An Improved OPP Problem Formulation for Distribution Grid Observability
Abstract
:1. Introduction
2. Improved OPP Problem Formulation for Distribution Grids
3. Solution Methods of the Formulated OPP Problem
3.1. Mixed Integer Linear Programming
3.2. Backtracking Search Algorithm
4. Results and Discussion
- (a)
- Full grid observability of the test feeders without considering ZIB and channel limits.
- (b)
- Full grid observability of the test feeders considering ZIB only.
- (c)
- Full grid observability of the test feeders considering both ZIB and PMU channel limitations.
4.1. Full Grid Observability without Considering ZIB and Channel Limits
4.2. Full Grid Observability Considering the Presence of ZIB Only
4.3. Full Grid Observability Considering Both ZIB and Channel Limits
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Nomenclature
α | Multiplying factor for the control parameter Fcp. |
ψ | Ratio of the required number of PMU to the total number of nodes. |
A | A binary network connectivity matrix. |
apq | Elements of the network connectivity matrix. |
Bmap | A binary integer-valued matrix. |
C | Installation cost of a single PMU. |
f | A vector whose non-zero entries indicate the observability of the corresponding nodes whereas zero entries indicate un-observability. |
fq | Observability constraint. |
Dm | Population dimension. |
Fcp | Control parameter. |
L | Number of channels. |
low & up | lower and upper boundaries of the parameters. |
NB | Number of nodes of the selected distribution grid. |
Npop | Population size. |
Pp | Penalty factor for un-observability of node ‘p’. |
Ppop | Initial population. |
Qp | Incentive factor for bus ‘p’ if it is observed more than required. |
Qhis | Historical population. |
& | Uniformly distributed random numbers in [0, 1]. |
Primary trial population. | |
Final trial population. | |
U | Uniform distribution. |
up | A binary number to indicate un-observability of node ‘p’ where the value ‘0’ indicates the observability and ‘1’ indicates un-observability. |
vp | Difference between expected and obtained values of fp. |
yp | A binary number to indicate the PMU installation at node ‘p’ where the value ‘1’ indicates the presence of PMU and ‘0’ indicates the absence of PMU. |
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Test Feeders | Passive Measurements/ZIB | Radial Node | Sectionalizing and Tie Switches | Connected to Incomers | ||
---|---|---|---|---|---|---|
No. | Location | No. | Location | |||
IEEE 13-node | 3 | 633, 680, 684 | 6 | 611, 634, 646, 652, 675, 680 | 1 | 1 |
IEEE 34-node | 5 | 812, 814, 850, 852, 888 | 9 | 810, 822, 826, 838, 840, 848, 856, 864, 890 | 0 | 1 |
IEEE 37-node | 11 | 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 775 | 15 | 712, 718, 722, 724, 725, 728, 729, 731, 732, 735, 736, 740, 741, 742, 775 | 0 | 1 |
IEEE 128-node | 37 | 3, 8, 13, 14, 15, 18, 21, 23, 25, 26, 27, 36, 40, 44, 54, 57, 61, 67, 72, 78, 81, 89, 91, 93, 97, 101, 105, 108, 110, 135, 149, 151, 152, 160, 250, 300, 450 | 37 | 2, 4, 6, 10, 11, 12, 16, 17, 20, 22, 24, 32, 33, 37, 39, 41, 43, 46, 48, 56, 59, 66, 71, 75, 79, 83, 85, 88, 90, 92, 96, 104, 107, 111, 114, 350, 610 | 11 | 5 |
