Matheuristics to optimize refueling and maintenance planning of nuclear power plants

Abstract

Planning the maintenance of nuclear power plants is a complex optimization problem, involving a joint optimization of maintenance dates, fuel constraints and power production decisions. This paper investigates Mixed Integer Linear Programming (MILP) matheuristics for this problem, to tackle large size instances used in operations with a time scope of 5 years, and few restrictions with time window constraints for the latest maintenance operations. Several constructive matheuristics and a Variable Neighborhood Descent local search are designed. The matheuristics are shown to be accurately effective for medium and large size instances. The matheuristics give also results on the design of MILP formulations and neighborhoods for the problem. Contributions for the operational applications are also discussed. It is shown that the restriction of time windows, which was used to ease computations, induces large over-costs and that this restriction is not required anymore with the capabilities of matheuristics or local searches to solve such size of instances. Our matheuristics can be extended to a bi-objective optimization extension with stability costs, for the monthly re-optimization of the maintenance planning in the real-life application.

Appendices

Appendix A: Relaxation of constraints

In this appendix, we justify the relaxation of modulation constraints CT12, and decreasing profile constraints CT6. Such constraints were often relaxed in the approaches for the EURO/ROADEF 2010 Challenge, inducing more complexity and difficulty in the optimization modeling and solving. This appendix provides justifications, regarding the industrial application and some numerical analyses.

1.1 A.1 Modulation constraints CT12

In this paper, modulation constraints, denoted CT6 in the Challenge description, are relaxed. In the Challenge, it is formulated as a maximal volume of energy generated when a unit i is not producing at the maximal power $\mathbf {\overline{P}}_{i,w}$ at time step w, the cumulated energy being counted for all cycle k when the fuel level is superior to $\mathbf {Bo}_{i,k}$. When the fuel level is inferior to $\mathbf {Bo}_{i,k}$, the production is at the maximal power defined by the CT6 decreasing profile.

Fig. 2

Illustration of modulation constraints for nuclear power plants

Actually, such modulation constraints are approximated and aggregated from technical constraints of nuclear power plants. The fine grain technical constraints are as following, similarly to Rajan and Takriti (2005), Dupin (2017), and illustrated Fig. 2:

• Min-up/min-down constraints: every unit u has a minimum up time $\varDelta ^{on}_{u}$ online and a minimum down time $\varDelta ^{off}_{u}$ offline.

• Max duration with a non maximal power constraints: if a nuclear power plant is online, there is a maximal duration to have modulated power, i.e. non maximal power.

• Min duration between consecutive modulations: between consecutive modulations, there is a minimal duration to have maximal power.

• Max number of modulation constraints: for each day, there is a maximal number of periods where the production is not maximal.

If a T2 unit is online, the three last constraints induce a maximal volume of modulated power. Actually, the CT12 modulation constraints consider that the T2 units are always on-line in a production cycle. This last assumption has its justification as T2 production is less expensive than the T1 production. However, modulation may be necessary with low demands (lower than the available T2 production). In such cases, T2 units can be set offline with the min-up/min-down constraints, defining a volume of modulation that is not bounded. In these cases, there will be minimal durations offline with Min-up/min-down constraints. These extreme modulations happen in Fig. 5 using the model of Sect. 4.

1.2 A.2 Decreasing profile for T2 units CT6

In the Challenge, the “stretch decreasing profile” CT6 are defined when the fuel stock level of a T2 unit $i \in {\mathcal {I}}$ is inferior to the level $\mathbf {Bo}_{i,k}$ in the cycle $k \in K_i$. The production of i is deterministically defined, following a decreasing profile, piece-wise linear function of the stock level, as in Fig. 3. More precisely:

CT6, “stretch” constraints: During every time step $w\in {\mathcal {W}}$ of the production campaign of cycle $k \in K_i$, if the current fuel stock of plant $i \in {\mathcal {I}}$ is inferior to the level $\mathbf {Bo}_{i,k}$ , the production of i is deterministic, following a decreasing profile, piece-wise linear function of the stock level, as in Fig. 3. $m \in {\mathcal {M}}_{i,k}= {\llbracket }1,\mathbf {Np}_{i,k}{\rrbracket }$ define the points of the CT6 profile ending the cycle (i, k). The points are denoted $(\mathbf {f}_{i,k,m},\mathbf {c}_{i,k,m})$ for each cycle (i, k), $\mathbf {f}_{i,k,m}$ denote the fuel levels, $\mathbf {c}_{i,k,m} \in [0,1]$ is defined as the loss coefficient in power compared to the maximal power.

