Multi-Objective Optimal Reactive Power Planning under Load Demand and Wind Power Generation Uncertainties Using ε-Constraint Method
Abstract
:1. Introduction
2. Uncertainty Modeling
2.1. Modeling the Load Demand Uncertainty
2.2. Modeling the Wind Power Generation Uncertainty
- Several scenarios for the load level are considered.
- The probability of each system loading scenario (level of the load) and its corresponding value using Equations (1) and (2) are calculated.
- Several scenarios for wind speed are considered.
- The probability of each wind speed scenario and its corresponding value using Equations (4) and (5) are calculated.
- The output power of the wind farm using the estimated wind speed in each scenario and Equation (6) is generated.
- The final number of combined load-wind scenarios is obtained by multiplying the number of load scenarios by the number of wind scenarios. By multiplying the probability of the load scenario by the probability of wind speed scenario, the probability of the combined load-wind scenarios () can be calculated as follows [19]:
3. Problem Formulation
3.1. Variables
3.2. Objective Functions
3.2.1. Minimization of Total VAR Investment Cost
- (1).
- The first part evaluates the expected cost of energy loss () during the generated scenarios and is expressed as follows [16,17,18,19,20,21,22,23,24,25,26]:
- (2).
3.2.2. Minimization of Voltage Stability Index
3.2.3. Maximization of the Loadability Factor
3.3. Constraints
3.3.1. Equality Constraints
3.3.2. Inequality Constraints
3.4. Other Considerations in the Problem Formulation
- The transformers tap settings and output reactive power of the VAR sources are treated as continuous variables. Therefore, the whole problem is stated as a probabilistic multi-objective nonlinear problem.
- Since the matrix of power systems is dependent on the transformers tap settings and due to the fact that the transformers tap settings are defined as scenario-dependent variables, the matrix should be calculated for each scenario separately.
- The -index value varies between 0 and 1 for power systems. It should be noted that except for the defined boundaries, the -index value should be obtained without any further restriction during the optimization procedure.
4. Optimization Method
4.1. Multi-Objective Optimization Using ε-Constraint Method
- Each objective function () is optimized with the existing constraints separately and the results are saved in a table, called the payoff table.
- According to the priority of the objective functions, one objective function is selected as the main objective function. Then, the rest of the objective functions are treated as new constraints and added to the main constraints. It should be noted that except for the main objective function, if the goal is to minimize and maximize all the objective functions, then, and , respectively. Also, is a variable parameter.
- In order to assign values to , the maximum () and minimum () values of each objective function should be considered, as shown in Equation (40). It should be noted that those values can be obtained from the payoff table.
- To generate different values for , Equations (41) and (42) are used to minimize and maximize the objective function, respectively. By dividing the domain of the objective function into equal parts using Equations (41) and (42), different values are obtained for . It should be noted that denotes the number of available generated values for .
- By using the obtained values from Step 4, it can be derived that or . For different values of , a set of solutions is obtained, which forms the Pareto front of the problem.
4.2. Fuzzy Decision Maker (FDM)
5. Simulation Results and Discussions
- A.
- Deterministic multi-objective RPP without considering the loadability factor (assessing the proficiency of ε-constraint method)
- B.
- Deterministic multi-objective RPP considering the loadability factor
- C.
- Probabilistic multi-objective RPP considering the load demand uncertainty
- D.
- Probabilistic multi-objective RPP considering the wind power generation uncertainty
- E.
