Stability Problems in Electrical Grids

The complexity of electricity grids

The introduction of new renewable energy generation technologies such as photovoltaic and wind power installations and storage systems, as well as the pursuit of greater energy efficiency, has shifted the generation paradigm. The energy model has transitioned from centralized (large generators) to decentralized (distributed across small renewable generators) close to consumers. On the one hand, the advantages [1] of this model include: smaller generators, greater adaptability, lower probability of chain failures, lower investment and lower energy losses in transmission lines. On the other hand, the complexity of the electricity grid increases due to the introduction of new players, more interconnections (coordination and stability) and dependence on intermittent resources.

Furthermore, financial and regulatory constraints have forced power generators to operate the electrical power system close its stability limits.

Therefore, the two factors above have created new types of stability problems that require greater resilience in control solutions to ensure quality standards, such as  frequency and voltage consistency and high reliability. Design and operational criteria must focus on ensuring that the electricity grid can withstand a wide variety of disturbances without load loss, so that normal contingencies do not result in uncontrolled cascading outages. Normal contingencies include faults that temporarily disconnect loads or generation.

The interface of new players: Power Electronic Converters

The connection of renewable generators involves the use of power electronic converters (PEC) as an interface to increase the quality of the energy delivered and control the energy flow. Therefore, research efforts are being made to control these devices appropriately in order to ensure a stable supply.

Power electronic converters are used in renewable energy applications such as wind, solar and photovoltaic, but also in others such as high-voltage direct current (HVDC) transmission and flexible alternating current transmission systems (FACTS). Therefore, the future of electrical grids will be permeated by power electronic converters that change the physical characteristics of the power system: electromagnetic induction by mechanical rotation vs. high frequency switched semiconductor devices. It should be noted that PECs have the following characteristics:

1. Fast and flexible control: Ability to operate across a wide dynamic range with digital controls that allow for great algorithmic freedom.
2. Reduced contribution to short circuits: PECs are composed of capacitors and coils with reduced energy storage capacity.
3. Harmonic emission: The switching of semiconductors leads to the emission of unwanted harmonics into the electrical grid.

The first point is the key to mitigating the stability problems of current and future electricity grids. Furthermore, stability analysis cannot be limited to the fundamental frequency at which energy transfer takes place (50 or 60 Hz in the case of AC) but must be extended to the entire frequency range.

Power System Stability

The stability of the electrical power system can be defined as [2,3]:

‘A system’s ability to remain in a state of equilibrium under normal operating conditions and to return to an acceptable state of equilibrium after being subjected to a disturbance’.

The electrical power system is a dynamic system, so this concept of stability is the same as the one applied to any other dynamic system. Fig. 2 shows a basic example of the concept of stability using what is known as a phase portrait. This geometric representation shows the trajectories of the two system states for two different parameterizations: stable and unstable. It can be seen that in the stable parameterization, the system reaches an equilibrium, i.e., a constant value over time. In the unstable case, however, the signals never reach a point of equilibrium and will grow progressively towards infinity.

Designing a large, interconnected electrical power system that ensures stability is a complex problem. From a control theory perspective, electrical power systems can be described as high-order multivariable systems (many states) with non-linear loads. In this scenario, identifying the key participants in such adverse interactions or instability is necessary to make reasonable simplifications for analysis. In other words, specific interaction mechanisms can be isolated and the system modelled with an appropriate level of detail to reproduce the phenomenon and investigate possible mitigation measures.

Therefore, categorizing instabilities into different classes makes analysis somewhat easier. In any case, we must not forget that the stability problem of the electrical power system is a joint problem and that no category should be given greater weight than others.

The classification of the phenomenon of stability is not immutable and also depends on the points of view employed by researchers of the corresponding era. Authors such as Kundur [2] established the first considerations for the classification of stability:

1. Physical nature of the mode resulting in instability: Requires identification of the main variables in the system.
2. Size of the disturbance: This influences the calculation method (small or large signal/disturbance).
3. Devices or processes involved and time interval: Consideration should be given to whether there are any energy or thermal effects.

