4.2.1
System description
G 3 4 5 6 7 8 9 10 2 1 Slack bus G 33/11kVBus with load Bus without load Transformer Circuit
Generator G
KEY:
Circuit (overloaded)
Fig. 4.2 Single line diagram of the 11 kV radial distribution system used in this work This system is based on network data for part of an 11 kV distribution network from the south east of England [70], which was used in previous AuRA-NMS work testing the PFM-OPF and PFM-CSP algorithms [29–31]. As shown in Figure 4.2, the system has an infeed at 33 kV, transformation down to 11 kV, and a single feeder with a radial topology. The system features 10 buses (6 with loads), 8 cable sections, a single transformer, and 2 generators, which are situated at the towards the remote end of the feeder. As per the descriptions within previous work using this system [29–31], the two circuits highlighted in red in Figure 4.2 (5-6 and 8-9) have had their ratings reduced so that they can potentially become overloaded due to power exported from the generators. Appropriate ratings to achieve this were determined by using the process outlined in Section 4.1.1.
4.2.2
Test states
10,000 system states were generated randomly for the 11 kV radial distribution system in order to test the power flow management algorithms. The following parameters were varied for each state:
Table 4.1 Baseline loadings for the overloaded branches within the 11 kV radial distribution system
Branch Rating
[MVA] loadingMin. [%] Mean loading [%] Max. loading [%] No. of overloads [count] Mean overload [%] Circuit 5-6 2.00 5.70 62.81 164.30 1472 117.42 Circuit 8-9 1.50 0.34 59.60 132.09 1872 112.79
• System load: all loads within the system were scaled by the same amount, between 0% and 100%. The scaling factor was drawn from a uniform random distribution.
• Generator output: the real power output of each of the generators within the system were scaled between 0% and 100%. Separate scaling factors were used for each generator, and these were drawn from independent uniform random distributions.
4.2.3
Baseline performance
Each of the 10,000 states was simulated for the system with no algorithm applied in order to understand the baseline performance of the system. 2462 (24.62%) of the states tested featured overloads, and 882 (35.82%) of these have both circuits overloaded. This gives a total of 3344 overloads, which have a total energy of 871.95 MVAh.
Table 4.1 provides a summary of the loadings along the two overloaded circuits within the 11 kV radial distribution system. Circuit 5-6 can become more heavily overloaded as it has both generators feeding in to it, although circuit 8-9 is more frequently overloaded.
Applying the relative error test (Section 4.1.1) for the baseline performance with respect to the number of overloads across the 10,000 tested states yields εµN =0.037, whereas for
overload energy εµN =0.048. Both of these values are below the target maximum relative
error (δ = 0.05) so generating additional states and testing the system’s behaviour on those states was not required. Curtailment was not considered for the relative error test as that performance measure had the same value (0.0 MWh) for all states.
4.2.4
Algorithm performance
Figure 4.3 summarises the performance of the power flow management algorithms when applied to the 10,000 test states in the 11 kV radial distribution system. Four of the algorithms are able to remove all the overloads, and thus have identical performance with respect to the number and energy of overloads. PFM-OPF is the only algorithm that fails to remove all overloads, leaving 2 overloads in the system with a total energy of 0.10 MVAh.
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773.33
Fig. 4.3 Overview of power flow management algorithm performance when applied to the 11 kV radial distribution system
Comparing the distribution of performance of PFM-OPF against any of the four other algorithms using the paired t-test yields p-values of 0.1573 and 0.1627 for the number and energy of overloads, respectively. The Benjamini-Hochberg correction for multiple comparisons (as described in Section 4.1.3) was applied to these p-values – along with the p-values of all the other pair-wise t-test comparisons in the family of hypotheses for this case study system – to determine their statistical significance. This revealed that the results of the t-tests comparing the overload performance of PFM-OPF to each of the other power flow management algorithms fail to reject the null hypothesis (described in Section 4.1.3) that the distributions of the algorithms’ performance are the same.
While there is no statistically significant difference in the algorithms’ performance with respect to removing overloads, the amount of curtailment applied varies significantly between the algorithms. PFM-OPF applies the least curtailment, with PFSF-LP applying just 0.44% more. PFSF-Egal applies 11.20% more curtailment than PFM-OPF, and PFSF-TMA applies 15.72% more. PFM-CSP is the worst performing algorithm, applying over double the amount of curtailment of any other algorithm and 175.53% more than PFM-OPF.
Although the difference in the amount of curtailment applied by PFM-OPF and PFSF-LP appears small, using the paired t-test to compare the distribution of curtailment performance for these two algorithms yields a p-value < 0.0001. Applying the same analysis to all other pair-wise combinations of algorithms reveals similar p-values (< 0.0001). After correcting for multiple comparisons using the Benjamini-Hochberg procedure, all of these p-values reject the null hypothesis that the distributions of performance are the same, so there are statistically significant differences in the distribution of curtailment applied by the algorithms.
Based on these results, PFM-OPF is the power flow management algorithm that gives the best performance for the 11 kV radial distribution system, as it applies the least curtailment whilst minimising the number and energy of overloads to a level where there is no statistically significant difference to the other algorithms. However, if the absolute number of overloads is considered, then the amount of curtailment, PFSF-LP is the most effective algorithm.