A Novel Technique to Improve the Online Calculation Performance of Nonlinear Problems in DC Power Systems
Abstract
:1. Introduction
1.1. The Novel Idea to Improve the Holistic Performance
1.2. The Development Trend of DC Power Systems
1.3. The Organization of This Paper
2. The Potentiality of the Holistic Performance
2.1. The Computation Load of the Power System Analysis Center
2.2. The Repeated Analysis Computation Load
- The derivation of the formulas is based on the power grid characters (such as topology, line parameters and bus types);
- Many power grid parameters are listed as constant coefficients of the formulas;
- The influence of each parameter variation on analysis results has to be calculated by solving nonlinear equations.
2.3. The Potentiality of the Holistic Performance
3. The Linear Relationship Based Nonlinear Problem
3.1. The Nonlinearity and Linearity of Power Systems
- The nonlinear equipment and components in power grids such as power electronic devices which are on the inside of power grids;
- The nonlinear characters of complicated loads and sources which are on the outside of power grids;
- The nonlinear mathematical relationships, for example, the relationship between powers and voltages.
3.2. The Nonlinearity and Linearity in Power System Equations
3.2.1. Establish the Relationship between Currents and Voltages
3.2.2. Get the Injected Powers Expression
3.3. The Definition of Linear Relationship Based Nonlinear Problem
4. Explanation of Basic Principles of the Proposed Technique
4.1. The Frequently Changed and Slowly Changed Computation Load
4.2. What the Proposed Technique Does
4.3. Separate the Computation Load with Different Variation Frequencies
4.4. How to Solve the Nonlinear Equations
4.5. Why the Proposed Technique Contributes to Deep Coupling Systems Analysis
5. Demonstration of the Linear Property
6. Derivations of the Topology Separation
6.1. The Free Variables in the View of Electrical Circuit
6.2. The Relationship between Injected Currents and Line Currents
6.3. The Method to Calculate the Matrix A
6.4. The Relationship between Injected Currents and Bus Voltage Drops
6.5. The Separation of Topology Analysis
7. The Derivation of Bus Types Separation
7.1. The Analysis of V Buses
7.2. The Discussion of P Buses
7.3. The Separation of V Buses
7.4. The Separation of I Buses and P Buses
7.5. The Separation of Bus Types Analysis
8. Discussion
8.1. The New Way to Understand a Power System
8.2. The Discussion of Application on AC Power Systems
9. Verification of the Proposed Technique
9.1. Test Systems
- The topology of each network is not changed;
- The PV buses are converted to V buses and PQ buses are converted to P buses;
- The resistances of the lines in DC systems are specified according to the reactances of the lines in AC system;
- In the bipolar DC systems, the positive and negative phases are thought to be balanced;
- Some P buses and V buses may be replaced by I buses.
- Each test system is solved with the Newton–Raphson method to validate the reasonability. The results show that the converted test systems can exist.
- The topology, components and parameters of each test system are established and tested in SIMULINK. The simulation results prove that the modification is reasonable.
