- freely available
Energies 2013, 6(3), 1439-1455; doi:10.3390/en6031439
- the criteria for optimization are evaluated on active power distribution systems (containing distributed generators connected directly to the main distribution system and microgrids operated in grid-connected mode);
- the original formulation of the optimization problem, as a Pareto optimal one, with two objective functions (active power losses and system average interruption frequency index);
- an original genetic algorithm (based on NSGA-II) to solve the problem (as a Pareto optimal one) in a non-prohibitive execution time.
2. Problem Formulation
2.1. Criteria for Optimization
2.1.1. Active Power Losses (ΔP)
2.1.2. Reliability of the Distribution System
- Reliability of a particular customer: e.g., the average number of interruptions to the power supply. This index can represent a possible objective and/or constraint in the optimization problem (because some customers can impose maximal/minimal limits in their supply contracts);
- N = 30 ∙ 6 = 180 (six nodes, 30 customers for each node);
- λt2∙ N2 = 0.5 ∙ 30 = 15;
- λt3∙ N3 = 0.5 ∙ 3 ∙ 0 = 0 (three branches, the microgrid is expected to operate as an island and customers from a microgrid will be supplied even if a fault has occurred in the main distribution system);
- λt4∙ N4 = 0.5 ∙ 3 ∙ 30 = 45 (three branches, if a fault occurs on the path 1-7-4-3, the DG unit will be switched off and the customers will be not supplied);
- λt5∙ N5 = λt6∙ N6 = 0.5 ∙ 2 ∙ 30 = 30 (two branches);
- λt7∙ N7 = 0.5 ∙ 3 ∙ 30 = 45;
- SAIFI = (15 + 0 + 45 + 30 + 30 + 45)/180 = 0.9167.
2.1.3. Other Criteria
- Node Voltages (Vi): Basically, each voltage r.m.s. value of the network nodes must be framed within the allowable limits.
- Branch Load Limits through Lines (Iij): a typical constraint on the reconfiguration problem.
- Safeguard of power supplies for all customers: The attached graph of the electric system should be connected (a tree or a forest).
- Configuration of the Distribution System: Generally, electrical distribution systems are operated in radial configuration. This condition can be expressed as follows:
2.2. Pareto Optimality Problem Formulation
- Objective function
3. Problem Solving
3.1. Genetic Encoding and Decoding of Power Distribution System Topology
3.1.1. Genetic Encoding
3.1.2. Genetic Decoding
3.2. Genetic Operators
4. Simulation Results
- System A (Figure 5a) [3,27]: There are four DG units installed on nodes 3, 6, 24 and 29 . In the base case, the total active power losses are 169.881 kW. By applying MOReco algorithm (after reconfiguration), the total active power losses are 115.748 kW. The evolution of the active power losses along the searching process is presented in Figure 6a.
- System B (Figure 5b) : In this case, there are eight DG units installed on nodes: 7, 12, 19, 28, 34, 71, 75 and 79 . Before the reconfiguration, the total active power losses are 425.131 kW (as in ). After reconfiguration, we obtained a better configuration than the one presented in . The total active power losses are 380.656 kW, therefore smaller than 383.524 kW. The evolution of the active power losses along the searching process is presented in Figure 6b.
|System||Configuration||Open branches (tie lines)||Active power losses||CPU runtime for 100 runs||Population/Generations|
|A||Base case||8–21, 9–15, 12–22, 18–33, 25–29||169.881||kW||-|
|MOReco||7–8, 9–10, 14–15, 28–29, 32–33||115.748||kW||Minimum (3 s: 804 ms)|
Maximum (3 s: 974 ms)
Average (3 s: 910 ms)
|B||Base case||5–55, 7–60, 11–43, 12–72, 13–76, 14–18, 16–26, 20–83, 28–32, 29–39, 34–46, 40–42, 53–64||425.131||kW||-|
|MOReco||6–7, 12–13, 32–33, 38–39, 41–42, 54–55, 62–63, 71–72, 82–83, 11–43, 14–18, 16–26, 28–32||380.656||kW||Minimum (6 s: 202 ms)|
Maximum (7 s: 108 ms)
Average (6 s: 675 ms)
|System||Open branches (tie lines)||Active power losses||SAIFI||CPU runtime for 100 runs||Population/Generations|
|B with DGs||6–7, 12–13, 32–33, 38–39, 41–42, 54–55, 62–63, 71–72, 82–83, 11–43, 14–18, 16–26, 28–32||380.656||kW||1.143||Minimum (6 s: 952 ms)|
Maximum (7 s: 279 ms)
Average (7 s: 40 ms)
|5–55, 6–7, 11–43, 12–72, 13–76, 14–18, 16–26, 20–83, 28–32, 29–39, 32–33, 40–42, 53–64||396.143||kW||0.751||-|
|5–55, 6–7, 11–43, 12–72, 13–76, 14–18, 16–26, 20–83, 28–32, 29–39, 34–46, 40–42, 53–64||409.526||kW||0.648||-|
|5–55, 7–60, 11–43, 12–72, 13–76, 14–18, 16–26, 20–83, 28–32, 29–39, 34–46, 40–42, 53–64||425.131||kW||0.472||-|
|B with microgrids||6–7, 12–13, 32–33, 38–39, 41–42, 54–55, 62–63, 71–72, 82–83, 11–43, 14–18, 16–26, 28–32||380.656||kW||0.988||Minimum (7 s: 114 ms)|
Maximum (7 s: 342 ms)
Average (7 s: 171 ms)
|5–55, 6–7, 11–43, 12–72, 13–76, 14–18, 16–26, 20–83, 28–32, 29–39, 32–33, 40–42, 53–64||396.143||kW||0.619||-|
|5–55, 7–60, 11–43, 12–72, 13–76, 14–18, 16–26, 20–83, 28–32, 29–39, 34–46, 40–42, 53–64||425.131||kW||0.427||-|
- System B with DG units: The proposed algorithm has obtained a Pareto front with four solutions (Figure 7a). In this case, the first non-dominated solution was obtained from initial population. After the first generation, the algorithm found the second non-dominated solution (the Pareto front contains two solutions). The searching process continued and the third non-dominated solution was found in generation 2 (at the end of generation 2, the Pareto front contains three solutions). The searching process continued, but without finding other non-dominated solutions until generation 9, where the fourth and final non-dominated solution was found. In the end, the Pareto front contains four non-dominated solutions.
- System B with microgrids (instead of DG units, as sources on the points of common coupling): The proposed algorithm obtained a Pareto front with three solutions (Figure 7b). As in the previous case, along the searching process, the Pareto front increases from one solution (from the initial population), to two solutions (generation 2) and, finally, to three solutions (in generation 9). We can observe that the SAIFI index is smaller in the case of existing microgrids than in the case where we consider just distributed generators.
|Branch||i||j||λ [year−1]||Branch||i||j||λ [year−1]||Branch||i||j||λ [year−1]|
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