Towards Quantum Stochastic Optimization for Energy Systems under Uncertainty: Joint Chance Constraints with Quantum Annealing

TL;DR

The paper addresses the challenge of solving chance-constrained unit commitment under uncertainty by exploring quantum-annealing-based methods. It develops a scenario-based MILP formulation, extends it to a binary-encoded BLP, and then to a QUBO for quantum annealing, comparing pure quantum, hybrid quantum–classical, and classical solvers. Key findings show that hybrid quantum–classical solvers can rival classical MILP performance on large scenario sets under tight time constraints, while current quantum hardware cannot embed realistic stochastic UCP QUBOs due to embedding and connectivity limits; adaptive penalty tuning improves feasibility in deterministic QUBO tests. The work lays a foundation for quantum-inspired optimization in power systems, highlighting both the potential gains and the hardware-driven limits, and points to hardware advances and transparent hybrid algorithms as necessary for further progress.

Abstract

Uncertainty is fundamental in modern power systems, where renewable generation and fluctuating demand make stochastic optimization indispensable. The chance constrained unit commitment problem (UCP) captures this uncertainty but rapidly becomes computationally challenging as the number of scenarios grows. Quantum computing has been proposed as a potential route to overcome such scaling barriers. In this work, we evaluate the applicability of quantum annealing platforms to the chance constrained UCP. Focusing on a scenario approximation, we reformulated the problem as a mixed integer linear program and solved it using DWave hybrid quantum classical solver alongside Gurobi. The hybrid solver proved competitive under strict runtime limits for large scenario sets (15,000 in our experiments), while Gurobi remained superior on smaller cases. QUBO reformulations were also tested, but current annealers cannot accommodate stochastic UCPs due to hardware limits, and deterministic cases suffered from embedding overhead. Our study delineates where chance constrained UCPs can already be addressed with hybrid quantum classical methods, and where current quantum annealers remain fundamentally limited.

Figures (9)

• Figure 1: Flow diagram of reformulations of deterministic and stochastic UCPs into different instances. Both problems in their MILP formulation were solved using Gurobi for benchmarking purposes. Black dashed box: QUBOs unsolved on D-Wave due to size limitations (see Section \ref{['sec:results_det_formualtion']}).
• Figure 2: Expected cost of the MILP reformulated stochastic UCP as a function of the reliability level $p$ of the joint chance constraints, comparing Gurobi and D-Wave’s hybrid CQM solver. Subfigures correspond to: (a) uncorrelated, (b) moderately correlated, and (c) strongly correlated Gaussian demand distributions. For each instance, 1000 scenarios are considered, and each run has a runtime limit of 20 seconds. For the hybrid solver, results are based on 5 runs, each of them producing an average of about 100 solutions; the reported values correspond to the average of the 5 lowest costs of those runs, with the shaded region indicating the range from best to worst among them. Each D-Wave run included 32 ms of QPU access time.
• Figure 3: Expected cost of the stochastic UCP with moderate correlations as a function of the number of scenarios. Results compare Gurobi with D-Wave’s hybrid solver (MILP and BLP formulations). The shaded areas correspond to error bars indicating variability across runs; missing points for the BLP reflect the absence of feasible solutions. Each run has a runtime limit of 35 seconds. For the hybrid solver, results are based on 4 runs, each of them producing an average of about 100 solutions. The QPU access time of the hybrid solver ranges from 0ms to 70ms (see Section \ref{['sec:discussion']} and Figure \ref{['fig:QPU_time']} for more details).
• Figure 4: Feasibility histogram of the stochastic UCP with 15,200 scenarios in its MILP formulation solved on D-Wave’s hybrid solver. The solver gives 404 solutions, 268 are feasible (orange) and 136 are infeasible (blue). Feasible solutions concentrate near the optimal cost of 254.8 $, while a large number of infeasible solutions are also returned. The best feasible solution found has expected cost 260.2 $.
• Figure 5: Induced QUBO graph for the stochastic UCP considering 10 scenarios. Each node corresponds to a binary decision variable and edges denote quadratic couplings arising from the cost function. The slack variables forming a circle around the graph correspond to those used for the demand inequality constraints.