Multi-Level Condition-Based Maintenance Planning for Railway Infrastructures Zhou Su Ali Jamshidi Alfredo N´ u˜ nez Simone Baldi Bart De Schutter Delft Center for Systems & Control Section Railway Engineering Delft University of Technology June 1 Zhou Su (DCSC) LCCC Focus Period June 1 1 / 38
Outline 1 Background 2 Multilevel Maintenance Planning High-Level Problem Low-Level Problem 3 Case Study 4 Conclusions & Future Work Zhou Su (DCSC) LCCC Focus Period June 1 2 / 38
1 Background 2 Multilevel Maintenance Planning High-Level Problem Low-Level Problem 3 Case Study 4 Conclusions & Future Work Zhou Su (DCSC) LCCC Focus Period June 1 3 / 38
Dutch Railway Network Overview • One of the most intensive railway networks in Europe • 6830 km of track, 388 stations, 7508 switches, 4500 km catenary • Maintenance managed by ProRail, performed by contractors Zhou Su (DCSC) LCCC Focus Period June 1 4 / 38
Track Defects & Interventions Ballast Defects& Tamping Squats & Grinding Zhou Su (DCSC) LCCC Focus Period June 1 5 / 38
Optimal Maintenance Planning Preliminaries • A railway network is divided into multiple sections • Deterioration dynamics of each section is stochastic & independent Performance Criteria • Cost-efficiency • Robustness • Scalability Zhou Su (DCSC) LCCC Focus Period June 1 6 / 38
1 Background 2 Multilevel Maintenance Planning High-Level Problem Low-Level Problem 3 Case Study 4 Conclusions & Future Work Zhou Su (DCSC) LCCC Focus Period June 1 7 / 38
Multilevel Scheme for Maintenance Planning High-Level Problem Long-term section-wise intervention plan Motivation Low-Level Problem • Computational tractability • Different time scales Short-term schedule & routes of maintenance crew Condition Infrastructure Network Zhou Su (DCSC) LCCC Focus Period June 1 8 / 38
1 Background 2 Multilevel Maintenance Planning High-Level Problem Low-Level Problem 3 Case Study 4 Conclusions & Future Work Zhou Su (DCSC) LCCC Focus Period June 1 9 / 38
High-level Intervention Planning Problem Description • Minimize expected condition deterioration & maintenance cost • Subject to: • Safety constraints • Resource constraints Model-Predictive Control (MPC) Zhou Su (DCSC) LCCC Focus Period June 1 10 / 38
Deterioration Dynamics Maintenance • Cannot restore to perfect Operational limit σ max condition Hazard Maintenance limit • Becomes less effective the σ r Condition more it is applied Renewal σ 0 • “As good as new” condition Renewal Maintenance Time • Expensive Zhou Su (DCSC) LCCC Focus Period June 1 11 / 38
Deterioration Model Notations � x con � Condition j , k x j , k = : state x aux “Memory” j , k u j , k ∈ { a 1 , a 2 . . . , a N } : N maintenance options ���� � �� � No maintenance Interventions θ j , k ∈ Θ j : bounded uncertain parameters with unknown probability distribution Stochastic Deterioration Model x j , k +1 = f j ( x j , k , u j , k , θ j , k ) f 1 j ( x j , k , θ j , k ) if u j , k = 1 Natural degradation f q = j ( x j , k , θ j , k ) if u j , k = q Effect of maintenance ∀ q ∈ { 2 , . . . , N − 1 } f N j ( θ j , k ) if u j , k = N Effect of renewal Zhou Su (DCSC) LCCC Focus Period June 1 12 / 38
