Date Published: March 21, 2019
Publisher: Public Library of Science
Author(s): Laurent Pagnier, Philippe Jacquod, Lei Chen.
Conventional generators in power grids are steadily substituted with new renewable sources of electric power. The latter are connected to the grid via inverters and as such have little, if any rotational inertia. The resulting reduction of total inertia raises important issues of power grid stability, especially over short-time scales. With the motivation in mind to investigate how inertia reduction influences the transient dynamics following a fault in a large-scale electric power grid, we have constructed a model of the high voltage synchronous grid of continental Europe. To assess grid stability and resilience against disturbance, we numerically investigate frequency deviations as well as rates of change of frequency (RoCoF) following abrupt power losses. The magnitude of RoCoF’s and frequency deviations strongly depend on the fault location, and we find the largest effects for faults located on the support of the slowest mode—the Fiedler mode—of the network Laplacian matrix. This mode essentially vanishes over Belgium, Eastern France, Western Germany, northern Italy and Switzerland. Buses inside these regions are only weakly affected by faults occuring outside. Conversely, faults inside these regions have only a local effect and disturb only weakly outside buses. Following this observation, we reduce rotational inertia through three different procedures by either (i) reducing inertia on the Fiedler mode, (ii) reducing inertia homogeneously and (iii) reducing inertia outside the Fiedler mode. We find that procedure (iii) has little effect on disturbance propagation, while procedure (i) leads to the strongest increase of RoCoF and frequency deviations. This shows that, beyond absorbing frequency disturbances following nearby faults, inertia also mitigates frequency disturbances from distant power losses, provided both the fault and the inertia are located on the support of the slowest modes of the grid Laplacian. These results for our model of the European transmission grid are corroborated by numerical investigations on the ERCOT transmission grid.
The short-time voltage angle and frequency dynamics of AC power grids is standardly modeled by the swing equations . The latter determine how local disturbances about the synchronous operational state propagate through the grid. They emphasize in particular how voltage angle and frequency excursions are partially absorbed on very short time scales by the inertia of rotating machines, before primary control sets in. With the energy transition, more and more new renewable energy sources (RES) such as solar photovoltaic units—having no inertia—and wind turbines—whose inertia is at this time essentially suppressed by inverters—substitute for conventional power generators. The resulting overall reduction in rotational inertia raises a number of issues related to system dynamics and stability [2, 3]. It is in particular desirable to determine how much inertia is sufficient and where to optimally locate it to guarantee short-time grid stability. Determining the optimal placement of inertia is of paramount importance at the current stage of the energy transition, as it would help determine where the substitution of conventional generators by RES crucially needs to be accompanied by the deployment of synchronous condensers or synthetic inertia.
We have imported and combined publicly available data to construct a geolocalized model of the high voltage synchronous grid of continental Europe. The geographical location and the electrical parameters of each bus is determined, including voltage level, dynamical parameters (inertia and damping coefficients), generator type and rated power. Line capacities are extracted from their length. They are compared with known values for a number of lines and found to be in good agreement. Different load situations are investigated using a demographically-based distribution of national loads, together with a dispatch based on a DC optimal power flow. Details of these procedures are given in S2 Appendix. To confirm our conclusions, we alternatively used a model of the Texas ERCOT transmission grid , where inertia and damping coefficients are obtained using the same procedure as for the European model.
Our numerical data monitor the voltage angle and frequency excursion following an abrupt power loss. Fig 1 shows two such events with series of snapshots illustrating the propagation of the disturbance over the continental European grid during the first 2.5 seconds after the contingency. The two events differ only by the location of the power loss. In the top row the faulted power plant is in Greece, while in the bottom row it is in Switzerland (fault locations are indicated by purple circles). In both instances, the lost power is ΔP = 900 MW and the grid, including loads and feed-ins, inertia distribution, damping parameters and electrical parameters of all power lines, is the same.
So far we have established that disturbances have strongly location-dependent magnitudes, and in particular that stronger disturbances originate from buses with large amplitude of the two slowest eigenmodes of the network Laplacian. We next investigate how rotational inertia influences this finding. To that end we modify inertia on the network following three different procedures where the inertia of a generator on bus #i is increased/decreased according to one of the following probability distributions
We have presented numerical investigations on disturbance propagation following a generator fault in the synchronous transmission grid of Continental Europe. The first step was to build up a numerical model, including all necessary parameters to perform dynamical calculations. To the best of our knowledge, no model of this kind is publicly available.