© European Physical Society, EDP Sciences, 2025

Our large overdamped harmonic oscillators: the power grid

Electric power systems use alternating current (AC), in which voltages and currents oscillate at frequencies of 50 Hz or 60 Hz, depending on the country. This power-grid frequency is an essential observable in power system control as it measures the balance of generation and load (Fig. 1(a)). This can be understood from the principle of energy conservation and the fact that the power-grid frequency is determined by the rotational frequency of synchronous generators in the grid. If load exceeds generation, where load being the total energy being consumed in the grid, rotational energy is converted to electric energy, and the generators slow down, decreasing the frequency of rotation. Similarly, if generation exceeds the load, generators accelerate and the frequency increases.

Fig. 1 (a) The power-grid frequency measures the balance of generation and load. (b) Synchronised power-grid frequency measured at different locations in the Continental European grid from July 2019. We observe a strong decrease in the power-grid frequency at 22:00 due to a rapid de-ramping of power generation. The inset shows a magnification of the time series for a 90-second interval. The frequency is almost the same for all locations in the grid; residual differences are damped out on the time scale of seconds. (c) The frequency in a microgrid at the main campus of the University of the Free State in South Africa. The microgrid operates close to the nominal frequency of 50 Hz. The actual frequency decreases after a loss of power in the municipality.

Let us formalise these relations to understand how the frequency evolves and how control comes into play [1]. We typically consider the difference to the grid’s reference frequency ∆ f(t) = f(t) – f_ref, 50 Hz in Europe. If is not too large, the dynamics is determined by the aggregated swing equation,

$$M\frac{d\mathrm{\Delta }f}{dt}=P\left(t\right),$$(1)

where M is proportional to the aggregated moment of inertia of all synchronous machines in the grid. Inertia is inextricably connected with the stability of the grid; We will return to it at the end of this article. For a detailed analysis, we decompose the momentary power imbalance P(t) into three parts,

$$P\left(t\right)=P_{imb}\left(t\right)+P_{fluct}\left(t\right)+P_{control}\left(t\right)\mathrm{.}$$

The impact of systemic imbalances P_imb(t) and fast, ambient fluctuations Pfluct(t) will be discussed in detail below. The control Pcontrol(t) balances these disturbances and restores the reference frequency. Typically, three control layers are implemented in large interconnected grids:

• Primary control is implemented in a decentralised manner: Dedicated power plants and battery energy storage systems measure the frequency and adapt their power output. In the simplest case, the response follows a proportional law Pprimarycontrol(t) = –α∆f(t). In practice, the control law may include a ‘deadband’ that will be discussed below.

• Secondary control is activated centrally on the basis of the integral of the frequency deviation. Hence, we have Psecondarycontrol(t) = –βθ(t) with dθ/dt = ∆f(t).

• Tertiary control can be activated manually. We will not consider it in the following.

From a physicist’s perspective, it is interesting to see that the equations of motion of the load-frequency control system are equivalent to a driven overdamped harmonic oscillator – one of the most iconic models of a mass-spring system in any introductory mechanics book. Differentiating Eq. (1) with respect to time and substituting the primary and secondary control yields

$$M\frac{d^2\mathrm{\Delta }f}{{dt}^2}=-\alpha \frac{d\mathrm{\Delta }f}{dt}-\beta \mathrm{\Delta }f-\frac{{dP}_{imb}}{dt}-\frac{{dP}_{fluct}}{dt}$$

The primary control corresponds to the damping, modulated by α, while the secondary control corresponds to the restoring force of the harmonic oscillator, modulated by β. The main complication arises from the external driving due to systemic imbalances and fluctuations – which show complex temporal patterns.

Sample power-grid frequency time series from the Continental European power grid are shown in Fig. 1(b). We see that the frequency is virtually the same for all measurement locations. Residual differences are damped out on the time scale of seconds. That is, all synchronous generators from Portugal to the Ukraine, from Denmark to Sicily, run with the same rotational frequency. Power grids are highly coupled oscillator systems with strong phase locking between oscillators. It is thus justified to neglect spatial differences and study ‘the’ grid frequency. Figure 1(c) shows the frequency in a microgrid at the main campus of the University of the Free State in South Africa after a loss of power in the entire municipality. The frequency decreases until it is stabilised by the control system.