- Probabilistic multi-objective RPP considering load demand and wind power generation uncertainties at the same time
5.1. Case Study Descriptions and Simulation Results
5.1.1. Case A: Deterministic Multi-Objective RPP without Considering the Loadability Factor
5.1.2. Case B: Deterministic Multi-Objective RPP Considering the Loadability Factor
5.1.3. Case C: Probabilistic Multi-Objective RPP Considering the Load Demand Uncertainty
5.1.4. Case D: Probabilistic Multi-Objective RPP Considering the Wind Power Generation Uncertainty
5.1.5. Case E: Probabilistic Multi-Objective RPP Considering Load Demand and Wind Power Generation Uncertainties
5.1.6. Case F: Probabilistic Multi-Objective RPP Considering Load Demand and Wind Power Generation Uncertainties Incorporating Reactive Power from Wind Farms
5.2. Discussions
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A
Parameter | Value |
---|---|
2 | |
10 | |
3 m/s | |
10.28 m/s | |
25 m/s | |
40 MW |
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Reference | Problem Framework | Load Demand Uncertainty | Wind Power Generation Uncertainty | Load Demand and Wind Power Generation Uncertainties | Objective Function | Solution Methodology |
---|---|---|---|---|---|---|
[3] | Deterministic, Multi-Objective | - | - | - | Active power losses, Total VAR cost, Voltage stability index | MOEA |
[4] | Deterministic, Multi-Objective | - | - | - | Investment cost, Short-term voltage stability level, Transient stability level | MOEA |
[5] | Deterministic, Single-Objective | - | - | - | Active power losses, Voltage deviations, Operating cost | WOA, DE, GWO, QODE, QOGWO |
[6] | Deterministic, Single-Objective | - | - | - | Active power losses, Operating cost | SPSO, APSO, EPSO |
[7] | Deterministic, Multi-Objective | - | - | - | Investment cost, Short-term voltage stability, Transient stability | MOEA |
[8] | Deterministic, Single-objective | - | - | - | Active power losses, Voltage deviations, Operating cost | WOA, DE, GWO, QODE, QOGWO |
[9] | Deterministic, Single-Objective | - | - | - | Active power losses, Operating cost | SPSO, APSO, EPSO |
[10] | Deterministic, Multi-Objective | - | - | - | Loadability factor, Active power losses, VAR investment cost | GA |
[11] | Deterministic, Single-Objective | - | - | - | Loadability factor | GA |
[12] | Probabilistic, Single-Objective | ✓ | ✓ | ✓ | Fuel cost, VAR cost, Total cost | Mathematical Programming |
[13] | Probabilistic, Single-Objective | - | ✓ | - | VAR investment cost | DE |
[14] | Probabilistic, Single-Objective | ✓ | - | - | VAR investment cost | Mathematical Programming |
[15] | Probabilistic, Multi-Objective | ✓ | - | - | Operating cost, VAR investment cost, Load shedding risk | Multi-Objective Mathematical Programming (ε-constraint method) |
[16] | Probabilistic, Single-Objective | ✓ | - | - | Operating cost, VAR investment cost | GA |
[17] | Probabilistic, Single- Objective | ✓ | - | - | VAR investment cost | Mathematical Programming |
Present Paper | Probabilistic, Multi-Objective | ✓ | ✓ | ✓ | Active power losses, Total VAR cost, Voltage stability index, Loadability factor | Multi-objective Mathematical Programming (ε-constraint method) |
