#
Transient Frequency Estimation Methods for the Fast Frequency Response in Converter Control^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- Implementation and parameter derivation for three methods to estimate the electrical frequency during transients online and offline.
- Sensitivity analysis of the implemented methods and determining the most robust method.
- Correlation between sensitive parameters in the power system and the parameters of the frequency estimation methods.
- Implementation and evaluation of the frequency estimation methods for the fast frequency response of a converter-based generator.

## 2. Electrical Frequency

#### 2.1. Definition of the Instantaneous Frequency

#### 2.2. Frequency in Power Systems

#### 2.3. Fast Frequency Response

_{T}that must be maintained constant for a defined time duration T

_{dur}and uses the frequency as a trigger to start providing the predefined amount of active power. The time delay T

_{delay}until the additional active power starts adapting and the slope of the active power change K

_{p}are predefined by the system operator. Different levels or tariffs of FFR take into account different time constants for the additional active power T

_{delay}, different frequency deviations as triggers, or different durations T

_{dur}[16].

## 3. Frequency Estimation Methods

#### 3.1. Phase-Locked Loop

#### 3.2. Zero-Crossing Method

**s**and

**t**for the signal and time values, respectively, are defined. The initial values K

_{1}= 0 for the sample number of the first zero crossing and j = 1 for the index for the continuous storage of the frequency estimation values are defined. The loop runs from i = 2 over all N time values. The starting value i = 2 is chosen in this method, as two consecutive values are compared at a time. Within the loop, the inequality $\mathit{s}\left(i\right)\xb7\mathit{s}(i-1)\le 0$ is used to check whether a zero crossing is present. If this is the case, the interpolated sample value of the zero crossing is determined according to Equation (7).

_{1}is updated with the calculated value. After the loop runs through all time values, the time course of the frequency estimation values ${\widehat{\mathit{f}}}_{\mathrm{R}\mathrm{L}}$ is filtered. Finally, the filtered vector $\widehat{\mathit{f}}$ of the frequency values is output. The online ZC method removes the outer loop in Figure 5 and passes the estimated frequency of each zero crossing directly to the low-pass filter.

#### 3.3. Gauss–Newton Method

**s**over a moving window of width m. This is done by changing the parameter vector

**x**in such a way that the sum of the error squares $E\left(\mathit{x}\right)$ in Equation (10) reaches its minimum.

**s**and

**t**for the signal and time values, respectively, are defined. The initial values are set as follows: ${\mathit{x}}_{0}={\left[12\pi \xb7500\right]}^{\mathrm{T}}$ for the initial values of the parameter vector

**x**, $m=40$ for the window width, $\epsilon =1\times {10}^{-30}$ for the accuracy to be achieved, and $maxIt=50$ for the maximum number of iterations before the loop is terminated. The first loop runs from $l=1$ over all time values to the final value $N\u2013m$. The next loop checks whether the number of iterations starting at $k=1$ is smaller than the maximum permissible value $maxIt$. At the beginning of each iteration, the sum of the error squares $E\left(\mathit{x}\right)$ is set to zero. The following loop considers all signals starting at $i=1$ within the window width $m$. Within this loop, the rows of the Jacobian matrix $\mathbf{J}\left(\mathit{x}\right)$, the error $\mathit{e}\left(\mathit{x}\right)$ and the sum of the error squares $E\left(\mathit{x}\right)$ are calculated. The calculation of the updated step $\mathsf{\Delta}\mathit{x}$ of the parameter vector is done by Equation (11) with the Jacobian matrix according to (12) with $n=3$ elements of the parameter vector $\mathit{x}$.

**x**and a new loop iteration is started. The frequency value estimated by the parameter vector is stored for the l-th time step. The loop runs from $h=1$ over the $N-1$ values of the signal and time vector. By assigning $\mathit{s}\left(i\right)=\mathit{s}\left(i+1\right)$ and $\mathit{t}\left(i\right)=\mathit{t}(i+1)$, the next pair of values of the input signal is used for the execution of the parameter estimation procedure. After running the procedure over all $N-m$ input values, the frequency values stored are corrected by a rate limit. The frequency values are filtered using the moving average method.

**x**and the associated matrix inversion, this method has a high computational cost. Due to the resulting computing time and the need for m subsequent signal values, the method is not suitable for real-time applications.