Stochastic Local MPC Problem Chance-Constrained MPC Problem u j , k , ˜ min x j , k E ˜ θ j , k [ J j (˜ x j , k , ˜ θ j , k )] u j , k , ˜ ˜ x j , k = ˜ u j , k , ˜ subject to: ˜ f j (˜ θ j , k ; x j , k ) � � u j , k , ˜ x con max ˆ j , k + l | k (˜ θ j , k ; x j , k ) ≤ x max ≥ 1 − ǫ j Chance Constraint P ˜ θ j , k l =1 ,..., N P where ǫ j is the violation level Robust MPC Problem u j , k , ˜ min x j , k max J j (˜ x j , k , ˜ θ j , k ) u j , k , ˜ ˜ θ j , k ∈ ˜ ˜ Θ j x j , k = ˜ u j , k , ˜ subject to: ˜ f j (˜ θ j , k ; x j , k ) u j , k , ˜ x con max max ˆ j , k + l | k (˜ θ j , k ; x j , k ) ≤ x max Robust Constraint l =1 ,..., N P θ j , k ∈ ˜ ˜ Θ j Zhou Su (DCSC) LCCC Focus Period June 1 13 / 38
Stochastic Local MPC Problem Chance-Constrained MPC Problem u j , k , ˜ min x j , k E ˜ θ j , k [ J j (˜ x j , k , ˜ θ j , k )] u j , k , ˜ ˜ x j , k = ˜ u j , k , ˜ subject to: ˜ f j (˜ θ j , k ; x j , k ) � � u j , k , ˜ x con max ˆ j , k + l | k (˜ θ j , k ; x j , k ) ≤ x max ≥ 1 − ǫ j Chance Constraint P ˜ θ j , k l =1 ,..., N P where ǫ j is the violation level Robust MPC Problem u j , k , ˜ min x j , k max J j (˜ x j , k , ˜ θ j , k ) u j , k , ˜ ˜ θ j , k ∈ ˜ ˜ Θ j x j , k = ˜ u j , k , ˜ subject to: ˜ f j (˜ θ j , k ; x j , k ) u j , k , ˜ x con max max ˆ j , k + l | k (˜ θ j , k ; x j , k ) ≤ x max Robust Constraint l =1 ,..., N P θ j , k ∈ ˜ ˜ Θ j We choose chance-constrained MPC in order to avoid conservatism Zhou Su (DCSC) LCCC Focus Period June 1 13 / 38
Scenario-based Approach Scenario-based Approach Approximate chance constraint by set of deterministic constraints : u j , k , ˜ θ ( h ) x con max ˆ j , k + l | k (˜ j , k ; x j , k ) ≤ x max ∀ h ∈ H j l =1 ,..., N P H j : set of random scenarios for section j ; ˜ θ ( h ) j , k : realization of ˜ θ ( h ) j , k in scenario h Remarks • Sufficiently large |H j | gives the probabilistic guarantee : � � � � u j , k , ˜ x con max ˆ j , k + l | k (˜ θ j , k ; x j , k ) ≤ x max ≥ 1 − ǫ j ≥ 1 − β j P h P ˜ θ j , k l =1 ,..., N P • Integer decision variables → non-convex chance-constrained MPC problem • Existing bounds on |H j | for non-convex chance-constrained problem are conservative Zhou Su (DCSC) LCCC Focus Period June 1 14 / 38
Scenario-based Approach Scenario-based Approach Approximate chance constraint by set of deterministic constraints : u j , k , ˜ θ ( h ) x con max ˆ j , k + l | k (˜ j , k ; x j , k ) ≤ x max ∀ h ∈ H j l =1 ,..., N P H j : set of random scenarios for section j ; ˜ θ ( h ) j , k : realization of ˜ θ ( h ) j , k in scenario h Remarks • Sufficiently large |H j | gives the probabilistic guarantee : � � � � u j , k , ˜ x con max ˆ j , k + l | k (˜ θ j , k ; x j , k ) ≤ x max ≥ 1 − ǫ j ≥ 1 − β j P h P ˜ θ j , k l =1 ,..., N P • Integer decision variables → non-convex chance-constrained MPC problem • Existing bounds on |H j | for non-convex chance-constrained problem are conservative We choose the two-stage approach from Margellos et al. (2014) Zhou Su (DCSC) LCCC Focus Period June 1 14 / 38
