8.2. Influence Sets and Dominance Sets
Comparing the numbers of the sets
AS,j and
AD,j j ∈ {1, 2, …, 14} for Case 0, Case 1, Case 2, and Case 3 (
Table 17), we can note that these numbers can be different.
Table 17 shows that as the significance level α increases, the number of the influence sets may increase. This is one of the consequences of the increasing number of CRs considered statistically significant. Another consequence is increasing the cardinality of the existing influence sets. In the case of the dominance sets, for each study case Case 0, Case 1, Case 2, and Case 3 taken separately, their number does not change when the constant α changes.
It should be noted that among sets
AS,j, as well as sets A
D,j j ∈ {1, 2, …, 14}, there are sets that contain only one element (
Table 18). Such sets are shown in
Table 19,
Table 20 and
Table 21. Taking into account the influence sets, when α = 0.01, it can be said that each one-element set
AS,j j ∈ {1, 2, …, 14} is associated with the nodal reactive power, which is correlated with the magnitude of the voltage at the node at which this power is determined and this correlation is relatively strong. As α increases, we have an increasing number of one-element influence sets, each of which is defined as
AS,i = {
Vj}
i ≠
j.
The strength of CRs, which are additionally taken into account for larger values of α, decreases with the increase in α. In the statistical sense, this tendency is related to the deterioration of the evaluation of the performed analyses along with the increase in the value of α.The situation is different with regard to one-element dominance sets. For each value of α, in the case of each one-element set AD,j j ∈ {1, 2, …, 14}, the set element is a value of the voltage magnitude Vj, being in the relationship crVj-Qj, where Qj is the nodal reactive power at node j and with which the considered set is associated; Vj is a magnitude of the voltage at node j. It should be noted that each of the aforementioned dominance sets contains the voltage magnitude that enters the CR with the highest statistical scores compared to the CRs between this voltage magnitude and the other nodal reactive powers.
Table 19.
One-Element Influence Sets for Considered Study Cases when α = 0.01, 0.02.
Table 19.
One-Element Influence Sets for Considered Study Cases when α = 0.01, 0.02.
α = 0.01 | α = 0.02 |
---|
Case 0 | Case 1 | Case 2 | Case 3 | Case 0 | Case 1 | Case 2 | Case 3 |
---|
| AS,1 = {V1} | AS,1 = {V1}, | AS,1 = {V1}, | | AS,1 = {V1}, | AS,1 = {V1}, | AS,1 = {V1}, |
| AS,2= {V2} | AS,2= {V2}, | AS,11= {V12} | AS,2= {V2}, | AS,2 = {V2}, |
| | AS,3= {V3} | | AS,7= {V1} | AS,3 = {V3}, |
| | | | | AS,7 = {V1}, |
| | | | | AS,14 = {V7} |
Table 20.
One-Element Influence Sets for Considered Study Cases when α = 0.05, 0.1.
Table 20.
One-Element Influence Sets for Considered Study Cases when α = 0.05, 0.1.
α = 0.05 | α = 0.1 |
---|
Case 0 | Case 1 | Case 2 | Case 3 | Case 0 | Case 1 | Case 2 | Case 3 |
---|
AS,14 = {V14} | AS,1 = {V1}, | AS,1 = {V1}, | AS,1 = {V1}, | AS,10 = {V6}, | | AS,7 = {V1}, | AS,7 = {V1}, |
| AS,7 = {V8}, | AS,7 = {V1}, | AS,7 = {V1} | AS,11 = {V3}, | AS,12 = {V1} | AS,10 = {V3} |
| AS,13 = {V13} | AS,12 = {V1} | | AS,14 = {V14} | | |
Table 21.
One-Element Dominance Sets for Considered Study Cases.
Table 21.
One-Element Dominance Sets for Considered Study Cases.