#### 3.4. Recursive Gauss–Newton Method

**s**and

**t**of the signal and time values, as well as the parameter vector

**x**and its initial value ${\mathit{x}}_{0}$ are identical to the ones in the GN method. Additionally, the initial values of ${\mathit{P}}_{0}={E\times 10}^{4}$ for the covariance matrix and ${\lambda}_{k}=0.86$ for the correction factor are set. The procedure consists of a loop over all time values, starting at $k=1$ to the final value $N$. Analogous to the GN procedure, one row of the Jacobian matrix is calculated. The covariance matrix is calculated using Equation (14). An estimated signal value according to Equation (15) is determined, and the frequency value

**x**(2) is stored. The parameter vector

**x**is updated using Equation (13). Finally, the stored frequency values are corrected with the rate limit and filtered using a moving average method.

_{k}for the model must be determined. The choice of the initial values ${\mathit{x}}_{0}$ and ${\mathit{P}}_{0}$ has less influence on the quality of the estimation. This procedure is used compared to the nonrecursive GN method with only one loop and without matrix inversion. As a result, significantly reduced computation time is required for execution, so online frequency estimation is possible with this method.

## 4. Modelling

#### 4.1. Medium-Voltage Testbench

_{onsite}that takes into account the overlying interconnected power system. An additional line with the impedance ${\underset{\_}{Z}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}}$ models the electrical distance between the external grid and the PCC. A disturbance in the form of a load step in the static load L is applied to the system, resulting in an active power mismatch and a dynamic frequency course, which is evaluated in the next chapter. The frequency estimation is done with a voltage measurement at the PCC for testing purposes and at the low-voltage side of the RES transformer to evaluate the local frequency to be used in the converter control. The two loads are modeled as constant power loads in order to focus solely on the dynamic behavior of the RES and SG. The line is modeled as a π-section equivalent circuit and the transformer as a T equivalent circuit. The installed power of the testbench components is given in Table 1.

#### 4.2. Converter Model

_{F}and the filter inductance L

_{F}. The voltage and current measurements are implemented on the low-voltage side of the transformer. The control of the VSC relies on this local measurement. Since the grid-supporting control is implemented in dq coordinates, the frequency and angle estimation play major roles. Here, the PLL is depicted. However, in the context of this study, different frequency estimation methods are applied. The estimated angle derived from the voltage measurement is passed to the coordinate transformation blocks at the start and the end of the cascaded control.

_{d}and i

_{q}, which are passed to the inner control. The low-level control or inner control repeats a similar control for the current, leading to the reference values for direct and quadrature voltage. Additionally, the cross coupling in the inner current control takes into account that due to the filter inductance, the current lags the voltage, which must be considered in dq coordinates. Further details about cross-coupling control can be found, e.g., in [21]. Limiting the reference current output of the outer control includes the maximum overcurrent capacity of real converters to the model. Finally, the pulse width modulation (PWM) controls the individual switches of the converter. Since the average value model does not include a detailed converter model, the PWM is assumed to be ideal, and the controlled voltage output of the inner control is directly fed to the three-phase voltage source that represents the electrical model.

#### 4.3. External Grid Model

_{ref}, P

_{m,T}, ω

_{r}, and ω

_{g}are the reference active power, the mechanical output power of the turbine, the rated frequency, and the measured grid frequency, respectively. K

_{Gov}is the proportional gain of the frequency control and the inverse of the governor droop d

_{Gov}. The turbine is considered as a time delay with the time constant T

_{T}.

_{r}of these plants, the inertia constant H, and the governor droop d

_{Gov}. According to the coherency method, the SG of the overlying high-voltage grid is assumed to be coherent. Therefore, the aggregate model parameters can be derived from Equations (22)–(24). Assuming that in future scenarios, not all of the power plants can provide an inertial response, the aggregate parameters are related to the total system load P

_{L}.

_{Gov,agg}and the aggregate inertia constant H

_{agg}are both calculated by the sum of the weighted values for each generator and referred to the total load of the system as shown in Equations (22) and (23). The installed power of the aggregate generator S

_{r,agg}is calculated using Equation (24). It takes into account the aggregate short-circuit power S

_{SC}and the R to X ratio of the corresponding system. The subtransient d-axis reactance ${x}_{\mathrm{d}}^{\u2033}$ is given in p.u. and the maximum voltage factor is assumed to be ${c}_{\mathrm{m}\mathrm{a}\mathrm{x}}=1.1$.

