A Unified Analytical Method to Quantify Three Types of Fast Frequency Response from Inverter-based Resources

With the increasing share of inverter-based resources (IBRs), power systems are experiencing lower frequency nadirs following disturbances, such as generation trips and large renewable power variations Gao2022tpwrs . Considering under-frequency load-shedding (UFLS) control, the lower frequency nadir might trip loads and cause a power outage. To tackle these challenges and keep the frequency nadir above the UFLS thresholds, one immediate solution is to leverage the capability of existing IBRs to provide fast frequency response (FFR). Here, FFR refers to the fast power injection in response to the frequency decline, aiming to increase the frequency nadir. Based on nerc2020ffr , IBRs can provide three main types of FFR: step response (Fig. 1(b)), proportional response (Fig. 1(c)), and derivative response (Fig. 1(d)).

With IBRs providing three types of FFR, one challenge for power system operators is how to quickly predict the frequency nadir after disturbances. This prediction method allows for determining whether power systems have enough FFR capacity to keep the frequency nadir above the UFLS threshold. Existing frequency nadir prediction methods mainly include simulation-based doherty2005frequency and analytical approaches xiong2022performance ; badesa2019simultaneous ; liu2020analytical ; guggilam2018optimizing . Among them, the simulation-based method in doherty2005frequency approximates the frequency nadir with a linear function; however, this requires creating a large database through numerous electromagnetic (EMT) simulations, which is known to be time-consuming and intractable for large-scale systems. Among analytical approaches, xiong2022performance is computationally inexpensive, but omits the dynamic details of the turbine governors of the synchronous generator (SG) and the IBR in the system model. The methods in badesa2019simultaneous and liu2020analytical approximate the output of the SG turbine governor with a parameterized ramp function and a polynomial, respectively. They allow for the formulation of a tractable frequency-constrained optimization problem. But the accuracy of this frequency nadir approximation depends on the values of the selected parameters. The method in guggilam2018optimizing considers a first-order turbine governor model and analytically characterizes frequency dynamics; however, it does not simultaneously consider the three types of FFR from IBRs. Indeed, to the best of our knowledge, few existing studies on fast frequency nadir prediction have fully considered the flexible combination of different types of FFR.

In this paper, we propose an accurate and efficient method to analytically predict the frequency nadir of power systems. The proposed method fully considers the impacts of three types of inverter-based FFR and provides a fast prediction speed, avoiding repetitive and time-consuming EMT simulation efforts.

The heterogeneity in FFR designs within IBRs poses a serious challenge to the studies of power system frequency dynamics. To tackle this challenge, this section first overviews the three representative types of FFRs provided by IBRs (see Fig. 1). As our first contribution, we develop the Laplace-domain models for these three types of FFRs to facilitate the system-level modeling and analysis of frequency dynamics.

We assume that the grid frequency drops as shown in Fig. 1(a) following the N-1 contingency at $t=0~{}\mathrm{s}$. Then as illustrated in Figs. 1(b)–(d), IBRs can provide three main types of FFR, i.e., step response $P_{\rm ffr1}$, proportional response $P_{\rm ffr2}$, and derivative response $P_{\rm ffr3}$, to contain the frequency deviation and avoid triggering UFLS. Below, we introduce these three types of FFR and propose their Laplace-domain models.

1) Step Response: As shown in Fig. 1(b), IBRs start to provide the step response after a time delay $t_{1}$ following the grid frequency drop at $t=0~{}\mathrm{s}$. Then the active power from IBRs ramps up until reaching the predefined saturation value $P_{\rm sus}$ at $t=t_{2}$. Here, as displayed in Fig. 2, we notice that the step response $P_{\rm ffr1}$ in Fig. 1(b) can be represented by the difference between two ramp functions with time delays $t_{1}$ and $t_{2}$, respectively. That is, we have the following time-domain decomposition for the step response

where $u(t)$ denotes the unit step function. Taking the Laplace transform of (1), we can obtain the Laplace-domain model of the step response as below:

Note that in this paper, we let $\Delta(\cdot)$ denote the perturbations in variable $(\cdot)$ after the disturbance.

