An Efficient Method for the Optimal Control of Microgrids Under Uncertainties using Local Reduction

Edoardo Scaccia; Eric C. Kerrigan; Anna Sadowska

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An Efficient Method for the Optimal Control of Microgrids Under Uncertainties using Local Reduction

Edoardo Scaccia, Eric C. Kerrigan, Anna Sadowska

arXiv:2606.12345v1Jun 10, 2026math.OC

TL;DR

Robust microgrid sizing and scheduling must handle uncertain demand, renewable generation, prices, and battery efficiencies. The paper compares MILP and smooth NLP formulations solved by local reduction, with both achieving average physical feasibility rates above 90% in 100,000-sample simulations.

Problem
Robust microgrid sizing and scheduling requires control decisions that remain feasible across uncertainties in demand, renewable generation, prices, efficiencies, and other system conditions.
Method
The paper compares MILP and smooth nonconvex NLP formulations with embedded logical constraints, solving both through an extended nested local reduction scheme.
Results
Above 90% average physical feasibility rates were achieved by both formulations in 100,000-sample Monte Carlo simulations.
Takeaways & Limitations
Local reduction obtained robust solutions using at most 10 generated scenarios, compared with 2^N = 2570 extreme scenarios for full box-set enumeration.
Takeaways & Limitations
The robustness of the MILP solutions remains an issue, and specialized solvers may be needed to improve computational speed.

Abstract

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The problem of optimal sizing and power scheduling in microgrids subject to uncertainties is well known to the control community. Commonly, the optimal control problem is cast as a mixed-integer program to model the logical constraints arising in energy storage systems, and is then solved approximately using numerical methods such as the scenario approach. In this paper, we propose and compare two formulations of a robust microgrid sizing and power scheduling optimal control problem with logical constraints and uncertainties in the user's power demand, solar power generation, grid electricity prices and battery efficiencies. The first formulation uses binary variables and big-M constraints, leading to a mixed-integer linear program. The second formulation casts the problem as a continuous nonlinear program through an exact smooth reformulation of the logical constraints, consisting of additional modelling variables and non-convex constraints. We then propose a novel local reduction algorithm, extending an existing method, to solve both problems. The two formulations are compared by evaluating the solutions returned by local reduction using 100,000-sample Monte Carlo simulations and achieve promising results, with both averaging feasibility rates above 90%.

1. INTRODUCTION

Microgrids integrate energy storage, renewable and conventional generation, and users to meet variable electricity demand while minimizing operating costs. Their optimal control is challenging because uncertainty across generation, demand, prices, system parameters, failures, degradation, and maintenance requires robust input sequences.

Motivation: 30% of global final energy consumption is attributed to residential and commercial buildings, while global electricity demand grows at an annual average rate of 3.4%.The passage links rising population and energy consumption with a shift toward renewable generation.
Microgrids: Microgrids use advanced control algorithms to generate, store, and deploy electricity for varying end-user demand while minimizing operational costs.Their components include energy storage systems, renewable and non-renewable energy sources, and users.
Microgrids: Grid-connected microgrids can trade power with the electricity market, whereas islanded microgrids rely solely on renewable and non-renewable sources to meet user demand.The passage describes these as the two operating modes of microgrids.
Control challenge: Robust microgrid optimal control must address uncertainties in renewable generation, user demand, electricity prices, system parameters, sensor readings, failures, capacity degradation, and maintenance schedules.The required input sequence must remain robust to every uncertainty realisation.

2. STATE OF THE ART

The state of the art in robust microgrid optimal control centers on system, uncertainty, and optimization modelling. Existing work commonly represents storage complementarity with binary variables, models uncertainty through min-max, robust adaptable, or probabilistic formulations, and uses duality or approximation methods for solution.

Research focus: Robust microgrid optimal control is organized around system modelling, uncertainty modelling, and optimization techniques.Energy storage systems receive substantial modelling attention because of their charge/discharge complementarity.
System and uncertainty modelling: Binary variables and mixed-integer programming commonly represent energy-storage charge/discharge complementarity.Renewable generation, users’ power demand, and electricity prices are the main uncertainty sources addressed in the literature.
System and uncertainty modelling: Uncertainty is modelled using min-max, robust adaptable or multi-stage, and probabilistic formulations.These approaches address uncertainty in renewable energy sources, users’ power demand, and electricity prices.
Optimization techniques: Strong duality converts the inner maximization of many linear and convex-concave quadratic min-max problems into standard minimization problems.Scenario and Monte Carlo methods also provide tractable approximate solutions for min-max and probabilistic problems.

3. CONTRIBUTIONS

The paper develops a robust microgrid sizing and scheduling framework under uncertainty using complementary MILP and continuous NLP formulations. It also incorporates efficiency uncertainty into ESS dynamics and extends local reduction to handle existence constraints through nested optimization.

Formulations: The paper proposes MILP and continuous NLP formulations for encoding logical constraints on battery and grid controls.The MILP uses binary variables and big-M constraints, while the NLP uses an exact smooth reformulation with auxiliary variables and non-convex constraints.
Uncertainty: Uncertainty is included directly in ESS dynamics through battery efficiencies, alongside demand, solar generation, and electricity price uncertainty.
Local reduction: An extension of local reduction handles existence constraints through a logical reformulation of the lower-level maximization problem, producing a nested local reduction scheme.

4. PROBLEM FORMULATION

The paper formulates a robust optimal-control problem for a simplified grid-connected microgrid with uncertain demand, solar generation, prices, and storage efficiencies. It models storage and grid operating logic through either MILP big-M constraints or an equivalent smooth NLP formulation, then discretizes the resulting problem over a one-day horizon.