_{L}is used for the calculation to take into account that not all generating plants provide inertial behavior.${N}_{\mathrm{G}}$, ${H}_{i},{K}_{\mathrm{G}\mathrm{o}\mathrm{v},i}$, ${P}_{\mathrm{r}\mathrm{G},i}$,${S}_{\mathrm{r}\mathrm{G},i}$ are the number of SGs in the overlying grid as well as the inertia constant, the governor gain, the installed active, and the apparent power of the i-th generator. For the parameter derivation, typical machine parameters and power system characteristics are used based on [1,5,22,23]. The aggregate parameters are given in Table 2. The detailed model parameters are given in Appendix A. The onsite load of the aggregated SG is assumed to be 10% of the installed power. In the following sections, the index agg is replaced by SG in order to name the components according to Figure 8.

## 5. Evaluation Criteria

- Main criterion: A large deviation of the estimated value from the real measured frequency can have far-reaching consequences for the power system. For this reason, the aim of the estimation methods is to achieve the lowest possible maximum deviation according to Equation (25).

- 2.
- Secondary criterion: The frequency curves of two estimation methods with the same maximum deviation can turn out very differently. For this reason, the error area according to Equation (26), is evaluated over the period of simulation.

- 3.
- Tertiary criterion: Since the frequency curve serves as the input parameter for the inverter control, it makes sense that it has a smooth curve. As a criterion, it qualitatively evaluates how smooth the estimated frequency curve is.

## 6. Results

#### 6.1. Comparison of Frequency Estimation Methods

#### 6.2. Linearization of Relevant Parameters

_{g}for the ZC and the RGN method, and additionally, the proportional factor ${k}_{\mathrm{p}}$ of the PID controller of the PLL. Figure 16 shows that the sensitive parameters of the methods show a linear relationship with the magnitude of the load step. The ZC method shows the best robustness to changes in the size of the load step, in particular, simulations show that the quality of the estimate is largely independent of the parameters of the method. However, there are only a few degrees of freedom in the tuning for this method, so the PLL provides better results in the tuning for a special design case.

_{g}for the RGN method is corrected from the interpolation result in such a way that a negative offset is eliminated in order to cope with the physical limits.

