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This work is organised as follows. In section 2 we shortly introduced the historical context situating the South African power grid, the ongoing load-shedding events, and we showcase the direct effects at the University of the Free State (UFS) in Bloemfontein. In section 3 we introduce both a low-level dynamical model of power grids affected by noise as well as Fokker–Planck approximation of the power-grid frequency statistics. From this basis, we present our theoretical and numerical results. In section 4 we discuss the outcomes and limitations of our data examination and present some thoughts on future work and the necessity to examine previously overlooked synchronous areas.
In this work, we focus on power-grid frequency recordings from the Continental European power grid, the Nordic Grid, and the South African power grid. It is the latter that interests us the most for this examination. The South African power grid is undergoing an unprecedented transformation.
The power sector in South Africa is dominated by the state-owned entity ESKOM, which both imports and exports energy from surrounding states with some input from nuclear and renewable energy (photovoltaic (PV), wind and concentrated solar power) [34]. Unfortunately, the South African power system has been experiencing an energy crisis since 2007 with electrical generation lagging the electrical demand, ultimately leading to implemented rolling blackouts in an effort to stabilise the national grid.
Due to extended periods of lack of maintenance and mismanagement resulted in an unpredictable and unreliable power system in South Africa, that during 2021 experienced load-shedding for 1169 h with 1775 GWh (with the upper limit of 2521 GWh) energy shed. The latter amounts to 13.3 of all h of 2021 being in the state of, mostly, Stage 2 load-shedding (for details see table 1 in section 3.5) [10]. The power system remained coal powered during 2021, but with contributed by renewable energy (of which were variable sources) [2]. Concernedly, 3.2 TWh of energy was generated by using diesel.
Table 1. Total estimated load shed energy as reported by the Council for Scientific and Industrial Research—Energy Centre of South Africa (see p 171 in [10]) at different load-shedding stages (St. 1 to St. 6). Shown as well are the total number of hours under load-shedding in South Africa in 2019, 2020 and 2021.
With critical low reserve margins and increasing load-shedding, some consumers are considering the implementation of self-sufficient microgrids with some renewable input. The latter is preceded by the digitisation of the existing electrical network topology (i.e. smart grids) that requires initial capital input [35]. South African university campuses are naturally forced to evolve according to the constraints of the national utility, the ability to swiftly react to demand reduction signals being the most critical, see figure 1. In cases of severe loss of supply, self-sufficient microgrids are utilised that synchronise with available renewable supply to ensure supply continuity on campus. The latter typically being hybrid PV-diesel-based microgrids.
Figure 1. Demand-side management (as response to utility load-shedding) at the University of the Free State (UFS) Campus in Bloemfontein, South Africa, in comparison with a typical day. This is one solution that public and private institutions have had to implement to be able to operate uninterruptedly. Displayed are the 19th and 20th of May 2022, the first of this a day with campus demand response totalling MWh, the second a typical day without the call for demand response.
An operating AC power-grid system is a set of highly synchronised oscillators operating at a rotational frequency multiple of 50 Hz (or 60 Hz). In a reduced format, we can consider a general model for power grids as coupled inertial oscillators in a network. A synchronous machine j, a generator or a load, is described by its rotor-angle and its angular frequency . It obeys the equations of motion [17, 38–40]
where Mj is the inertial mass of the jth rotating machine, Dj its damping factor, comprises the self-generated mechanical power (or if negative it acts as a load), and Pj is the exchanged power with the other oscillators in the network. The innocently looking exchange power Pj embodies the very complex structure of power-grid systems, i.e. Pj is given by
where Ej is the transient voltage at jth machine, which is our reduced swing equation model (1) is static. The parameters and denote the real and imaginary parts of the nodal admittance matrix and encode the network structure. The conductance comprises all the resistive terms of the network which we will not deal with, i.e. we will consider the case of a purely lossless system . Conductance effects can be disregarded as they are necessarily balanced in a functional power system where the generated power is higher than the consumed one to compensate for these losses. These models are also known as second-order/inertial Kuramoto models [30, 31, 38, 41].
Returning to our noise-disturbed swing equation (1) we can now focus on writing a Fokker–Planck equation for the probability density for a single node [29, 30, 32]. Let us consider the noise is delta correlated , with B the diffusion constant. Now considering precisely the previous argument that our models are descriptive of , we can take the ''mean-field'' approximation yielding [29, 30]
We assume that and . We can separate the probability density and focus solely on the statistics of the angular frequency , following Acebrón and Spigler [29] (cf their equation (8)). This results in
The static solution at (and having appropriate decay conditions of at ) is the well known Gaussian distribution
To best understand the distribution of power-grid frequency in the various grid, we can directly examine the skewness s and kurtosis κ, given respectively by
A Gaussian distribution has skewness s = 0 and a kurtosis κ = 3. That is, it is purely symmetric and does not have any heavy (or light) tails. Being facetious, Gaussian distributions are very normal. What we observe is that none of the recordings are symmetric (s ≠ 0) and they all show heavy tails (κ > 3). The aforementioned changes in the dispatch/market activity in Continental Europe and the Nordic Grid explain this behaviour. But what about South Africa?
We have introduced a Fokker–Planck equation describing the probability density , wherein we can directly evaluate the effects of the inertia m in the system and the presence of noise, mediated by its amplitude B. We can now test, using our frequency recordings, how much our simplistic Fokker–Planck model (4) can describe the data. In a more general setting, we write a Fokker–Planck equation as [28, 63, 64]
where is known as the drift and the diffusion. In general, are the Kramers–Moyal coefficients. If we compare this with our Fokker–Planck equation, we see that
First, we notice the linear response of the drift , which is a linear function of ω. Second, we note that given out choice of noise ξ mediated by B, the diffusion is not a function of the angular frequency.
We are now interested in obtaining the drift D1 and diffusion D2 directly from the frequency data. We can make use of (Nadaraya–Watson) non-parametric estimators [65, 66] to extract the functional form of D1 and D2 from the data [56, 67–71]. For a time series x(t), like power-grid frequency recordings, the Kramers–Moyal coefficients can be estimated using a kernel density estimation [68]
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