Two-Stage Approach Stage 1 Generate H j satisfying � 1 � �� e Θ j | − 1 + ln 1 2 | ˜ |H j | ≥ ǫ j · e − 1 β j and solve the convex scenario-based optimization problem | ˜ Θ j | � min τ i − τ i | ˜ Θj | { ( τ i , τ i ) } i =1 i =1 subject to: (˜ θ j , k ) ( h ) ∀ h ∈ H , ∀ i ∈ { 1 , . . . , | ˜ ∈ [ τ i , τ i ] Θ j |} i to obtain the smallest hyperbox B ∗ j , k covering all scenarios in H j . Probabilistic Guarantee � � � � θ j , k ∈ B ∗ ˜ ≥ 1 − ǫ j ≥ 1 − β j P h P ˜ j , k θ j , k Zhou Su (DCSC) LCCC Focus Period June 1 15 / 38
Two-Stage Approach Stage 2 Solve the resulting robust optimization problem 1 � x ( h ) min J j (˜ j , k , ˜ u j , k ) |H j | x ( h ) u j , k , ˜ ˜ h ∈H j j , k x con u j , k , ˜ subject to: max max ˆ j , k + l | k (˜ θ j , k ; x j , k ) ≤ x max Robust Constraint l =1 ,..., N P θ j , k ∈B ∗ ˜ j , k ∩ ˜ Θ j x ( h ) j , k = ˜ u j , k , ˜ θ ( h ) ˜ f j (˜ j , k ; x j , k ) ∀ h ∈ H j Remarks • Less conservative than standard robust approach • Tractability depends on the robust problem Zhou Su (DCSC) LCCC Focus Period June 1 16 / 38
Worst-Case Scenario Worst-Case Scenario • Define the worst-case scenario ˜ θ ( w ) u j , k , ˜ x con j , k ∈ arg max max ˆ j , k + l | k (˜ θ j , k ; x j , k ) l =1 ... N P θ j , k ∈B ∗ ˜ j , k ∩ ˜ Θ j • ˜ θ ( w ) j , k + l | k is concave w.r.t. ˜ x con j , k i is easy to obtain if ˆ θ j , k . Zhou Su (DCSC) LCCC Focus Period June 1 17 / 38
Worst-Case Scenario Worst-Case Scenario • Define the worst-case scenario ˜ θ ( w ) u j , k , ˜ x con j , k ∈ arg max max ˆ j , k + l | k (˜ θ j , k ; x j , k ) l =1 ... N P θ j , k ∈B ∗ ˜ j , k ∩ ˜ Θ j • ˜ θ ( w ) j , k + l | k is concave w.r.t. ˜ x con j , k i is easy to obtain if ˆ θ j , k . Sufficient Condition on Concavity � f con � ( x j , k , u j , k , θ j , k ) j , if f con and f aux For f j ( x j , k , u j , k , θ j , k ) = are f aux j j ( x j , k , u j , k , θ j , k ) j • concave in ˜ θ j , k • concave and non-decreasing in every dimension of x j , k j , k + l | k is concave in ˜ x con then ˆ θ j , k for any l = 1 , . . . , N P . Zhou Su (DCSC) LCCC Focus Period June 1 17 / 38
Scenario-based Robust MPC Deterministic MPC Problem 1 � x ( h ) min J j (˜ j , k , ˜ u j , k ) |H j | x ( h ) u j , k , ˜ ˜ h ∈H j j , k x ( w ) subject to: P j ˜ j , k ≤ x max ∀ w ∈ W j x ( s ) j , k = ˜ u j , k , ˜ θ ( s ) ∀ s ∈ H j ∪ { ˜ θ ( w ) ˜ f j (˜ j , k ; x j , k ) j , k } � �� � Deterministic prediction model for scenario s Zhou Su (DCSC) LCCC Focus Period June 1 18 / 38
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