/Data-Collection-Algorithm

Optimization-Based Exploration of the Feasible Power Flow Space for OPF Data Collection

Primary LanguageJulia

Optimization-Based Exploration of the Feasible Power Flow Space for Rapid Data Collection

Test functions statement

Following is a subset of the functions which were experimentally tested in this paper. In the following, $\Gamma$ is the set of all previously collected data points, $G$ is the set of dispatchable generator buses, and $N$ is the set of all buses. Lower case variables are decision variables (e.g., $p_{i}$), while upper case variables are numerical data points (e.g., $P_{i,j}$). In our tests, we also attempted to normalize all variables using statistical metrics from the exhaustive set. This normalization aimed at scaling all variable distances between zero and one. However, this strategy is omitted from this paper, since it did not present significant improvements in the numerical study.

$$f_{1}= \sum_{j\in\Gamma}\Bigg(\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{2}+\sum_{i\in{\mathcal G}}\Big(q_{i}-Q_{i,j}\Big)^{2}\Bigg)$$

$$f_{2}= \sum_{j\in\Gamma}\Bigg(\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{3}+\sum_{i\in{\mathcal G}}\Big(q_{i}-Q_{i,j}\Big)^{3}\Bigg)$$

$$f_{3}= \sum_{j\in\Gamma}\Bigg(\log\bigg(\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{2}\bigg)+\log\bigg(\sum_{i\in{\mathcal G}}\Big(q_{i}-Q_{i,j}\Big)^{2}\bigg)\Bigg)$$

$$f_{4}= \sum_{j\in\Gamma}\Bigg(\log_{10}\bigg(\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{2}\bigg)+\log_{10}\bigg(\sum_{i\in{\mathcal G}}\Big(q_{i}-Q_{i,j}\Big)^{2}\bigg)\Bigg)$$

$$f_{5}= \sum_{j\in\Gamma}\Bigg(\log_{2}\bigg(\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{2}\bigg)+\log_{2}\bigg(\sum_{i\in{\mathcal G}}\Big(q_{i}-Q_{i,j}\Big)^{2}\bigg)\Bigg)$$

$$f_{6}= \sum_{j\in\Gamma}\Bigg(\exp\bigg(\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{2}\bigg)+\exp\bigg(\sum_{i\in{\mathcal G}}\Big(q_{i}-Q_{i,j}\Big)^{2}\bigg)\Bigg)$$

$$f_{7}= \sum_{j\in\Gamma}\Bigg(\exp_{10}\bigg(\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{2}\bigg)+\exp_{10}\bigg(\sum_{i\in{\mathcal G}}\Big(q_{i}-Q_{i,j}\Big)^{2}\bigg)\Bigg)$$

$$f_{8}= \sum_{j\in\Gamma}\Bigg(\log\bigg(\sum_{i\in{\mathcal G}}\left|p_{i}-P_{i,j}\right|\bigg)+\log\bigg(\sum_{i\in{\mathcal G}}\left|q_{i}-Q_{i,j}\right|\bigg)\Bigg)$$

$$f_{9}= \sum_{j\in\Gamma}\Bigg(\exp\bigg(\sum_{i\in{\mathcal G}}\left|p_{i}-P_{i,j}\right|\bigg)+\exp\bigg(\sum_{i\in{\mathcal G}}\left|q_{i}-Q_{i,j}\right|\bigg)\Bigg)$$

$$f_{10}= \sum_{j\in\Gamma}\Bigg(\exp\bigg(\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{3}\bigg)+\exp\bigg(\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{3}\bigg)\Bigg)$$

$$f_{11}= \sum_{j\in\Gamma}\Bigg(\sqrt{\sum_{i\in{\mathcal G}}\left|p_{i}^{2}-P_{i,j}^{2}\right|}+\sqrt{\sum_{i\in{\mathcal G}}\left|q_{i}^{2}-Q_{i,j}^{2}\right|}\Bigg)$$