#### 6.3. Fast-Frequency Response with Optimized Local Frequency Estimation

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

EMT | Electro-Magnetic Transients |

FFR | Fast Frequency Response |

GN | Gauss–Newton (Method) |

PCC | Point of Common Coupling |

PI | Proportional Integral (Controller) |

PLL | Phase-Locked Loop |

PSS | Power System Stabilizer |

PWM | Pulse Width Modulation |

RES | Renewable Energy Source |

RGN | Recursive Gauss–Newton |

RL | Rate Limit |

RoCoF | Rate of Change of Frequency |

SG | Synchronous Generator |

SRF | Synchronous Reference Frame |

VCO | Voltage-Controlled Oscillator |

VSC | Voltage-Source Converter |

ZC | Zero-Crossing (Method) |

## Appendix A. Medium-Voltage Testbench

Parameter | Symbol | Value |
---|---|---|

Nominal active power | ${P}_{\mathrm{n}}$ | 30.8 MW |

Nominal voltage | ${V}_{\mathrm{n}}$ | 20 kV |

Nominal frequency | ${f}_{\mathrm{n}}$ | 50 Hz |

Synchronous reactance in d-axis | ${X}_{\mathrm{d}}$ | 1.8 p.u. |

Transient reactance in d-axis | ${X}_{\mathrm{d}}^{\prime}$ | 0.3 p.u. |

Subtransient reactance in d-axis | ${X}_{\mathrm{d}}^{\u2033}$ | 0.25 p.u. |

Synchronous reactance in q-axis | ${X}_{\mathrm{q}}$ | 1.7 p.u. |

Transient reactance in q-axis | ${X}_{\mathrm{q}}^{\prime}$ | 0.55 p.u. |

Subtransient reactance in q-axis | ${X}_{\mathrm{q}}^{\u2033}$ | 0.25 p.u. |

Transient time constant in d-axis | ${T}_{\mathrm{d}}^{\prime}$ | 8 s |

Subtransient time constant in d-axis | ${T}_{\mathrm{d}}^{\u2033}$ | 0.03 s |

Transient time constant in q-axis | ${T}_{\mathrm{q}}^{\prime}$ | 0.4 s |

Subtransient time constant in q-axis | ${T}_{\mathrm{q}}^{\u2033}$ | 0.05 s |

stator resistance | ${R}_{\mathrm{s}}$ | 0.0379 p.u. |

inertia constant | $H$ | 0.2–10 s |

Friction coefficient | $F$ | 0 |

## Appendix B. Frequency Estimation Methods

Method | Parameter | Value |
---|---|---|

PLL | Rate limit | 12 Hz·s^{−1} |

Filter frequency f_{g} | 25 Hz | |

PID controller k_{p}, k_{i}, k_{d} | 180, 3200 s, 1 s^{−1} |

## References

- Balu, N.J.; Lauby, M.G.; Kundur, P. (Eds.) Power System Stability and Control; McGraw-Hill Inc.: New York, NY, USA, 1994. [Google Scholar]
- Fang, J.; Li, H.; Tang, Y.; Blaabjerg, F. On the Inertia of Future More-Electronics Power Systems. IEEE J. Emerg. Sel. Top. Power Electron.
**2019**, 7, 2130–2146. [Google Scholar] [CrossRef] - Milano, F.; Dorfler, F.; Hug, G.; Hill, D.J.; Verbic, G. Foundations and Challenges of Low-Inertia Systems (Invited Paper). In Proceedings of the 20th Power Systems Computation Conference: PSCC2018 Dublin, Dublin, Ireland, 11–15 June 2018; University College Dublin: Dublin, Ireland, 2018; pp. 1–25. [Google Scholar]
- Dong, D.; Wen, B.; Boroyevich, D.; Mattavelli, P.; Xue, Y. Analysis of Phase-Locked Loop Low-Frequency Stability in Three-Phase Grid-Connected Power Converters Considering Impedance Interactions. IEEE Trans. Ind. Electron.
**2015**, 62, 310–321. [Google Scholar] [CrossRef] - Rocabert, J.; Luna, A.; Blaabjerg, F.; Rodríguez, P. Control of Power Converters in AC Microgrids. IEEE Trans. Power Electron.
**2012**, 27, 4734–4749. [Google Scholar] [CrossRef] - Ortega, A.; Milano, F. Comparison of different PLL implementations for frequency estimation and control. In Proceedings of the ICHQP 2018: 18th International Conference on Harmonics and Quality of Power, Ljubljana, Slovenia, 13–16 May 2018; pp. 1–6. [Google Scholar]
- Machowski, J.; Lubosny, Z.; Bialek, J.; Bumby, J.R. Power System Dynamics: Stability and Control; John Wiley: Hoboken, NJ, USA, 2020. [Google Scholar]
- Eriksson, R.; Modig, N.; Elkington, K. Synthetic inertia versus fast frequency response: A definition. IET Renew. Power Gener.
**2018**, 12, 507–514. [Google Scholar] [CrossRef] - Poolla, B.K.; Gros, D.; Dorfler, F. Placement and Implementation of Grid-Forming and Grid-Following Virtual Inertia and Fast Frequency Response. IEEE Trans. Power Syst.
**2019**, 34, 3035–3046. [Google Scholar] [CrossRef][Green Version] - Riepnieks, A.; Strickland, D.; White, D.R.; Kirkham, H. “Frequency” and the PMU standard. In Proceedings of the To Measure Is to Know: I2MTC 2021: IEEE International Instrumentation and Measurement Technology Conference, Virtual Conference: 2021 Conference Proceedings, Glasgow, UK, 17–21 May 2021; pp. 1–6. [Google Scholar]
- Boashash, B. Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals. Proc. IEEE