The well-defined, but difficult to manage, deterministic frequency deviations

Frequency recordings from the Continental European grid show recurrent, precisely timed patterns. In Fig. 1(b), we observe a strong decrease of the frequency around 22:00, leading to a sag of 60 mHz below the reference and a total drop of 110 mHz. Similar sags and spikes are observed at all full hours in the morning and evening.

These deterministic frequency deviations are caused by the structure of the European electricity market. Electricity is traded in blocks of an hour or one-quarter hour. Hence, power plant operators ramp up or down generators rapidly at the beginning of each hour. This leads to a rapid change of the power imbalance P_imb(t) and thus to a rapid change of the frequency according to Eq. (1). In the morning, P_imb(t) switches from negative to positive, leading to positive spikes in ∆ f(t). In the evening, P_imb(t) switches from positive to negative, leading to negative spikes or sags in ∆ f(t).

The noise is the signal

Power grids are a good example of complex systems. Each action we take in our daily lives affects the grid – every time we turn on or off a light, more or less power is needed. So, apart from large deterministic frequency deviations, power grid frequency is filled with fluctuations, or what we might call ‘noise.’ Interestingly, the frequency dynamics differ considerably between power grids. In Fig. 2, top row, nine recordings from various power grids around the globe are shown. Herein, we find fluctuations of low amplitude, e.g., in SGP, IND, MYS, IRL, PM_ESP, as well as more abrupt ones, particularly in ISL (see Fig. 2 for acronyms). Interestingly, we can harness these fluctuations as a measurement device [3] to learn a lot about the dynamics of a power grid and the operation of its load-frequency control system.

Fig. 2 Upper panels: Power-grid frequency time series for Singapore (SGP), Indonesia (IND), Malaysia (MYS), Ireland (IRL), Australia (AUS), Iceland (ISL), and Mallorca Island of Spain (PMESP). Lower panels: Estimated drift coefficient of the power-grid frequency time series. The black dashed lines show a piecewise-linear drift function with a deadband, corresponding to an idealised primary control law [2].

We first revisit the primary control. We can neglect the secondary control for the time being because it is activated on longer timescales. The impact of systemic imbalances and ambient fluctuations, not easily distinguishable experimentally, is lumped into a ‘noise’ term with amplitude σ(∆ f ). Focusing on very small time scales, the aggregated swing equation Eq. (1) thus simplifies to

$$Md\mathrm{\Delta }f=-D_1\left(\mathrm{\Delta }f\right)dt+\alpha \left(\mathrm{\Delta }f\right)d{W}_t\mathrm{.}$$

Here, we have used the language of stochastic differential equations, where one avoids the time derivative d∆f/dt because it is not well defined. The term D₁(∆f ) generalises the simple proportional law α∆t for the activation of the primary control. As it directly affects the change of the frequency, it is called drift. The term W_t accounts for the motion of a stochastic process, such as Brownian motion, and models the noise. However, the precise nature of this stochastic process is intricate and might well include memory effects that are difficult to capture mathematically.

The drift D₁(∆f ) can be inferred from the changes in the frequency ∆f (t + τ) – ∆f (t) for a small time step τ [4], leading to the results shown in Fig. 2. In many grids, primary control is activated only when the frequency deviation exceeds a certain limit and then increases linearly. We thus have to generalise the simple linear relation α∆ f(t) by a piecewise linear function D₁(∆f ) as shown in black dashed lines in Fig. 2. The area in the middle where D₁(∆f ) = 0 is commonly referred to as a ‘deadband.’

We observe that different power grids have distinct statutory limits (cf. vertical dashed lines for SGP, IND, MYS, and IRL) and different proportionality constants. Yet, what we observe experimentally often contrasts with each power grid’s requirements. On the one hand, we have SGP as an example where the statutory frequency limits to restore frequency are set at ±50 mHz, meaning a positive or negative change in frequency around 50 Hz.