$$f_{12}= \sum_{j\in\Gamma}\Bigg(\log\bigg(\sqrt{\sum_{i\in{\mathcal G}}\left|p_{i}^{2}-P_{i,j}^{2}\right|}\bigg)+\log\bigg(\sqrt{\sum_{i\in{\mathcal G}}\left|q_{i}^{2}-Q_{i,j}^{2}\right|}\bigg)\Bigg)$$

$$f_{13}= \sum_{j\in\Gamma}\Bigg(\log_{10}\bigg(\sqrt{\sum_{i\in{\mathcal G}}\left|p_{i}^{2}-P_{i,j}^{2}\right|}\bigg)+\log_{10}\bigg(\sqrt{\sum_{i\in{\mathcal G}}\left|q_{i}^{2}-Q_{i,j}^{2}\right|}\bigg)\Bigg)$$

$$f_{14}= \sum_{j\in\Gamma}\Bigg(\exp\bigg(\sqrt{\sum_{i\in{\mathcal G}}\left|p_{i}^{2}-P_{i,j}^{2}\right|}\bigg)+\exp\bigg(\sqrt{\sum_{i\in{\mathcal G}}\left|q_{i}^{2}-Q_{i,j}^{2}\right|}\bigg)\Bigg)$$

$$f_{15}= \sum_{j\in\Gamma}\Bigg(\exp_{10}\bigg(\sqrt{\sum_{i\in{\mathcal G}}\left|p_{i}^{2}-P_{i,j}^{2}\right|}\bigg)+\exp_{10}\bigg(\sqrt{\sum_{i\in{\mathcal G}}\left|q_{i}^{2}-Q_{i,j}^{2}\right|}\bigg)\Bigg)$$

$$f_{16}= \sum_{j\in\Gamma}\Bigg(\sqrt{\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{2}}+\sqrt{\sum_{i\in{\mathcal G}}\Big(q_{i}-Q_{i,j}\Big)^{2}}\Bigg)$$

$$f_{17}= \sum_{j\in\Gamma}\Bigg(\log\sqrt{\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{2}}+\log\sqrt{\sum_{i\in{\mathcal G}}\Big(q_{i}-Q_{i,j}\Big)^{2}}\Bigg)$$

$$f_{18}= \sum_{j\in\Gamma}\Bigg(\log_{10}\sqrt{\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{2}}+\log_{10}\sqrt{\sum_{i\in{\mathcal G}}\Big(q_{i}-Q_{i,j}\Big)^{2}}\Bigg)$$

$$f_{19}= \sum_{j\in\Gamma}\Bigg(\log_{2}\sqrt{\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{2}}+\log_{2}\sqrt{\sum_{i\in{\mathcal G}}\Big(q_{i}-Q_{i,j}\Big)^{2}}\Bigg)$$

$$f_{20}= \sum_{j\in\Gamma}\Bigg(\exp\sqrt{\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{2}}+\exp\sqrt{\sum_{i\in{\mathcal G}}\Big(q_{i}-Q_{i,j}\Big)^{2}}\Bigg)$$

$$f_{21}= \sum_{j\in\Gamma}\Bigg(\exp_{10}\sqrt{\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{2}}+\exp_{10}\sqrt{\sum_{i\in{\mathcal G}}\Big(q_{i}-Q_{i,j}\Big)^{2}}\Bigg)$$

$$f_{22}= \sum_{j\in\Gamma}\Bigg(\exp_{2}\sqrt{\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{2}}+\exp_{2}\sqrt{\sum_{i\in{\mathcal G}}\Big(q_{i}-Q_{i,j}\Big)^{2}}\Bigg)$$

$$f_{23}= \sum_{j\in \Gamma}\left(\Bigg(1-\frac{\sum p_{i}P_{i,j}}{\sqrt{\sum p_{i}^{2}}\sqrt{\sum P_{i,j}^{2}}}\Bigg)+\Bigg(1-\frac{\sum q_{i}Q_{i,j}}{\sqrt{\sum q_{i}^{2}}\sqrt{\sum Q_{i,j}^{2}}}\Bigg)\right)$$