9.2. 14 Buses Case
9.3. 30 Buses Case
9.4. 118 Buses Case
10. Conclusions
Supplementary Materials
Supplementary File 1Author Contributions
Funding
Conflicts of Interest
Abbreviations
LRBNP | Linear relationship based nonlinear problem |
AC | Alternative current |
RES | Renewable energy resources |
DC | Direct current |
GGDF | Generalized generation shift distribution factor |
GSDF | generation shift distribution factor |
ZBD | Z-bus distribution factor |
PTDF | Power transfer distribution factor |
JBDF | Jacobian based distribution factor |
PV | Constant power and voltage |
PQ | Constant active power and reactive power |
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1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | −0.8380 | −0.7465 | −0.6675 | −0.6106 | −0.6299 | −0.6573 | −0.6573 | −0.6519 | −0.6480 | −0.6391 | −0.6317 | −0.6330 | −0.6437 |
2 | −0.1620 | −0.2535 | −0.3325 | −0.3894 | −0.3701 | −0.3427 | −0.3427 | −0.3481 | −0.3520 | −0.3609 | −0.3683 | −0.3670 | −0.3563 |
3 | 0.0273 | −0.5320 | −0.1514 | −0.1031 | −0.1195 | −0.1427 | −0.1427 | −0.1382 | −0.1348 | −0.1273 | −0.1210 | −0.1221 | −0.1311 |
4 | 0.0572 | −0.1434 | −0.3168 | −0.2157 | −0.2501 | −0.2987 | −0.2987 | −0.2891 | −0.2822 | −0.2664 | −0.2532 | −0.2556 | −0.2745 |
5 | 0.0774 | −0.0711 | −0.1994 | −0.2918 | −0.2604 | −0.2159 | −0.2159 | −0.2246 | −0.2310 | −0.2454 | −0.2575 | −0.2553 | −0.2381 |
6 | 0.0273 | 0.4680 | −0.1514 | −0.1031 | −0.1195 | −0.1427 | −0.1427 | −0.1382 | −0.1348 | −0.1273 | −0.1210 | −0.1221 | −0.1311 |
7 | 0.0800 | 0.3071 | 0.5033 | −0.3016 | −0.0279 | 0.3589 | 0.3589 | 0.2830 | 0.2278 | 0.1021 | −0.0034 | 0.0158 | 0.1662 |
8 | 0.0029 | 0.0111 | 0.0182 | −0.0109 | −0.2171 | −0.6342 | −0.6342 | −0.4513 | −0.4097 | −0.3151 | −0.2356 | −0.2501 | −0.3633 |
9 | 0.0017 | 0.0064 | 0.0104 | −0.0062 | −0.1246 | −0.1661 | −0.1661 | −0.2590 | −0.2351 | −0.1808 | −0.1352 | −0.1435 | −0.2085 |
10 | −0.0045 | −0.0174 | −0.0286 | 0.0171 | −0.6584 | −0.1997 | −0.1997 | −0.2897 | −0.3552 | −0.5041 | −0.6292 | −0.6065 | −0.4282 |
11 | −0.0027 | −0.0105 | −0.0172 | 0.0103 | 0.2057 | −0.1202 | −0.1202 | −0.1744 | −0.2846 | −0.5350 | 0.1757 | 0.1522 | −0.0316 |
12 | −0.0004 | −0.0015 | −0.0025 | 0.0015 | 0.0302 | −0.0177 | −0.0177 | −0.0256 | −0.0157 | 0.0069 | −0.5201 | −0.1687 | −0.0882 |
13 | −0.0014 | −0.0054 | −0.0088 | 0.0053 | 0.1057 | −0.0618 | −0.0618 | −0.0896 | −0.0549 | 0.0240 | −0.2849 | −0.5900 | −0.3084 |
14 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | −1.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
15 | 0.0029 | 0.0111 | 0.0182 | −0.0109 | −0.2171 | 0.3658 | 0.3658 | −0.4513 | −0.4097 | −0.3151 | −0.2356 | −0.2501 | −0.3633 |
16 | 0.0027 | 0.0105 | 0.0172 | −0.0103 | −0.2057 | 0.1202 | 0.1202 | 0.1744 | −0.7154 | −0.4650 | −0.1757 | −0.1522 | 0.0316 |
