4.4.1
System description
Figure 4.6 shows the version of the IEEE 14-bus system used in this work. The network topology and electrical parameters of transformers and circuits are taken from a common repository for network data [73]. However, the generators and the branch ratings have been modified using the process defined in Section 4.1.1 in order to create a system in which the generators can cause overloads and thus power flow management algorithms can be applied.
The IEEE 14-bus system is originally derived from a US transmission system and comprises 14 buses, 15 circuits, and 5 transformers (if the 3-winding transformer between buses 4, 8, and 9 is treated as a set of 2-winding transformers). Due to its transmission origin, the circuits are mainly overhead lines whose X/R ratio is between 2 and 4; furthermore, the original data does not include ratings for the circuits and transformers. The slack bus is located at bus 1, and the original configuration from [73] features a single generator (at
bus 2) and three synchronous condensers (at buses 3, 6, and 8). Loads are spread amongst 11 of the buses, with bus 9 also featuring a shunt capacitor.
The original configuration of a single generator and branches without ratings does not present a power flow management problem, creating the need to modify the system. As shown in Figure 4.6, the generator and synchronous condensers have been removed, and instead four generators have been added at buses 10, 11, 12, and 14. The generators are assumed to operate with a unity power factor, and their sizes were determined using the process described in Section 4.1.1. That process was also used to determine ratings for the branches within the system, and the 7 branches whose ratings allow them to become overloaded due to generator output are highlighted red in the figure.
4.4.2
Test states
The same method as used for the 11 kV radial distribution system was used to randomly generate 10,000 test states for the IEEE 14-bus system, with random variables drawn from independent uniform distributions used to scale:
• All loads by a common scaling factor.
• Each of the four generators, by separate scaling factors.
4.4.3
Baseline performance
Out of the 10,000 states tested using the IEEE 14-bus system, 4,008 (40.08%) result in overloads. As shown in Figure 4.7, a number of states feature more than one overload, giving 5345 overloads, with a total energy of 9116.25 MVAh.
Table 4.3 tabulates a summary of the loadings for each of the seven circuits within the IEEE 14-bus system that can become overloaded. No circuit is overloaded for more than 25% of all states. Circuits 9-10 and 10-11 are the most heavily loaded, in terms of: 1) the number of overloads, with these two circuits appearing in 58.71% and 37.62% of the overloaded states, respectively; and 2) the magnitudes of the overloads, with these two circuits having both the largest mean overload (120.79% and 113.24%, respectively) and maximum loading (190.58% and 157.00%, respectively).
Applying the relative error test (Section 4.1.1) yields εµN =0.027 for the number of
overloads and εµN =0.035 for overload energy. Both of these values are below the target
1
2
3
4
Number of simultaneous overloads
0
500
1000
1500
2000
2500
3000
Number of states [count]
2839
1012
146
11
Fig. 4.7 Frequency of simultaneous overloads for the IEEE 14-bus system
Table 4.3 Baseline loadings for the overloaded branches within the IEEE 14-bus system Branch Rating
[MVA] loadingMin. [%] Mean loading [%] Max. loading [%] No. of overloads [count] Mean overload [%] Circuit 3-4 30.00 10.28 60.82 116.32 558 104.55 Circuit 6-11 10.00 6.91 45.14 133.72 147 107.48 Circuit 9-10 10.00 2.44 75.14 190.58 2353 120.79 Circuit 9-14 15.00 0.62 50.74 142.03 738 112.10 Circuit 10-11 10.00 10.32 66.14 157.00 1508 113.24 Circuit 12-13 10.00 1.61 40.08 101.98 2 101.53 Circuit 13-14 10.00 2.28 36.95 117.62 39 104.18
Table 4.4 Statistical significance of differences in the distribution of the number of overloads for the power flow management algorithms applied to the IEEE 14-bus system
p-values (bold if not statistically significant) Algorithm Performance
[count] Baseline PFM-CSP PFM-OPF PFSF-Egal PFSF-TMA PFSF-LP Baseline 5345 – 0.0000 0.0000 0.0000 0.0000 0.0000 PFM-CSP 45 – 0.0366 0.0462 0.1432 0.0000 PFM-OPF 24 – 0.0001 0.0007 0.0000 PFSF-Egal 66 – 0.3546 0.0000 PFSF-TMA 60 – 0.0000 PFSF-LP 1172 –
4.4.4
Algorithm performance
Figure 4.8 summarises the performance of the power flow management algorithms when applied to the 10,000 test states in the IEEE 14-bus system.
In terms of minimising the number of overloads, PFM-OPF is the most effective algorithm as it leaves only 24 overloads, compared with 45 or more remaining after the application of any of the other algorithms. The overload performance for PFM-CSP, PFSF-Egal, and PFSF-TMA are within the same order of magnitude, whereas the number of overloads remaining after the application of PFSF-LP is two orders of magnitude higher.
