Optimal Allocation of Energy Storage System Considering Multi-Correlated Wind Farms
Abstract
:1. Introduction
2. Problem Formulation
2.1. Modeling Multiple Dependent Wind Probability
2.2. Discretizing Joint Distribution
- (1)
- Calculate the first few moments of wind joint distribution when one of the wind farms is at its limits either zero or full wind power;
- (2)
- Find the remaining correlated finite points representing both wind farms generating power not at their limits. The procedure of the discretization are described as follows.
2.2.1. The Boundary Points
- (1)
- (2)
- Calculate the joint probabilities () when the output power of wind farm 1 is 0 MW combining with L different discretized wind powers in wind farm 2. L points in boundary are obtained.
- (3)
- Calculate the probabilities () when the output power in wind farm 2 is 0 MW combining with K different discretized wind powers in wind farm 1. K points in boundary can be obtained.
- (4)
- Calculate the probabilities () when the wind power in wind farm 1 achieves maximum combining with L different discretized wind power in wind farm 2. L boundary points are attained.
- (5)
- Calculate the probabilities () when the wind power in wind farm 2 achieves maximum combining with K different discretized wind power in wind farm 1. K boundary points are attained.
2.2.2. The Interior Points
2.3. Objective Function
2.4. Problem Constraints
2.4.1. Equality Constraints
2.4.2. Inequality Constraints
3. Solution Method
3.1. Discretizing Multi-Correlated Wind Distribution
3.1.1. Wind Distribution
3.1.2. Discretizing Wind Power Joint Distribution
- (1)
- The boundary points
- Calculate the probabilities (, ) when the output power of wind farm 1 is 0 MW or reaches the maximum combining with five different discretized wind powers of wind farm 2. Consequently, 10 discrete points are obtained;
- Apply the same algorithm into wind farm 2 and the probabilities that are and are determined. 10 discrete points are obtained herein.
- (2)
- The interior points
- The interior discrete probability points of the joint distribution are decided by Equations (3)–(6) and (13). Three points are selected for each wind farm distribution in the calculation and a total of 9 discrete probability points are obtained.
3.1.3. Combination Method for High-Dimensional Correlated Wind Distribution
3.2. Hybrid MOPSO for Economic Emission Dispatch
- (1)
- Randomly generate population P with N particles for initializing all generators’ voltage, output power and position and size of ESS. The random selections of swarm of particles considering constraints and corresponding velocity for each particle are initialized.
- (2)
- Model joint probability of multi-wind farms by Clayton-Copula method.
- (3)
- Discretize the joint wind power distribution into 25-point distribution by the new proposed estimation method, which has been discussed in Section 3, 25 scenarios in two different load conditions for each study case are created.
- (4)
- Through probabilistic power flow, evaluate the particles by fitness function and recall their best positions associated with the best fitness value.
- (5)
- Check and preserve the and , if the algorithm has not yet converged, update the and .
- (6)
- Duplicate population P to population Q to form a combined population R and update the position and velocity of each particle.
- (7)
- Sort the members in population R through NSGA-II with elitism algorithm for selecting N best solutions to renew population P.
- (8)
- Repeat Steps 4–7 until all the scenarios are considered.
4. Result and Discussion
4.1. System Configuration
4.2. Economic and Emission Analysis
5. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