Test Feeder | Method | No. of PMU | Locations of the PMU | Measurement Redundancy | Execution Time (second) | |
---|---|---|---|---|---|---|
Required | Achieved | |||||
IEEE 13-node | MILP | 6 | 634, 646, 650, 675, 680, 684 | 13 | 14 | 0.0312 |
BSA | 6 | 632, 633, 645, 671, 684, 692 | 23 | 6.0843 | ||
IEEE 34-node | MILP | 12 | 802, 810, 814, 820, 824, 836, 838, 842, 848, 854, 864, 888 | 34 | 35 | 0.0468 |
BSA | 12 | 802, 808, 814, 820, 824, 834, 836, 846, 854, 858, 862, 888 | 42 | 6.0684 | ||
IEEE 37-node | MILP | 12 | 702, 705, 707, 709, 710, 711, 714, 725, 732, 734, 744, 799 | 37 | 43 | 0.0936 |
BSA | 12 | 701, 702, 705, 706, 707, 708, 709, 710, 711, 714, 734, 744 | 47 | 6.2244 | ||
IEEE 128-node | MILP | 48 | 2, 4, 6, 8, 14, 15, 20, 22, 24, 27, 28, 32, 37, 39, 41, 43, 45, 48, 50, 52, 56, 58, 62, 65, 67, 70, 74, 78, 82, 85, 86, 88, 90, 92, 94, 95, 99, 102, 104, 106, 110, 114, 135, 150, 250, 300, 451, 610 | 128 | 132 | 0.1404 |
BSA | 48 | 1, 3, 6, 8, 14, 15, 18, 19, 21, 23, 28, 31, 33, 36, 38, 40, 42, 45, 47, 51, 52, 54, 55, 58, 61, 63, 65, 67, 70, 74, 76, 78, 82, 85, 87, 89, 91, 95, 99, 101, 103, 106, 110, 113, 149, 250, 300, 450 | 169 | 45.6927 |
Method | IEEE 13-Node | IEEE 34-Node | IEEE 37-Node | IEEE 128-Node | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
RNP 1 | RMR 2 | AMR 3 | RNP | RMR | AMR | RNP | RMR | AMR | RNP | RMR | AMR | |
Proposed (BSA) | 6 | 13 | 23 | 12 | 34 | 42 | 12 | 37 | 47 | 48 | 128 | 169 |
MILP | 6 | 13 | 14 | 12 | 34 | 35 | 12 | 37 | 43 | 48 | 128 | 132 |
Two-stage [31] | 6 | - | - | 13 | - | - | - | - | - | 49 | - | - |
ILP [31] | 6 | - | - | 12 | - | - | - | - | - | 48 | - | - |
Test Feeder | Method | No. of PMU | Locations of the PMU | Measurement Redundancy | |
---|---|---|---|---|---|
Required | Achieved | ||||
IEEE 13-node | MILP | 7 | 632, 634, 645, 650, 671, 675, 684 | 20 | 23 |
BSA | 7 | 632, 633, 645, 650, 671, 684, 692 | 25 | ||
IEEE 34-node | MILP | 19 | 800, 802, 808, 812, 814, 816, 818, 822, 824, 830, 832, 834, 836, 844, 846, 854, 858, 862, 890 | 59 | 62 |
BSA | 19 | 800, 802, 808, 812, 814, 816, 818, 820, 824, 830, 832, 834, 836, 844, 846, 854, 858, 862, 888 | 64 | ||
IEEE 37-node | MILP | 18 | 701, 702, 704, 705, 707, 708, 709, 710, 711, 718, 720, 725, 727, 730, 734, 738, 744, 799 | 59 | 64 |
BSA | 18 | 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 714, 720, 727, 734, 738, 744, 799 | 67 | ||
IEEE 128-node | MILP | 73 | 1, 3, 6, 8, 9, 14, 15, 18, 20, 21, 24, 26, 27, 28, 29, 32, 34, 35, 36, 39, 41, 43, 44, 46, 47, 50, 51, 52, 54, 55, 58, 59, 60, 62, 64, 65, 67, 69, 70, 73, 74, 77, 78, 81, 82, 85, 86, 87, 90, 92, 93, 95, 97, 98, 100, 102, 103, 106, 107, 108, 110, 112, 114, 149, 150, 152, 195, 250, 251, 300, 450, 451, 610 | 219 | 230 |
BSA | 73 | 1, 3, 5, 8, 9, 14, 15, 18, 19, 21, 23, 26, 27, 28, 29, 31, 34, 35, 36, 38, 40, 42, 44, 45, 47, 50, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 65, 67, 69, 70, 73, 74, 76, 77, 78, 81, 82, 84, 87, 89, 91, 93, 95, 97, 98, 100, 102, 103, 105, 106, 108, 110, 112, 113, 149, 150, 152, 195, 250, 251, 300, 450, 451 | 253 |