Generator Voltage Magnitude | |
(p.u.) | 1.050 |
(p.u.) | 1.044 |
(p.u.) | 1.023 |
(p.u.) | 1.025 |
(p.u.) | 1.050 |
(p.u.) | 1.050 |
Transformer Tap Settings | |
(p.u.) | 0.950 |
(p.u.) | 1.100 |
(p.u.) | 1.025 |
(p.u.) | 1.050 |
VAR Compensator | |
(MVAR) | 0.000 |
(MVAR) | 0.000 |
(MVAR) | 0.000 |
(MVAR) | 0.000 |
(MVAR) | 0.000 |
Objective | |
(MW) | 5.4970 |
($) | 2.8892 × 106 |
0.1635 |
Control Variable | Value |
---|---|
(p.u.) | 0.900 |
(p.u.) | 1.100 |
(p.u.) | 0.900 |
(p.u.) | 1.100 |
(MVAR) | 0.000 |
(MVAR) | 35.00 |
1 | 3.0334 × 106 | 0.1241 | 0.0000 | 1.0000 | 0.0000 |
2 | 2.9064 × 106 | 0.1241 | 0.3033 | 0.9286 | 0.3033 |
3 | 2.8182 × 106 | 0.1246 | 0.5139 | 0.8571 | 0.5139 |
4 | 2.7532 × 106 | 0.1249 | 0.6692 | 0.7857 | 0.6692 |
5 | 2.7084 × 106 | 0.1252 | 0.7762 | 0.7143 | 0.7143 |
6 | 2.6802 × 106 | 0.1255 | 0.8437 | 0.6429 | 0.6429 |
7 | 2.6611 × 106 | 0.1257 | 0.8892 | 0.5714 | 0.5714 |
8 | 2.6496 × 106 | 0.1260 | 0.9166 | 0.5000 | 0.5000 |
9 | 2.6415 × 106 | 0.1263 | 0.9360 | 0.4286 | 0.4286 |
10 | 2.6350 × 106 | 0.1266 | 0.9516 | 0.3571 | 0.3571 |
11 | 2.6294 × 106 | 0.1269 | 0.9649 | 0.2857 | 0.2857 |
12 | 2.6247 × 106 | 0.1271 | 0.9762 | 0.2143 | 0.2143 |
13 | 2.6207 × 106 | 0.1274 | 0.9858 | 0.1429 | 0.1429 |
14 | 2.6174 × 106 | 0.1277 | 0.9936 | 0.0714 | 0.0714 |
15 | 2.6147 × 106 | 0.1280 | 1.0000 | 0.0000 | 0.0000 |
Method | ||||
---|---|---|---|---|
ε-constraint Method | Base Case | 5.4970 | 2.8892 × 106 | 0.16350 |
BCS | 4.9813 | 2.7084 × 106 | 0.12520 | |
Reduction (%) | 9.3815 | 6.2578 | 23.4251 | |
MODE Algorithm | Base Case | 4.9630 | 2.6085 × 106 | 0.19780 |
BCS | 4.8300 | 2.5387 × 106 | 0.12040 | |
Reduction (%) | 2.6798 | 2.6759 | 39.1304 |
Control Variable | Optimal Value |
---|---|
(p.u.) | 1.06940 |
(p.u.) | 1.06150 |
(p.u.) | 1.04110 |
(p.u.) | 1.04260 |
(p.u.) | 1.10000 |
(p.u.) | 1.05550 |
(p.u.) | 1.03640 |
(p.u.) | 0.92960 |
(p.u.) | 0.97700 |
(p.u.) | 0.99910 |
(MVAR) | 20.9529 |
(MVAR) | 1.84190 |
(MVAR) | 2.34290 |
(MVAR) | 3.28580 |
(MVAR) | 0.00000 |
1 | 2.9015 × 106 | 0.1272 | 0.0000 | 1.0000 | 1.0000 | 0.0000 | 0.0000 |
2 | 3.1145 × 106 | 0.1309 | 0.0236 | 0.9511 | 0.9286 | 0.0724 | 0.0724 |
3 | 3.3885 × 106 | 0.1347 | 0.0472 | 0.8881 | 0.8571 | 0.1448 | 0.1448 |
4 | 3.7008 × 106 | 0.1384 | 0.0708 | 0.8164 | 0.7857 | 0.2172 | 0.2172 |
5 | 4.0028 × 106 | 0.1421 | 0.0944 | 0.7470 | 0.7143 | 0.2896 | 0.2896 |
6 | 4.3051 × 106 | 0.1459 | 0.1181 | 0.6776 | 0.6429 | 0.3620 | 0.3620 |
7 | 4.6107 × 106 | 0.1496 | 0.1417 | 0.6074 | 0.5714 | 0.4344 | 0.4344 |
8 | 4.9361 × 106 | 0.1533 | 0.1653 | 0.5326 | 0.5000 | 0.5068 | 0.5000 |
9 | 5.2798 × 106 | 0.1571 | 0.1889 | 0.4537 | 0.4286 | 0.5792 | 0.4286 |
10 | 5.6345 × 106 | 0.1608 | 0.2125 | 0.3722 | 0.3571 | 0.6515 | 0.3571 |
11 | 5.9691 × 106 | 0.1645 | 0.2361 | 0.2953 | 0.2857 | 0.7239 | 0.2857 |
12 | 6.2864 × 106 | 0.1683 | 0.2597 | 0.2224 | 0.2143 | 0.7963 | 0.2143 |
13 | 6.6184 × 106 | 0.1720 | 0.2833 | 0.1462 | 0.1429 | 0.8687 | 0.1429 |
14 | 6.9648 × 106 | 0.1757 | 0.3070 | 0.0666 | 0.0714 | 0.9411 | 0.0666 |
