^{1}

^{2}

^{1}

^{2}

^{1}

^{2}

Planning primary electric power distribution involves solving an optimization problem using nonlinear components, which makes it difficult to obtain the optimum solution when the problem has dimensions that are found in reality, in terms of both the installation cost and the power loss cost. To tackle this problem, heuristic methods have been used, but even when sacrificing quality, finding the optimum solution still represents a computational challenge. In this paper, we study this problem using genetic algorithms. With the help of a coding scheme based on the dandelion code, these genetic algorithms allow larger instances of the problem to be solved. With the stated approach, we have solved instances of up to 40,000 consumer nodes when considering 20 substations; the total cost deviates 3.1% with respect to a lower bound that considers only the construction costs of the network.

Electric power distribution must satisfy user needs within a given geographic area, and for this purpose, an electric network that can identify a primary and a secondary distribution network is used. In the primary network, electric power is distributed in a tree topology, an activity in which electric companies invest approximately 50% of the transport network’s capital [

The feeders allow power to be transferred from the substations to places close to the final consumers, using a voltage level that can satisfy the requirements of each consumer. The power flows radially along conductors, which, because of their nature, offer resistance to the flow, causing part of this power to be dissipated as heat. These conductors have different costs depending on their cross sections and types of materials, and, therefore, the flow must be adjusted to minimize the loss. This paper considers the design problem that also involves minimizing the amount of materials and installation costs.

Energy distribution from a substation to the consumption points can be represented by a tree in which the substation corresponds to the root and the trees nodes represent the consumers. This tree can be coded as an array of integer numbers that represent the labels of the nodes. If we consider the bijection between a Cayley tree [

The optimum generation of electric power distribution trees is a problem that has been studied in the literature and is known as the distribution tree problem (DTP) [

Various authors have studied the DTP by considering genetic algorithms (GAs), which are stochastic search methods based on principles of evolution and natural selection [

There are several ways of representing trees within a GA, but there is little consensus as to which of these representations is “the best” [

The second section of the paper describes the main components of the model, and the third section presents the results. The last section presents the conclusion of the study.

Let

To represent a distribution tree, all of the substations are labeled with natural numbers from 1 to

As an example, consider an electric distribution system with

Function

Electric distribution tree corresponding to code

Any set of electric distribution trees that solves the DTP can be represented by the proposed model, with

The initial population is generated using a modified Prim algorithm [

The construction cost between the

To perform the experiment, the test instances are generated first, which have the following input data: active power

Conductor data.

Conductor | Impedance | Resistance | Current | Cost |

(Ohm/Km) | (Ohm/Km) | Capacity (A) | (dollars) | |

1 | 0.00010 | 0.0016 | 0.08429 | 8000 |

2 | 0.00010 | 0.0008 | 0.12644 | 9000 |

3 | 0.00009 | 0.0005 | 0.14232 | 10000 |

4 | 0.00008 | 0.0004 | 0.18071 | 12000 |

5 | 0.00008 | 0.0008 | 0.21073 | 13000 |

6 | 0.00007 | 0.0004 | 0.24624 | 17000 |

The main parameters of a GA are: population size, the cross-over probability, and the mutation probability. In this paper, we will use the parameters proposed by Grefenstette [

Although it is recommended to use a low mutation probability [

The calibration process was executed 5 times with an instance of 500 nodes and 20 substations. The network’s data are shown in Table

Network data.

Base power (kVA) | 1000 |

Available power (kVA) | 21000 |

Required power (kVA) | 17000 |

Operating time (years) | 10 |

Base voltage (kV) | 12 |

Number of consumers | 500 |

Number of substations | 20 |

Study of the mutation.

Mutation (%) | 0.01 | 0.01 | 0.02 | 0.04 | 0.08 | 0.10 |
---|---|---|---|---|---|---|

Av. installation cost (US$) | 167895 | 169552 | 187395 | 170235 | 171649 | 161885 |

Av. loss cost (US$) | 39757425 | 18641845 | 6069982 | 189316 | 519619 | 1462243 |

Av. total cost (US$) | 39925320 | 18811398 | 6257377 | 359551 | 691268 | 1624,128 |

Mutation (%) | 0.12 | 0.14 | 0.18 | 0.20 | 0.22 | |

Av. installation cost (US$) | 160996 | 166901 | 166722 | 157471 | 156557 | |

Av. loss cost (US$) | 1570343 | 157148 | 2839201 | 215340 | 1684931 | |

Av. total cost (US$) | 1731339 | 324050 | 3005924 | 372810 | 1841488 |

The equipment used had a 2-Quad-Core CPU 2.00 GHz Intel Xeon with 16 Gbytes of RAM, and the operating system was Ubuntu 10.04.2 LTS Kernel 2.6.32-28-generic, compiler GNU C (4.4.3).

For each instance, the following input data were considered: 20 substations, base power 1000 kVA, base voltage 12 kV, and an operating time of 10 years.

Figure

Cost versus number of generations.

To validate the results obtained with the GA, the results are compared with a lower bound. To find such bound, use is made of a Prim algorithm with which the minimum construction cost of the network is sought. In this way, comparisons of the results are established that validate the results obtained with the GA. This can be observed in all of the test instances (Table

Results.

Number of nodes | 35000 | 40000 | 45000 |
---|---|---|---|

Total cost (average) (US$) | 1021351 | 1069387 | 1132179 |

Minimum cost (US$) | 1013664 | 1068609 | 1123047 |

Minimum bound (US$) | 970082 | 1036867 | 1102479 |

Average time (HH:MM: SS) | 87:17:00 | 110:54:46 | 127:00:18 |

Minimum time (HH:MM:SS) | 84:58:47 | 105:3330 | 118:59:16 |

Available power (kVA) | 20000 | 21000 | 22000 |

Required power (kVA) | 16000 | 17000 | 18000 |

This paper proposes a solution for the DTP that has a real application in the optimization of electric distribution networks. The problem is approached using GAs, proposing a model that considers construction and power loss costs. To perform the search in the distribution tree space, the dandelion code is used. The proposed approach shows that the code used is efficient in the representation of trees, and its use allows real problems to be solved using GAs.

The approach used also allows one to find solutions to problems with extremely large instances, e.g., for the instance of 45,000 nodes, which requires a computing time of 118:59:16.