**1992**, 80, 520–538. [Google Scholar] [CrossRef] - Milano, F. Rotor Speed-Free Estimation of the Frequency of the Center of Inertia. IEEE Trans. Power Syst.
**2018**, 33, 1153–1155. [Google Scholar] [CrossRef][Green Version] - Pfendler, A.; Steppan, R.; Hanson, J. Comparison of Transient Frequency Estimation Methods for Evaluating the Frequency Gradient in Active Distribution Grids: Accepted for publication. In Proceedings of the 57th International Universities Power Engineering Conference (UPEC), Istanbul, Turkey, 30 August–2 September 2022. [Google Scholar]
- Hatziargyriou, N.; Milanovic, J.; Rahmann, C.; Ajjarapu, V.; Canizares, C.; Erlich, I.; Hill, D.; Hiskens, I.; Kamwa, I.; Pal, B.; et al. Definition and Classification of Power System Stability–Revisited & Extended. IEEE Trans. Power Syst.
**2021**, 36, 3271–3281. [Google Scholar] [CrossRef] - Fernandez-Munoz, D.; Guisandez, I.; Perez-Diaz, J.I.; Chazarra, M.A. Femandez-Espina, and F. Burke Fast Frequency Control Services in Europe. In Proceedings of the 2018 15th International Conference on the European Energy Market (EEM), Lodz, Poland, 27–29 June 2018; pp. 1–5. [Google Scholar]
- Paul, D.; Mai, T.; Kenyon, R.W.; Kroposki, B.; O’Malley, M. Inertia and the Power Grid: A Guide Without the Spin. Golden, 2020. Available online: https://www.nrel.gov/docs/fy20osti/73856.pdf (accessed on 18 January 2023).
- ENTSO-E. Technical Requirements for Fast Frequency Reserve Provision in the Nordic Synchronous Area–External Document; ENTSO-E: Brussels, Belgium, 2021. [Google Scholar]
- Martin-Martinez, S.; Gomez-Lazaro, E.; Molina-Garcia, A.; Fuentes, J.A.; Vigueras-Rodriguez, A.; Plata, S.A. A New Three-Phase DPLL Frequency Estimator Based on Nonlinear Weighted Mean for Power System Disturbances. IEEE Trans. Power Deliv.
**2013**, 28, 179–187. [Google Scholar] [CrossRef] - Carcelen-Flores, A.; Fuentes, J.A.; Molina-Garcia, A.; Gomez-Lazaro, E.; Vigueras-Rodriguez, A. Comparison of Instantaneous Frequency Estimation Algorithms under Power System Disturbances. In Proceedings of the IEEE Power and Energy Society General Meeting, San Diego, CA, USA, 22–26 July 2012. [Google Scholar] [CrossRef]
- Terzija, V.V. Improved recursive newton-type algorithm for frequency and spectra estimation in power systems. IEEE Trans. Instrum. Meas.
**2003**, 52, 1654–1659. [Google Scholar] [CrossRef][Green Version] - Zhou, S.; Liu, J.; Zhou, L.; She, H. Cross-coupling and decoupling techniques in the current control of grid-connected voltage source converter. In Proceedings of the IEEE Applied Power Electronics Conference and Exposition (APEC), Charlotte, NC, USA, 15–19 March 2015; Charlotte Convention Center: Charlotte, NC, USA, 2015; pp. 2821–2827. [Google Scholar]
- Tayyebi, A.; Dörfler, F.; Kupzog, F.; Miletic, Z.; Hribernik, W. Grid-Forming Converters–Inevitability, Control Strategies and Challenges in Future Grids Application; AIM: Montréal, QC, Canada, 2018. [Google Scholar]
- Korunović, L.M.; Stojanović, D.P.; Milanović, J.V. Identification of static load characteristics based on measurements in medium-voltage distribution network. IEEE Trans. Power Syst.
**2008**, 2, 227. [Google Scholar] [CrossRef] - Technische Regeln für den Anschluss von Kundenanlagen an das Mittelspannungsnetz und Deren Betrieb: Technical Regulation for the Connection of Customer Installations to the Medium-Voltage Grid and Their Operation; VDE-AR-N 4110; VDE: Frankfurt am Main, Germany, 2018.
- NERC Inverter-Based Resource Performance Task Force (IRPTF), Fast Frequency Response Concepts and Bulk Power System Reliability Needs: White Paper. 2020. Available online: www.nrel.gov/grid/ieee-standard-1547/bulk-power-reliability-needs.html (accessed on 19 January 2023).
- Fradley, J. Frequency Containment Using Voltage Source Converters in Future Power Systems. Ph.D. Thesis, The University of Manchester, Manchester, UK, 2020. [Google Scholar]

**Figure 2.**Implementation of the fast frequency response (FFR) as (

**a**) linear dependency on the grid frequency and (

**b**) constant additional power infeed according to [16].

**Figure 3.**Basic structure of a phase-locked loop with rate limiter and low-pass filter, based on [14].

**Figure 10.**Overview of the synchronous generator control, based on [10].

**Figure 12.**Frequency curve obtained by the Phase-Locked-Loop with standard parameters according to Table A2 (standard PLL) at the PCC and rotational speed of the synchronous generator (SG).