15 | 7.2547 × 106 | 0.1795 | 0.3262 | 0.0000 | 0.0000 | 1.0000 | 0.0000 |
Control Variable | Optimal Value |
---|---|
(p.u.) | 1.06300 |
(p.u.) | 1.05310 |
(p.u.) | 1.07610 |
(p.u.) | 1.05420 |
(p.u.) | 1.10000 |
(p.u.) | 1.08520 |
(p.u.) | 1.02690 |
(p.u.) | 0.90000 |
(p.u.) | 1.00600 |
(p.u.) | 0.97310 |
(MVAR) | 0.00000 |
(MVAR) | 0.00000 |
(MVAR) | 0.00000 |
(MVAR) | 3.03920 |
(MVAR) | 2.66360 |
Scenario | Level of the Load | Probability | Duration of the Load (h) |
---|---|---|---|
0.95 | 0.1 | 2920 | |
1.00 | 0.8 | 4380 | |
1.05 | 0.1 | 1460 |
1 | 1.3129 × 106 | 0.1294 | 0.0000 | 1.0000 | 1.0000 | 0.0000 | 0.0000 |
2 | 1.4223 × 106 | 0.1339 | 0.0237 | 0.9443 | 0.9197 | 0.0722 | 0.0722 |
3 | 1.5418 × 106 | 0.1383 | 0.0473 | 0.8834 | 0.8395 | 0.1443 | 0.1443 |
4 | 1.6647 × 106 | 0.1428 | 0.0710 | 0.8207 | 0.7592 | 0.2165 | 0.2165 |
5 | 1.7954 × 106 | 0.1472 | 0.0946 | 0.7542 | 0.6790 | 0.2887 | 0.2887 |
6 | 1.9328 × 106 | 0.1517 | 0.1183 | 0.6841 | 0.5987 | 0.3609 | 0.3609 |
7 | 2.0774 × 106 | 0.1561 | 0.1420 | 0.6105 | 0.5185 | 0.4330 | 0.4330 |
8 | 2.2510 × 106 | 0.1606 | 0.1688 | 0.5220 | 0.4382 | 0.5148 | 0.4382 |
9 | 2.3848 × 106 | 0.1637 | 0.1893 | 0.4538 | 0.3818 | 0.5774 | 0.3818 |
10 | 2.5365 × 106 | 0.1673 | 0.2131 | 0.3765 | 0.3172 | 0.6501 | 0.3172 |
11 | 2.7000 × 106 | 0.1713 | 0.2379 | 0.2932 | 0.2447 | 0.7256 | 0.2447 |
12 | 2.9942 × 106 | 0.1783 | 0.2852 | 0.1433 | 0.1187 | 0.8700 | 0.1187 |
13 | 3.1909 × 106 | 0.1828 | 0.3143 | 0.0431 | 0.0376 | 0.9586 | 0.0376 |
14 | 3.2627 × 106 | 0.1844 | 0.3252 | 0.0065 | 0.0072 | 0.9919 | 0.0065 |
15 | 3.2754 × 106 | 0.1848 | 0.3278 | 0. 0000 | 0. 0000 | 1. 000 | 0.0000 |
Control Variable | Expected Value | |||
---|---|---|---|---|
(p.u.) | 1.0621 | 1.0632 | 1.0640 | 1.0632 |
(p.u.) | 1.0530 | 1.0532 | 1.0533 | 1.0532 |
(p.u.) | 1.0749 | 1.0783 | 1.0794 | 1.0780 |
(p.u.) | 1.0550 | 1.0522 | 1.0514 | 1.0524 |
(p.u.) | 1.1000 | 1.1000 | 1.1000 | 1.1000 |
(p.u.) | 1.0828 | 1.0903 | 1.0927 | 1.0898 |
(p.u.) | 1.0285 | 1.0249 | 1.0234 | 1.0251 |
(p.u.) | 0.9000 | 0.9000 | 0.9000 | 0.9000 |
(p.u.) | 1.0036 | 1.0143 | 1.0171 | 1.0135 |
(p.u.) | 0.9721 | 0.9500 | 0.9492 | 0.9521 |
(MVAR) | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
(MVAR) | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
(MVAR) | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
(MVAR) | 1.8307 | 0.0000 | 0.0000 | 0.1831 |
(MVAR) | 3.1448 | 0.0000 | 0.1993 | 0.3344 |
Scenario | Wind Power Generation (MW) | Probability | Level of the Load | Duration of the Load (h) |
---|---|---|---|---|
0.00000 | 0.0861 | 1 | 1460 | |
5.27050 | 0.1212 | 1 | 1460 | |
15.0917 | 0.1492 | 1 | 1460 | |
24.9726 | 0.1546 | 1 | 1460 | |
34.8784 | 0.1413 | 1 | 1460 | |
40.0000 | 0.3476 | 1 | 1460 |
1 | 3.7924 × 105 | 0.1222 | 0.0000 | 1.0000 | 1.0000 | 0.0000 | 0.0000 |
2 | 4.1835 × 105 | 0.1282 | 0.0300 | 0.9538 | 0.9096 | 0.0724 | 0.0724 |
3 | 4.6485 × 105 | 0.1335 | 0.0600 | 0.8988 | 0.8293 | 0.1448 | 0.1448 |
4 | 5.1663 × 105 | 0.1386 | 0.0900 | 0.8376 | 0.7523 | 0.2172 | 0.2172 |
5 | 5.7321 × 105 | 0.1442 | 0.1200 | 0.7707 | 0.6683 | 0.2896 | 0.2896 |