Appendix A
HMOPSO | Hybrid multi-objective particle swarm optimization |
EED | Economic emission dispatch |
ESS | Energy storage system |
FOSMM | First-order second-moment method |
CM | Cumulant method |
PEM | Point estimation methods |
MOPSO | Multi-objective particle swarm optimization |
NSGA-II | Nondominated Sorting Genetic Algorithms-II |
MOGA | Multi-objective genetic algorithm |
Joint cumulative distribution function | |
Correlation coefficient | |
Bivariate probability density function | |
and | The rated wind power of wind farm 1 and wind farm 2 |
The mean of (X, Y) | |
The standard deviation of (X, Y) | |
The central moment of (X, Y) | |
The probability of interior points | |
The total probability of interior points | |
z | The standardized value of (X, Y) |
The probability of peak-load condition | |
The probability of off peak-load condition | |
The probability of i-th discrete point for the joint distribution of two wind farm | |
Total operation cost at the i-th or k-th discrete point ($/h) with the peak load or off peak load | |
The total emission at the i-th or k-th discrete point (kg/h) with the peak load or with the off peak load | |
The number of generators | |
The fuel cost of generator j ($/h) | |
ECOx (PGj) | The COx emission of generator j (kg/h) |
Cw | The cost of wind power generator ($/h) |
Cs | The cost of ESS ($/h) |
aj, bj, cj | The fuel cost coefficients of generator j |
θj, δj, γj | The COx emission coefficients of generator j |
Copw | The operation cost of wind power generator ($/MWh) |
Cops | The operation cost of ESS ($/MWh) |
Pwind | The power of wind power generator (MW) |
Pstorage | The capacity of installed ESS (MW) |
k | Shape parameter of distribution |
λ | Scale parameter of distribution |
W | The injected power |
X | Actual wind speed |
M | The maximum power of wind turbine |
N | The total number of points of the discretized wind power |
α and β | The linear coefficients |
Vci, Vco and Vno | The cut-in wind speed, cut-out wind speed and normal wind speed |
p11-33 | The sum of the probabilities of the interior points |
The expected voltage | |
The maximum of voltage deviation |
Appendix B
Appendix B.1. MOPSO
Appendix B.2. NSGA-II
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Wind Farm | k | λ | α | β | Max Power |
---|---|---|---|---|---|
1 | 2.5034 | 10.0434 | −15.75 | 4.5 | 45 MW |
2 | 2.4 | 11.5 | −115.5 | 33 | 330 MW |
Point | Probability (%) | Wind Farm 1 (MW) | Wind Farm 2 (MW) |
---|---|---|---|
1 | 0.759 | 0 | 0 |
2 | 1.015 | 0 | 46.57 |
3 | 3.457 | 0 | 173.96 |
4 | 1.330 | 0 | 293.18 |
5 | 0.334 | 0 | 330 |
6 | 0.072 | 45 | 0 |
7 | 1.199 | 45 | 46.57 |
8 | 0.932 | 45 | 173.96 |
9 | 1.037 | 45 | 293.18 |
10 | 9.043 | 45 | 330 |
11 | 0.934 | 16.46 | 0 |
12 | 2.815 | 22.23 | 0 |
13 | 1.014 | 39.09 | 0 |
14 | 3.710 | 16.46 | 330 |
15 | 6.402 | 22.23 | 330 |
16 | 3.518 | 39.09 | 330 |
17 | 5.897 | 15.96 | 47.51 |
18 | 7.320 | 15.96 | 175.28 |
19 | 2.546 | 15.96 | 294.23 |
20 | 8.019 | 21.47 | 47.51 |
21 | 14.690 | 21.47 | 175.28 |
22 | 8.574 | 21.47 | 294.23 |
23 | 1.496 | 38.24 | 47.51 |
24 | 8.574 | 38.24 | 175.28 |
25 | 5.350 | 38.24 | 294.23 |
No. | Peak Load Condition | Off Peak-Load Condition | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Case 1 | Case 2 | Case 3 | Case 1 | Case 2 | Case 3 | |||||||
Cost 104 $/h | Em 104 kg/h | Cost 104 $/h | Em 104 kg/h | Cost 104 $/h | Em 104 kg/h | Cost 104 $/h | Em 104 kg/h | Cost 104 $/h | Em 104 kg/h | Cost 104 $/h | Em 104 kg/h | |
1 | 10.88 | 8.62 | 10.13 | 7.70 | 7.35 | 5.34 | 3.53 | 2.43 | 3.09 | 1.81 | 2.69 | 1.62 |
2 | 9.92 | 7.62 | 8.82 | 6.44 | 7.25 | 5.09 | 3.2 | 2.03 | 3.13 | 1.78 | 2.4 | 1.28 |