α = 0.01, 0.02 | α = 0.05, 0.1 |
---|
Case 0 | Case 1 | Case 2 | Case 3 | Case 0 | Case 1 | Case 2 | Case 3 |
---|
AD,1 = {V1} | AD,1 = {V1}, | AD,1 = {V1}, | AD,1 = {V1}, | AD,1 = {V1} | AD,1 = {V1}, | AD,1 = {V1}, | AD,1 = {V1} |
| | AD,2 = {V2}, | | | | AD,2 = {V2}, | |
| AD,3 = {V3} | AD,3 = {V3}, | AD,3 = {V3} | | AD,3 = {V3} | AD,3 = {V3}, | |
| | AD,6 = {V6} | | | | AD,6 = {V6} | |
In the case of influence sets, in Case 0 for α = 0.01, 0.02, and Case 1 for α = 0.1, there are no one-element sets. In each of the study cases: Case 1, Case 2, and Case 3 for α = 0.01, 0.02, 0.05 there is a set containing V1, which is associated with the power Q1. In Case 2, as well as in Case 3, for α = 0.01, 0.02 there is a one-element set containing V2, which is associated with the power Q2. It should be noted that node 1 and node 2 are generation nodes.
In the case of dominance sets, in each of the considered study cases (i.e., in Case 0, Case 1, Case 2, and Case 3), regardless of the value of α, also there is the one-element set containing
V1, which is associated with the power
Q1 (
Table 21). The one-element set
AS,2 = {
V2,} is in Case 2, regardless of the value of α. One should pay attention to the one-element set containing
V3, which is associated with the power
Q3. Such a set is in Case 1, Case 2, and Case 3 for α = 0.01, 0.02, and in Case 1 and Case 2 for α = 0.05, 0.1.
The analysis of the influence sets in the study cases under consideration shows that regardless of the value of α in each study case, the nodal reactive powers, with which these influence sets are associated, are
Q1,
Q2,
Q3,
Q6,
Q8, and
Q9. Additionally, in Case 0 and Case 2, there are influence sets associated with the power
Q5. It turns out that the power system nodes with the mentioned powers are in the first part of the ranking list based on the index defined as follows:
where
c =0.39 (coefficient
c is determined experimentally);
j—a number of a data item of quantity
X;
m—the number of all data of quantity
X; and
dXj—
j-th data item of quantity
X.
is a measure of the variability of the quantity X. , , and are standardized measures of the variability of Vi, Qi, and δi, respectively.
For the considered study cases, the ranking lists of the test-system nodes, when the index
Z is taken into account, are in
Table 22. In that table, some TS nodes are distinguished by:
- (i)
The shading, when at the nodes, there are the nodal reactive powers with which the existing influence sets are associated;
- (ii)
The darker shading, when at the nodes, there are nodal reactive powers with which the existing dominance sets are associated.
Table 22.
The ranking list of the TS nodes, when a ranking criterion is the index Z, for considered study cases.
Table 22.
The ranking list of the TS nodes, when a ranking criterion is the index Z, for considered study cases.