**Figure 13.**Frequency deviation obtained by offline estimation with Gauss–Newton (GN), Zero-Crossing (ZC), and recursive Gauss–Newton (RGN) method at the PCC and a load step $\mathsf{\Delta}{P}_{\mathrm{L}}=3\mathrm{M}\mathrm{W}$.

**Figure 14.**Frequency deviation obtained by Phase-Locked-Loop (PLL), Zero-Crossing (ZC), and recursive Gauss–Newton (RGN) online estimation at the PCC and a load step $\mathsf{\Delta}{P}_{\mathrm{L}}=3\mathrm{M}\mathrm{W}$.

**Figure 15.**Frequency course of the Phase-Locked Loop estimation with parametrization according to Table 3 and the corresponding synchronous generator (SG) speed for different load steps.

**Figure 16.**Tuned parameters for (

**a**) Phase-Locked Loop (PLL), (

**b**) Zero-Crossing (ZC), and (

**c**) Recursive Gauss–Newton (RGN) method for different load steps $\mathsf{\Delta}{P}_{\mathrm{L}}=2\dots 5\mathrm{M}\mathrm{W}$.

**Figure 17.**Online frequency estimation results for different load steps $\mathsf{\Delta}{P}_{\mathrm{L}}=2\dots 5\mathrm{M}\mathrm{W}$ using (

**a**) Phased-Locked-Loop (PLL), (

**b**) Zero-Crossing (ZC), and (

**c**) recursive Gauss–Newton (RGN) method.

**Figure 18.**Active power output results of the linear fast frequency response (FFR) result using the linearized Phased-Locked-Loop (lin. PLL), the linearized Zero-Crossing (lin. ZC) method, the linearized recursive Gauss–Newton (lin. RGN) method, the Phase-Locked Loop in standard parametrization (standard PLL), and the synchronous generator (SG) speed for a load step of $\mathsf{\Delta}{P}_{\mathrm{L}}=2\mathrm{M}\mathrm{W}$.

**Table 1.**Testbench parameters of a synchronous generator (SG), renewable energy source (RES), and load L.

Parameter | Symbol | Value |
---|---|---|

Rated power SG | ${S}_{\mathrm{r},\mathrm{S}\mathrm{G}}$ | 30.8 MVA |

Rated power load L | ${S}_{\mathrm{r},\mathrm{L}}$ | 15 MVA |

Load step of load L | ∆${P}_{\mathrm{L}}$ | 5 MW |

Rated power RES | ${S}_{\mathrm{r},\mathrm{R}\mathrm{E}\mathrm{S}}$ | 2.5 MVA |

Parameter | Symbol | Value |
---|---|---|

Governor droop | ${d}_{\mathrm{G}\mathrm{o}\mathrm{v},\mathrm{a}\mathrm{g}\mathrm{g}}$ | 5% |

Inertia constant | ${H}_{\mathrm{a}\mathrm{g}\mathrm{g}}$ | 5 s |

Rated power | ${S}_{\mathrm{r},\mathrm{a}\mathrm{g}\mathrm{g}}$ | 30.8 MVA |

Turbine time constant | ${T}_{\mathrm{T}}$ | 0.3 s |

Onsite power load | ${S}_{\mathrm{r},\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}$ | 3.08 MVA |

**Table 3.**Tuned parameters for best frequency estimation when applying a load step of $\mathsf{\Delta}{P}_{\mathrm{L}\mathrm{o}\mathrm{a}\mathrm{d}}=3\mathrm{M}\mathrm{V}\mathrm{A}$. Underlined parameters are sensitive to tuning.