$$f_{24}= \sum_{j\in\Gamma}\Bigg(\log\bigg(\max_{i\in{\mathcal G}}(p_{i}-P_{i,j})\bigg)+\log\bigg(\max_{i\in{\mathcal G}}(q_{i}-Q_{i,j})\bigg)\Bigg)$$

$$f_{25}= \sum_{j\in\Gamma}\Bigg(\sqrt{\sum_{k\in{\mathcal N}}\big(v_{k}-V_{k,j}\Big)^{2}}\Bigg)$$

$$f_{26}= \sum_{j\in\Gamma}\Bigg(\log\bigg(\sqrt{\sum_{k\in{\mathcal N}}\big(v_{k}-V_{k,j}\Big)^{2}}\bigg)\Bigg)$$

$$f_{27}= \sum_{j\in\Gamma}\Bigg(\log_{10}\bigg(\sqrt{\sum_{k\in{\mathcal N}}\big(v_{k}-V_{k,j}\Big)^{2}}\bigg)\Bigg)$$

$$f_{28}= \sum_{j\in\Gamma}\Bigg(\log_{2}\bigg(\sqrt{\sum_{k\in{\mathcal N}}\big(v_{k}-V_{k,j}\Big)^{2}}\bigg)\Bigg)$$

$$f_{29}= \sum_{j\in\Gamma}\Bigg(\exp\bigg(\sqrt{\sum_{k\in{\mathcal N}}\big(v_{k}-V_{k,j}\Big)^{2}}\bigg)\Bigg)$$

$$f_{30}= \sum_{j\in\Gamma}\Bigg(\log\bigg(\sum_{k\in{\mathcal N}}\left|v_{k}-V_{k,j}\right|\bigg)\Bigg)$$

$$f_{31}= \sum_{j\in\Gamma}\Bigg(\log_{10}\bigg(\sum_{k\in{\mathcal N}}\left|v_{k}-V_{k,j}\right|\bigg)\Bigg)$$

$$f_{32}= \sum_{j\in\Gamma}\Bigg(\log_{2}\bigg(\sum_{k\in{\mathcal N}}\left|v_{k}-V_{k,j}\right|\bigg)\Bigg)$$

$$f_{33}= \sum_{j\in\Gamma}\Bigg(\log\bigg(\sqrt{\sum_{k\in{\mathcal N}}\Big(v_{k}-V_{k,j}\Big)^{2}}\bigg)+\log\bigg(\sqrt{\sum_{k\in{\mathcal N}}\Big(\theta_{k}-\Theta_{k,j}\Big)^{2}}\bigg)\Bigg)$$

$$f_{34}= \sum_{j\in\Gamma}\Bigg(\log_{2}\bigg(\sqrt{\sum_{k\in{\mathcal N}}\Big(v_{k}-V_{k,j}\Big)^{2}}\bigg)+\log_{2}\bigg(\sqrt{\sum_{k\in{\mathcal N}}\Big(\theta_{k}-\Theta_{k,j}\Big)^{2}}\bigg)\Bigg)$$

$$f_{35}= \sum_{j\in\Gamma}\Bigg(\exp\bigg(\sum_{k\in{\mathcal N}}\Big(v_{k}-V_{k,j}\Big)^{2}\bigg)\Bigg)^2+\sum_{j\in\Gamma}\Bigg(\exp\bigg(\sum_{k\in{\mathcal N}}\Big(\theta_{k}-\Theta_{k,j}\Big)^{2}\bigg)\Bigg)^2$$

$$f_{36}= \sum_{j\in\Gamma}\Bigg(\log\bigg(\sqrt{\sum_{k\in{\mathcal N}}\Big(v_{k}-V_{k,j}\Big)^{2}}\bigg)+\log\bigg(\sqrt{\sum_{k\in{\mathcal N}}\Big(\theta_{k}-\Theta_{k,j}\Big)^{2}}\bigg)\Bigg)+\sum_{j\in\Gamma}\Bigg(\log\bigg(\sqrt{\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{2}}\bigg)+\log\bigg(\sqrt{\sum_{i\in{\mathcal G}}\Big(q_{i}-Q_{i,j}\Big)^{2}}\bigg)\Bigg)$$