6 | 6.3071 × 105 | 0.1480 | 0.1500 | 0.7028 | 0.6093 | 0.3620 | 0.3620 |
7 | 6.9212 × 105 | 0.1525 | 0.1800 | 0.6302 | 0.5418 | 0.4344 | 0.4344 |
8 | 7.5741 × 105 | 0.1570 | 0.2100 | 0.5530 | 0.4740 | 0.5068 | 0.4740 |
9 | 8.2287 × 105 | 0.1615 | 0.2400 | 0.4757 | 0.4058 | 0.5792 | 0.4058 |
10 | 8.8964 × 105 | 0.1661 | 0.2700 | 0.3968 | 0.3355 | 0.6515 | 0.3355 |
11 | 9.4995 × 105 | 0.1706 | 0.3000 | 0.3255 | 0.2680 | 0.7239 | 0.2680 |
12 | 1.0169 × 106 | 0.1752 | 0.3300 | 0.2463 | 0.1984 | 0.7963 | 0.1984 |
13 | 1.0876 × 106 | 0.1798 | 0.3600 | 0.1628 | 0.1285 | 0.8687 | 0.1285 |
14 | 1.1621 × 106 | 0.1845 | 0.3900 | 0.0747 | 0.0583 | 0.9411 | 0.0583 |
15 | 1.2253 × 106 | 0.1883 | 0.4144 | 0.0000 | 0.0000 | 1.0000 | 0.0000 |
Control Variable | Expected Value | ||||||
---|---|---|---|---|---|---|---|
(p.u.) | 1.0630 | 1.0627 | 1.0619 | 1.0611 | 1.0604 | 1.0599 | 1.0611 |
(p.u.) | 1.0531 | 1.0530 | 1.0527 | 1.0523 | 1.0520 | 1.0518 | 1.0523 |
(p.u.) | 1.0767 | 1.0778 | 1.0773 | 1.0769 | 1.0766 | 1.0765 | 1.0769 |
(p.u.) | 1.0536 | 1.0525 | 1.0528 | 1.0531 | 1.0533 | 1.0534 | 1.0531 |
(p.u.) | 1.1000 | 1.1000 | 1.1000 | 1.1000 | 1.1000 | 1.1000 | 1.1000 |
(p.u.) | 1.0867 | 1.0893 | 1.0887 | 1.0883 | 1.0882 | 1.0883 | 1.0883 |
(p.u.) | 1.0264 | 1.0266 | 1.0289 | 1.0311 | 1.0330 | 1.0338 | 1.0310 |
(p.u.) | 0.9000 | 0.9000 | 0.9000 | 0.9000 | 0.9000 | 0.9000 | 0.9000 |
(p.u.) | 1.0084 | 1.0121 | 1.0094 | 1.0071 | 1.0052 | 1.0043 | 1.0070 |
(p.u.) | 0.9657 | 0.9499 | 0.9490 | 0.9480 | 0.9468 | 0.9462 | 0.9491 |
(MVAR) | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
(MVAR) | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
(MVAR) | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
(MVAR) | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
(MVAR) | 3.8586 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.3321 |
Scenario | Wind Power Generation (MW) | Level of the Load | Duration of the Load (h) | Probability |
---|---|---|---|---|
0.00000 | 0.95 | 400 | 0.0086 | |
5.27050 | 0.95 | 400 | 0.0121 | |
15.0917 | 0.95 | 400 | 0.0149 | |
24.9726 | 0.95 | 400 | 0.0155 | |
34.8784 | 0.95 | 400 | 0.0141 | |
40.0000 | 0.95 | 400 | 0.0348 | |
0.00000 | 1.00 | 730 | 0.0689 | |
5.27050 | 1.00 | 730 | 0.0970 | |
15.0917 | 1.00 | 730 | 0.1194 | |
24.9726 | 1.00 | 730 | 0.1237 | |
34.8784 | 1.00 | 730 | 0.1130 | |
40.0000 | 1.00 | 730 | 0.2781 | |
0.00000 | 1.05 | 330 | 0.0086 | |
5.27050 | 1.05 | 330 | 0.0121 | |
15.0917 | 1.05 | 330 | 0.0149 | |
24.9726 | 1.05 | 330 | 0.0155 | |
34.8784 | 1.05 | 330 | 0.0141 | |
40.0000 | 1.05 | 330 | 0.0348 |
1 | 2.0134 × 105 | 0.1172 | 0.0011 | 1.0000 | 1.0000 | 0.0000 | 0.0000 |
2 | 2.1190 × 105 | 0.1224 | 0.0307 | 0.9718 | 0.9277 | 0.0717 | 0.0717 |
3 | 2.2563 × 105 | 0.1277 | 0.0607 | 0.9352 | 0.8555 | 0.1441 | 0.1441 |
4 | 2.4456 × 105 | 0.1330 | 0.0907 | 0.8847 | 0.7832 | 0.2165 | 0.2165 |
5 | 2.6724 × 105 | 0.1382 | 0.1207 | 0.8242 | 0.7110 | 0.2889 | 0.2889 |
6 | 2.9247 × 105 | 0.1435 | 0.1507 | 0.7569 | 0.6387 | 0.3614 | 0.3614 |
7 | 3.1866 × 105 | 0.1487 | 0.1807 | 0.6871 | 0.5667 | 0.4338 | 0.4338 |
8 | 3.4683 × 105 | 0.1539 | 0.2107 | 0.6119 | 0.4946 | 0.5062 | 0.4946 |
