Surrogate Models for LED Performance Mapping

• Surrogate-based performance mapping is a method that replaces detailed simulations with efficient models to estimate system-level LED performance.
• It integrates Gamma process degradation, Bayesian calibration with Arrhenius scaling, and a linear surrogate to achieve rapid and reliable maintenance evaluations.
• The approach cuts computation time dramatically, supporting thousands of multi-objective Monte Carlo policy evaluations and Pareto-front analyses for cost-effective scheduling.

Surrogate-based performance mapping denotes the practice of replacing computationally intensive physical simulations with efficient surrogate models to project system-level metrics, specifically for facilitating scalable optimization and policy evaluation. Within the context of maintenance optimization for large-scale LED lighting systems, the approach addresses the challenge of rapidly evaluating the spatio-temporal illuminance performance in response to stochastic degradation and failure, with negligible @@@@1@@@@ loss compared to full ray-tracing simulation. This enables thousands of policy evaluations in multi-objective discrete-event Monte Carlo frameworks, supporting decision support through Pareto-front analysis of maintenance costs and illumination compliance (Shi et al., 14 Jan 2026).

1. System Overview and Degradation Modeling

LED lighting systems experience both gradual degradation of package lumen output and abrupt driver outages. Package degradation is modeled as a non-homogeneous Gamma process: for package $j$, the cumulative loss-of-light $X_j(t)$ at time $t$ is defined as $X_j(t) = 1 - P_{j, \mathrm{out}}(t)$, with $0 \leq P_{j, \mathrm{out}}(t) \leq 1$ and failure at $L_{70}$ corresponding to $X_j(t) > 0.30$. The increments $$X_j(t) - X_j(s) \sim \mathrm{Gamma}(\alpha(t) - \alpha(s), \beta)$$ are independent, encoding both the systematic exponential lumen-maintenance trend ($\alpha(t) = A e^{b t}$) and the stochastic path uncertainty crucial for risk-aware decision-making. The mean and variance are $\mu(t) = \alpha(t)/\beta = A e^{b t}/\beta$ and $\mathrm{Var}[X_j(t)] = A e^{b t}/\beta^2$, respectively. Driver failures are described by a Weibull model with scale $\lambda$ and shape $\eta$, yielding a two-mode degradation/failure representation suitable for competing-risk analysis.

2. Bayesian Calibration and Uncertainty Incorporation

The parameterization $(A, b, \beta)$ for package degradation, and $(\lambda, \eta)$ for driver Weibull models, is calibrated via Bayesian inference on LM-80 accelerated degradation test (ADT) data. The Gamma-process parameters' dependence on temperature is modeled with an Arrhenius law: $\beta(T) = C \exp( E_a / (k_B T) )$, with $C$ the pre-exponential factor, $E_a$ activation energy, and $k_B$ Boltzmann’s constant. Observed increments $\Delta x_k$ are used in a likelihood construction over the data $D = \{ (x_{k-1}, x_k, t_{k-1}, t_k, T_k) \}$, with priors reflecting physical monotonicity and activation constraints. Markov chain Monte Carlo (CmdStanPy) draws are used both for epistemic parameter samples and aleatory path trajectories, thus fully propagating model and process uncertainty into system-level evaluations.

3. Surrogate Mapping: Illuminance and Deficiency Metrics

System-level performance is defined by compliance with spatio-temporal illuminance and uniformity requirements on a working plane. The current state vector $L(t)$ (across luminaires) is mapped to normalized output scales $Q(t) = 1_J - L(t)$. Performance indices are computed on a spatial grid: average illuminance $E_\mathrm{avg}(t) = (1/N) \sum_{i=1}^N E_i(t)$ and uniformity $U(t) = E_\mathrm{min}(t) / E_\mathrm{avg}(t)$. Thresholds $S_E$ (illuminance) and $S_U$ (uniformity) define durations $T_\mathrm{defi}(k)$ in each sampling interval where performance criteria are unfulfilled:

$$T_\mathrm{defi}(k) = \max\{ \Delta t \cdot I[E_\mathrm{avg} < S_E],\; \Delta t \cdot I[U < S_U] \}$$

The overall deficiency ratio $$R_\mathrm{DR} = \sum_{k=1}^K T_\mathrm{defi}(k) / T_\mathrm{over}$$ quantifies the fraction of the operational horizon spent in noncompliance.