$$f_{37}= \sum_{j\in\Gamma}\Bigg(\log\bigg(\sqrt{\sum_{k\in{\mathcal N}}\Big(v_{k}-V_{k,j}\Big)^{2}}\bigg)\Bigg)+\sum_{j\in\Gamma}\Bigg(\log\bigg(\sqrt{\sum_{i\in{\mathcal G}}\Big(p_{i}-P_{i,j}\Big)^{2}}\bigg)+\log\bigg(\sqrt{\sum_{i\in{\mathcal G}}\Big(q_{i}-Q_{i,j}\Big)^{2}}\bigg)\Bigg)$$

$$f_{38}= \sum_{j\in\Gamma}\Bigg(\log\bigg(\sum_{i\in{\mathcal G}}\left|p_{i}-P_{i,j}\right|\bigg)+\log\bigg(\sum_{i\in{\mathcal G}}\left|q_{i}-Q_{i,j}\right|\bigg)+\sum_{j\in\Gamma}\Bigg(\log\bigg(\sum_{k\in{\mathcal N}}\left|v_{k}-V_{k,j}\right|\bigg)\Bigg)$$

$$f_{39}= \sum_{j\in\Gamma}\Bigg(\sum_{i\in{\mathcal G}}\log\left(\Big(p_{i}-P_{i,j}\Big)^{2}\right)+\sum_{i\in{\mathcal G}}\log\left(\Big(q_{i}-Q_{i,j}\Big)^{2}\right)\Bigg)$$

$$f_{40}= \sum_{j\in\Gamma}\Bigg(\sum_{i\in{\mathcal G}}\log\left(|p_{i}-P_{i,j}|\right)+\sum_{i\in{\mathcal G}}\log\left(|q_{i}-Q_{i,j}|\right)\Bigg).$$

N dimension Hausdorff of 2 dimension sets

This section shows in tables the multi-dimensional Hausdorff distances after 301 iterations computed after the euclidean distance of two variables is calculated. This numerical study considers two combinations: the complex power distance and the active power with the voltage magnitude distance.

The N dimension Hausdorff distance algorithm is applied to all the functions listed above in five test systems of the IEEE library: 3-bus, 5-bus, 14-bus, 30-bus, and 57-bus.

The solver chosen for this task is the Ipopt, an interior point optimizer. It is an open-source software for large-scale non-linear optimization. This optimizer works swiftly with the data of this report. However, there are some functions which are not capable of optimizing. Hence the results stated as "DNF" mean that the function could not be solved with the used software. It might be solvable with a more powerful and more expensive optimizer, but they will not qualify for the post-analysis in this report.

Furthermore, the "DNF" status encloses a wide range of different errors. From maximum iterations exceeded to an error in the step computation, to steps size becomes too small. The reasons they cannot be solved will not be considered, and they will just be labelled as "DNF".

Complex power values

The following table shows the values of the complex power Hausdorff distance.