The statistical analysis in Tables 4.4, 4.5, and 4.6 show the p-values of the pair-wise t-test comparisons of the algorithms’ performance for each of the performance measures. Values that fail to reject the null hypothesis of no difference in algorithm performance, following the application of the Benjamini-Hochberg procedure, are shown in bold. The remaining p-values indicate statistically significant differences in performance. With respect to the differences in performance for the number of overloads, Table 4.4 shows that the differences between PFSF-TMA and both of PFSF-Egal and PFM-CSP are not statistically significant; however, the differences in performance between all other pair-wise combinations of algorithms are statistically significant.
Although PFM-OPF removes the most overloads, it is not the most effective algorithm at reducing the total overload energy. Rather, PFSF-TMA is the most effective, and is able to reduce overload energy by an additional 96.68% compared with PFM-OPF. When compared with the baseline performance of the system, this represents a 99.98% reduction in overload energy. PFSF-Egal reduces overload energy by a similar amount as PFSF-TMA, and the analysis in Table 4.5 indicates that the difference in performance for these two algorithms is not statistically significant. The performance of PFM-CSP sits between PFSF-TMA (and PFSF-Egal) and PFM-OPF, while PFSF-LP gives the worst performance, with overload
0
1000
2000
3000
4000
5000
6000
Number of overloads [count]
Baseline
PFM-CSP
PFM-OPF
PFSF-Egal
PFSF-TMA
PFSF-LP
5345
45
24
66
60
1172
0
2000
4000
6000
8000
10000
Overload energy [MVAh]
Baseline
PFM-CSP
PFM-OPF
PFSF-Egal
PFSF-TMA
PFSF-LP
9116.25
11.01
49.97
1.75
1.57
105.55
0
10000
20000
30000
40000
50000
60000
Total curtailment [MWh]
Baseline
PFM-CSP
PFM-OPF
PFSF-Egal
PFSF-TMA
PFSF-LP
0.00
58855.10
19544.04
33900.57
21797.45
19503.16
Fig. 4.8 Overview of power flow management algorithm performance when applied to the IEEE 14-bus system
Table 4.5 Statistical significance of differences in the distribution of overload energy for the power flow management algorithms applied to the IEEE 14-bus system
p-values (bold if not statistically significant) Algorithm Performance
[MVAh] Baseline PFM-CSP PFM-OPF PFSF-Egal PFSF-TMA PFSF-LP Baseline 9116.25 – 0.0000 0.0000 0.0000 0.0000 0.0000 PFM-CSP 11.01 – 0.0211 0.0000 0.0000 0.0000 PFM-OPF 49.97 – 0.0040 0.0039 0.0014 PFSF-Egal 1.75 – 0.3197 0.0000 PFSF-TMA 1.57 – 0.0000 PFSF-LP 105.55 –
Table 4.6 Statistical significance of differences in the distribution of the amount of curtailment for the power flow management algorithms applied to the IEEE 14-bus system
p-values (bold if not statistically significant) Algorithm Performance
[MWh] Baseline PFM-CSP PFM-OPF PFSF-Egal PFSF-TMA PFSF-LP Baseline 0.00 – 0.0000 0.0000 0.0000 0.0000 0.0000 PFM-CSP 58855.10 – 0.0000 0.0000 0.0000 0.0000 PFM-OPF 19544.04 – 0.0000 0.0000 0.6957 PFSF-Egal 33900.57 – 0.0000 0.0000 PFSF-TMA 21797.45 – 0.0000 PFSF-LP 19503.16 –
energy of over the double that of PFM-OPF, although PFSF-LP still removes 98.84% of overload energy when compared with the system baseline performance.
PFSF-LP applies the least curtailment out of the five algorithms; however, Table 4.6 shows there is no statistically significant difference in the curtailment it applies compared with PFM-OPF. This is despite the PFM-OPF being the most effective algorithm at reducing the number of overloads, while PFSF-LP is the least effective for the same performance measure. The differences in the amount of curtailment applied by the other three algorithms is stark, however, and is surprising given their similar performance at reducing the number and energy of overloads. PFM-CSP applies the most curtailment, 73.61% higher than the next worst algorithm, PFSF-Egal. PFSF-Egal, in turn, applies 55.53% more curtailment than the next worst algorithm, PFSF-TMA. The amount of curtailment applied by PFSF-TMA is of a similar order to that applied by PFM-OPF, although it is 11.53% higher and, as Table 4.6 indicates, this is a statistically significant difference in performance.
Out of the algorithms tested on the IEEE 14-bus system, which algorithm performs most effectively depends on whether the number or energy of overloads is of most interest. If the
objective is minimising the number of overloads, while also minimising the curtailment ap- plied, then PFM-OPF is the most effective algorithm. However, if the objective is minimising the energy of overloads, while minimising the curtailment applied, then PFSF-TMA is the most effective algorithm.