Fig. 3

Illustration of the production domain for T2 power plants in the challenge ROADEF

Fig. 4

Illustration of the production domain for T2 power plants with light CT6 constraints

We will consider a lighter constraint , imposing only the UB constraints as illustrated in Fig. 4. This MILP formulation was already introduced in Dupin and Talbi (2020a). New fuel variables $x_{i,w}$ are introduced, representing the residual fuel at week w for T2 unit i. These variables require the following constraints to enforce $x_{i,w}$ to be the residual fuel:

$$\begin{aligned} \forall i,k,w,\; \; \; x_{i,w} \leqslant x_{i,k}^{i} - \sum _{w'\leqslant w} \mathbf {F}_{w'} \, p_{i,k,w'} + M_i \; (1 - d_{i,k,w} + d_{i,k-1,w}) \end{aligned}$$

(33)

where $M_i = \max _k \mathbf {\overline{S}}_{i,k}$, such that $M_i$ verifies $x_{i,t} \leqslant M_i$. Indeed, if $d_{i,k,w} - d_{i,k-1,w}=1$, week w happens in cycle k, the active constraint is $x_{i,t} \leqslant x_{i,k,s}^{i} - \sum _{w'\leqslant w} \mathbf {F}_{w'} \, p_{i,k,w'}$, otherwise we have $x_{i,t} \leqslant M_i$, which is always true. The UB of the CT6 constraints, as illustrated Fig. 4, are given with:

$$\begin{aligned} \forall i,k,w,m>0, \; \; \; \frac{p_{i,k,w}}{\mathbf {\overline{P}}_i^t} \leqslant \frac{\mathbf {c}_{i,k,m-1} - \mathbf {c}_{i,k,m}}{\mathbf {f}_{i,k,m-1} - \mathbf {f}_{i,k,m}} (x_{i,w} - \mathbf {f}_{i,k,m}) + \mathbf {c}_{i,k,m} \end{aligned}$$

(34)

If ( $\mathbf {c}_{i,k,m},\mathbf {f}_{i,k,m-1}$) do not depend on indexes k, we can have an equivalent MILP formulation using aggregated constraints, writing the constraints for the global production power $\sum _k {p_{i,k,w}}$:

$$\begin{aligned} \forall i,w,m>0 \;\;\; \sum _k \frac{p_{i,k,w}}{\mathbf {\overline{P}}_{i,w}} \leqslant \frac{\mathbf {c}_{i,m-1} - \mathbf {c}_{i,m}}{\mathbf {f}_{i,m-1} - \mathbf {f}_{i,m}} (x_{i,w} - \mathbf {f}_{i,m}) + \mathbf {c}_{i,m} \end{aligned}$$

(35)

Adding the lighter CT6 constraints, it makes MILP solving difficult. Computations were not possible for the datasets B and X, inducing memory errors. We note that the aggregated formulation (35) provided improvements in the computation times compared to (34), but the formulation is still untractable for the real size instances. It can be explained in terms of number of variables, the extension adds $I \times W$ continuous variables $x_{i,w}$, whereas $I \times W \times K$ continuous variables were already in the model with T2 productions $p_{i,k,w}$. The main difference appears in the number of constraints, there were mainly $I \times W \times K$ constraints in the MILP of Sect. 4 with constraints (7), (33–35) require to add $I \times W \times (K+N_p)$ constraints, whereas (34–33) require to add $I \times W \times (N_p+1) \times K$ constraints. This slows down very significantly the LP computations, when computable.

Fig. 5

Illustration of an imposed modulation with stretch constraints on instance A3_3_210

Adding CT6 constraints (33–34) or (33–35) has furthermore few impact in the quality of the LP relaxation, which was already noticed in Dupin and Talbi (2020a). We analyze here whether it is insightful in searching primal solutions. Actually, decreasing profile phases are rarely activated in our model. An explanation is that the upper bounds restrict the T2 production capacities. The T2 units having the lowest marginal costs of production, the optimization avoids the over-costs of production with mainly maximal T2 productions. Maximizing the T2 production available tends to avoid the decreasing profile phases, preferring to have earlier outages to produce at the maximum power in the production cycles. Optimal uses of decreasing profile occur when it is imposed by time windows constraints to have longer cycles than the duration to reach $\mathbf {Bo}_{i,k}$ fuel levels, which is the situation met in Fig. 5. In such cases, the optimization tends to produce at the upper bound of production thanks to the difference in marginal costs between T1 and T2 units, which justifies the lighter formulation of CT6 constraints.

The over-costs to project the optimal (or best known) solutions computed thanks to the MILP of Sect. 4.2 into a MILP with stretch constraints is given for the small instances in Table 14. Little over-costs are observed, around $0.3\%$ in average. The full relaxation of CT6 constraints allows to generate solutions of very good quality after repairing of CT6. Figure 5 suggests that some local modifications around the CT6 situations can improve the previous projected solution. Table 14 gievs also the improvements of the projected solutions with only two iterations of local search with MILP neighborhoods ${\mathcal {N}}_{(0,2)}^{TW}$ using (34), which are tractable computations. These very local modifications around the CT6 situations improve significantly the projected costs.

These conclusions justify the relaxation of CT6 constraints in a matheuristic approach, as developed in this paper, but also in the approaches of Lusby et al. (2013).

Appendix B: Intermediate results

See Tables 10, 11, 12, 13 and 14.

Table 10 Comparison of termination time of B&B to optimality: without and with the exact pre-processing of Dupin and Talbi (2020a), pre-processing and B&B warmstart with the optimal (or best known) solution, and with penalisation costs to the baseline solution given by Sect. 5.1

Table 11 Comparison of the gaps to the BKS for the POPMUSIC-VND and for the VND with single types of large neighborhoods

Table 12 Result comparison for constructive primal math-heuristics

Table 13 Quality of local minimums considering one single type of neighborhoods in the VND of Algorithm 3

Table 14 Over-costs of the outage solutions with the MILP of Sect. 4.2 in the model adding CT6 constraints, and local improvements with two iterations of VND with neighborhoods ${\mathcal {N}}_{(0,2)}^{TW}$