Full ray-tracing simulation for every policy evaluation is computationally prohibitive given stochastic paths. A trained linear surrogate,

$E(t) = b_E + C_E Q(t),$

replaces ray-tracing, yielding per-snapshot efficiency gain from minutes to milliseconds and enabling massive Monte Carlo policy search with negligible performance loss compared to high-fidelity models.

4. Maintenance Policy Optimization Framework

Performance-driven maintenance optimization is formulated as a multi-objective discrete-event Monte Carlo problem. Two central decision variables control the opportunistic policy: preventive-maintenance interval $T_\mathrm{PM}$, and the opportunistic threshold $H_\mathrm{OM}$. At each corrective maintenance (CM) or scheduled preventive visit, luminaires with remaining time-to-PM less than $H_\mathrm{OM} T_\mathrm{PM}$ are also replaced, reducing site visits at the cost of potentially increased replacement volume. The framework evaluates (for $n_\mathrm{path} \times n_\mathrm{ps}$ model trajectories) the trade-offs across deficiency ratio $R_\mathrm{DR}$, number of visits, and number of replacements per policy setting, with Pareto-front analysis supporting operational decision-making.

5. Key Parameters: Physical Interpretation and Impact

Parameter impacts are critical for mapping degradation and optimizing policies:

Parameter	Physical Role	Impact on Performance Mapping
$A$	Early-time amplitude	Larger $A$ accelerates early degradation variability
$b$	Exponential growth rate	Larger $b$ increases both mean degradation and variance
$\beta$	Rate (reciprocal units)	Larger $\beta$ (less severe) lowers both mean/variance at given $t$
$C$, $E_a$	Stress acceleration	Larger $E_a$ increases temperature sensitivity
$\lambda$, $\eta$	Weibull parameters	Control rate and shape of abrupt driver failures

The configuration of $(A, b, \beta)$ governs the stochastic evolution of lumen depreciation and thus directly influences the deficiency ratio and associated maintenance costs. The Weibull model for drivers encodes hard failure timing. Maintenance policy trade-offs arise from these physical/statistical underpinnings: higher $A$ or $b$ bias toward faster preventive cycles, while stress-acceleration (especially via $E_a$) can make temperature control pivotal.

6. Assumptions, Limitations, and Possible Extensions

The surrogate-based performance mapping framework rests on several structural assumptions:

• Linear acceleration: Only $\beta$ (not $\alpha(t)$) is varied across stress levels via Arrhenius scaling.
• Exponential shape function: $\alpha(t) = A e^{b t}$ models mean lumen loss; does not encompass saturation or initial transient behavior.
• Field heterogeneity: All packages assumed drawn from a single posterior parameter set; real-world variation might require clustered or hierarchical inference.
• Full renewal: Only full luminaire replacements are modeled; no partial repairs or component-level interventions.
• Extensions may include alternative semi-physical shape functions (e.g., two-phase kinetics), step-stress protocols, explicit degradation-to-threshold hazard modeling, and more detailed cost structures (inventory, capacity, downtime penalties).

A plausible implication is that, while the surrogate approach dramatically accelerates policy search and uncertainty propagation, the modeling choices limit its accuracy in regimes exhibiting strong non-exponential degradation or field heterogeneity.

7. Significance in Maintenance Optimization and System Reliability

Surrogate-based mapping offers an effective method for propagating stochastic component-level degradation and failures into actionable system-level performance metrics, suitable for maintenance optimization in large-scale, sensor-rich systems. By recasting illuminance mapping as a linear surrogate, the framework achieves massive efficiency gain and supports comprehensive decision support analyses such as Pareto-optimal trade-off evaluation. This principled integration of semi-physical Gamma process modeling, Arrhenius stress translation, Bayesian parameter learning, and surrogate-based performance mapping constitutes a robust methodological foundation for risk-aware, cost-efficient maintenance scheduling in engineered systems requiring spatio-temporal compliance (Shi et al., 14 Jan 2026).