17 | 0.0018 | 0.0069 | 0.0114 | −0.0068 | −0.1359 | 0.0794 | 0.0794 | 0.1152 | 0.0706 | −0.0308 | −0.1951 | −0.2413 | −0.6034 |
18 | 0.0027 | 0.0105 | 0.0172 | −0.0103 | −0.2057 | 0.1202 | 0.1202 | 0.1744 | 0.2846 | −0.4650 | −0.1757 | −0.1522 | 0.0316 |
19 | −0.0004 | −0.0015 | −0.0025 | 0.0015 | 0.0302 | −0.0177 | −0.0177 | −0.0256 | −0.0157 | 0.0069 | 0.4799 | −0.1687 | −0.0882 |
20 | −0.0018 | −0.0069 | −0.0114 | 0.0068 | 0.1359 | −0.0794 | −0.0794 | −0.1152 | −0.0706 | 0.0308 | 0.1951 | 0.2413 | −0.3966 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | −0.0496 | −0.0442 | −0.0395 | −0.0361 | −0.0373 | −0.0389 | −0.0389 | −0.0386 | −0.0383 | −0.0378 | −0.0374 | −0.0375 | −0.0381 |
2 | −0.0442 | −0.1495 | −0.0695 | −0.0565 | −0.0609 | −0.0671 | −0.0671 | −0.0659 | −0.0650 | −0.0630 | −0.0613 | −0.0616 | −0.0640 |
3 | −0.0395 | −0.0695 | −0.0954 | −0.0742 | −0.0814 | −0.0916 | −0.0916 | −0.0896 | −0.0881 | −0.0848 | −0.0820 | −0.0825 | −0.0865 |
4 | −0.0361 | −0.0565 | −0.0742 | −0.0869 | −0.0825 | −0.0764 | −0.0764 | −0.0776 | −0.0785 | −0.0805 | −0.0822 | −0.0819 | −0.0795 |
5 | −0.0373 | −0.0609 | −0.0814 | −0.0825 | −0.2485 | −0.1268 | −0.1268 | −0.1506 | −0.1680 | −0.2075 | −0.2407 | −0.2347 | −0.1874 |
6 | −0.0389 | −0.0671 | −0.0916 | −0.0764 | −0.1268 | −0.2242 | −0.2242 | −0.1839 | −0.1738 | −0.1507 | −0.1313 | −0.1348 | −0.1625 |
7 | −0.0389 | −0.0671 | −0.0916 | −0.0764 | −0.1268 | −0.2242 | −0.4003 | −0.1839 | −0.1738 | −0.1507 | −0.1313 | −0.1348 | −0.1625 |
8 | −0.0386 | −0.0659 | −0.0896 | −0.0776 | −0.1506 | −0.1839 | −0.1839 | −0.2336 | −0.2188 | −0.1853 | −0.1572 | −0.1623 | −0.2024 |
9 | −0.0383 | −0.0650 | −0.0881 | −0.0785 | −0.1680 | −0.1738 | −0.1738 | −0.2188 | −0.2793 | −0.2246 | −0.1720 | −0.1752 | −0.1998 |
10 | −0.0378 | −0.0630 | −0.0848 | −0.0805 | −0.2075 | −0.1507 | −0.1507 | −0.1853 | −0.2246 | −0.3140 | −0.2058 | −0.2044 | −0.1937 |
11 | −0.0374 | −0.0613 | −0.0820 | −0.0822 | −0.2407 | −0.1313 | −0.1313 | −0.1572 | −0.1720 | −0.2058 | −0.3738 | −0.2778 | −0.2099 |
12 | −0.0375 | −0.0616 | −0.0825 | −0.0819 | −0.2347 | −0.1348 | −0.1348 | −0.1623 | −0.1752 | −0.2044 | −0.2778 | −0.3116 | −0.2276 |
13 | −0.0381 | −0.0640 | −0.0865 | −0.0795 | −0.1874 | −0.1625 | −0.1625 | −0.2024 | −0.1998 | −0.1937 | −0.2099 | −0.2276 | −0.3656 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
---|---|---|---|---|---|---|---|---|---|
1 | 0.3669 | −0.3962 | 0.1573 | 0.1453 | 0.1195 | 0.0608 | 0.0115 | 0.0205 | 0.0907 |
2 | 0.1128 | 0.0706 | −0.2981 | −0.1743 | −0.1433 | −0.0729 | −0.0138 | −0.0245 | −0.1088 |
3 | 0.0448 | 0.0281 | −0.0631 | −0.1593 | −0.1310 | −0.0666 | −0.0126 | −0.0224 | −0.0995 |
