Inertia location and slow network modes determine disturbance propagation in large-scale power grids

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.

A New Approach to the On-line Estimation of the Loss of Generation Size in Power Systems

Rasoul Azizipanah-Abarghooee, Mario Paolone

Following an unintended disconnection of a synchronous generator (SG) from the power system, what is also known as a loss of generation (LoG), it is not trivial to precisely estimate the post-event power system's inertia and the LoG size. One of reasons for that is that both of them are a function of the unknown inertia reduction. To solve this challenging problem, this paper presents an analytical method based on the rate-of-change-of-frequency (RoCoF). The method relies on a modified swing equation, allowing a simultaneous estimation of both unknowns. To this end, the values of mechanical starting time, apparent power and loading of lost generator are formulated for the power system under study. In a practical application, the method can use RoCoF measured by phasor measurement units (PMUs). The paper discusses the impact of various frequency estimation approaches to the proposed LoG estimation. Furthermore, a new method for LoG size estimation, based on the interpolated estimated inertial response, is proposed. The efficiency of the proposed approach is validated through extensive simulations with Matlab/Simulink using a simple power system and the IEEE 39-bus test network.

2018

Optimal Placement of Inertia and Primary Control: A Matrix Perturbation Theory Approach

Philippe Jacquod

The increasing penetration of new renewable sources of electrical energy reduces the overall mechanical inertia available in power grids. This raises a number of issues regarding grid stability over short to medium time scales. A number of approaches have been proposed to compensate for this inertia reduction by deploying substitution inertia in the form of synchronous condensers, flywheels or powerelectronic-based synthetic inertia. These resources are limited and expensive; therefore, a key issue is to determine how to optimally place them in the power grid, for instance, to mitigate voltage angle and frequency disturbances following an abrupt power loss. Performance measures in the form of H-2-norms have recently been introduced to evaluate the overall magnitude of such disturbances. However, despite the mathematical convenience of these measures, analytical results can only be obtained under rather unrealistic assumptions of a uniform damping-to-inertia ratio or a homogeneous distribution of the inertia and/or primary control. Here, we introduce and apply matrix perturbation theory to obtain analytical results for an optimal inertia and primary control placement in the case where both are heterogeneous. This powerful method allows us to construct two simple algorithms that independently optimize the geographical distribution of the inertia and primary control. The algorithms are then implemented for a numerical model of the synchronous transmission grid of continental Europe with different initial configurations. We find that an inertia redistribution has little effect on the grid performance but that the primary control should be redistributed on the slow modes of the network, where the intrinsic grid dynamic requires more time to damp frequency disturbances. For a budget-constraint optimization, we show that increasing the amount of primary control in the periphery of the grid, without changing the inertia distribution, achieves 90 % or more of the maximal possible optimization, already for relatively moderate budgets.

2019

Optimal Placement of Inertia and Primary Control: A Matrix Perturbation Theory Approach

Philippe Jacquod

2019