No. | Case 0 | Case 1 | Case 2 | Case 3 |
---|
| i | Zi | i | Zi | i | Zi | i | Zi |
---|
1 | 1 | 6.99 | 6 | 9.62 | 6 | 7.63 | 2 | 6.19 |
2 | 2 | 5.05 | 9 | 4.58 | 2 | 3.76 | 1 | 5.34 |
3 | 6 | 3.05 | 2 | 3.25 | 1 | 3.48 | 9 | 3.18 |
4 | 9 | 2.97 | 8 | 1.81 | 9 | 2.91 | 6 | 3.11 |
5 | 8 | 2.37 | 3 | 1.64 | 8 | 2.53 | 8 | 2.58 |
6 | 3 | 1.92 | 1 | 1.59 | 3 | 1.94 | 3 | 1.66 |
7 | 5 | 1.467 | 12 | 1.401 | 5 | 1.42 | 12 | 1.52 |
8 | 12 | 1.466 | 13 | 1.396 | 12 | 1.41 | 13 | 1.48 |
9 | 13 | 1.45 | 14 | 1.374 | 13 | 1.39 | 5 | 1.47 |
10 | 4 | 1.43 | 10 | 1.371 | 4 | 1.383 | 4 | 1.38 |
11 | 14 | 1.38 | 11 | 1.31 | 10 | 1.378 | 14 | 1.37 |
12 | 10 | 1.37 | 5 | 1.28 | 14 | 1.35 | 10 | 1.33 |
13 | 11 | 1.35 | 4 | 1.26 | 11 | 1.3 | 11 | 1.32 |
14 | 7 | 1.20 | 7 | 1.19 | 7 | 1.21 | 7 | 1.14 |
Indeed, the index Zi refers to CR crVi-Qi i ∈ {1, 2, …,14}. However, any such CR, so long as it is SSCR, relates to a voltage magnitude that is in some influence set. In the considered case, the voltage magnitude is Vi, and the mentioned influence set is AS,i, with which the reactive power Qi is associated; i.e., this nodal reactive power is at the same node as the voltage magnitude Vi. Therefore, the aforementioned ranking list can be associated with the existing influence sets. It should be emphasized that there is no influence set without the magnitude of the voltage at the node at which there is the nodal reactive power associated with the mentioned set. In most cases, KRCC of crVi-Qi is maximal or close to the maximal value of KRCCs of CRs crVi-Qj i = i1, i2, …, icj, where IS,j = {i1, i2, …, icj}; IS,j—a set of indices of the voltages whose magnitudes are in the influence set AS,j; cj = |AS,j|.
It should be noted that in addition to the previously mentioned influence sets associated with Q1, Q2, Q3, Q6, Q8, and Q9 and eventually with Q5, for α > 0.01, influence sets associated with other nodal reactive powers can occur. However, the statistical evaluation of those sets is inferior compared to the sets associated with Q1, Q2, Q3, Q6, Q8, Q9, and Q5.
The same nodal-voltage magnitude can be in more than one influence set AS,j j ∈ {1, 2, …, 14}. This is a consequence of the fact that more than one nodal reactive power can have a significant influence on a given nodal-voltage magnitude. This voltage magnitude is in the set AD,j j ∈ {1, 2, …, 14}, which is associated with the nodal reactive power having the greatest influence on the considered voltage magnitude.
If significance level α changes from 0.01 to 0.1, we can observe that:
In the case of some influence sets, their cardinalities do not change—such sets are (i) AS,2 for Case 0, (ii) AS,3 for Case 0 and Case 1, and (iii) AS,9 for Case 1.
Taking into account the ranking list of influence sets, when a ranking criterion is the cardinality of a set (RAS,c where c stands for the cardinality of the set AS), we can state that:
In Case 0, influence sets that are ranked for α = 0.01 do not change rank for α > 0.01; there is no such regularity in other study cases, i.e., in Case 1, Case 2, or Case 3;
In Case 0, Case 1, and Case 3, the relation between the numbers of the positions in the ranking list RAS,c taken by the sets associated with the dominant nodal reactive powers does not change with changes in α; this statement also applies to Case 2, provided that the dominant nodal reactive powers, with which single-element dominance sets are associated, are omitted.
Taking into account the ranking list of influence sets, when a ranking criterion is the index κI (RAS,κI), we can state that:
In Case 0, the influence sets distinguished for α = 0.01 being in the first five positions of the ranking list RAS,κI, are in the same position in the ranking list RAS,κI for each α satisfying the condition α > 0.01; the same can be seen in Case 2 for the first three positions and in Case 3 for the first two positions in the ranking list RAS,κI.
In Case 0, the relation between the numbers of the positions in the ranking list taken by the sets associated with the dominant nodal reactive powers does not change with changes in α; the statement does not apply in Case 1, Case 2, and Case 3.
Generally, for the same case and the same significance level α, both previously considered ranking lists of the influence sets (i.e., RAS,c and RAS,κI) are different. When two influence sets are considered, the higher position of one of them on the ranking list RAS,c does not mean that it will occupy a higher position in relation to the second set on the ranking list RAS,κI.
