# A new benchmarking method for the study of dynamic interactions in island power systems with hybrid conventional and inverter-based resources controlled in grid-forming mode

Pamela Zoghby, Bogdan Marinescu, Antoine Rossé

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## To cite this version:

Pamela Zoghby, Bogdan Marinescu, Antoine Rossé. A new benchmarking method for the study of dynamic interactions in island power systems with hybrid conventional and inverter-based resources controlled in grid-forming mode. 2nd International Conference on Renewable Energies and Smart Technologies (REST 2024), Jun 2024, Prishtina, Kosovo. hal-04670312

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Submitted on 12 Aug 2024

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# A new benchmarking method for the study of dynamic interactions in island power systems with hybrid conventional and inverter-based resources controlled in grid-forming mode

Pamela Zoghby

Electricité de France (EDF R&D)

Palaiseau, France

Ecole Centrale Nantes

CNRS, LS2N UMR 6004, Nantes

pamela.zoghby@edf.fr

Bogdan Marinescu

Nantes université

Ecole Centrale Nantes

CNRS, LS2N UMR 6004

Nantes, France

bogdan.marinescu@ec-nantes.fr

Antoine Rossé

Électricité de France (EDF R&D)

Palaiseau, France

antoine.rosse@edf.fr

**Abstract**—This paper presents a novel methodology for detecting and analyzing oscillation modes resulting from interactions between synchronous machines (SMs) and inverter-based resources (IBRs). The methodology utilizes a conceptual benchmark specifically designed to identify various oscillation modes, particularly those involving SMs and IBRs in a minimal order model for any power system. These phenomena can be studied by adapting the benchmark to represent any power system - in this paper a French island power system - where the IBR is controlled in grid-forming mode. In this system, oscillations among its generation sources raise concerns for the grid operator, necessitating comprehensive studies to understand potential stability and safety risks. The benchmark is linearised around an operating point and the type of each detected mode is identified through calculation of participation factors. The impact of key parameters on the damping of the modes is studied through sensitivity analyses. Nonlinear time-domain simulations, based on electromagnetic transient (EMT) modeling using EMTP-RV software are employed to validate the linearised model of the benchmark. Finally, a power oscillation damping (POD) controller installed on the inverter is proposed to mitigate critical oscillation modes.

**Index Terms**—Grid-forming control, Modal Analysis, Synchro- nous Machine, Conceptual Benchmark, Inverter Based Re- sources, Interactions

## I. INTRODUCTION

As power systems transition towards cleaner energy sources, integrating both synchronous machines (SMs) and inverter- based resources (IBRs) like photovoltaics (PV), battery energy storage systems (BESS), and wind turbines, the dynamics of SMs may interact with the inverter controls, introducing new modes of oscillation and potentially threatening system stability.

Inverters can operate in either current control mode, com- monly referred to as grid-following mode, or in voltage control mode, known as grid-forming mode. Grid-following sources use a phase-locked loop (PLL) to measure the grid voltage angle, enabling synchronisation and the injection of active and reactive power into the ac grid [1]. Nevertheless, this control

mode lacks the inertia contribution provided by conventional SM [2] and may encounter stability challenges in weak grids. To address these concerns, grid-forming controls in which both the magnitude and phase at the point of common coupling (PCC) are imposed, have been explored in the literature [3]. Numerous grid-forming strategies exist in the literature, among which droop control stands out as a well established and robust approach, essentially replicating the droop behavior seen in traditional power plants [4]. Additionally, various methods have been developed to not only replicate damping but also emulate the inertial effect of SMs. These include virtual synchronous machine control (VSIMA) [5], virtual synchronous generator (VSG) [6], synchronverter [7], and matching control [8]. The distinction between these strategies lies in their ability to capture the dynamics of SMs with varying levels of detail. In previous works, many studies have been conducted to examine the stability limits of power systems comprising SMs and IBRs controlled in grid-forming mode [9]–[13]. Coupling modes due to the interaction between SMs and IBRs have been highlighted in low inertia power systems, as well as the impact of key parameters such as short circuit ratio and IBR penetration level on small-signal stability [14] [15].

All these studies were conducted on a simplified academic representation consisting of an inverter vs SM model that does not capture the influence or dynamics of the rest of the system. Having a model for the rest of the system is essential as the dynamics of the ac system can significantly impact the stability of the power system, as demonstrated later in this paper. This paper proposes a methodology to comprehensively investigate novel coupling modes resulting from interactions between SMs and IBRs, as well as the influence of IBR control parameters on existing modes. This methodology, based on a conceptual benchmark introduced in [16] originally proposed for the synthesis of new IBR controls, represents a multi-source system comprising a SM, an IBR,
<figure><img src="image_3.png" /><figcaption>Fig. 1: A benchmark representing a real power system.</figcaption></figure>

and an equivalent grid to model the rest of the system. Derived from previous works [17] [18], where it was designed to study oscillation modes in any power system in a reduced-order modeling, the benchmark is adapted in this study to replicate the characteristics of a real French island power system.