3 | 7.8 | 5.36 | 5.93 | 3.70 | 5.7 | 3.52 | 2.61 | 1.22 | 2.33 | 1.09 | 2.28 | 0.84 |
4 | 8.37 | 3.77 | 5.44 | 2.79 | 4.68 | 2.40 | 2.38 | 0.77 | 2.29 | 0.66 | 2.27 | 0.50 |
5 | 7.02 | 3.36 | 5.11 | 2.39 | 4.55 | 2.13 | 2.36 | 0.69 | 2.35 | 0.46 | 2.32 | 0.39 |
6 | 10.65 | 8.35 | 8.43 | 6.22 | 6.84 | 4.97 | 3.44 | 2.30 | 3.36 | 2.03 | 2.49 | 1.63 |
7 | 9.7 | 7.37 | 7.75 | 5.49 | 6.36 | 4.29 | 3.12 | 1.92 | 2.89 | 1.51 | 2.38 | 1.29 |
8 | 7.63 | 5.16 | 6.49 | 3.97 | 5.38 | 3.28 | 2.56 | 1.15 | 2.51 | 0.90 | 2.35 | 0.85 |
9 | 6.24 | 3.61 | 5.32 | 2.66 | 4.53 | 2.32 | 2.37 | 0.73 | 2.31 | 0.47 | 2.28 | 0.47 |
10 | 5.9 | 3.21 | 5.23 | 2.53 | 4.63 | 2.12 | 2.36 | 0.65 | 2.29 | 0.42 | 2.28 | 0.42 |
11 | 10.56 | 8.26 | 10.48 | 7.97 | 7.48 | 5.54 | 3.4 | 2.26 | 2.47 | 1.56 | 2.43 | 1.50 |
12 | 9.63 | 7.28 | 8.4 | 5.96 | 7.32 | 5.06 | 3.1 | 1.88 | 2.62 | 1.43 | 2.53 | 1.32 |
13 | 7.57 | 5.09 | 6.48 | 3.85 | 5.15 | 3.03 | 2.55 | 1.13 | 2.43 | 0.84 | 2.3 | 0.78 |
14 | 6.19 | 3.55 | 5.26 | 2.62 | 4.6 | 2.16 | 2.37 | 0.71 | 2.28 | 0.52 | 2.27 | 0.44 |
15 | 5.86 | 3.16 | 5.25 | 2.41 | 4.54 | 2.05 | 2.36 | 0.64 | 2.3 | 0.37 | 2.29 | 0.30 |
16 | 10.33 | 7.99 | 7.87 | 5.68 | 7.84 | 5.68 | 3.31 | 2.14 | 3.13 | 1.78 | 2.41 | 1.44 |
17 | 9.41 | 7.04 | 8.63 | 6.12 | 6.48 | 4.35 | 3.02 | 1.78 | 2.5 | 1.21 | 2.35 | 1.19 |
18 | 7.39 | 4.89 | 6.49 | 3.95 | 5.32 | 3.15 | 2.51 | 1.06 | 2.32 | 0.98 | 2.31 | 0.80 |
19 | 6.06 | 3.39 | 5.55 | 2.76 | 4.73 | 2.09 | 2.36 | 0.68 | 2.35 | 0.58 | 2.24 | 0.44 |
20 | 5.74 | 3.01 | 4.91 | 2.13 | 4.49 | 2.03 | 2.37 | 0.61 | 2.31 | 0.32 | 2.3 | 0.29 |
21 | 10.25 | 7.90 | 9.83 | 7.28 | 6.71 | 4.78 | 3.28 | 2.10 | 3.18 | 1.87 | 2.43 | 1.52 |
22 | 9.33 | 6.95 | 9.06 | 6.47 | 6.07 | 4.04 | 3 | 1.74 | 2.88 | 1.39 | 2.47 | 1.15 |
23 | 7.34 | 4.82 | 5.88 | 3.46 | 5.34 | 3.05 | 2.5 | 1.03 | 2.43 | 0.76 | 2.3 | 0.69 |
24 | 6.01 | 3.33 | 5.11 | 2.37 | 4.44 | 2.09 | 2.36 | 0.67 | 2.31 | 0.43 | 2.3 | 0.41 |
25 | 5.7 | 2.96 | 4.99 | 2.21 | 4.37 | 1.98 | 2.37 | 0.60 | 2.33 | 0.47 | 2.31 | 0.42 |
Generator | a | b | c | θ (10−2) | δ (10−2) | γ (10−2) |
---|---|---|---|---|---|---|
1 | 0 | 20 | 0.0775795 | 3.965 | −5.876 | 7.632 |
3 | 0 | 20 | 0.25 | 2.543 | −6.047 | 5.638 |
6 | 0 | 40 | 0.01 | 4.258 | −5.094 | 4.586 |
9 | 0 | 40 | 0.01 | 4.258 | −5.094 | 4.586 |
12 | 0 | 20 | 0.0322581 | 4.872 | −4.663 | 5.449 |
Cost, Emissions and Loss | Case 1 | Case 2 | Case 3 | Case 4 | |
---|---|---|---|---|---|
Peak load | Cost ($/h) | 74,169.85 | 64,138.60 | 53,709.76 | 57,871.49 |
Emission (kg/h) | 48,861.97 | 38,261.51 | 31,038.75 | 33,525.19 | |
Power loss (MW) | 67.3 | 53.94 | 35.35 | 40.87 | |
Off peak-load | Cost ($/h) | 26,137.06 | 24,480.15 | 23,334.71 | 23,637.90 |
Emission (kg/h) | 11,280.01 | 8646.78 | 7617.79 | 8353.37 | |
Power loss (MW) | 15.01 | 12.19 | 9.18 | 10.23 | |
Total operation cost ($/h) | 50,153.46 | 44,309.38 | 38,522.24 | 40,754.70 | |
Total operation emission (kg/h) | 30,070.99 | 23,454.15 | 19,328.27 | 20,939.28 | |
Total size of ESS (MW) | 0 | 84.08 | 72.38 | 80.78 |
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Wen, S.; Lan, H.; Fu, Q.; Yu, D.C.; Hong, Y.-Y.; Cheng, P. Optimal Allocation of Energy Storage System Considering Multi-Correlated Wind Farms. Energies 2017, 10, 625. https://doi.org/10.3390/en10050625
Wen S, Lan H, Fu Q, Yu DC, Hong Y-Y, Cheng P. Optimal Allocation of Energy Storage System Considering Multi-Correlated Wind Farms. Energies. 2017; 10(5):625. https://doi.org/10.3390/en10050625
Chicago/Turabian StyleWen, Shuli, Hai Lan, Qiang Fu, David C. Yu, Ying-Yi Hong, and Peng Cheng. 2017. "Optimal Allocation of Energy Storage System Considering Multi-Correlated Wind Farms" Energies 10, no. 5: 625. https://doi.org/10.3390/en10050625