2) Proportional Response: As shown in Fig. 1(c), IBRs can also provide proportional response $P_{\rm ffr2}$ following a grid frequency drop. Note that the proportional response here is reminiscent of the P/f droop control or primary frequency control in that the IBR output change is proportional to the frequency deviation, as follows:

where $R_{\rm ibr}$ is the IBR droop coefficient, $f$ is the grid frequency, and $f_{\rm n}$ is the rated frequency. By taking the Laplace transform of (3), we obtain the Laplace-domain expression of the proportional response as follows:

Note that we assume $f_{\rm n}$ remain unchanged, and thus $\Delta f_{\rm n}=0$.

3) Derivative Response: The derivative response $P_{\rm ffr3}$ in Fig. 1(d) allows IBRs to provide "synthetic inertia" to power systems. To achieve this, $P_{\rm ffr3}$ is controlled to be proportional to the time derivative of the measured grid frequency, as follows:

in which $H_{\rm ibr}$ denotes the IBR emulated inertia. In the Laplace domain, the derivative response can be expressed as

In this section, we first improve the conventional system frequency response (SFR) model by including the three types of FFR in Section 2. Our improved second-order SFR model can accurately predict the frequency dynamics of power systems with high penetration of IBRs. Thereafter, by solving this second-order SFR model, we obtain the analytical expression of the post-disturbance system frequency. This expression of frequency nadir allows us to directly predict the post-disturbance frequency nadir without onerous simulation. In this way, we can easily see whether the current FFR settings avoid triggering the ULFS under a given disturbance.

This section validates both the accuracy and the efficiency of our proposed frequency nadir prediction method with the modified 39-bus test system, as shown in Fig. 4. Note that the rated capacities of the SGs or IBRs connected to buses 38 and 39 and buses 31–37 are, respectively, $1500$ and $1000~{}\mathrm{MVA}$. All SGs are equipped with DC1A exciters and the IEEEG1 steam governor model, with the parameters reported in Table 1 and sauer1998power . The IBRs can represent inverter-interfaced batteries or hybrid photovoltaic (PV) plants that combine PV and batteries. We assume that all IBRs adopt conventional GFL controllers.

1) Verification of Prediction Accuracy: We consider cases I–V (summarized in Table 2) to show the accuracy of the proposed prediction method by comparing the simulated and predicted frequency nadir $f_{\rm nadir}$. In case I, we keep the SGs at buses 32, 33, and 36, whereas in cases II–V, we replace these three SGs with IBRs providing different combinations of FFR types (see Table 1 for FFR parameters). With these settings in place, we simulate the 39-bus system in PSCAD/EMTDC and trip the SG connected to Bus 39 at $t=2~{}\mathrm{s}$ to get the actual $f_{\rm nadir}$ of the center-of-inertia frequency $f_{\rm COI}$. Also, we predict $f_{\rm nadir}$ with our proposed method in Section 3.2. As shown in Fig. 5, the predicted $f_{\rm nadir}$ (blue dotted trace) is close to the simulated one (red point A) in cases I–V. As reported in Table 2, all prediction errors in cases I–V are within $0.06~{}\mathrm{Hz}$.

2) Verification of Prediction Efficiency: We further leverage cases V–XI to show the efficiency of the proposed prediction method. As reported in Table 3, our verification is achieved by comparing the simulation and prediction time in each case. Note that in cases V–XI, we have the same FFR settings but trip different SGs. Taking case XI in Table 3 as an example, when tripping the SG at Bus 38, we need 988.55 s to obtain $f_{\rm nadir}$ via simulation in PSCAD/EMTDC, but our proposed method takes only $0.15~{}\mathrm{ms}$, and the prediction error is within $0.06~{}\mathrm{Hz}$. This is also true for other cases in Table 3; thus, our prediction method is $10^{6}$ times faster than the simulation-based method.

This paper proposes an analytical method to predict the frequency nadir of power systems with three types of FFR provided by IBRs. The nonlinear dynamics of the turbine governor’s response and the IBR FFR following generation trip events are captured in our derivation of the nadir formulation. Our prediction method not only demonstrates high prediction accuracy but also exhibits extra-fast prediction speed compared to traditional time-domain simulations.

Compelling future directions include frequency nadir constraint formulation in generation scheduling optimization problems (e.g., unit commitment and economic dispatch), evaluation of FFR capacity adequacy from IBRs, and the application of the prediction method in real-time power system security monitoring.

This work was authored in part by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. This material is based upon work supported by the U.S. Department of Energy’s Office of Energy Efficiency and Renewable Energy (EERE) under the Solar Energy Technologies Office Award Number 37772. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for U.S. Government purposes. The views expressed herein do not necessarily represent the views of the U.S. Department of Energy or the United States Government.