Taking into account the ranking list of dominance sets, when a ranking criterion is the cardinality of a set (RAD,c), as well as when a ranking criterion is the index κD (RAD,κD), we can state that in each of the cases: Case 0, Case 1, and Case 2, both the ranking lists are independent from α. For the ranking list RAD,κD, in Case 3, the three first positions of the ranking list are also independent from α.
Comparing both aforementioned ranking lists of the dominance sets (i.e., RAD,c and RAD,κD), we can observe the identity of these lists in each of the cases: Case 0, Case 1, and Case 2. In Case 3, the differences between those lists are in the first two positions.
8.3. Evaluation of Dominant Nodal Reactive Powers with the Use of the Index κ
This subsection considers the dominant nodal reactive powers (for the IEEE 14-node TS); i.e., these powers with which dominance sets are associated, in the context of their evaluation with the use of the index κ. The influence sets related to the mentioned powers, which are taken into account when determining the ranking of dominant nodal reactive powers, are included in the analysis.
CRs between nodal-voltage magnitudes and dominant nodal reactive powers, which are the strongest from the point of view of individual powers, are given in
Table 23. Taking into account the rules given in
Table 2, it can be concluded that in
Table 23, there is only one relationship in each of the cases: Case 0, Case 2, and Case 3, in which the strength of the association between the considered quantities can be evaluated as large. In the case of the remaining relationships, the strength of association between the quantities taken into account is medium or low. In
Table 23, the latter relationships are the fewest. These are CRs: in Case 0—cr
V3-Q3; in Case 2—cr
V2-Q2, cr
V3-Q3, and cr
V11-Q5.
In each of the study cases: Case 0, Case 1, Case 2, and Case 3, the strongest CR is crV8-Q8; i.e., the CR between the magnitude of the nodal-voltage V8 and the nodal reactive power Q8. In effect, the voltage magnitude V8 is not only in the set AS,8, but also in the set AD,8.
The cardinality of the influence sets associated with the dominant nodal reactive powers in Case 0, Case 1, Case 2, and Case 3 are in
Table 24.
Table 25 shows indices κ
S,j, where
j is an element of the set of indices of the nodes at which there are the dominant nodal reactive powers.
Table 23.
Nodal Reactive Powers and Nodal-Voltage Magnitudes between which There Are the Strongest CRs (from the Point of View of the Considered Power) and KRCCs Characterizing These CRs for Case 0, Case 1, Case 2, and Case 3.
Table 23.
Nodal Reactive Powers and Nodal-Voltage Magnitudes between which There Are the Strongest CRs (from the Point of View of the Considered Power) and KRCCs Characterizing These CRs for Case 0, Case 1, Case 2, and Case 3.
Case 0 | Case 1 | Case 2 | Case 3 |
---|
Qj | Vi | tk_Vi-Qj | Qj | Vi | tk_Vi-Qj | Qj | Vi | tk_Vi-Qj | Qj | Vi | tk_Vi-Qj |
---|
Q8 | V8 | 0.517 | Q8 | V8 | 0.462 | Q8 | V8 | 0.500 | Q8 | V8 | 0.613 |
Q1 | V1 | 0.474 | Q9 | V9 | 0.457 | Q9 | V9 | 0.444 | Q1 | V1 | 0.455 |
Q9 | V9 | 0.397 | Q1 | V1 | 0.395 | Q1 | V1 | 0.396 | Q6 | V6 | 0.451 |
Q6 | V6 | 0.385 | Q3 | V3 | 0.336 | Q6 | V6 | 0.302 | Q9 | V9 | 0.367 |
Q3 | V3 | 0.249 | Q6 | V6 | 0.330 | Q3 | V3 | 0.278 | Q3 | V3 | 0.338 |
| | | | | | Q5 | V11 | 0.267 | | | |
| | | | | | Q2 | V2 | 0.243 | | | |
Table 24.