Test Feeder | Method | No. of PMU | Locations of the PMU | Measurement Redundancy | |
---|---|---|---|---|---|
Required | Achieved | ||||
IEEE 13-node | MILP | 13 | 611, 632, 633, 634, 645, 646, 650, 652, 671, 675, 680, 684, 692 | 26 | 37 |
BSA | 13 | 611, 632, 633, 634, 645, 646, 650, 652, 671, 675, 680, 684, 692 | 37 | ||
IEEE 34-node | MILP | 27 | 800, 802, 808, 810, 814, 818, 820, 822, 824, 826, 830, 832, 834, 836, 838, 840, 844, 846, 848, 850, 854, 856, 858, 862, 864, 888, 890 | 68 | 78 |
BSA | 27 | 800, 802, 808, 810, 812, 814, 816, 820, 822, 824, 826, 830, 832, 834, 836, 838, 840, 842, 846, 848, 854, 856, 858, 862, 864, 888, 890 | 79 | ||
IEEE 37-node | MILP | 31 | 701, 702, 703, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 718, 722, 724, 725, 728, 729, 731, 732, 734, 735, 736, 738, 740, 741, 742, 744, 775, 799 | 74 | 89 |
BSA | 31 | 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 714, 718, 722, 724, 725, 728, 729, 731, 732, 734, 735, 736, 737, 740, 741, 742, 744, 775, 799 | 90 | ||
IEEE 128-node | MILP | 103 | 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 27, 28, 29, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 50, 51, 52, 53, 55, 56, 58, 59, 60, 61, 63, 64, 65, 66, 67, 69, 70, 71, 72, 74, 75, 76, 78, 79, 81, 82, 83, 84, 85, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 103, 104, 105, 106, 107, 109, 110, 111, 113, 114, 135, 149, 150, 195, 250, 251, 300, 350, 450, 451, 610 | 256 | 300 |
BSA | 103 | 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 63, 64, 65, 66, 67, 69, 70, 71, 72, 74, 75, 76, 78, 79, 81, 82, 83, 84, 85, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 99, 100, 101, 103, 104, 106, 107, 108, 110, 111, 113, 114, 149, 150, 195, 250, 251, 300, 350, 450, 451, 610 | 306 |
Test Feeder | Method | No. of PMU | Locations of the PMU | Measurement Redundancy | |
---|---|---|---|---|---|
Required | Achieved | ||||
IEEE 13-node | MILP | 4 | 632, 645, 684, 692 | 10 | 15 |
BSA | 4 | 632, 645, 671, 692 | 16 | ||
IEEE 34-node | MILP | 10 | 800, 808, 820, 824, 836, 844, 848, 854, 858, 862 | 29 | 33 |
BSA | 10 | 802, 808, 820, 824, 834, 836, 846, 854, 858, 862 | 36 | ||
IEEE 37-node | MILP | 8 | 701, 709, 711, 714, 733, 734, 744, 799 | 26 | 28 |
BSA | 8 | 701, 702, 709, 710, 711, 714, 734, 744 | 32 | ||
IEEE 128-node | MILP | 28 | 1, 5, 8, 13, 19, 30, 31, 38, 42, 45, 48, 49, 54, 55, 58, 62, 65, 70, 74, 76, 82, 84, 87, 95, 100, 103, 106, 113 | 91 | 95 |
BSA | 28 | 1, 5, 8, 18, 19, 30, 31, 38, 42, 45, 47, 50, 54, 55, 58, 62, 65, 70, 74, 76, 82, 84, 87, 95, 100, 103, 106, 113 | 97 |
Test Feeder | Method | No. of PMU | Locations of the PMU | Measurement Redundancy | |
---|---|---|---|---|---|
Required | Achieved | ||||
IEEE 13-node | MILP | 6 | 632, 645, 650, 675, 684, 692 | 18 | 19 |
BSA | 6 | 632, 645, 650, 671, 684, 692 | 22 | ||
IEEE 34-node | MILP | 15 | 800, 802, 808, 818, 820, 824, 828, 832, 834, 836, 844, 846, 854, 858, 862 | 50 | 51 |
BSA | 15 | 800, 802, 808, 818, 820, 824, 828, 832, 834, 836, 844, 846, 854, 858, 862 | 51 | ||
IEEE 37-node | MILP | 14 | 701, 702, 703, 707, 708, 709, 710, 711, 714, 727, 734, 738, 744, 799 | 49 | 52 |
