Energy-Aware Integrated Proactive Maintenance Planning and Production Scheduling

1. Problem Statement

The paper addresses the complex trade-off manufacturers face when implementing Demand-Side Energy Management (DSM) strategies, specifically Real-Time Pricing (RTP), in the presence of machine degradation.

• The Conflict: RTP allows manufacturers to shift production to low-price electricity hours to save costs. However, machine health degrades over time due to wear and stress, leading to reduced processing capacity and increased energy consumption.
• The Dilemma:
  • Proactive Maintenance (PM) restores machine health but requires production downtime.
  • Scheduling PM during low-price hours sacrifices potential energy savings.
  • Deferring PM to low-price hours leads to operating degraded machines, which increases energy consumption and risks capacity losses.
• The Gap: Existing literature often treats production scheduling and maintenance planning as decoupled problems or assumes static machine health. There is a lack of integrated frameworks that jointly optimize PM and production scheduling while accounting for the bidirectional relationship between production load, machine degradation, and energy efficiency.

2. Methodology

The authors propose a Hierarchical Bi-Level Control Framework that coordinates long-term maintenance planning with short-term production scheduling.

A. System Modeling

The system is modeled as a discrete-time state-space network involving buffers, machines, and product flows. Key models include:

1. Manufacturing Network: A state-space model tracking buffer inventory and material flow.
2. Machine Degradation: A linear model where machine health ($h$) decreases with production throughput ($u$) and increases with maintenance actions ($m$).
3. Performance-Degradation Integration:
   • Capacity: Effective processing capacity decreases linearly as health declines.
   • Energy: Power consumption increases linearly as health declines.
4. Energy Costs: Calculated using RTP, distinguishing between day-ahead forecasts ($\rho_{est}$) for planning and realized real-time prices ($\rho_{act}$) for execution.

B. Hierarchical Control Architecture

The framework operates on two levels:

1. Higher-Level (Offline PM Planning):

   • Objective: Determine the optimal daily PM schedule ($m$) to minimize total operational costs (energy + maintenance labor/opportunity costs).
   • Mechanism: Formulated as a Bi-Level Optimization problem. The upper level selects a PM plan, while the lower level (embedded as a sub-problem) evaluates the feasibility and cost of that plan.
   • Constraints: Crew availability, permissible maintenance windows, and ensuring end-of-day machine health remains above a safety threshold.

2. Lower-Level (Runtime Production Scheduling):

   • Objective: Execute closed-loop production scheduling to minimize energy costs while meeting delivery targets and respecting the PM plan.
   • Mechanism: A Model Predictive Control (MPC) controller that runs hourly.
   • Optimization: It solves a convex Quadratic Program (QP). To handle the bilinear term between machine health and processing rate in the energy cost function, the authors use McCormick envelopes to create a convex relaxation.

C. Solution Algorithm

To solve the bi-level Mixed-Integer Linear Programming (MILP) problem, the authors employ Generalized Benders Decomposition (GBD):

• The Master Problem (Higher Level) proposes a PM plan.
• The Subproblem (Lower Level L-MPC) evaluates the plan's cost and returns dual multipliers (subgradients).
• Benders Cuts are generated from the dual solutions to refine the Master Problem's search space iteratively until convergence.

3. Key Contributions

1. Unified State-Space Model: A novel model linking buffer/machine dynamics, health evolution, and energy use, capturing how degradation impacts both capacity and energy intensity.
2. Hierarchical Bi-Level Framework: A control architecture that explicitly embeds the runtime MPC as a sub-problem within the offline PM planning, ensuring maintenance decisions are "production-aware."
3. Convex Relaxation for Energy Cost: The application of McCormick envelopes to linearize the bilinear interaction between machine health and processing rate, enabling efficient convex optimization within the MPC.
4. Case Study Validation: Demonstration of the framework on a lithium-ion battery pack assembly line, showing significant cost reductions compared to baseline fixed-schedule maintenance.

4. Results

The framework was tested on a 5-day (120-hour) simulation of a battery assembly line using PJM Chicago RTP data. It was compared against a Baseline Controller using a fixed, calendar-based PM schedule (1 hour daily for specific machines).

• Cost Reductions:
  • Total Operational Cost: Reduced by 23.70% ($3,232.91 $\to$ $2,466.62).
  • Energy Cost: Reduced by 5.38%.
  • Maintenance Cost: Reduced by 35.00% (due to fewer maintenance hours: 20 $\to$ 13).
  • Cost per Unit: Reduced by 5.41%.
• Operational Improvements:
  • Strategic Scheduling: The framework shifted PM to high-price RTP hours (sacrificing production during expensive times) and utilized low-price hours for production, unlike the baseline which interrupted production regardless of price.
  • Machine Health: Achieved a 3.45% improvement in average system health, preventing premature degradation.
  • Production Targets: Both controllers met daily delivery targets, but the proposed framework tracked targets more tightly and recovered faster post-maintenance.
• Computational Performance:
  • The 5-day simulation completed in 36.32 seconds.
  • The hourly MPC solve time averaged 0.25 seconds, confirming the approach is computationally tractable for online deployment.

5. Significance

This paper provides a critical advancement in Smart Manufacturing and Industrial Energy Management.

• Bridging the Gap: It successfully bridges the gap between long-term asset management (maintenance) and short-term operational efficiency (production scheduling), a challenge often overlooked in Industry 4.0 applications.
• Economic Viability: It demonstrates that integrating machine health dynamics into energy-aware scheduling yields substantial economic benefits without requiring new capital investment, simply by optimizing decision-making.
• Scalability: While the master problem scales exponentially with the number of machines, the use of decomposition and convex relaxation makes the approach viable for current industrial systems, with clear pathways for handling larger networks through aggregation.

In conclusion, the proposed framework offers a robust, mathematically rigorous method for manufacturers to navigate the complexities of Real-Time Pricing while maintaining equipment reliability, ultimately leading to a more sustainable and cost-effective production environment.