The Optimal Allocation of Distributed Generators Considering Fault Current and Levelized Cost of Energy Using the Particle Swarm Optimization Method
Abstract
:1. Introduction
2. Problem Statement
3. Distributed Generators and Their Fault Analysis
3.1. Distributed Generators
- Rotational DGs such as wind power and gas turbines;
- Stationary DGs using inverters such as solar power and variable-speed wind power.
3.2. Distribution Generator Modeling Method
3.2.1. Voltage Source
3.2.2. Current Source
3.3. Fault Analysis
3.3.1. Unsymmetrical Electrical Fault
3.3.2. General Analysis Method of Single Line-to-Ground Fault
3.4. Short-Circuit Current Calculation Example
3.4.1. Fault Current Contributed by a Voltage Source
3.4.2. Current Injection Method
3.4.3. Effect of Current Source
4. Proposed Methods
4.1. Particle Swarm Optimization
- Inertia: the velocity of particles at the previous step.
- Cognitive force: the distance from the known best position of each particle, which is also called the individual best.
- Social force: the distance from the known best position of the swarm, which is also called swarm best.
- = velocity in jth step
- = weight of inertia
- = acceleration
- = random number
- = individual best position
- = swarm best position
- = position of each particles in jth step
- = cognitive component (i.e., individual best, cognitive force)
- = social component (i.e., swarm best, social force)
4.2. Mathematical Optimization Problem
4.2.1. Fault Current and Levelized Cost of Energy
4.2.2. Objective Function
4.2.3. Variable Normalization
5. Optimization and Results
5.1. Conducting PSO in Detail
Algorithm 1 PSO Algorithm | |
1: | Set up PSO parameters |
2: | Initialize position of particles and velocity |
3: | Set iteration counter |
4: | For each particle do |
5: | Calculate initial fitness of particle |
6: | End |
7: | Find index of the best particle of population and best fitness |
8: | Select and |
9: | While current iteration maximum iterations do |
10: | Calculate inertial weight |
11: | For each particle do |
12: | For each dimension do |
13: | Pick random numbers and |
14: | Update the particle’s velocity |
15: | Update the particle’s position |
16: | End |
17: | Evaluate fitness of particles |
18: | If then |
19: | Update of population |
20: | Update best fitness |
21: | End |
22: | End |
23: | Find index of the best particle and best fitness |
24: | If then |
25: | Update of population |
26: | Update best particle |
27: | Update best fitness |
28: | End |
29: | Increase iteration counter |
30: | End |
31: | Print out index of best particle and |
GA Details
- (a)
- The production of initial population members,
- (b)
- The evaluation of the fitness function of each individual member,
- (c)
- Selection,
- (d)
- Crossover,
- (e)
- Mutation.
- (a)
- A time limit is reached;
- (b)
- A fixed number of generations is reached;
- (c)
- Sufficient fitness is achieved, or;
- (d)
- A combination of these conditions [42].
5.2. Results
5.2.1. Simulation Results
5.2.2. Index for Objective Function Selection and DG Installation Property
(a) Voltage Profile Index (IVD)
(b) Fault Current Level Index (FCI)
(c) Real and Reactive Power Losses (ILP and ILQ)
5.2.3. Index Results by Function
5.2.4. Evaluation of Swarm Intelligence
6. Discussion
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
DG | distributed generator |
DLG | double line-to-ground |
IBDG | inverter based distributed generator |
GA | genetic algorithm |
LL | line-to-line |
PSO | particle swarm optimization |
SCADA | supervisory control and data acquisition |
SLG | single line-to-ground |
LCOE | levelized cost of energy |
Thevenin’s equivalent voltage source | |
Thevenin’s equivalent impedance | |
Norton’s equivalent current source | |
Norton’s equivalent admittance | |
zero-, positive-, and negative-sequence fault voltages | |
zero-, positive-, and negative-sequence fault currents | |
zero-, positive-, and negative-sequence fault impedances | |
pre-fault voltage source | |
fault impedance | |