4 | 0.1025 | 0.1687 | 0.0440 | 0.0406 | 0.0334 | 0.0170 | 0.0032 | 0.0057 | 0.0253 |
5 | −0.0344 | −0.0215 | −0.1111 | −0.2207 | −0.3592 | −0.6740 | −0.0174 | −0.0311 | −0.1378 |
6 | −0.0051 | −0.0032 | −0.0163 | −0.0324 | −0.0267 | −0.0136 | −0.5484 | −0.1956 | −0.1038 |
7 | −0.0177 | −0.0111 | −0.0571 | −0.1134 | −0.0933 | −0.0474 | −0.3841 | −0.6842 | −0.3630 |
8 | 0.1005 | 0.0629 | 0.4544 | 0.2998 | 0.2465 | 0.1254 | 0.0237 | 0.0422 | 0.1872 |
9 | 0.0123 | 0.0077 | 0.2475 | −0.4741 | −0.3898 | −0.1983 | −0.0375 | −0.0667 | −0.2960 |
10 | 0.0344 | 0.0215 | 0.1111 | 0.2207 | −0.6408 | −0.3260 | 0.0174 | 0.0311 | 0.1378 |
11 | 0.0227 | 0.0142 | 0.0734 | 0.1458 | 0.1199 | 0.0610 | −0.0675 | −0.1202 | −0.5333 |
12 | 0.0344 | 0.0215 | 0.1111 | 0.2207 | 0.3592 | −0.3260 | 0.0174 | 0.0311 | 0.1378 |
13 | −0.0051 | −0.0032 | −0.0163 | −0.0324 | −0.0267 | −0.0136 | 0.4516 | −0.1956 | −0.1038 |
14 | −0.0227 | −0.0142 | −0.0734 | −0.1458 | −0.1199 | −0.0610 | 0.0675 | 0.1202 | −0.4667 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
---|---|---|---|---|---|---|---|---|---|
1 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
2 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
3 | −0.0413 | −0.0258 | −0.0177 | −0.0164 | −0.0134 | −0.0068 | −0.0013 | −0.0023 | −0.0102 |
4 | −0.0258 | −0.0425 | −0.0111 | −0.0102 | −0.0084 | −0.0043 | −0.0008 | −0.0014 | −0.0064 |
5 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
6 | −0.0177 | −0.0111 | −0.0800 | −0.0528 | −0.0434 | −0.0221 | −0.0042 | −0.0074 | −0.0330 |
7 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
8 | −0.0164 | −0.0102 | −0.0528 | −0.1050 | −0.0863 | −0.0439 | −0.0083 | −0.0148 | −0.0655 |
9 | −0.0134 | −0.0084 | −0.0434 | −0.0863 | −0.1405 | −0.0715 | −0.0068 | −0.0121 | −0.0539 |
10 | −0.0068 | −0.0043 | −0.0221 | −0.0439 | −0.0715 | −0.1341 | −0.0035 | −0.0062 | −0.0274 |
11 | −0.0013 | −0.0008 | −0.0042 | −0.0083 | −0.0068 | −0.0035 | −0.1403 | −0.0500 | −0.0265 |
12 | −0.0023 | −0.0014 | −0.0074 | −0.0148 | −0.0121 | −0.0062 | −0.0500 | −0.0891 | −0.0473 |
13 | −0.0102 | −0.0064 | −0.0330 | −0.0655 | −0.0539 | −0.0274 | −0.0265 | −0.0473 | −0.2097 |
1 | 0.2535 | 0.0150 |
2 | 0.0425 | 0.0500 |
3 | 0.1768 | 0.0132 |
4 | −0.0102 | 0.0095 |
5 | −0.0317 | −0.0100 |
6 | −0.2152 | −0.0083 |
7 | −0.0881 | −0.0300 |
8 | −0.1027 | −0.0060 |
9 | −0.0345 | −0.0067 |
10 | −0.0773 | −0.0083 |
11 | 0.0084 | −0.0097 |
12 | 0.0012 | −0.0094 |
13 | 0.0043 | −0.0075 |
14 | −0.1233 | - |
15 | 0.0206 | - |
16 | −0.0084 | - |
17 | −0.0055 | - |
18 | −0.0084 | - |
19 | 0.0012 | - |
20 | 0.0055 | - |
Line | Newton Method | Topology Separation | Buses Types Separation |
---|---|---|---|
1 | 0.25350684468481 | 0.25350684468693 | 0.25350684468481 |
2 | 0.34171461636364 | 0.34171461636478 | 0.34171461636414 |
