These three algorithms all use PFSFs and share some common code, with PFSF-Egal and PFSF-TMA only differing on a few key lines of their process (see Section 2.3.4). As these lines have a common effect on their performance, these algorithms are analysed together in this section. This analysis makes reference to Table 4.9, which tabulates a number of metrics regarding the performance of these three algorithms across the four case study systems, including the number of states where the algorithms report errors during their process. The tabulated metrics are numbered for ease of reference.
PFSFs linearise the power flow equations around a particular operating point and can be manipulated using basic arithmetic operations, so can speed up execution when compared with algorithms that use a full non-linear representation of the system. However, because PFSFs are a linearisation of a non-linear system around one state, they are approximations and can become inaccurate when used to calculate power flows for states that deviate from the initial state. The effect of this is that the PFSF-based algorithms, which use a common set of PFSFs that are calculated offline for a single state in each system, may overestimate the change in power flows for particular curtailment values, and thus leave overloads remaining in the system under control. Similarly, the algorithms may also underestimate the effect of some curtailments, applying more curtailment than necessary to remove an overload.
The algorithms employ various design features to mitigate the approximation inherent in using PFSFs. The first is to adjust ratings to 99% within their internal processes so that
Table 4.9 Additional metrics related to the performance of the PFSF-based algorithms 1. Number of overloads
Algorithm 11 kV radial 33 kV meshed IEEE 14-bus IEEE 57-bus
PFSF-Egal 0 3594 66 24823
PFSF-TMA 0 1903 60 24247
PFSF-LP 0 1819 1172 12863
2. States with overloads
Algorithm 11 kV radial 33 kV meshed IEEE 14-bus IEEE 57-bus
PFSF-Egal 0 3594 66 5165
PFSF-TMA 0 1903 60 4995
PFSF-LP 0 1819 1029 5262
3. States with overloads and curtailment Algorithm 11 kV radial 33 kV meshed IEEE 14-bus IEEE 57-bus
PFSF-Egal 0 3406 56 503
PFSF-TMA 0 1715 50 951
PFSF-LP 0 1631 1019 5239
4. States with overloads but no curtailment Algorithm 11 kV radial 33 kV meshed IEEE 14-bus IEEE 57-bus
PFSF-Egal 0 188 10 4662
PFSF-TMA 0 188 10 4044
PFSF-LP 0 188 10 23
5. States with overloads, no curtailment, and algorithm error Algorithm 11 kV radial 33 kV meshed IEEE 14-bus IEEE 57-bus
PFSF-Egal 0 0 0 4639
PFSF-TMA 0 0 0 4021
PFSF-LP 0 0 0 0
6. States with overloads, no curtailment, but no algorithm error Algorithm 11 kV radial 33 kV meshed IEEE 14-bus IEEE 57-bus
PFSF-Egal 0 188 10 23
PFSF-TMA 0 188 10 23
PFSF-LP 0 188 10 23
7. States with no overloads but with curtailment Algorithm 11 kV radial 33 kV meshed IEEE 14-bus IEEE 57-bus
PFSF-Egal 2546 3186 4017 297
PFSF-TMA 2546 4877 4023 467
PFSF-LP 2546 4961 3054 200
8. Mean remaining overload energy versus baseline (%) for overloaded states with curtailment
Algorithm 11 kV radial 33 kV meshed IEEE 14-bus IEEE 57-bus
PFSF-Egal – 10.25 17.78 6.95
PFSF-TMA – 17.14 19.69 3.86
the algorithms will tend to overestimate the amount of curtailment needed, which can help remove overloads when the PFSFs overestimate the effect of curtailments. The second design feature is to run a full non-linear load flow to validate the curtailment values derived using PFSFs, which is used within PFSF-Egal and PFSF-TMA.
One design feature shared by these algorithms is that the logic used to detect overloads calculates the apparent power flowing through a branch from the average real and reactive power measured at both ends, rather than calculating the apparent power at both ends and comparing the maximum of these to the branch rating. This feature allows each branch to be associated with single values of real and reactive power, which simplifies the algorithms’ logic and speeds up execution. For example, PFSF-LP could be adapted to consider power flow constraints at both ends of a branch, rather than looking at a constraint based on average flows; however, this would double the number of variables in the LP.