BSA | 14 | 701, 702, 704, 705, 708, 709, 710, 711, 714, 727, 734, 738, 744, 799 | 52 | ||
IEEE 128-node | MILP | 48 | 1, 3, 5, 8, 13, 18, 19, 25, 26, 28, 29, 31, 35, 38, 42, 45, 47, 49, 51, 54, 55, 58, 60, 62, 64, 65, 67, 69, 70, 73, 74, 76, 78, 81, 83, 85, 87, 95, 98, 99, 102, 103, 106, 112, 113, 195, 250, 450 | 162 | 166 |
BSA | 48 | 1, 3, 5, 8, 13, 18, 19, 21, 25, 28, 29, 31, 38, 42, 44, 45, 47, 49, 50, 54, 55, 58, 60, 62, 64, 65, 67, 69, 70, 73, 74, 76, 78, 81, 82, 84, 87, 95, 98, 99, 102, 103, 106, 112, 113, 195, 250, 450 | 168 |
Test Feeder | Method | No. of PMU | Locations of the PMU | Measurement Redundancy | |
---|---|---|---|---|---|
Required | Achieved | ||||
IEEE 13-node | MILP | 7 | 632, 645, 646, 650, 675, 684, 692 | 20 | 21 |
BSA | 7 | 632, 645, 646, 650, 675, 684, 692 | 21 | ||
IEEE 34-node | MILP | 23 | 800, 802, 808, 810, 818, 820, 822, 824, 826, 828, 834, 836, 838, 840, 844, 846, 848, 854, 856, 858, 862, 864, 888 | 58 | 66 |
BSA | 23 | 800, 802, 808, 810, 816, 820, 822, 824, 826, 828, 832, 834, 836, 838, 840, 844, 846, 848, 854, 856, 858, 862, 864 | 68 | ||
IEEE 37-node | MILP | 16 | 701, 702, 704, 705, 708, 709, 710, 711, 714, 718, 728, 729, 734, 738, 744, 799 | 52 | 55 |
BSA | 16 | 701, 702, 703, 705, 709, 710, 711, 714, 718, 720, 728, 729, 734, 738, 744, 799 | 55 | ||
IEEE 128-node | MILP | 62 | 1, 2, 5, 6, 7, 8, 13, 14, 18, 19, 20, 28, 29, 31, 32, 38, 39, 42, 43, 45, 46, 47, 48, 49, 51, 54, 55, 56, 58, 59, 62, 63, 65, 66, 67, 68, 70, 71, 74, 75, 81, 82, 83, 84, 85, 86, 87, 88, 95, 96, 98, 100, 103, 104, 106, 107, 113, 114, 195, 250, 300, 450 | 182 | 184 |
BSA | 62 | 1, 2, 5, 6, 7, 8, 13, 18, 19, 20, 28, 29, 31, 32, 35, 38, 39, 42, 43, 45, 46, 47, 48, 50, 51, 54, 55, 56, 58, 59, 60, 62, 63, 65, 66, 67, 69, 70, 71, 74, 75, 76, 78, 82, 83, 84, 85, 87, 88, 95, 96, 98, 100, 103, 104, 106, 107, 113, 114, 195, 250, 450 | 186 |
Test Feeder | Method | No. of PMU | Locations of the PMU | Measurement Redundancy | |
---|---|---|---|---|---|
Required | Achieved | ||||
IEEE 13-node | MILP | 10 | 632, 632, 645, 646, 650, 671, 671, 675, 680, 692 | 20 | 20 |
BSA | 10 | 632, 633, 645, 646, 650, 671, 675, 684, 684, 692 | 20 | ||
IEEE 34-node | MILP | 30 | 800, 802, 802, 806, 808, 810, 816, 818, 820, 822, 824, 826, 828, 830, 834, 836, 838, 840, 842, 844, 846, 848, 854, 856, 858, 860, 862, 864, 888, 890 | 58 | 60 |
BSA | 30 | 800, 802, 802, 806, 808, 810, 818, 820, 820, 822, 824, 826, 828, 830, 834, 836, 838, 840, 842, 844, 846, 848, 852, 854, 856, 858, 860, 862, 864, 888 | 60 | ||
IEEE 37-node | MILP | 27 | 701, 708, 708, 708, 709, 709, 709, 709, 711, 714, 718, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 740, 741, 744, 744, 775, 799 | 52 | 54 |
BSA | 27 | 701, 708, 708, 708, 709, 709, 709, 709, 711, 714, 718, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 740, 741, 744, 744, 775, 799 | 54 | ||
IEEE 128-node | MILP | 91 | 1, 2, 5, 6, 19, 20, 28, 29, 31, 32, 38, 39, 40, 40, 41, 42, 42, 42, 43, 44, 44, 44, 45, 45, 46, 47, 47, 48, 49, 50, 52, 55, 56, 58, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 74, 75, 76, 81, 82, 82, 83, 84, 84, 85, 86, 87, 88, 95, 95, 96, 98, 99, 103, 104, 106, 107, 111, 112, 113, 114, 135, 135, 149, 149, 151, 151, 152, 152, 160, 160, 195, 250, 251, 300, 300, 300, 350, 450, 451, 610 | 182 | 182 |