F IEEE3 IEEE5 IEEE 14 IEEE 30 IEEE 57
1 0,627106 6,569731 0,494571 0,335585 4,477464
2 DNF DNF DNF DNF DNF
3 0,480737 3,4464 DNF 0,250789 2,424641
4 0,511438 3,320618 0,222556 0,23554 1,925264
5 0,473523 3,336494 DNF 0,245588 DNF
6 0,737029 DNF 0,269667 0,32102 DNF
7 0,737029 DNF 0,384685 0,335581 DNF
8 0,847258 3,033554 0,251332 0,2223 DNF
9 0,627106 7,335899 0,428832 0,309139 3,572925
10 0,983041 DNF 0,466129 0,374483 DNF
11 0,479627 4,209596 0,237468 DNF DNF
12 0,649108 3,321969 DNF 0,250216 1,610526
13 0,607738 3,25027 DNF 0,224168 1,493927
14 DNF DNF DNF DNF DNF
15 DNF DNF DNF 0,283906 DNF
16 DNF DNF DNF DNF DNF
17 0,459662 3,554565 0,257766 0,234971 2,340729
18 0,51359 3,253236 0,218473 0,223962 1,86068
19 0,474246 3,47093 0,248185 0,224916 DNF
20 DNF DNF DNF DNF 4,477464
21 DNF DNF DNF 0,32102 4,477464
22 0,737029 DNF 0,428832 0,335585 DNF
23 DNF DNF 0,528889 0,335585 DNF
24-25 DNF DNF DNF DNF DNF
26 0,734689 4,434955 0,377998 0,301497 1,9555
27 0,731707 4,48439 0,31687 0,303438 2,024717
28 0,710965 4,43401 0,342646 0,271446 1,912997
29 DNF DNF DNF DNF DNF
30 0,737253 4,504636 0,417803 0,418933 2,581535
31 0,737448 4,400133 0,428267 0,421751 2,698014
32 0,737264 4,311681 0,437301 0,419949 2,797347
33 0,382181 4,192798 0,368557 0,294004 1,883026
34 0,431283 3,811446 0,304908 0,292869 1,813766
35 0,737029 DNF 0,428844 0,32102 DNF
36 0,457824 3,626734 0,211565 0,27462 1,791248
37 0,460138 3,280765 0,2183 0,23497 1,808853
38 0,799807 3,121847 0,249082 DNF 1,390772
39-40 DNF DNF DNF DNF DNF

Active power and voltage magnitude values

The following table compares a multi-dimensional profile with the euclidean distances of voltage magnitude and active power generation.

F IEEE3 IEEE5 IEEE14 IEEE30 IEEE57
1 0,450036 2,825208 0,390939 0,131712 4,413723
2 DNF DNF DNF DNF DNF
3 0,238171 2,242697 DNF 0,089379 1,470119
4 0,267379 1,573723 0,138653 0,098549 1,582602
5 0,223993 1,674841 DNF 0,093355 DNF
6 0,505872 DNF 0,172858 0,131712 DNF
7 0,505872 DNF 0,333461 0,182413 DNF
8 0,862205 1,056312 0,188589 0,10423 DNF
9 0,450036 3,980856 0,387271 0,135768 3,472789
10 0,933686 DNF 0,391108 0,183072 DNF
11 0,230415 1,881679 0,174417 DNF DNF
12 0,267379 0,984608 DNF 0,119454 1,075534
13 0,3285 1,066166 DNF 0,115684 0,903991
14 DNF DNF DNF DNF DNF
15 DNF DNF DNF 0,143334 DNF
16 DNF DNF DNF DNF DNF
17 0,248731 1,73463 0,17264 0,095667 1,49698
18 0,261035 1,504097 0,155999 0,097071 1,493332
19 0,248731 1,511583 0,164225 0,095299 DNF
20 DNF DNF DNF DNF 4,413723
21 DNF DNF DNF 0,131712 4,413723
22 0,505872 DNF 0,380753 0,131712 DNF
23 DNF DNF 0,407348 0,179383 DNF
24-25 DNF DNF DNF DNF DNF
26 0,564326 1,691745 0,140759 0,111681 1,450164
27 0,5289 1,865098 0,143931 0,114069 1,543377
28 0,516241 1,59901 0,132596 0,113742 1,371806
29 DNF DNF DNF DNF DNF
30 0,571863 1,418356 0,199889 0,164211 2,34488
31 0,567696 1,804293 0,177851 0,164212 2,528032
32 0,571863 1,561311 0,137783 0,164212 2,604332
33 0,201794 1,509811 0,139263 0,085951 1,143824
34 0,21418 1,578466 0,136489 0,09786 1,047962
35 0,505872 DNF 0,377743 0,131712 DNF
36 0,249232 1,624081 0,12902 0,071779 1,267764
37 0,272594 1,597604 0,151409 0,074652 1,187258
38 0,611335 1,192883 0,126662 DNF 0,88341
39-40 DNF DNF DNF DNF DNF