The inverter is controlled in grid-forming mode with the VSM control strategy. Modes are identified and examined using modal analysis and participation factors. Additionally, sensitivity analyses are conducted to identify key parameters influencing system stability. Finally, a power oscillation damping (POD) controller that aims to improve the damping of critical oscillation modes is proposed.

The remainder of this paper is structured as follows: section II explains the concept of the benchmark and its adaptation to represent the island power system of interest. In section III, the modes of oscillation obtained from the modal analysis are identified and analysed using participation factors. Section IV presents sensitivity analyses by adjusting essential parameters and examining their effect on the oscillation modes. Section V proposes a corrective measure to damp critical modes of oscillation. Section VI is devoted to the conclusion.

## II. A CONCEPTUAL BENCHMARK FOR THE STUDY OF INTERACTIONS IN A REAL FRENCH ISLAND POWER SYSTEM

For a comprehensive investigation and quantification of coupling modes resulting from the interactions between SMs and IBRs controlled in grid-forming mode, the benchmark proposed in [17] and [18] is employed to ensure a thorough study. This benchmark, shown in Fig. 1, is simple yet sufficient in its design; it includes dynamic elements of interest required for analysis, and ensures that every component is modeled in Electromagnetic Transient (EMT), thus avoiding the risk of oversight of critical dynamics.

Modifications have been applied to the benchmark to accurately represent the French island power system, divided into two regions: an eastern area encompassing production and load, represented by the EquGrid block, and a western area incorporating a mix of a traditional SM and an IBR unit (PV or BESS) linked to a 20 kV medium-voltage grid. These generators in the western region represent upcoming projects planned for grid integration. A 90 kV high-voltage transmission line denoted by $Z_{cc}$ interconnects the two regions.

Equivalent line impedance values derived from the detailed grid data are as follows: $Z_1 = Z_2 = 0.12 + j0.1 \Omega/km$ and $Z_{cc} = 0.14 + j0.4 \Omega/km$ with distances $l_1 = l_2 = 15 km$ and $l_{cc} = 200 km$.

A diesel generator, with an exciter, prime mover, and governor as described in [17], has been employed to model the conventional generation within the medium-voltage grid of the western area.

### A. Inverter model and grid-forming control

The IBR in the western area is modeled as a voltage source converter (VSC) using an ideal average model, neglecting the high-frequency switching of the inverter. The DC link is modeled as a constant voltage source without energy limitations. This study focuses on the inverter operating in grid-forming mode, whereas previous research primarily investigated the inverter under grid-following control mode [17] [18].

As shown in Fig. 2, a cascaded voltage and current loop configuration is used to regulate the voltage at the synchronisation point. As discussed in the previous section, different grid-forming strategies have been documented in the literature, each distinguished by its external control mechanisms. It is noteworthy that the filtered droop-based strategy and the VSM strategy are equivalent [15]. Additionally, the droop strategy without a filter represents a specific instance of the VSM strategy with zero inertia as demonstrated in [19]. In the present study, the VSM grid-forming control structure is adopted, as depicted in Fig. 2. In this form, the relationship between the filter time constant $\tau_\omega$ and the equivalent inertia constant $H_{VSM}$ is provided in equation (1).

$$ \tau_{\omega} = 2H_{VSM}R_p \quad (1) $$

<figure><img src="image_4.png" /><figcaption>Fig. 2: Grid-forming control block diagram.</figcaption></figure>
## B. Equivalent grid model

In the island power system under examination, primary frequency control is performed by a combination of BESS and conventional power plants equipped with SM including hydroelectric and thermal technologies. To accommodate this, and better represent the overall aggregate dynamics of the system, enhancements have been made to the equivalent grid model proposed in [17] by incorporating a lead-lag filter in the feedback loop of the swing equation as proposed in [20] and illustrated in Fig. 3.

<figure><img src="image_5.png" /><figcaption>Fig. 3: Equivalent grid block diagram.</figcaption></figure>

$P_{EG}$ denotes the active power generated within the grid, while $P_l$ represents the power consumed by the system's loads. The equivalent grid parameters have been tuned using the island system's measured frequency response to a 3 MW load step. As shown in Fig. 4, this response has been approximated by a second-order response, whose transfer function was used to identify the parameters of the equivalent grid model shown in Fig. 3. The identification results in: $H_e = 3.78$ s, $K_g=10.24$, $T_{e1} = -0.5056$ s, and $T_{e2} = 1.21$ s.