3 | 0.17679446380765 | 0.17679446380819 | 0.17679446380765 |
4 | 0.53923167081852 | 0.53923167081961 | 0.53923167081938 |
5 | 0.35205905241400 | 0.35205905241475 | 0.35205905241465 |
6 | 0.35126777874479 | 0.35126777874530 | 0.35126777874569 |
7 | −0.80411541593387 | −0.80411541593531 | −0.80411541593479 |
8 | 0.01533113318128 | 0.01533113318117 | 0.01533113318090 |
9 | 0.16968716980752 | 0.16968716980739 | 0.16968716980731 |
10 | −0.34209994458276 | −0.34209994458297 | −0.34209994458320 |
11 | 0.57015340375823 | 0.57015340375882 | 0.57015340375880 |
12 | 0.29128015861995 | 0.29128015861975 | 0.29128015861975 |
13 | 0.73685125815450 | 0.73685125815460 | 0.73685125815459 |
14 | −0.81341683093722 | −0.81341683093724 | −0.81341683093762 |
15 | 0.82874796411849 | 0.82874796411805 | 0.82874796411816 |
16 | −0.14939008574679 | −0.14939008574677 | −0.14939008574676 |
17 | 0.11339815651291 | 0.11339815651279 | 0.11339815651280 |
18 | −0.46038925417374 | −0.46038925417335 | −0.46038925417333 |
19 | 0.10745065050639 | 0.10745065050670 | 0.10745065050670 |
20 | 0.42849525446940 | 0.42849525446939 | 0.42849525446938 |
Bus | Newton Method | Topology Separation | Buses Types Separation |
---|---|---|---|
1 | 0.01500000000000 | 0.01500000000013 | 0.01500000000000 |
2 | 0.05000000000000 | 0.05000000000023 | 0.05000000000000 |
3 | 0.11007732819872 | 0.11007732819904 | 0.11007732819887 |
4 | 0.07621602803375 | 0.07621602803400 | 0.07621602803386 |
5 | −0.01000000000000 | −0.00999999999980 | −0.01000000000000 |
6 | 0.11328337476959 | 0.11328337476988 | 0.11328337476966 |
7 | −0.03000000000000 | −0.02999999999971 | −0.03000000000000 |
8 | 0.20445393830227 | 0.20445393830251 | 0.20445393830230 |
9 | 0.19183047605666 | 0.19183047605691 | 0.19183047605670 |
10 | 0.10340351200751 | 0.10340351200783 | 0.10340351200763 |
11 | 0.06451237737657 | 0.06451237737672 | 0.06451237737652 |
12 | 0.08598961339979 | 0.08598961340000 | 0.08598961339980 |
13 | 0.23511453186023 | 0.23511453186044 | 0.23511453186023 |
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Xu, Q.; Wang, Y.; Cao, M.; Zheng, J. A Novel Technique to Improve the Online Calculation Performance of Nonlinear Problems in DC Power Systems. Electronics 2018, 7, 115. https://doi.org/10.3390/electronics7070115
Xu Q, Wang Y, Cao M, Zheng J. A Novel Technique to Improve the Online Calculation Performance of Nonlinear Problems in DC Power Systems. Electronics. 2018; 7(7):115. https://doi.org/10.3390/electronics7070115
Chicago/Turabian StyleXu, Qingshan, Yuqi Wang, Minjian Cao, and Jiaqi Zheng. 2018. "A Novel Technique to Improve the Online Calculation Performance of Nonlinear Problems in DC Power Systems" Electronics 7, no. 7: 115. https://doi.org/10.3390/electronics7070115