The Cardinality of the Influence Sets Associated with the Dominant Nodal Reactive Powers in Case 0, Case 1, Case 2, and Case 3.
Table 24.
The Cardinality of the Influence Sets Associated with the Dominant Nodal Reactive Powers in Case 0, Case 1, Case 2, and Case 3.
| α = 0.01 | α = 0.02 | α = 0.05 | α = 0.1 |
---|
Set | Case 0 | Case 1 | Case 2 | Case 3 | Case 0 | Case 1 | Case 2 | Case 3 | Case 0 | Case 1 | Case 2 | Case 3 | Case 0 | Case 1 | Case 2 | Case 3 |
---|
AS.1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 3 | 2 | 3 | 2 |
AS.2 | | | 1 | | | | 1 | | | | 3 | | | | 3 | |
AS.3 | 13 | 2 | 3 | 1 | 13 | 2 | 4 | 1 | 13 | 2 | 9 | 8 | 13 | 2 | 10 | 10 |
AS.5 | | | 10 | | | | 10 | | | | 11 | | | | 12 | |
AS.6 | 12 | 5 | 7 | 6 | 12 | 6 | 9 | 6 | 14 | 9 | 11 | 8 | 13 | 10 | 12 | 11 |
AS.8 | 13 | 5 | 10 | 4 | 13 | 8 | 11 | 5 | 14 | 12 | 11 | 6 | 14 | 13 | 12 | 10 |
AS.9 | 8 | 5 | 4 | 2 | 9 | 5 | 5 | 2 | 12 | 5 | 6 | 4 | 12 | 5 | 6 | 8 |
Table 24 shows that in each of the study cases: Case 0, Case 1, and Case 2, among the considered influence sets, there is no set of greater cardinality than that of the set
AS,8. In Case 3, only the cardinality of
AS,6 is greater than |
AS,8|. The power
Q8 therefore has an influence on the relatively large area of TS. The power
Q8 has also a relatively large influence on the voltage magnitudes in the mentioned area. This observation results from the analysis of
Table 25. It takes place that (i) κ
S.8 > κ
S.i i = 1, 3, 6, 9 for Case 0; (ii) κ
S.8 > κ
S.i i = 1, 3, 6 for Case 1; (iii) κ
S.8 > κ
S.i i = 1, 2, 3, 5, 6, 9 for Case 2; and (iv) κ
S.8 > κ
S.i i = 1, 3, 9 for Case 3. Such a large influence of the power
Q8 on the voltages in TS can be explained by the location of node 8. Note that node 8 is connected to the third winding of the transformer, which is between the higher-voltage part of TS and the lower-voltage part of TS.
As in cr
V8-Q8, in cr
V6-Q6 and cr
V9-Q9, there are the nodal reactive powers (
Q6 and
Q9) at the nodes to which transformers are connected. Those transformers are between the higher-voltage part of TS and the lower-voltage part of this system. The influence sets associated with the powers
Q6 and
Q9 have high cardinalities (
Table 24) and are also characterized by high values of the indices κ
S.6 and κ
S.9, respectively (
Table 25). It can therefore be concluded that the mentioned powers have a significant influence on the nodal-voltage magnitudes in TS. It should be noted that in Case 0, Case 1, and Case 3, for α ≠ 0.05, the powers
Q6,
Q8, and
Q9 are in the first three positions of the ranking list
RDr,κ; i.e., the ranking list of dominant nodal reactive powers when a ranking criterion is the index κ (Equation (18)) (
Table 26). In Case 2, the power
Q5 is among the first three dominant powers in the ranking list
RDr,κ, which in addition to that power are the powers
Q8 and
Q9. The power
Q5 is in the third position of that ranking list. The power
Q6 is in the fourth position of the mentioned ranking list. In Case 3 for α = 0.05, the power
Q3 is in the third position of the considered ranking list and the power
Q9 is in the fourth position of this list.
Table 25.
