Two-stage monitoring design and budgeted condition-based maintenance for LED lighting systems: a gamma-process-based ensemble Kalman filter approach

Technical Summary: Two-Stage Monitoring Design and Budgeted Condition-Based Maintenance for LED Lighting Systems

Problem Statement
Light-emitting diode (LED) lighting systems degrade over time, reducing working plane (WP) illuminance and increasing the risk of inadequate indoor lighting. While stochastic degradation models can predict general trends, they often lack accuracy for individual installed luminaires due to unit-to-unit variability and uncharacterized operating conditions. Direct measurement of individual luminaire degradation states (e.g., via integrating spheres) is impractical post-installation. Conversely, in situ WP illuminance measurements are practical but present a complex inverse problem: a single WP measurement reflects the superposed contributions of all luminaires in a zone rather than a local sensor-luminaire relationship.

This creates a dual challenge:

1. Inverse State Estimation: Inferring high-dimensional latent degradation states from sparse, globally coupled WP measurements requires an informative measurement layout that satisfies identifiability constraints (the number of measurements must at least equal the number of luminaires for a standalone estimate).
2. Budgeted Maintenance Optimization: Large-scale systems cannot afford to measure the full set of required points at every monitoring visit. There is a need to jointly optimize the visit schedule, the subset of measurements taken at each visit, and the preventive replacement policy under a fixed monitoring budget to minimize total downtime and replacement costs.

Methodology
The paper proposes a two-stage framework integrating design-stage layout optimization with runtime-stage sequential filtering and condition-based maintenance (CBM).

• Stage 1: Design-Stage Reference Layout Optimization

  • State-to-Illuminance Modeling: A nonhomogeneous gamma process models the monotonic degradation of each luminaire. A high-fidelity Radiance simulator generates forward illuminance data, which is approximated by a linear-Gaussian surrogate model ($E = Cx + b + \epsilon$). This explicit observation model links latent degradation states ($x$) to WP illuminance measurements ($E$), quantifying both model and measurement uncertainties.
  • Identifiability-Constrained D-Optimal Design: To enable inverse state estimation, the framework formulates the selection of WP measurement points as an optimal experimental design problem. It enforces a minimum measurement count ($m \ge J$, where $J$ is the number of luminaires) to ensure the Fisher Information Matrix is nonsingular. A D-optimal criterion (maximizing $\log \det(H^T \Omega^{-1} H)$) is used to select an informative "reference layout" ($\mathcal{S}_{ref}$) from a candidate grid. Heuristic strategies, including Hybrid Greedy Search (HG) and Genetic Algorithms (GA), are employed to solve this NP-hard combinatorial problem.

• Stage 2: Runtime Monitoring and Budgeted CBM

  • Sequential Filtering: An Ensemble Kalman Filter (EnKF) is used to track degradation states. Unlike standard Kalman filters, the EnKF handles the non-Gaussian nature of gamma process increments by propagating an ensemble of trajectories. It assimilates sparse WP measurements collected at runtime, combining them with prior information propagated from previous visits. This allows for accurate state estimation even when the number of measurements per visit is below the single-epoch identifiability limit.
  • Measurement Design and Policy Optimization: The runtime stage determines when to visit and which subset of the reference layout to measure. A "zone cyclic" rule partitions the reference layout into $n_{zone}$ zones, measuring one point from each zone per visit.
  • Risk-Based Maintenance: The posterior predictive distribution of degradation states is used to calculate the probability of a luminaire exceeding a failure threshold before the next visit. A preventive replacement is triggered if this probability exceeds a risk threshold ($p_{th}$).
  • Joint Optimization: The framework optimizes the visit schedule (first visit time, interval), the active measurement count ($n_{zone}$), and the risk threshold ($p_{th}$) simultaneously. This is solved using a Sequential Grid Search with Paired Screening (SGS-PS) algorithm to minimize expected total downtime and replacement costs subject to a monitoring budget ($B_{mon}$).

Key Contributions

1. Explicit Observation Model: Development of a linear-Gaussian surrogate model derived from Radiance simulations to map latent degradation states to WP illuminance, enabling inverse state estimation with quantified uncertainties.
2. Identifiability-Constrained Layout Design: Formulation of the design-stage measurement layout as a D-optimal experimental design problem constrained by the minimum measurement count required for single-epoch identifiability, resulting in an informative reference layout.
3. Sequential EnKF for Sparse Monitoring: Demonstration that an EnKF can effectively track degradation states using sparse measurements (fewer than the number of luminaires) by leveraging temporal information, thereby reducing runtime measurement effort.
4. Budgeted Joint Optimization: A unified framework that jointly optimizes the monitoring visit schedule, active measurement selection, and CBM risk threshold under a budget constraint, explicitly addressing the trade-off between temporal update frequency and spatial measurement information.

Results
The framework was evaluated in an office lighting zone case study with 76 luminaires and a 50-year horizon.

• Layout Selection: The Hybrid Greedy Search (HG) layout outperformed Nadir-point and Genetic Algorithm layouts in terms of the D-optimal objective, providing a reference layout that minimized generalized estimation uncertainty.
• Policy Performance: Under a base monitoring budget, the optimal policy utilized an intermediate active measurement count ($n_{zone}=20$) with a 3-month visit interval and a risk threshold of 0.12. This configuration prioritized frequent state updates over measuring the full reference layout.
• Comparison with Benchmarks: Compared to a block-replacement (time-based) benchmark, the proposed budgeted CBM policy reduced total downtime and replacement costs by 10.2% under baseline uncertainty and by 40.5% under enlarged hitting-time uncertainty.
• Sensitivity Analysis:
  • Uncertainty: Increased uncertainty in degradation hitting times shifted the optimal policy toward more aggressive risk thresholds and higher active measurement counts, as the value of condition information increased.
  • Budget: Under tight budgets, the framework prioritized visit frequency (temporal updates). As the budget increased, the optimal policy shifted toward measuring the full reference layout ($n_{zone}=76$) while maintaining frequent visits, indicating that spatial information becomes beneficial only when temporal frequency is not compromised.

Significance
The paper claims that its significance lies in providing a systematic approach to the inverse problem of inferring luminaire degradation from indirect, globally coupled measurements. By separating the design of an informative reference layout from the runtime selection of sparse measurements, the framework overcomes the economic and logistical barriers of full-system monitoring. The integration of sequential filtering with budget-constrained optimization demonstrates that reliable condition monitoring and cost-effective maintenance are achievable even when measurement resources are limited, offering a scalable solution for large-scale LED lighting infrastructure.