In 2004, the electrical system was still dominated by 50/60 Hz rotating synchronous machines, so all variables refer to the phasor corresponding to that frequency:

1. Rotor angle stability: Maintain synchronism between generators (rotor angles return to stable equilibrium without losing coherence) after a disturbance (fault or sudden load change).
2. Frequency stability: Maintain frequency within limits following a disturbance in the balance between active power generation and demand.
3. Voltage stability: Maintain acceptable voltage levels at all nodes following a disturbance (e.g. load change) by balancing reactive power demand and supply.

It does not take into account distortions in sub-/supersynchronous frequencies that cannot be distinguished in the RMS (Root Mean Square) values at the fundamental frequency of voltage and current. It would be more convenient to use the instantaneous values in the rotating reference frame corresponding to the fundamental frequency.

In 2020, the ‘IEEE Technical Report PES-TR77’ [5,6] highlighted that the dynamics associated with PECs were not fully captured within the previous classes, so it added two new classes:

1. Converter-driven stability: Ability to maintain stability in the interaction between the converter and the network in ‘slow’ (< 10 Hz) and ‘fast’ (> 10 Hz) dynamics.
2. Resonant stability: Ability to maintain stability in the face of interactions (energy exchange) at different sub-synchronous frequencies.

The definition of ‘resonant stability’ is ambiguous because resonances can be stable and are inherent in the presence of magnetic (inductive element) and electrical (capacitive element) energy storage. Hence, their presence does not pose a stability problem if after the disturbance the system returns to a point of equilibrium. Furthermore, some of these interactions could also fall within the definition of ‘converter-driven stability’.

The ambiguity issues arising from these new definitions have recently been identified by CIGRE researchers [6] who compiled a list of classic terms, identified in Fig. 3.

Most engineers think the new classification proposed by CIGRE is more in line with the new paradigm of PEC presence acting over a wide range of frequencies on instantaneous voltage and current signals. Then, the approach used for the classification in Fig. 4 breaks down stability issues by frequency ranges.

While stability problems close to the fundamental frequency can be analyzed using RMS phasor models, more distant phenomenon must be analysed using EMT (Electromagnetic Transients) models. Problems with the RMS model arisen when moving away from the fundamental frequency due system’s impedance matrix (the relationship between currents and voltages) of the system, which is assumed to be constant in that reference frame in order to simplify the analysis. This impedance matrix is frequency-dependent (it changes with the signal frequency), so modes associated with sub-/supersynchronous frequency ranges can only be analyzed with this model.

The selection of the ±3 Hz range around the fundamental frequency is based on the usual range of local and inter-area modes [3], as well as torsional modes of the wind turbine drive train.

This new classification of stability phenomena highlights the redefinition of certain concepts:

1. Angle stability: This is a generalization of the original definition referring to the rotor angle and its relationship with generator synchronism. Nowadays, PECs do not have rotating elements, but they include power synchronization algorithms for the Grid Forming (GFM) control strategy that are similar to this concept.
2. Sub-/supersynchronous stability: This relates to modes in the frequency range from 0 Hz to twice the fundamental frequency due to the impedance of devices connected to the grid. This class typically includes scenarios related to subsynchronous interaction between transmission lines and grid-following (GFL) control strategies in doubly fed induction generators (DFIGs).
3. Harmonic stability: This encompasses all stability phenomena in the remaining frequencies due to the interaction of PEC controllers with the grid. It should be noted that if these interactions are stable but not damped, they can become a power quality issue.

All the terms discussed so far refer to AC electrical grids, but new types of stability can be added for future multi-terminal HVDC grids, as indicated in [6]. PECs connect AC and DC grids, so they couple the dynamics of both sides, and part of the stability phenomena could be included in the respective frequency ranges analyzed above.

Stability Analysis Methods

Dynamic systems, such as electrical grids, are modelled using differential equations in the time domain. This modelling is essential for identifying stability issues or designing controllers that ensure stable operation of the PECs when they are connected to the grid.

Although there are methods for analysing stability in non-linear representations (Lyapunov functions), such as the electrical power system, power electronics researchers have preferred to approach the problem by approximating a linear and time-invariant (LTI) system at each power transfer operating point. This transformation is known as a small-signal model.