Indices κS,j Characterizing Influence Sets Associated with Dominant Nodal Reactive Powers in Case 0. Case 1. Case 2 and Case 3.
Table 25.
Indices κS,j Characterizing Influence Sets Associated with Dominant Nodal Reactive Powers in Case 0. Case 1. Case 2 and Case 3.
| α = 0.01 | α = 0.02 | α = 0.05 | α = 0.1 |
---|
Index | Case 0 | Case 1 | Case 2 | Case 3 | Case 0 | Case 1 | Case 2 | Case 3 | Case 0 | Case 1 | Case 2 | Case 3 | Case 0 | Case 1 | Case 2 | Case 3 |
---|
κS.1 | 0.60 | 0.40 | 0.40 | 0.45 | 0.60 | 0.40 | 0.40 | 0.45 | 0.68 | 0.40 | 0.40 | 0.45 | 0.68 | 0.54 | 0.66 | 0.60 |
κS.2 | | | 0.24 | | | | 0.24 | | | | 0.55 | | | | 0.55 | |
κS.3 | 2.15 | 0.57 | 0.74 | 0.34 | 2.15 | 0.57 | 0.94 | 0.34 | 2.15 | 0.57 | 1.76 | 1.47 | 2.15 | 0.57 | 1.90 | 1.74 |
κS.5 | | | 2.35 | | | | 2.35 | | | | 2.51 | | | | 2.66 | |
κS.6 | 2.59 | 1.42 | 1.67 | 1.98 | 2.59 | 1.61 | 2.05 | 1.98 | 2.66 | 2.10 | 2.37 | 2.31 | 2.66 | 2.24 | 2.51 | 2.72 |
κS.8 | 2.90 | 1.46 | 2.69 | 1.30 | 2.90 | 2.01 | 2.88 | 1.48 | 2.97 | 2.65 | 2.88 | 1.66 | 2.97 | 2.77 | 3.02 | 2.21 |
κS.9 | 1.57 | 1.68 | 1.35 | 0.65 | 1.65 | 1.68 | 1.53 | 0.65 | 1.86 | 1.68 | 1.69 | 0.98 | 1.86 | 1.68 | 1.69 | 1.51 |
Table 26.
Ranking Lists of Dominate Nodal Reactive Powers when a Ranking Criterion is the Index κ for Different Study Cases and Different Values of Level α.
Table 26.
Ranking Lists of Dominate Nodal Reactive Powers when a Ranking Criterion is the Index κ for Different Study Cases and Different Values of Level α.
α = 0.01 | α = 0.02 | α = 0.05 | α = 0.1 |
---|
Case 0 | Case 1 | Case 2 | Case 3 | Case 0 | Case 1 | Case 2 | Case 3 | Case 0 | Case 1 | Case 2 | Case 3 | Case 0 | Case 1 | Case 2 | Case 3 |
---|
Q8 | Q9 | Q8 | Q6 | Q8 | Q9 | Q8 | Q6 | Q8 | Q8 | Q8 | Q6 | Q8 | Q8 | Q8 | Q6 |
Q6 | Q8 | Q9 | Q8 | Q6 | Q8 | Q9 | Q8 | Q6 | Q9 | Q9 | Q8 | Q6 | Q9 | Q9 | Q8 |
Q9 | Q6 | Q5 | Q9 | Q9 | Q6 | Q5 | Q9 | Q9 | Q6 | Q5 | Q3 | Q9 | Q6 | Q5 | Q9 |
Q3 | Q3 | Q6 | Q1 | Q3 | Q3 | Q6 | Q1 | Q3 | Q3 | Q6 | Q9 | Q3 | Q1 | Q6 | Q3 |
Q1 | Q1 | Q3 | Q3 | Q1 | Q1 | Q3 | Q3 | Q1 | Q1 | Q3 | Q1 | Q1 | Q3 | Q3 | Q1 |
| | Q1 | | | | Q1 | | | | Q1 | | | | Q1 | |
| | Q2 | | | | Q2 | | | | Q2 | | | | Q2 | |
For middle values of the active power losses in TS (i.e., for Case 2), the cardinality of the set AS,5 is equal to the maximum value of cardinalities of the sets AS,i i = 1, 2, 3, 6, 8, 9, or it is only one lower than this value depending on the value α. The value of the index κS,5 is lower only than the value of the index κS,8. Due to the index κD,5, in the set DR, the power Q5 is in the third position in the ranking list RDr,κD. The situation is completely different in the other cases of the active power losses in TS, i.e., in Case 0, Case 1, and Case 3. In each of those cases, there is (i) a different relation between the cardinality of the set AS,5 and the cardinalities of other influence sets, (ii) a different relation between the index κS,5 and the indices κS,j j ∈ IAS j ≠ 5, characterizing other influence sets; and (iii) there is no set AD,5 and, therefore, power Q5 is not on the ranking list RDr,c nor on the ranking list RDr,κD. It should be added to the presented considerations that the power Q5 is at the node connected to the higher-voltage winding of the transformer, being between the higher-voltage part of TS and the lower-voltage part of this system. As the analyses show, this fact plays an important role when the system active-power losses are of a middle value.