The measured response from examining power-grid data from SGP shows an active response starting at ±25 mHz deviations. On the other hand, measurements in IND and ISL show no frequency deadband whatsoever.

A second look at Fig. 2 reveals that even a piecewise linear function is not always sufficient to describe the drift term; either because some power grids show an active, positively-signed drift within the deadband, or because the statutory deadband appears nonexistent in the estimation.

A future without synchronous generators

The transition to renewable energies, such as photovoltaic or wind power, poses a challenge to the frequency stability of power grids [5]. Renewable sources are connected to the grid via power electronic inverters that provide no intrinsic inertia. Hence, the aggregated inertia M on the left-hand side of Eq. (1) will decrease. Just like in an overdamped harmonic oscillator system, a setup with a smaller inertia, impacted by an identical deterministic imbalance, will induce a larger swing of the oscillator. This, in a power grid, leads to larger deterministic frequency excursions as well as larger fluctuations, which have to be abated to ensure a power grid remains connected. Failures in synchronisation can lead to large-scale blackouts, most notably, the recent complete power outage in the Iberian Peninsula on April 28, 2025.

A wealth of information about different power grids is still to be explored by examining the nature of the fluctuations, or noise, directly. Moreover, small-scale independent power grids, known as microgrids, have become commonplace in Europe and the rest of the world. Some are coupled and depend directly on the larger power-grid systems they are connected to for frequency control, yet their primary goal is to function independently from large power grids, most often relying only on volatile renewable energies. These microgrids often rely on a mixture of inertia provided by flywheels or gas or diesel generators in combination with solar or wind power, and their frequency dynamics can differ considerably from mechanical responses as that of the wellknown and studied overdamped harmonic oscillator.

About the Authors

Leonardo Rydin Gorjão works at the Department of Environmental Sciences, Open University of the Netherlands, and at the Norwegian University of Life Sciences, Norway. He is primarily interested in the application of statistical physics in energy systems.

Jacques M. Maritz leads the Grid Related Research Group at the University of the Free State, South Africa, where they investigate the physics of microgrids in connection with large solar generation.

Benjamin Schäfer leads the research group on data-driven analysis of complex systems for a sustainable future at the Karlsruhe Institute of Technology, Germany.

Dirk Witthaut leads the research group on networks and complex systems at Forschungszentrum Jülich, Germany, and is a professor of theoretical physics at the University of Cologne, Germany.

Acknowledgements

The data collection and analysis, along with the development of equipment to capture power-grid frequency with a high temporal resolution, have been carried out by various colleagues (in alphabetical order): Ellen Förstner, Richard Jumar, Heiko Maaß, Ulrich J. Oberhofer, Sebastian Pütz, Xinyi Wen, and G. Cigdem Yalcin. The illustration was kindly provided by Matin Mahmood. The authors would like to emphasise the enormous effort in the open science and open data community, driving projects such as PyPSA, PyPSA-Eur, PyPSA-Earth, and its various derivatives, as well as the Open Street Maps, in mapping power grids worldwide.

References

All Figures

Fig. 1 (a) The power-grid frequency measures the balance of generation and load. (b) Synchronised power-grid frequency measured at different locations in the Continental European grid from July 2019. We observe a strong decrease in the power-grid frequency at 22:00 due to a rapid de-ramping of power generation. The inset shows a magnification of the time series for a 90-second interval. The frequency is almost the same for all locations in the grid; residual differences are damped out on the time scale of seconds. (c) The frequency in a microgrid at the main campus of the University of the Free State in South Africa. The microgrid operates close to the nominal frequency of 50 Hz. The actual frequency decreases after a loss of power in the municipality.

In the text

Fig. 2 Upper panels: Power-grid frequency time series for Singapore (SGP), Indonesia (IND), Malaysia (MYS), Ireland (IRL), Australia (AUS), Iceland (ISL), and Mallorca Island of Spain (PMESP). Lower panels: Estimated drift coefficient of the power-grid frequency time series. The black dashed lines show a piecewise-linear drift function with a deadband, corresponding to an idealised primary control law [2].

In the text