4.1 Model Assumptions and Parameters: The simplified model contains a single grid-connected user, an energy-storage system, and a solar photovoltaic renewable-energy source.The formulation is intended to remain valid for more general microgrid models.
4.1 Model Assumptions and Parameters: Demand, photovoltaic generation, grid purchase and sale prices, and storage charging and discharging efficiencies are uncertain.Storage dynamics are linear, while simultaneous charging/discharging and simultaneous grid buying/selling are prohibited.
4.2 Microgrid Dynamics: The microgrid dynamics comprise an ODE for storage energy and an algebraic power-balance equation with grid power split into positive and negative components.The decision variables s and b scale photovoltaic generation and storage capacity and maximum powers, respectively.
4.2 Microgrid Dynamics: The formulation imposes bounds on storage energy, charging and discharging powers, and positive and negative grid powers.The grid-trading fraction is represented by g, with the base grid-power limit set to peak demand over the considered horizon.
4.2 Microgrid Dynamics: The discontinuous logical constraints are reformulated either with binary variables and big-M constraints as an MILP or with additional continuous variables and non-convex smooth constraints as an NLP.The NLP uses variables λ1(t) and λ2(t) constrained to [0,1], while retaining constraints (4)–(7).
4.3 Robust Optimal Control Problem: The dynamics are discretized over a one-day scheduling window from t0 = 00:00 to tf = 24:00 using 15-minute intervals, yielding N = 95 grid points.The objective minimizes initial investment and operational cost.
4.3 Robust Optimal Control Problem: The uncertainty set is polyhedral, although the proposed method remains valid for any compact uncertainty set, and the robust formulation is an existence-constrained semi-infinite program.The MILP and NLP cases use X and Λ, respectively, for their logic-variable sets.

5. THE LOCAL REDUCTION ALGORITHM

The local reduction algorithm iteratively builds finite uncertainty sets by solving master problems and searching for violating scenarios. Its nested formulation handles the disjunctive lower-level problem through epigraph and big-M reformulations, repeating scenario additions until convergence or iteration limits.

Upper-level local reduction: The algorithm discretizes the uncertainty set with finite scenarios, solves the resulting master problem, and searches for a violating scenario at the current solution.This process iteratively approximates the original uncountable uncertainty set.
Lower-level reformulation: The lower-level disjunction is represented using an epigraph reformulation and binary variables with big-M constraints.At least one element of the constraint vector must exceed the lower-bound cost variable.
Nested local reduction: The reformulated lower-level problem is itself a nested ECSIP solved by local reduction over a finite set of added scenarios.Violating scenarios are found by maximizing the lower-level constraint expression at the current solution.
Stopping conditions: Each newly found violating scenario is added to the master problem, and the procedure repeats until no violations remain or the maximum scenario count is reached.The complete routine is summarized in Algorithm 1 using upper- and lower-level local reduction loops.

6. RESULTS

Results show that MILP and NLP achieve similar feasibility, while MILP is substantially faster and NLP yields lower average constraint violations. The NLP formulation also produces more infeasible sizing cases and more balanced battery–PV allocations than MILP.

Experimental setup: The numerical study implements Algorithm 1 in Julia with JuMP, Ipopt, and Gurobi on a Microsoft Surface Studio 2 laptop.The implementation and data used to generate the numerical results are publicly available on GitHub.
Evaluation: 100,000 uniformly random scenarios assess solution quality across 25 combinations of grid power cap and initial SoC.The cap factor g and initial state of charge SoC0 are independently varied from 20% to 100%.
Comparative performance: MILP and NLP achieve similar feasibility scores, but MILP is substantially faster.The speed difference is attributed to Gurobi solving linear MILPs, whereas a general-purpose interior-point solver handles non-convex bilinear NLPs.
Constraint violations: 0.099% of total constraints violated for NLP compared with 0.56% for MILP.The most violated constraints are cost and grid bounds for MILP and NLP, respectively; MILP returns no SoC-bounds violations.
Sizing decisions: NLP returns more infeasible sizing cases than MILP and produces a more balanced number of battery packs and PV units.MILP instead selects substantially more batteries relative to PV units, especially in the upper-left part of the grid.

7. CONCLUSIONS

The paper applies local reduction with MILP and exact smooth NLP formulations to robust microgrid sizing and scheduling under uncertainty. Both formulations exceed 90% average physical feasibility in 100,000-sample Monte Carlo simulations, while several extensions remain for improving robustness, modeling, and comparison.

Conclusions: Both the proposed MILP and NLP formulations achieve average physical feasibility rates above 90% in 100,000-sample Monte Carlo simulations.The NLP formulation embeds logical constraints through an exact smooth reformulation and is solved via local reduction.
Conclusions: The paper is the first to formulate a microgrid optimal control problem under uncertainty as an NLP with embedded logic and solve it via local reduction.The approach targets robust microgrid sizing and scheduling with logical constraints.
Future research: Future work should address the robustness of the MILP solutions and solve the same problem over longer horizons with a nonlinear battery model.These extensions target limitations of the current formulations and broader operating conditions.
Future research: Future work could optimize SoC0 with a terminal cost penalizing deviations from a final reference and develop specialized interior point solvers for the NLP formulation.The proposed solvers would exploit the resulting bilinear structure to improve NLP efficiency.
Future research: These extensions would enable a more meaningful comparison between the NLP and MILP formulations.The comparison would follow improvements in robustness, horizon length, battery modeling, initialization, and solver specialization.