9 | 3.7578 × 105 | 0.1592 | 0.2407 | 0.5347 | 0.4224 | 0.5786 | 0.4224 |
10 | 4.0375 × 105 | 0.1644 | 0.2707 | 0.4601 | 0.3503 | 0.6511 | 0.3503 |
11 | 4.3143 × 105 | 0.1697 | 0.3007 | 0.3863 | 0.2782 | 0.7235 | 0.2782 |
12 | 4.6080 × 105 | 0.1749 | 0.3307 | 0.3079 | 0.2061 | 0.7959 | 0.2061 |
13 | 4.9207 × 105 | 0.1801 | 0.3607 | 0.2245 | 0.1351 | 0.8684 | 0.1351 |
14 | 5.2537 × 105 | 0.1850 | 0.3910 | 0.1357 | 0.0672 | 0.9414 | 0.0672 |
15 | 5.7623 × 105 | 0.1899 | 0.4153 | 0.0000 | 0.0000 | 1.0000 | 0.0000 |
Control Variable | (p.u.) | (p.u.) | (p.u.) | (p.u.) | (p.u.) | (p.u.) | (p.u.) | (p.u.) | (p.u.) | (p.u.) | (MVAR) | (MVAR) | (MVAR) | (MVAR) | (MVAR) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.0622 | 1.0530 | 1.0766 | 1.0534 | 1.1000 | 1.0868 | 1.0271 | 0.9000 | 1.0103 | 0.9522 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |
1.0618 | 1.0528 | 1.0759 | 1.0539 | 1.1000 | 1.0854 | 1.0288 | 0.9000 | 1.0072 | 0.9563 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.1342 | |
1.0610 | 1.0525 | 1.0754 | 1.0545 | 1.1000 | 1.0849 | 1.0311 | 0.9000 | 1.0047 | 0.9555 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.1557 | |
1.0602 | 1.0521 | 1.0751 | 1.0545 | 1.1000 | 1.0846 | 1.0333 | 0.9000 | 1.0026 | 0.9546 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.1916 | |
1.0594 | 1.0518 | 1.0747 | 1.0547 | 1.1000 | 1.0846 | 1.0351 | 0.9000 | 1.0007 | 0.9541 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.3510 | |
1.0594 | 1.0517 | 1.0751 | 1.0545 | 1.1000 | 1.0856 | 1.0353 | 0.9000 | 1.0008 | 0.9532 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.4686 | |
1.0631 | 1.0532 | 1.0775 | 1.0529 | 1.1000 | 1.0884 | 1.0257 | 0.9000 | 1.0114 | 0.9569 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.6385 | |
1.0627 | 1.0530 | 1.0772 | 1.0530 | 1.1000 | 1.0881 | 1.0270 | 0.9000 | 1.0100 | 0.9558 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.4814 | |
1.0619 | 1.0526 | 1.0768 | 1.0533 | 1.1000 | 1.0874 | 1.0294 | 0.9000 | 1.0073 | 0.9551 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.5306 | |
1.0611 | 1.0523 | 1.0764 | 1.0536 | 1.1000 | 1.0870 | 1.0315 | 0.9000 | 1.0050 | 0.9540 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.5171 | |
1.0603 | 1.0519 | 1.0760 | 1.0538 | 1.1000 | 1.0869 | 1.0335 | 0.9000 | 1.0029 | 0.9534 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.6548 | |
1.0599 | 1.0518 | 1.0758 | 1.0540 | 1.1000 | 1.0868 | 1.0344 | 0.9000 | 1.0018 | 0.9535 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.8457 | |
1.0639 | 1.0533 | 1.0787 | 1.0521 | 1.1000 | 1.0910 | 1.0240 | 0.9000 | 1.0141 | 0.9575 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 2.2814 | |
1.0635 | 1.0531 | 1.0786 | 1.0521 | 1.1000 | 1.0908 | 1.0253 | 0.9000 | 1.0130 | 0.9555 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.8830 | |
1.0628 | 1.0528 | 1.0781 | 1.0524 | 1.1000 | 1.0901 | 1.0277 | 0.9000 | 1.0102 | 0.9544 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.8246 | |
1.0620 | 1.0524 | 1.0775 | 1.0527 | 1.1000 | 1.0895 | 1.0299 | 0.9000 | 1.0076 | 0.9544 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 2.0862 | |