BSA | 91 | 1, 2, 3, 5, 6, 13, 13, 13, 14, 18, 19, 19, 20, 21, 23, 25, 26, 27, 27, 29, 30, 31, 32, 36, 38, 39, 40, 42, 43, 44, 45, 46, 47, 47, 48, 50, 51, 53, 54, 55, 56, 58, 59, 60, 60, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 74, 75, 76, 76, 77, 81, 82, 82, 83, 84, 85, 86, 87, 88, 89, 93, 94, 95, 95, 96, 97, 99, 100, 103, 104, 106, 107, 109, 113, 114, 149, 195, 250, 450 | 182 |
Test Feeder | Channel Limit | Method | Cases | |||||
---|---|---|---|---|---|---|---|---|
Normal Operation | Single Line Outage | Single PMU Loss | ||||||
RNP 1 | AMR 1 | RNP | AMR | RNP | AMR | |||
IEEE 13-node | 1 | MILP | 5 | 10 | 9 | 18 | 10 | 20 |
BSA | 5 | 10 | 9 | 18 | 10 | 20 | ||
2 | MILP | 4 | 12 | 6 | 18 | 8 | 22 | |
BSA | 4 | 12 | 6 | 18 | 8 | 22 | ||
3 | MILP | 4 | 14 | 5 | 18 | 7 | 22 | |
BSA | 4 | 14 | 5 | 18 | 7 | 22 | ||
4 | MILP | 4 | 15 | 6 | 19 | 7 | 21 | |
BSA | 4 | 16 | 6 | 22 | 7 | 21 | ||
5 | MILP | 4 | 15 | 6 | 19 | 7 | 21 | |
BSA | 4 | 16 | 6 | 22 | 7 | 21 | ||
IEEE 34-node | 1 | MILP | 15 | 30 | 26 | 52 | 30 | 60 |
BSA | 15 | 30 | 26 | 52 | 30 | 60 | ||
2 | MILP | 11 | 33 | 17 | 50 | 23 | 64 | |
BSA | 11 | 33 | 17 | 50 | 23 | 65 | ||
3 | MILP | 10 | 33 | 15 | 51 | 23 | 66 | |
BSA | 10 | 36 | 15 | 51 | 23 | 68 | ||
4 | MILP | 10 | 33 | 15 | 51 | 23 | 66 | |
BSA | 10 | 36 | 15 | 51 | 23 | 68 | ||
5 | MILP | 10 | 33 | 15 | 51 | 23 | 66 | |
BSA | 10 | 36 | 15 | 51 | 23 | 68 | ||
IEEE 37-node | 1 | MILP | 14 | 28 | 25 | 50 | 27 | 54 |
BSA | 14 | 28 | 25 | 50 | 17 | 54 | ||
2 | MILP | 9 | 27 | 17 | 50 | 19 | 55 | |
BSA | 9 | 27 | 17 | 50 | 19 | 55 | ||
3 | MILP | 8 | 28 | 14 | 50 | 16 | 53 | |
BSA | 8 | 30 | 14 | 50 | 16 | 53 | ||
4 | MILP | 8 | 28 | 14 | 52 | 16 | 55 | |
BSA | 8 | 32 | 14 | 52 | 16 | 55 | ||
5 | MILP | 8 | 28 | 14 | 52 | 16 | 55 | |
BSA | 8 | 32 | 14 | 52 | 16 | 55 | ||
IEEE 128-node | 1 | MILP | 48 | 96 | 81 | 162 | 91 | 182 |
BSA | 46 | 92 | 81 | 162 | 91 | 182 | ||
2 | MILP | 31 | 93 | 54 | 162 | 66 | 183 | |
BSA | 31 | 93 | 4 | 162 | 66 | 183 | ||
3 | MILP | 28 | 92 | 50 | 167 | 63 | 185 | |
BSA | 28 | 93 | 49 | 164 | 63 | 186 | ||
4 | MILP | 28 | 95 | 48 | 166 | 62 | 184 | |
BSA | 28 | 97 | 48 | 168 | 62 | 186 | ||
5 | MILP | 28 | 95 | 48 | 166 | 62 | 184 | |
BSA | 28 | 97 | 48 | 168 | 62 | 186 |
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Share and Cite
Shafiullah, M.; Abido, M.A.; Hossain, M.I.; Mantawy, A.H. An Improved OPP Problem Formulation for Distribution Grid Observability. Energies 2018, 11, 3069. https://doi.org/10.3390/en11113069
Shafiullah M, Abido MA, Hossain MI, Mantawy AH. An Improved OPP Problem Formulation for Distribution Grid Observability. Energies. 2018; 11(11):3069. https://doi.org/10.3390/en11113069
Chicago/Turabian StyleShafiullah, Md, M. A. Abido, Md Ismail Hossain, and A. H. Mantawy. 2018. "An Improved OPP Problem Formulation for Distribution Grid Observability" Energies 11, no. 11: 3069. https://doi.org/10.3390/en11113069