Improving reliability with optimal allocation of maintenance resources: an application to power distribution networks

Abstract

Power distribution networks should strive for reliable delivery of energy. In this paper, we support this endeavor by addressing the Maintenance Resources Allocation Problem (MRAP). This problem consists of scheduling preventive maintenance plans on the equipment of distribution networks for a planning horizon, seeking the best trade-offs between system reliability and maintenance budgets. We propose a novel integer linear programming (ILP) formulation to effectively model and solve the MRAP for a single distribution network. The formulation also enables flexibility to suit new developments, such as different reliability metrics and smart-grid innovations. Then we develop a straightforward ILP formulation to address the MRAP for several distribution networks which takes the advantages of exchanging maintenance information between local agents and upper management. Using a general-purpose ILP solver, we performed computational experiments to assess the performance of the proposed approaches. Optimal maintenance trade-offs were achieved with the new formulations for real-scale distribution networks within short running times. To the best of our knowledge, this is the first time that the MRAP is optimally solved using ILP, for single or multiple distribution networks.

Appendix. A biased random-key genetic algorithm for the MRAP

The BRKGA proposed in Gonçalves and Resende (2011) is an evolutionary metaheuristic in which a population of individuals evolves through the Darwinian principle of the survival of the fittest. Each individual of the population is represented by a chromosome Q, encoded as a vector with m alleles. An allele is a random key uniformly drawn over the interval [0, 1]. A chromosome Q is mapped into a solution R through a decoder algorithm, which is the problem-specific component of the metaheuristic.

The BRKGA starts by generating an initial population with p random chromosomes. Throughout g generations, the population goes through a selective pressure environment in which the fittest individuals are more likely to endure and produce offspring. A distinguished feature of the BRKGA evolutionary strategy is that it partitions the population into elite and non-elite sets. As the name suggests, the elite set comprises the fittest individuals; the non-elite set contains all the remaining individuals, including the so-called mutants, which are randomly generated chromosomes to promote diversification in the search process.

In a nutshell, at each generation, the BRKGA performs the following steps:

• 1. Decode each chromosome Q into its corresponding solution R;

• 2. Evaluate the fitness of each solution R;

• 3. Identify the elite set (i.e., the best $p_e$ individuals);

• 4. Copy the elite set to the next generation;

• 5. Include $p_m$ new random chromosomes (mutants) in the next generation;

• 6. Produce $p - (p_e + p_m)$ offspring using crossover operators, and insert them in the next generation.

The crossover generates a new individual by sampling each allele from one of its parents. Both parents are from the current generation, one from the elite set and the other from the non-elite set. An allele comes from the elite parent with probability $\rho _e$. After g generations, the BRKGA returns the best chromosome $Q^*$ and its decoded solution $R^*$. The following items describe the problem-specific aspects of using BRKGA to tackle the local-level MRAP.

Chromosome Each allele of a chromosome Q corresponds to a pair (e, t) of equipment e and period t with key Q(e, t).

Decoder Algorithm 1 details the BRKGA decoder for solving the MRAP of a single distribution network. It receives as input a chromosome Q and returns the fitness of its corresponding solution R. The decoding process starts with an initial empty solution, meaning no maintenance on any equipment in any period. Then, in decreasing order of the chromosome key values $Q(e^*,t^*)$, the maintenance of equipment $e^*$ in period $t^*$ is included in the solution until it becomes feasible with respect to the SAIFI target.

Fitness The fitness of a solution is given by the sum of preventive and corrective costs, in the present value, calculated according to Eq. (8a).

Table 5 gives the BRKGA parameters for the experiments in Sect. 5.

Table 5 BRKGA parameters