1.0610 | 1.0518 | 1.0764 | 1.0527 | 1.1000 | 1.0905 | 1.0316 | 0.9000 | 1.0075 | 0.9524 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.9060 | |
1.0606 | 1.0515 | 1.0758 | 1.0528 | 1.1000 | 1.0910 | 1.0324 | 0.9000 | 1.0074 | 0.9518 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.9321 | |
Expected Value | 1.0611 | 1.0523 | 1.0764 | 1.0536 | 1.1000 | 1.0874 | 1.0314 | 0.9000 | 1.0053 | 0.9543 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.6427 |
Scenario | Wind Power Generation (MVAR) |
---|---|
0.0000 | |
1.0702 | |
3.0645 | |
5.0709 | |
7.0824 | |
8.1223 | |
0.0000 | |
1.0702 | |
3.0645 | |
5.0709 | |
7.0824 | |
8.1223 | |
0.0000 | |
1.0702 | |
3.0645 | |
5.0709 | |
7.0824 | |
8.1223 |
1 | 2.0015 × 105 | 0.1164 | 0.0010 | 1.0000 | 1.0000 | 0.0000 | 0.0000 |
2 | 2.1125 × 105 | 0.1215 | 0.0308 | 0.9706 | 0.9279 | 0.0719 | 0.0719 |
3 | 2.2251 × 105 | 0.1267 | 0.0608 | 0.9407 | 0.8559 | 0.1443 | 0.1443 |
4 | 2.4128 × 105 | 0.1319 | 0.0908 | 0.8910 | 0.7838 | 0.2167 | 0.2167 |
5 | 2.6295 × 105 | 0.1370 | 0.1208 | 0.8336 | 0.7118 | 0.2892 | 0.2892 |
6 | 2.8786 × 105 | 0.1422 | 0.1508 | 0.7676 | 0.6397 | 0.3616 | 0.3616 |
7 | 3.1517 × 105 | 0.1474 | 0.1808 | 0.6953 | 0.5677 | 0.4340 | 0.4340 |
8 | 3.4345 × 105 | 0.1525 | 0.2108 | 0.6203 | 0.4959 | 0.5064 | 0.4959 |
9 | 3.7331 × 105 | 0.1577 | 0.2408 | 0.5412 | 0.4240 | 0.5789 | 0.4240 |
10 | 4.0216 × 105 | 0.1629 | 0.2708 | 0.4648 | 0.3521 | 0.6513 | 0.3521 |
11 | 4.3098 × 105 | 0.1680 | 0.3008 | 0.3884 | 0.2802 | 0.7237 | 0.2802 |
12 | 4.6004 × 105 | 0.1732 | 0.3308 | 0.3115 | 0.2083 | 0.7961 | 0.2083 |
13 | 4.9099 × 105 | 0.1782 | 0.3609 | 0.2295 | 0.1377 | 0.8685 | 0.1377 |
14 | 5.2389 × 105 | 0.1831 | 0.3909 | 0.1423 | 0.0696 | 0.9410 | 0.0696 |
15 | 5.7760 × 105 | 0.1881 | 0.4153 | 0.0000 | 0.0000 | 1.0000 | 0.0000 |
Control Variable | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.0622 | 1.0530 | 1.0766 | 1.0534 | 1.1000 | 1.0868 | 1.0271 | 0.9000 | 1.0103 | 0.9522 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |
1.0618 | 1.0528 | 1.0757 | 1.0541 | 1.1000 | 1.0845 | 1.0301 | 0.9000 | 1.0063 | 0.9568 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.0974 | |
1.0609 | 1.0524 | 1.0748 | 1.0547 | 1.1000 | 1.0821 | 1.0350 | 0.9000 | 1.0020 | 0.9570 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.0878 | |
1.0601 | 1.0521 | 1.0739 | 1.0553 | 1.1000 | 1.0800 | 1.0397 | 0.9000 | 0.9980 | 0.9572 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.1175 | |
1.0594 | 1.0517 | 1.0732 | 1.0559 | 1.1000 | 1.0782 | 1.0441 | 0.9000 | 0.9945 | 0.9576 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.2400 | |
1.0593 | 1.0516 | 1.0732 | 1.0558 | 1.1000 | 1.0782 | 1.0458 | 0.9000 | 0.9936 | 0.9576 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.4022 | |
1.0631 | 1.0531 | 1.0773 | 1.0530 | 1.1000 | 1.0881 | 1.0258 | 0.9000 | 1.0109 | 0.9584 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 2.0174 | |
1.0627 | 1.0530 | 1.0770 | 1.0532 | 1.1000 | 1.0871 | 1.0284 | 0.9000 | 1.0090 | 0.9563 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.4457 | |