<figure><img src="image_6.png" /><figcaption>Fig. 4: Frequency response of the island power system after a sudden load variation of 3 MW.</figcaption></figure>

## III. MODAL ANALYSIS RESULTS

To explore oscillation modes resulting from the interactions between SMs and IBRs in grid-forming mode, both nonlinear time-domain simulations and modal analysis are employed. The latter enables the capture of highly damped modes and provides insights into all the different mode types. The modal analysis approach involves developing and linearising the differential-algebraic equations of the system as explained in [17] around an operating point and expressing the system equations in state-space representation as in (2). Subsequently, the eigenvalues of matrix A, representing the modes of oscillation, are analysed. Additionally, from matrix A, participation factors can be computed to gain insights into the states involved in each mode. Consequently, this provides an understanding of the nature of each mode.

$$ \dot{x} = Ax \quad (2) $$

The critical modes of oscillation detected are presented in Table I. The droop value $R_p$ is set at 4% in this case, and the inertia $H_{VSM}$ of the VSM is equal to 2.5 s.

<table><caption>TABLE I: Relevant modes of oscillation</caption><thead><tr><th>Mode</th><th>λ = σ ± jω</th><th>Frequency (Hz)</th><th>Damping (%)</th></tr></thead><tbody><tr><td>Inter-area Mode</td><td>-0.68 ± 8.8j</td><td>1.4</td><td>7.71</td></tr><tr><td>Frequency Mode</td><td>-0.27 ± 1.12j</td><td>0.18</td><td>23</td></tr><tr><td>VSM Mode</td><td>-1.78 ± 14j</td><td>2.23</td><td>12.6</td></tr></tbody></table>

The main participation factors of the first mode identified in Table I are $Δω_{SM}$ and $Δθ_e$, the rotor speed of the SM and equivalent grid angle respectively, with little participation of the grid-forming control angle $Δθ_m$. As the states are related to the swing equation of both SM and equivalent grid, the detected mode is identified as the well-known inter-area mode [21] between the equivalent grid and the SM.

The frequency mode is the one at very low frequency, representing the global frequency dynamics of the system. The states involved in this mode are the SM's speed deviation $Δω_{SM}$, the SM's governor-related state $ΔP_{gov}$, the equivalent grid's dynamics ($Δω_e, ΔT_e$) and the grid-forming control's speed $Δω_m$.

The VSM mode is the one introduced by the VSM control strategy of the grid-forming inverter. The main participation factors in this mode are the dynamics related to the VSM control strategy ($Δω_m, Δθ_m$), the angle of the equivalent grid swing equation $Δθ_e$ and a minor involvement of the SM's rotor speed $Δω_{SM}$. This mode is eliminated when $τ_ω$ equals zero, indicating that it is introduced by the VSM control strategy. This particular situation corresponds to the unfiltered droop strategy as explained in Section II.A.

## IV. SENSITIVITY ANALYSIS

To evaluate the influence of key parameters such as the droop value $R_p$, the VSM's inertia $H_{VSM}$ and the equivalent grid parameters ($H_e$ and $K_g$), a sensitivity analysis has been performed to observe the behavior of the identified modes.

### A. Sensitivity to the VSM inertia

Given that the inertia constant of the VSM is purely a control parameter with no physical constraints, unlike actual SM-based generators, any inertia value can be chosen for the VSM. In the sensitivity analysis performed here, it has been varied across a wide range, spanning from 1 to 30 s. As shown in Fig. 5, the damping of the inter-area and VSM modes decreases drastically when increasing the VSM inertia, and the inter-area mode even becomes unstable for values greater than 10 s. In contrast, there is a slight increase in the damping of the frequency mode with the increase in $H_{VSM}$.

### B. Sensitivity to the VSM droop control gain

To evaluate the influence of the primary frequency control gain on the system stability, $R_p$ has been varied from 1% to 10%. As illustrated in Fig. 6, a reduction in $R_p$ from 10% to 1% improves the overall system stability as the damping of the three modes increases.
<figure><img src="image_7.png" /><figcaption>Fig. 5: Modes sensitivity to the VSM inertia.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6: Modes sensitivity to $R_p$</figcaption></figure>

### C. Sensitivity to the equivalent grid parameters

As depicted in Fig. 7, increasing the inertia of the equivalent grid $H_e$ results in increasing the damping of the frequency mode. This effect is consistent with increasing the inertia of the VSM as shown in Fig. 5, as both contribute to the overall inertia of the grid, thus influencing the in-phase oscillation of all states in the system's global dynamic response which the frequency mode represents. In contrast, the inter-area and VSM modes involve oscillations between elements in the grid, depending on the inertia ratio of different sources. Here, the damping is primarily influenced by the inertia ratio ($H_{VSM}/H_e$), not the total grid inertia. Thus, increasing $H_e$ enhances damping by reducing this ratio, while increasing $H_{VSM}$ diminishes damping by raising the ratio. This aligns with Fig. 5 and Fig. 7, where lower $H_{VSM}/H_e$ ratios correspond to improved damping in these modes.

