Distributed Chance-Constrained Model Predictive Control for Condition-Based Maintenance Planning for Railway Infrastructures

Abstract

We develop a Model Predictive Control (MPC) approach for condition-based maintenance planning under uncertainty for railway infrastructure systems composed of multiple components. Piecewise-affine models with uncertain parameters are used to capture both the nonlinearity and uncertainties in the deterioration process. To keep a balance between robustness and optimality, we formulate the MPC optimization problem as a chance-constrained problem, which ensures that the constraints, e.g., bounds on the degradation level, are satisfied with a given probabilistic guarantee. Two distributed algorithms, one based on Dantzig-Wolfe decomposition and the other derived from a constraint-tightening technique, are proposed to improve the scalability of the MPC approach. Computational experiments show that the distributed method based on Dantzig-Wolfe decomposition performs the best in terms of computational time and convergence to global optimality. By comparing the chance-constrained MPC approaches with deterministic approach, and traditional time-based maintenance approach, we show that despite their high computational requirements, chance-constrained MPC approaches are cost-efficient and robust in the presence of uncertainties.

Appendix

1.1 Parameters for Case Study

See Table 3.

Table 3 Parameters of the functions $f_j^{\text{Deg}}$ and $f_j^{\text{Gr}}$ for five different models. Both the nominal values and the 95% nonsimultaneous confidence bounds (given in the square brackets) are provided for all uncertain parameters

1.2 Cyclic Approach

Let t _0,j denote the time instant of the first replacement on section j. Grinding is performed every T _Gr,j after the first replacement for section j. Furthermore, we assume that replacement is performed after r consecutive grindings since the last replacement on section j. Let k _end denote the planning horizon. Then the offline optimization problem of the cyclic maintenance approach can be formulated as:

$$\displaystyle \begin{aligned} \min_{t_0,\,T_{\text{Gr}},\,r}\sum_{k=1}^{k_{\mathrm{end}}}\sum_{j=1}^{n}x_{j,k}^{\mathrm{con}}+\lambda\sum_{q=2}^{3}c_{q,j}^{\text{Maint}}I_{u_{j,k}=q}{} \end{aligned} $$

(41)

subject to

$$\displaystyle \begin{aligned} x_{j,k+1}&=f_j(x_{j,k},\,u_{j,k};\,\mathbb{E}(\theta_{j,k}))\quad \forall j\in\{1,\dots,n\},\,\forall k\in\{0,\dots,\,k_{\mathrm{end}}-1\}{} \end{aligned} $$

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$$\displaystyle \begin{aligned} x^{\mathrm{con}}_{j,k} & \leq x_{\max}^{\mathrm{con}},\quad x^{\mathrm{aux}}_{j,k} \leq x^{\mathrm{aux}}_{\max}\quad \forall j\in\{1,\dots,n\},\,\forall k\in\{1,\dots,\,k_{\mathrm{end}}\}{} \end{aligned} $$

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$$\displaystyle \begin{aligned} u_{j,k}&=\begin{cases} 2,\, \text{if }(k-t_{0,j})\text{mod } \text{round}(T_{\text{Gr},j})=0\\ 3,\, \text{if } k=t_{0,j}\text{ or }(k-t_{0,j}) \text{mod } \text{round}(rT_{\text{Gr},j})=0\\ 1,\, \text{otherwise} \end{cases}{} \end{aligned} $$

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$$\displaystyle \begin{aligned} &\forall j\in\{1,\dots,n\},\,\forall k\in\{1,\dots,\,k_{\mathrm{end}}\}\notag\\ & 1\leq t_{0,j} \leq T_{\max}\quad \forall j\in\{1,\dots,n\}{} \end{aligned} $$

(45)

$$\displaystyle \begin{aligned} & 1 \leq T_{j,\text{Gr}}\leq T_{\max}\quad \forall j\in\{1,\dots,n\}{} \end{aligned} $$

(46)

$$\displaystyle \begin{aligned} &1 \leq \mu_j\leq \mu_{\max}\quad \forall j\in\{1,\dots,n\}.{} \end{aligned} $$

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