1.0619 | 1.0526 | 1.0761 | 1.0538 | 1.1000 | 1.0846 | 1.0333 | 0.9000 | 1.0045 | 0.9565 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.4365 | |
1.0611 | 1.0522 | 1.0753 | 1.0544 | 1.1000 | 1.0824 | 1.0380 | 0.9000 | 1.0005 | 0.9565 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.3856 | |
1.0603 | 1.0519 | 1.0744 | 1.0550 | 1.1000 | 1.0805 | 1.0425 | 0.9000 | 0.9967 | 0.9569 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.4942 | |
1.0598 | 1.0517 | 1.0740 | 1.0553 | 1.1000 | 1.0795 | 1.0447 | 0.9000 | 0.9948 | 0.9572 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.5764 | |
1.0639 | 1.0533 | 1.0787 | 1.0521 | 1.1000 | 1.0910 | 1.0240 | 0.9000 | 1.0141 | 0.9574 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 2.2599 | |
1.0635 | 1.0531 | 1.0783 | 1.0523 | 1.1000 | 1.0898 | 1.0266 | 0.9000 | 1.0120 | 0.9559 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.8262 | |
1.0628 | 1.0527 | 1.0774 | 1.0529 | 1.1000 | 1.0872 | 1.0315 | 0.9000 | 1.0073 | 0.9559 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.7669 | |
1.0620 | 1.0524 | 1.0766 | 1.0535 | 1.1000 | 1.0849 | 1.0363 | 0.9000 | 1.0030 | 0.9561 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.7580 | |
1.0612 | 1.0520 | 1.0758 | 1.0541 | 1.1000 | 1.0828 | 1.0408 | 0.9000 | 0.9990 | 0.9564 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.8356 | |
1.0608 | 1.0518 | 1.0754 | 1.0544 | 1.1000 | 1.0819 | 1.0431 | 0.9000 | 0.9971 | 0.9566 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.9071 | |
Expected Value | 1.0610 | 1.0522 | 1.0753 | 1.0544 | 1.1000 | 1.0826 | 1.0380 | 0.9000 | 1.0006 | 0.9569 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.5296 |
Case | ||||
---|---|---|---|---|
B | 4.9361 × 106 | 0.1533 | 0.1653 | 9.3494 |
C | 2.2510 × 106 | 0.1606 | 0.1688 | 9.5049 |
D | 7.5741 × 105 | 0.1570 | 0.2100 | 8.5777 |
E | 3.4683 × 105 | 0.1539 | 0.2107 | 8.5575 |
F | 3.4345 × 105 | 0.1525 | 0.2108 | 8.4807 |
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Share and Cite
Shojaei, A.H.; Ghadimi, A.A.; Miveh, M.R.; Mohammadi, F.; Jurado, F. Multi-Objective Optimal Reactive Power Planning under Load Demand and Wind Power Generation Uncertainties Using ε-Constraint Method. Appl. Sci. 2020, 10, 2859. https://doi.org/10.3390/app10082859
Shojaei AH, Ghadimi AA, Miveh MR, Mohammadi F, Jurado F. Multi-Objective Optimal Reactive Power Planning under Load Demand and Wind Power Generation Uncertainties Using ε-Constraint Method. Applied Sciences. 2020; 10(8):2859. https://doi.org/10.3390/app10082859
Chicago/Turabian StyleShojaei, Amir Hossein, Ali Asghar Ghadimi, Mohammad Reza Miveh, Fazel Mohammadi, and Francisco Jurado. 2020. "Multi-Objective Optimal Reactive Power Planning under Load Demand and Wind Power Generation Uncertainties Using ε-Constraint Method" Applied Sciences 10, no. 8: 2859. https://doi.org/10.3390/app10082859
APA StyleShojaei, A. H., Ghadimi, A. A., Miveh, M. R., Mohammadi, F., & Jurado, F. (2020). Multi-Objective Optimal Reactive Power Planning under Load Demand and Wind Power Generation Uncertainties Using ε-Constraint Method. Applied Sciences, 10(8), 2859. https://doi.org/10.3390/app10082859