In the set
DR of each of the cases: Case 0, Case 1, Case 2, and Case 3, there is power
Q3. Analyzing
Table 24, one can note that (i) |
AS.3| = |
AS.8|; i.e., |
AS.3| is equal to the maximum value of cardinalities of the considered influence sets in Case 0 when α = 0.01, 0.02 and Case 3 when α = 0.05, (ii) |
AS.3| is one less than the maximum value of cardinalities of the considered influence sets in Case 0 when α = 0.05, 0.1 and Case 3 when α = 0.1, and (iii) |
AS.3| is significantly smaller than the maximum value of cardinalities of the considered influence sets in other cases and when values of the level α are other than mentioned above. In the ranking list
RDr,κ
D, the power
Q3 is in:
Fourth position in Case 0 for α = 0.01, 0.02, 0.05, 0.1 and Case 3 for α = 0.05, 0.1;
Fifth position in Case 1 for α = 0.01, 0.02, 0.05, 0.1 and Case 3 for α = 0.01, 0.02;
Sixth position in Case 2 for α = 0.01, 0.02, 0.05, 0.1.
In effect, in ranking list RDr,κ, power Q3 is in:
Third position in Case 3 for α = 0.05;
Fourth position in Case 0, Case 1 for α = 0.01, 0.02, 0.05, Case 0 for α = 0.1, and Case 3 for α = 0.1;
Fifth position in Case 1 for α = 0.1, Case 2 for α = 0.01, 0.02, 0.05, 0.1, and Case 3 for α = 0.01, 0.02.
Thus, in general, the influence of the power Q3 on the magnitudes of the voltages in TS is smaller than the power Q6, Q8, and Q9. This is understandable due to the location of nodes 6, 8, and 9 in TS.
We can see in
Table 23 that among the strongest CRs, there are also cr
V1-Q1 (Case 0, Case 1, Case 2, and Case 3), and cr
V2-Q2 (Case 2). In these CRs, there are nodal reactive powers at the generator nodes. These powers have a relatively strong influence on the magnitudes of the voltages at the nodes where they are, and possibly at neighboring nodes. We can see that
Q1 in Case 0, and
Q2 in Case 0 and Case 1 significantly influence the magnitudes of the voltages at one of the nodes adjacent to node 1 or 2, respectively. The low cardinality of
AS,1 and a relatively low position of the power
Q1 in the ranking list
RDr,κ
D; i.e.,
The last position in Case 0, for α = 0.01, 0.02, 0.05, 0.1, and Case 3 for α = 0.05, 0.1;
The fourth position in Case 2, for α = 0.01, 0.02, 0.05, 0.1;
The one before the last position in other cases than those mentioned above means that the power Q1 is at the end of the ranking list RDr,κ; i.e.,
In the last position in Case 0, Case 1 for α = 0.01, 0.02, 0.05, Case 0 for α = 0.1, and Case 3 for α = 0.05, 0.1;
In the one before the last position in other cases than those mentioned above.