This design feature of the overload detection logic does not cause an issue for many of the tested states, due to the branch loading limit of 99% assumed to mitigate the approximation inherent in using PFSFs. This lowers the detection threshold, so the algorithms are more likely to correctly detect an overload, although, conversely, they may falsely detect an overload and apply curtailment when none is needed. This is quantified by metric 7 in Table 4.9, which shows the number of states within each system for which the algorithms falsely detect overloads and then apply curtailment. However, under certain conditions a branch may be overloaded at one or both ends, but the apparent power calculated from the average real and reactive powers at both ends may still be within the 99% limit used within the algorithms. Under these conditions, this design feature of the algorithms becomes a flaw as they will fail to detect overloads and thus no curtailment will be applied to remove the overloads. As shown in Table 4.9 (metric 6), for the 33 kV meshed distribution system, there are 188 states in which the PFSF-based algorithms fail to detect overloads; for the IEEE 14-bus system there are 10 such states; and for the IEEE 57-bus system there are 23. All overloads in the 11 kV radial distribution system are correctly detected.
For the 33 kV meshed distribution system and the IEEE 14-bus system, failure to detect overloads accounts for all the states where the PFSF-based algorithms do not apply any curtailment despite there being overloads. However, for the IEEE 57-bus system, there is a different aspect of designs of PFSF-Egal and PFSF-TMA that leads to these algorithms failing to apply curtailment when there are overloads, which is the iteration limit that ensures the algorithms execute in a reasonable time (described in Section 2.3.3). For a significant number of states (metric 5 in Table 4.9), the algorithms reach the iteration limit within their processes before the overloads within their internal models are removed, and then exit, reporting an error and applying no curtailment.
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Initial number of overloads
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% of states with errors
PFSF-Egal
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Fig. 4.15 Percentage of initially overloaded states in the IEEE 57-bus system where the PFSF-Egal and PFSF-TMA algorithms report errors
As can be seen in Figure 4.15, there is a link between the initial number of overloads in the system and the propensity for the algorithms to reach their iteration limit and report an error. PFSF-TMA is less likely than PFSF-Egal to reach its iteration limit and report an error for states with three or fewer overloads initially; however, for all states with 5 overloads or more, both algorithms always reach their iteration limits and report errors. The design of the algorithms is such that when the iteration limit is reached they will report an error and exit without applying curtailment. The consequence of this is that all overloads will remain in the system, resulting in the significant number of overloads for PFSF-Egal and PFSF-TMA on the IEEE 57-bus system, and making that the only system where these two algorithms are outperformed by PFSF-LP in terms of the reducing the number and energy of overloads.
Aside from the states in which overloads remain due to missed detection (metric 6 in Table 4.9) or algorithm error (metric 5), for the majority of states with overloads remaining the algorithms operate correctly according to their designs and apply some curtailment to the generators within the systems (metric 3). In these cases, the algorithms determine curtailments that are able to remove all overloads within their internal system models, but leave one or more overloads when applied to the actual system under control.
For the PFSF-LP algorithm, the difference between the internal solution and the external reality stems from the linearised form of the system response (the PFSFs) within the LP that are used to determine the curtailments. The curtailments calculated from the linearised
system representation are directly applied to the non-linear system under control, so may incorrectly estimate the effect on power flows that the curtailments aim to achieve.
For the PFSF-Egal and PFSF-TMA algorithms, the curtailments calculated using a linearised representation of the system are then validated by performing an AC load flow using a full non-linear representation of the system under control. While this validation step should correctly determine the change in power flows due to the curtailments, the overload detection logic used in these algorithms may fail to detect marginal overloads that may remain in the internal models. As no overloads are detected in the internal model, the curtailments are then applied to the system under control, but overloads may still remain. However, overload energy will be reduced, as shown by metric 8 in Table 4.9.