Analysis of algorithm performance: PFM-CSP

PFM-CSP is on average the second most effective algorithm at removing overloads. It failed to remove overloads on all three systems with a meshed network topology, due to a feature of its design rather than execution errors.

PFM-CSP uses an internal network model to check candidate generator output limits during the CSP solution process. This internal model is updated to match the initial state of the system under control, including the current output set points of the generators. When the internal model is used during the CSP solution process, candidate output limits are applied as new set points to the generators within the model to assess whether overloads would occur if the generators were operating at those limits. This ignores the available power from each generator, so, for example, a generator with only 50% of its maximum power available would have a output limit of 100% tested in the internal model by setting the appropriate generator to 100% output, rather than 50% output. This allows the maximum output limits to be determined, which the generators can increase their output to match, but means that PFM-CSP does not check that the overloads are removed when the output limits are applied to the generators in the system under control. A consequence of this is that PFM-CSP may fail to remove overloads within meshed systems, as explained below.

To explain how this phenomena occurs, consider the simple power system illustrated in Figure 4.14a. This example system has a meshed topology consisting of four buses, of which two feature generators and one is the slack bus. It is assumed that the generators are the same size and the parameters of all branches (except their ratings) are identical. With this configuration, the power flowing through circuit 3-4 combines the output of both generators, while power will only flow along circuit 1-2 if the output of the generators are different.

The state space for this system comprises two variables, which are the output levels of the two generators. Therefore, the state space can be represented in two dimensions, as shown in Figure 4.14b. Assuming that the ratings of circuits 1-2 and 3-4 are such as to cause a power flow management problem due to export of power from the generators, the operating

G G 2 1

3 4

(a) System schematic

A B 50% 100% 0% 0% 50% 100% Circuit 1-2 limits Circuit 3-4 limit Generator 1 output G en er a to r 2 o ut put Operation within limits

(b) System state space

Fig. 4.14 Simple power system for illustrating overload performance weaknesses of PFM- CSP algorithm

conditions that lead to overloads can be represented within the state space, as shown by the shaded areas in the figure. Also represented in the figure (by the nine coloured dots) are the output limits that the PFM-CSP algorithm would check within its internal model if an overload condition occurs. It is obvious that the only valid output limits that the algorithm would find are the points highlighted in green, which represent an output limit of 0% applied to both generators, and an output limit of 50% applied to both generators.

In Figure 4.14b point A is an example of an operating state in which circuit 1-2 is overloaded, due to the generators at bus 1 and 2 operating at different output levels (20% and 80%, respectively). The PFM-CSP algorithm would detect the overload and use its internal model to assess what output limits would lead to the overload being removed, by applying the nine operating points (represented by the coloured dots: (0%, 0%), (50%, 0%), etc.) to the generators in the internal model. Due to the algorithm’s preference constraint of minimising curtailment (refer to Section 2.3.1), the operating limits from point (50%, 50%) would be determined to be both a valid and the preferred solution. Applying a 50% limit to both generators would result in the output of the generator at bus 1 staying the same (as 20% < 50 %), while the output of the generator at bus 2 would fall from 80% to 50%. Although the system would be in a new state (point B) within the output limits defined by the PFM-CSP algorithm, this new state is still inside the region of the state space in which circuit 1-2 is overloaded. Therefore, although PFM-CSP may find a solution that removes all overloads when applied as output limits to its internal model, when the same limits are applied to the system under control, overloads may remain.

Aside from the failure of PFM-CSP to remove certain overloads, it applies the most curtailment of all the algorithms on each of the case study systems. This is due to the discrete

domains used in CSP process for the output limits, which can lead to more curtailment being applied than necessary. For example, if a generator operating at 100% (full output) only needed to operate at 99% output in order to remove an overload, PFM-CSP would not recognise this and would apply an output limit based on the closest value from the discrete domain that was less than or equal to that value, which in this case would be 50%.

The failure of PFM-CSP to remove certain overloads is down to a fundamental aspect of the algorithm’s design and re-designing the algorithm to overcome that issue – for example, by determining an operating “envelope” for the generators to operate within rather than just output limits – is outside the scope of this work. The issue of PFM-CSP applying excessive curtailment can be resolved more simply by increasing the number of values in the domain to make the output limits less coarse. This has a significant performance drawback, however, due to the poor scalability characteristics of PFM-CSP to increased domain sizes and numbers of generators, as noted in Section 2.2.3. For example, for the IEEE 14-bus system with four generators, changing the domains from {0%,50%,100%} to {0%,25%,50%,75%,100%} increases the worst-case execution time by a factor of 54_/34_{≈ 7.72.}