line-to-ground voltage of phases a, b, and c | |
zero-, positive-, negative-sequence impedances connected to the slack bus | |
three-winding transformer impedances | |
neutral grounding resistors | |
zero-, positive-, negative-sequence transformer impedances | |
Norton’s equivalent impedance | |
fault current caused by the voltage source | |
fault current caused by the current source | |
velocity in the jth step | |
weight of inertia | |
acceleration | |
random number | |
individual best position | |
swarm best position | |
position of each particles in the jth step |
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Rosenbrock Function | Himmelblau Function | Beale Function | |||
---|---|---|---|---|---|
Bus No. | Capacity (MVA) | Bus No. | Capacity (MVA) | Bus No. | Capacity (MVA) |
8 | 54 | 1 | 100 | 2 | 37 |
9 | 16 | 3 | 100 | 3 | 100 |
12 | 83 | 4 | 100 | 4 | 18 |
13 | 28 | 17 | 100 | 5 | 44 |
22 | 48 | 18 | 100 | 9 | 22 |
26 | 74 | 21 | 100 | 14 | 26 |
Goldstein-Price Function | Ackley Function | ||
---|---|---|---|
Bus No. | Capacity (MVA) | Bus No. | Capacity (MVA) |
6 | 22 | 9 | 1 |
16 | 20 | 18 | 1 |
21 | 1 | 20 | 1 |
26 | 1 | 26 | 1 |
Rosenbrock Function | Himmelblau Function | Beale Function | |||
---|---|---|---|---|---|
Bus No. | Capacity (MVA) | Bus No. | Capacity (MVA) | Bus No. | Capacity (MVA) |
2 | 100 | 1 | 100 | 2 | 50 |
5 | 90 | 4 | 100 | 12 | 50 |
13 | 99 | 5 | 100 | 15 | 50 |
15 | 100 | 6 | 100 | 16 | 50 |
28 | 100 | 10 | 100 | 22 | 50 |
29 | 78 | 13 | 100 | 24 | 51 |
Goldstein-Price Function | Ackley Function | ||
---|---|---|---|
Bus No. | Capacity (MVA) | Bus No. | Capacity (MVA) |
2 | 20 | 1 | 0 |
7 | 16 | 2 | 0 |
11 | 20 | - | - |
13 | 17 | - | - |
Bus No./Capacity | IVD | FCI | ILP | ILQ |
---|---|---|---|---|
Bus 8/54 MVA | 0 | 0.017 | 0.67 | 0.40 |
Bus 9/16 MVA | 0 | 0.004 | 0.88 | 0.75 |
Bus 12/83 MVA | 0 | 0.024 | 0.65 | 0.1 |
Bus 13/28 MVA | 0 | 0.001 | 0.84 | 0.61 |
Bus 22/48 MVA | 0 | 0.014 | 0.71 | 0.29 |
Bus 26/74 MVA | 0 | 0.019 | 1.09 | 1.23 |
Bus No./Capacity | IVD | FCI | ILP | ILQ |
---|---|---|---|---|
Bus 6/22 MVA | 0 | 0.007 | 0.99 | 0.98 |
Bus 16/20 MVA | 0 | 0.005 | 1.00 | 0.98 |
Bus 21/1 MVA | 0 | 0.004 | 0.99 | 0.99 |
Bus 26/1 MVA | 0 | 0.000 | 0.99 | 0.99 |
Bus No./Capacity | IVD | FCI | ILP | ILQ |
---|---|---|---|---|
Bus 2/100 MVA | 0.008 | 0.027 | 1.00 | 1.00 |
Bus 5/90 MVA | 0.007 | 0.014 | 1.00 | 0.99 |
Bus 13/99 MVA | 0.002 | 0.007 | 0.99 | 0.99 |
Bus 15/100 MVA | 0 | 0.007 | 1.16 | 1.21 |
Bus 28/100 MVA | 0 | 0.032 | 1.07 | 1.07 |
Bus 29/78 MVA | 0 | 0.025 | 1.40 | 1.56 |
Bus No./Capacity | IVD | FCI | ILP | ILQ |
---|---|---|---|---|
Bus 2/20 MVA | 0.008 | 0.019 | 1.00 | 1.00 |
Bus 7/16 MVA | 0.006 | 0.005 | 1.00 | 0.98 |
Bus 11/20 MVA | 0.003 | 0.005 | 1.00 | 1.00 |
Bus 13/17 MVA | 0.002 | 0.005 | 1.00 | 0.99 |
Goldstein-Price Function | Rosenbrock Function | |
---|---|---|
p-value |
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Kim, B.; Rusetskii, N.; Jo, H.; Kim, I. The Optimal Allocation of Distributed Generators Considering Fault Current and Levelized Cost of Energy Using the Particle Swarm Optimization Method. Energies 2021, 14, 418. https://doi.org/10.3390/en14020418
Kim B, Rusetskii N, Jo H, Kim I. The Optimal Allocation of Distributed Generators Considering Fault Current and Levelized Cost of Energy Using the Particle Swarm Optimization Method. Energies. 2021; 14(2):418. https://doi.org/10.3390/en14020418
Chicago/Turabian StyleKim, Beopsoo, Nikita Rusetskii, Haesung Jo, and Insu Kim. 2021. "The Optimal Allocation of Distributed Generators Considering Fault Current and Levelized Cost of Energy Using the Particle Swarm Optimization Method" Energies 14, no. 2: 418. https://doi.org/10.3390/en14020418
APA StyleKim, B., Rusetskii, N., Jo, H., & Kim, I. (2021). The Optimal Allocation of Distributed Generators Considering Fault Current and Levelized Cost of Energy Using the Particle Swarm Optimization Method. Energies, 14(2), 418. https://doi.org/10.3390/en14020418