A Control Co-Design Framework to Achieve Solution Feasibility in Energy System Optimization Problems

Abstract: This work explores methods to identify energy system designs for infeasible control co-design optimization problems. Control co-design, or CCD, has been recognized as a powerful tool to maximize energy system capabilities through simultaneous determination of plant and controller parameters. However, due to the inherent nonlinearities, complexity, and conflicting criteria of energy systems, CCD optimization problems are susceptible to infeasibility and can lack potential solutions. While transforming the optimization problem by relaxing constraints has been developed for optimal control infeasibility challenges, solution feasibility for CCD is relatively unexplored. This paper proposes a framework to convert infeasible optimization problems into solvable forms for a class of CCD problems. The framework introduces a procedure to rank metric bounds from least likely to most likely to cause infeasibility. This provides guidance to algorithmically relax a limited number of constraints, leaving others intact. The proposed framework is applied to a CCD problem for designing a battery within a microgrid. Comparison against a baseline approach for relaxing optimization problems shows the framework requires only a reduced number of iterations to determine a solution.

• The paper proposes a systematic CCD framework that selectively relaxes featured metric constraints to achieve feasibility in energy system optimization.
• It employs an offline ranking and iterative strategy to minimize relaxed constraints, enhancing computational efficiency.
• Benchmarking in a microgrid battery design highlights significant improvements over baseline approaches in maintaining engineering rigor.

Solution Feasibility in Control Co-Design for Energy Systems: A Framework-Based Approach

Introduction

Control co-design (CCD) has advanced as a critical methodology for the simultaneous optimization of plant and controller parameters, offering the potential for substantial improvements in energy system performance compared to traditional sequential design paradigms. Notwithstanding these benefits, the inherent nonlinearities and conflicting performance metrics in energy systems render CCD optimization problems susceptible to infeasibility—instances where no solution satisfies all imposed constraints. Despite significant developments in constrained optimization and infeasibility management for control design, the specific issue of solution feasibility within energy system CCD is under-investigated. This paper presents a structured framework to systematically relax a minimal set of constraints, facilitating solution feasibility for infeasible CCD problems without broadly sacrificing design rigor (2604.13215).

Problem Formulation

The generalized CCD problem is formalized as a nonlinear constraint satisfaction problem, involving a set of decision variables: plant states, control inputs, plant design variables, controller parameters, and a vector of featured metrics (e.g., metrics capturing energy performance, degradation, or emissions). The constraints comprise system dynamics, control laws, and upper/lower bounds on all variables, notably the featured metrics. The essential challenge arises when the bounds, especially those governing featured metrics critical to system viability, are sufficiently restrictive to preclude any feasible solution.

A salient observation is that relaxing equality constraints or bounds on plant and control variables often invalidates physical and engineering considerations; thus, the practical approach is constrained to relaxing featured metrics, with the objective of relaxing as few as necessary.

Motivating Example: Microgrid Battery CCD

The framework is instantiated in the context of a microgrid, where the CCD problem centers on joint battery sizing and controller optimization. A simplified equivalent circuit model of the microgrid is employed, capturing the dynamics of battery charge/discharge and the influence of control input (e.g., buck/boost converters). Two core metrics are originally considered: cumulative state tracking error with terminal cost, and total control effort. The under-constrained nature of this system, with tightly coupled and potentially conflicting metric bounds, leads to infeasibility under realistic parameter selections.

The standard remedy—relaxing all constraints by augmenting the cost function with slack variables weighted by tuning parameters—is shown to lack principled guidance for which constraints to relax and how to parameterize the penalties, resulting in suboptimal or even physically implausible solutions.

The Proposed CCD Feasibility Framework

A multi-step iterative framework is developed to address infeasibility efficiently and systematically:

1. Reformulation with Slack Variables and Selection Binaries: The optimization is augmented with slack variables corresponding to the featured metrics, and binary selection variables ($z_m$) that dictate which constraints are hard (unrelaxed) or soft (relaxed).
2. Ranking Procedure for Constraint Relaxation: An offline ranking step employs a short grid search over design variable space to empirically estimate which metrics are most prone to violate their bounds across candidate designs. Constraints are then prioritized for relaxation based on their degree of historical infeasibility.
3. Iterative Solution Process: The outer loop of the algorithm solves the reformulated problem with an initial set of hard constraints. Upon infeasibility, metrics are relaxed sequentially following the ranking, minimizing the set of violated constraints required to render the problem feasible.

This framework generalizes to higher-order systems and arbitrary numbers and types of metrics.

Evaluation and Baseline Comparison

The extended microgrid example incorporates two additional metrics representing battery degradation and mass (as a proxy for GHG emissions). The ranking step, using parameter sweeps, empirically determines that some metrics (e.g., state tracking error, battery degradation) are more frequently violated and hence should be candidates for early relaxation.

The framework is benchmarked against a baseline approach that relaxes all metric constraints and attempts to tune penalty weights in the objective. A complete combinatorial sweep of penalty weights confirms that the baseline overwhelmingly fails to minimize the number of relaxed constraints and is inefficient, requiring substantially more optimization iterations to match the feasible solutions found by the proposed framework. Only under 5% of baseline trials yielded solutions with the minimal necessary set of hard constraints, confirming that principled constraint prioritization via the proposed ranking is more computationally efficient and structurally aligned with engineering goals.

Implications and Future Directions

This approach offers tangible improvements for energy systems design, especially where trade-offs among technical, economic, and environmental objectives are present and regulatory or physical bounds induce infeasibility. The constraint-ranking methodology enables targeted relaxation, ensuring that solutions maintain engineering significance and only the least critical design metrics are sacrificed for feasibility.

On a theoretical level, the explicit separation of constraints into hard and relaxable via binary selection and empirical ranking offers an extensible template for multi-objective and multi-criteria optimal control, where infeasibility is not uncommon.

Anticipated future research directions include:

• Extension to multi-level, hierarchical optimization for large-scale energy systems with numerous interacting metrics.
• Integration of mandatory, nonzero objectives in tandem with constraint satisfaction, extending the framework's generality.
• Automated, adaptive sampling for the ranking procedure, potentially leveraging surrogate modeling or probabilistic methods.
• Application to real-world industrial energy systems beyond microgrids, such as distributed wind and hybrid vehicle platforms.

Conclusion

The presented framework enables systematic attainment of solution feasibility in nonlinear, constrained CCD problems for energy systems, leveraging a principled ranking of constraint violations and selective relaxation of featured metrics. Numerical experiments underscore its efficiency and efficacy compared to baseline relaxation strategies, with practical implications for advanced, high-performance energy system design. This methodology sets the stage for robust, theoretically justified extensions to large-scale and multi-objective CCD challenges.