In the case of three-phase AC networks, the variables and therefore dynamic equations are transformed to the rotating (phasor) reference frame at the fundamental frequency. In this way, AC signals at the fundamental frequency of the system are represented as continuous signals. Ultimately, this results in an LTI model with the following generalized structure called state space:

Where x ( t ) is the state vector, y ( t ) is the output vector, u ( t ) is the input vector, and matrices A, B, C, and D have corresponding dimensions and define the relationship between signals. The stability and response of this system are studied using the eigenvalues of matrix A, which are singular points where the transfer functions (input-output relationship) have unlimited magnitude.

These eigenvalues are called poles in the Laplace domain (signals in the frequency domain or ‘s-domain’) and depending on their location in the complex plane they will have natural frequency and damping characteristics that will define the behaviour of the system. If these poles have a positive real part, the system is said to be unstable.

Knowing the location of the poles is not usually an easy task due to the uncertainties of the model, as connecting and disconnecting equipment changes the analysis scenarios. Therefore, in many cases, the decision is made to identify these poles (also called modes) at different working points, and those that cause the system to behave in an oscillating and poorly damped manner are candidates for monitoring. The way to monitor them is by identifying the states that ‘participate’ in that pole/mode, which is called modal analysis [3]. In this way, it is possible to know which generator or load is negatively affecting the electrical power system.

The above analysis methodology works with operating points and models with a reduced number of states. Therefore, in order to deal with grid uncertainty impedance-based stability [7] is often used. The advantage of this method is that it tackles the problem by analysing the function that relates the PEC interconnection, i.e. the impedance. Fig. 6 shows a typical representation of impedance-based stability analysis. It is the relationship between the transfer functions Z₍ᵥ₎(s) and Yᵢ(s) that defines the stability and behaviour of each new generator/load connected to the grid.

There are several criteria for evaluating this interaction, but one of the most interesting and one that has received the most attention recently is the criterion of passivity [7]. This criterion is quite intuitive because it is a property related to the energy of the system: a system is passive if it delivers less (or the same amount of energy) than was supplied. In other words, it dissipates energy, and any system that dissipates energy is always stable because the electrical variables will tend towards a point of equilibrium.

This concept of passive behaviour is associated with the damping of certain modes in the sub-/supersynchronous frequency range or at higher frequencies. Therefore, currently, PECs connected to the grid are required to meet certain passivity requirements so as not to contribute energy to known modes of the grid.

The conclusion

Changes in the electrical power system towards a model with increasing penetration of power converters require an approach adapted to this new scenario. It is essential to identify the challenges presented by the new paradigm, but also its opportunities.

Anchoring our vision of the electricity grid from the perspective of traditional generators would be a mistake. We must take advantage of the flexibility that power converters give us to manage energy and use the most appropriate algorithms to ensure system stability.

References

1. G. Pepermans, J. Driesen, D. Haeseldonckx, R. Belmans, and W. D’haeseleer, “Distributed generation: Definition, benefits and issues,” Energy Policy, vol. 33, no. 6, pp. 787–798, apr 2005.
2. P. Kundur, J. Paserba, V. Ajjarapu, G. Andersson, A. Bose, C. Canizares, N. Hatziargyriou, A. Hill, A. Stankovic, C. Taylor, T. Van Cutsem and V. Vittal, “Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions,” EEE Transactions on ower Systems, 2004.
3. P. Kundur, “Power System Stability And Control”. McGraw-Hill, Inc, 1994.
4. IEEE, “PES TR-77: Stability definitions and characterization of dynamic behavior in systems with high penetration of power electronic interfaced technologies,” IEEE, 2020.
5. N. Hatziargyriou, J. Milanovic, C. Rahmann, V. Ajjarapu, C. Canizares, I. Erlich, D. Hill, I. Hiskens, I. Kamwa, B. Pal, P. Pourbeik, J. Sanchez-Gasca, A. Stankovic, T. Van Cutsem, V. Vittal and C. Vournas, “Definition and Classification of Power System Stability Revisited & Extended,” EEE Transactions on Power Systems, 2020.
6. M. Lindner, H. Abele, C. John, J. Lehner, K. Vennemann, T. Hennig, R. Dimitrovski, N. Klötzl, H. Just, R. Stornowski, “Suitable Classification of Power System Stability Phenomena”, CSE N°37 – June 2025.
7. Javier Serrano Delgado, “Contributions to Impedance Shaping Control Techniques for Power Electronic Converters, Thesis 2021.