Only in Case 3 is the power
Q2 among the dominant nodal reactive powers (
Table 23). In Case 3, the power
Q2 is in the last position of the ranking list of the dominant nodal reactive powers (
Table 26).
It should be noted that for each dominant reactive power, there is the CR between that power and a magnitude of the voltage at the node where this power is present. Except for the power Q5, the KRCC value for the earlier-mentioned CR is the largest, when we take into account the set of CRs of the power under consideration.
The list of the dominant reactive powers is different for the distinguished cases: Case 0, Case 1, Case 2, and Case 3. In each of the mentioned cases, this list includes the powers: Q1, Q3, Q6, Q8, and Q9. It should be noted that:
The listed powers are ordered differently in each of the cases;
In Case 2, there are also Q2 and Q5 in the list under consideration.
For each value of α and each of the study cases: Case 0, Case 1, and Case 2, the ranking list RDr,κ is different from the ranking list RDr,κD. In Case 3, independently from α, the ranking list RDr,κ is the same as the ranking list RDr,κD. The presented facts are a consequence of taking into account not only the evaluation of the dominance sets, but also the evaluation of the influence sets when establishing the ranking list RDr,κ. It should be underlined that taking into account the evaluation of the influence sets may or may not change ranking list RDr,κ in relation to ranking list RDr,κD.
In Case 0, as well as Case 2, the ranking list RDr,κ does not depend on significance level α. In Case 3, only the two first positions of the ranking list RDr,κ do not depend on level α. Note that also in each of the study cases: Case 1 and Case 3, the ranking list RDr,κ will not change when α = 0.02 is taken instead of α = 0.01. It is obvious, from a statistical point of view, that the results of the analyses are rated higher for α = 0.01 or α = 0.02 than for α > 0.02.
8.6. Computational Complexity
The presented method does not require complex calculations. The expected calculations include performing such operations as a comparison, addition/subtraction, multiplication/division, or changing the sign of scalar values.
The method assumes that for each pair (
Vi,
Qj)
i,
j ∈ {1, 2,...,
n}, the coefficient
tk is known; the definition of which is given in
Section 3. A number of operations performed to calculate and test the statistical significance of that coefficient are as follows:
because the numbers of additions/subtractions, multiplications/divisions, and comparisons are as follows: 1.5
m (
m − 1) + 2, 0.5
m (
m − 1) +3, 0.5
m (
m − 1) + 1, respectively.
For PS with
n nodes and nodal reactive powers considered at each node of PS, the number of all possible pairs (
Vi,
Qj)
i,
j ∈ {1, 2, …,
n} (and also KRCCs) is equal to
n2. Thus, the number of operations performed to calculate KRCCs of all possible CRs between nodal voltage magnitudes and nodal reactive powers in PS is equal to:
The absolute values of the respective KRCCs are taken to calculate κS as well as κD. In the extreme case, it is possible to change the sign for all KRCCs, which means that the number of these changes is n2.
Calculation of κ
S indices requires at most
Nκ
S operations, where
NκS is a number of addition operations.
In the extreme case, in order to calculate κ
D indices, operations whose number is equal to
Nκ
D (comparison operations: 1.5
n·(
n − 1), addition operations:
n − 1) should be performed:
The upper limit of the number of operations when calculating and ranking the κ indices is as follows:
because (i) the number of comparison operations is equal to
n·(
n − 1), and (ii) the number of multiplication operations is equal to
n.
In fact, the numbers
Nκ
S, Nκ
D, and
Nκ are much smaller than those results from the formulas given above, because only a relatively small part of all CRs is statistically significant (
Table 3,
Table 16).
Finally, the upper limit of the number of all operations required by the proposed method is as follows:
Taking into account all operations realized when the proposed method is utilized, it can be stated that the computational complexity of the proposed method is O((m ∗ n)2).