<figure><img src="image_9.png" /><figcaption>Fig. 7: Modes sensitivity to the equivalent grid inertia.</figcaption></figure>

## V. POWER OSCILLATION DAMPING CONTROL

As indicated by the sensitivity analyses conducted in the preceding section, it has been observed that under certain critical conditions, the three low-frequency oscillation modes identified in this study may exhibit insufficient damping characteristics. To ensure enhanced stability of the power system, a POD controller, depicted in Fig. 8, is proposed and added to the IBR control.

This controller takes the frequency $\omega_m$ of the VSM as an input and its output $P^*$ is subtracted from $P_0$ of the external power loop in Fig. 2. A washout filter with a time constant $T_{POD}$ set to 5 s is used to eliminate the steady-state value,

<table><caption>TABLE II: Modes of oscillation with and without POD</caption><thead><tr><th>Mode</th><th>1</th><th>2</th><th>3</th></tr></thead><tbody><tr><td>Without POD</td><td>f = 1.4 Hz<br>D = 6.5 %</td><td>f = 0.236 Hz<br>D = -2.99 %</td><td>f = 2.25 Hz<br>D = 3.15 %</td></tr><tr><td>With POD</td><td>f = 1.4 Hz<br>D = 8.37 %</td><td>f = 0.38 Hz<br>D = 14.5 %</td><td>f = 3.11 Hz<br>D = 9 %</td></tr></tbody></table>

<figure><img src="image_10.png" /><figcaption>Fig. 8: POD used to stabilise modes of oscillation.</figcaption></figure>

and a low-pass filter with a time constant of $T_{filt} = 0.2$ s is used to suppress the high frequencies. The gain $K_{POD}$ is set to 200 pu. To test the effectiveness of the proposed controller, $R_p$ is set at 10%. Furthermore, adjusting $K_g$ to 20 enables to intentionally destabilize the initial condition. In this case the frequency mode is unstable as indicated by a negative damping value and the two other modes have a very low damping as shown in Table II where 1, 2 and 3 are the inter-area, frequency and VSM modes respectively. The

<figure><img src="image_11.png" /><figcaption>Fig. 9: Active power measured after the inverters filter after a step of 0.05 pu at $P_0$</figcaption></figure>

nonlinear time-domain simulations depicted in Fig. 9 validate the conclusions drawn from the modal analysis outlined in Table II. Specifically, the VSM mode is observed immediately following the step change, while the frequency mode emerges as the diverging oscillation that can be observed toward the end of the simulation result. While the inter-area mode oscillation is also present, it is not visible on the active power curves shown in Fig. 9 as its amplitude is insignificant compared to the other two depicted modes. Moreover, in instances where additional damping for the inter-area mode is needed, the installation of a Power System Stabilizer (PSS) on the SM can also be considered [21].

## VI. CONCLUSION

This paper primarily proposes a new methodology to examine the dynamic stability of any real power system. This methodology is based on a conceptual benchmark designed to analyze various modes of oscillation between targeted dynamic elements of a power system, achieved by associating IBR and SM with a dynamic equivalent grid model. The structure of the benchmark allows for the representation of any real power system in a minimal order model as illustrated by the French island power system studied in this paper.
The analysis of participation factors has revealed that the VSM control of IBR introduces a low-frequency mode, in addition to the well-known inter-area mode between SM and the equivalent grid, and the frequency mode governing system-wide frequency behavior.

Sensitivity analyses reveal that higher VSM inertia reduces damping of certain critical modes, while a lower droop value (corresponding to a greater primary frequency regulation contribution) enhances it, consistent with prior research. However this study goes further by investigating the impact of equiva-lent grid model parameters on system dynamics, finding that increasing its inertia increases damping of critical modes, con- trary to VSM inertia. Additionally, this methodology enables the identification of the frequency mode and its effect on the system stability and provides a reduced order model for the power system of interest.

A POD installed on the external loop of the grid-forming control has been proposed and proven to be effective in increasing the damping of all three detected low-frequency modes.

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