUC is guaranteed feasible-region exact only for fast-ramping clusters, while PCUC and TUA were developed to extend aggregation to slow-ramping clusters.The cited passage identifies PCUC and TUA as subsequent approaches for the slow-ramping set.Source: C. Feasible-Region Exactness of Unit… · p. 2
D. PCUC Formulation
PCUC augments clustered unit commitment with ordered virtual units, intending to represent individual-unit ramping flexibility.
D. PCUC Formulation
- PCUC augments CUC with ordered virtual-unit status and above-minimum-output variables.Together with the specified equations, these variables constitute the PCUC formulation.Source: D. PCUC Formulation · p. 2
- The ordering constraint requires each virtual unit's status to be no greater than the preceding unit's status.It applies across clusters and periods.Source: D. PCUC Formulation · p. 2
- PCUC was claimed to represent individual-unit ramping flexibility accurately and overcome CUC's flexibility overestimation.This claim is attributed to.Source: D. PCUC Formulation · p. 2
E. TUA Formulation
TUA treats clusters in two subsets differently: it applies a PCUC-inspired treatment to one and specialized ramping constraints to the other.
E. TUA Formulation
- TUA partitions KS into KR and KL, applying a PCUC-inspired treatment to KL and a separate treatment to KR.For KR, it specifies startup and shutdown ramping horizons.Source: E. TUA Formulation · p. 2
- For clusters in KR, TUA defines total output as above-minimum output plus minimum output times the online-unit count.The multi-period output bound applies to this total output.Source: E. TUA Formulation · p. 2
- TUA includes ramping constraints for clusters in KR.The constraints are attributed to.Source: E. TUA Formulation · p. 3
- The KR constraints (32) and (33) are not equivalent to the corresponding constraints (19) and (20).The cited formulation comprises constraints (12)–(15) and (31)–(33) for KR.Source: E. TUA Formulation · p. 3
- Reference claims every feasible aggregate solution can be disaggregated feasibly, including optimal solutions without loss of optimality.Source: E. TUA Formulation · p. 3
A. A Common Counterexample to PCUC and TUA
A 10-period, two-unit example exposes an aggregate trajectory admitted by both PCUC and TUA that has no feasible unit-level realization.
A. A Common Counterexample to PCUC and TUA
- The example uses two identical units over 10 periods.Source: A. A Common Counterexample to PCUC and… · p. 3
- PCUC and TUA admit the aggregate trajectory (0, 10, 20, 20, 10, 20, 20, 10, 0, 0), but it has no feasible unit-level realization.The result means both formulations fail feasible-region exactness and overestimate individual-unit ramping flexibility.Source: A. A Common Counterexample to PCUC and… · p. 3
- The proof sketch assigns the second unit zero output at periods 4 and 5, then 20 at period 6, exceeding its ramp-up limit.The period-5-to-6 increase is 20 while RU is 10.Source: A. A Common Counterexample to PCUC and… · p. 3
- At period 5, the transition from period 4 requires above-minimum output allocation (10, 0), while the next transition requires (0, 10).Both allocations yield the same aggregate above-minimum output, but no single period-5 allocation satisfies both transitions.Source: A. A Common Counterexample to PCUC and… · p. 3
B. Further Computational Evidence
Disaggregation tests on 20 replication instances found infeasible unit-level realizations among optimal PCUC and TUA aggregate solutions.
B. Further Computational Evidence
- Both formulations were solved to optimality on every one of the 20 replication instances.The instances were reported in,.Source: B. Further Computational Evidence · p. 3
- PCUC solutions lacked feasible unit-level realizations in 5 instances, compared with all 20 TUA instances.Table I reports the disaggregation-test results for both formulations.Source: B. Further Computational Evidence · p. 3, Table I · p. 3
- An aggregate solution was tested for realizability by checking whether unit-level variables could satisfy the unit operating constraints.The test treated δ*=0, with tolerance 10^-4, as the criterion for a feasible realization.Source: B. Further Computational Evidence · p. 3
IV. CONCLUSION
Counterexamples establish that neither PCUC nor TUA is exact, and replication-case tests show both can produce optimal aggregate solutions without feasible unit-level realizations. Efficiently solvable exact aggregation remains an open challenge.
IV. CONCLUSION
- Counterexamples establish that neither PCUC nor TUA is exact.Both formulations admit aggregate solutions that cannot be realized at the unit level.Source: §IV · p. 3
- Replication-case tests, show that both formulations can yield optimal aggregate solutions with no feasible unit-level realization.Source: §IV · p. 3
- Developing exact aggregation models that are efficiently solvable remains a challenge.Future work will investigate such models.Source: §IV · p. 3