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