Method | Parameter | Value | |
---|---|---|---|

online | PLL | Rate limit | 0.85 Hz·s^{−1} |

Filter frequency f_{g} | 25 Hz | ||

PID controller k_{p}, k_{i}, k_{d} | 30, 1 s, 0.04 s^{−1} | ||

ZC | Rate limit | 0.75 Hz·s^{−1} | |

Filter frequency f_{g} | 25 Hz | ||

RGN | Initial covariance matrix P_{0} | 10^{4} E | |

Memory factor ${\lambda}_{k}$ | 0.9 | ||

Rate limit | 0.8 Hz·s^{−1} | ||

Filter frequency f_{g} | 20 Hz | ||

offline | GN | Maximum iteration maxit | 50 |

Accuracy $\epsilon $ | 10^{−30} | ||

Window width m | 40 samples | ||

Rate limit | 0.75 Hz·s^{−1} | ||

Moving average filter | 120 samples | ||

ZC | Rate limit | 1.9 Hz·s^{−1} | |

Moving average filter | 8 samples | ||

RGN | Initial covariance matrix P_{0} | 10^{4} E | |

Memory factor ${\lambda}_{k}$ | 0.86 | ||

Rate limit | 0.72 Hz·s^{−1} | ||

Moving average filter | 20 samples |

Method | Δ_{max} in mHz ^{1} | Δ_{A} in mHz·s ^{2} | Smooth Curve | |
---|---|---|---|---|

Online | PLL | 8.384 | 6.163 | Yes |

ZC | 13.360 | 9.540 | Yes | |

RGN | 11.560 | 7.516 | No | |

Offline | GN | 12.220 | 8.940 | Yes |

ZC | 10.690 | 7.160 | Yes | |

RGN | 11.720 | 8.363 | No |

^{1}Maximum deviation from synchronous generator frequency.

^{2}Area error between estimated frequency and synchronous generator frequency curve.

**Table 5.**Comparison of Rate of Change of Frequency (RoCoF) and frequency nadir ${f}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ for the different scenarios with linearized parametrization as input for the fast frequency response (FFR).

Linear FFR | Constant FFR | ||||
---|---|---|---|---|---|

Load Step | Method | RoCoF in Hz/s | f_{min} in Hz | RoCoF in Hz/s | f_{min} in Hz |

$\mathsf{\Delta}{P}_{\mathrm{L}}=2\mathrm{M}\mathrm{W}$ | PLL | 0.4434 | 49.4790 | 0.4628 | 49.4569 |

ZC | 0.4738 | 49.4796 | 0.4900 | 49.4576 | |

RGN | 0.4673 | 49.4797 | 0.4791 | 49.4577 | |

SG | 0.4770 | 49.4797 | 0.4894 | 49.4579 | |

$\mathsf{\Delta}{P}_{\mathrm{L}}=3\mathrm{M}\mathrm{W}$ | PLL | 0.6679 | 49.2195 | 0.6979 | 49.1928 |

ZC | 0.6952 | 49.2199 | 0.7159 | 49.1933 | |

RGN | 0.6876 | 49.2201 | 0.7189 | 49.1935 | |

SG | 0.7023 | 49.2200 | 0.7283 | 49.1935 | |

$\mathsf{\Delta}{P}_{\mathrm{L}}=4\mathrm{M}\mathrm{W}$ | PLL | 0.8771 | 48.9692 | 0.9113 | 48.9391 |

ZC | 0.9003 | 48.9695 | 0.9162 | 48.9394 | |

RGN | 0.8908 | 48.9697 | 0.9255 | 48.9396 | |

SG | 0.9127 | 48.9695 | 0.9443 | 48.9395 | |

$\mathsf{\Delta}{P}_{\mathrm{L}}=5\mathrm{M}\mathrm{W}$ | PLL | 1.0680 | 48.7286 | 1.0943 | 48.6968 |

ZC | 1.0864 | 48.7289 | 1.0795 | 48.6970 | |

RGN | 1.0750 | 48.7291 | 1.0952 | 48.6973 | |

SG | 1.1041 | 48.7288 | 1.1357 | 48.6970 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pfendler, A.; Steppan, R.; Hanson, J.
Transient Frequency Estimation Methods for the Fast Frequency Response in Converter Control. *Electronics* **2023**, *12*, 1862.
https://doi.org/10.3390/electronics12081862

**AMA Style**

Pfendler A, Steppan R, Hanson J.
Transient Frequency Estimation Methods for the Fast Frequency Response in Converter Control. *Electronics*. 2023; 12(8):1862.
https://doi.org/10.3390/electronics12081862

**Chicago/Turabian Style**

Pfendler, Anna, Rafael Steppan, and Jutta Hanson.
2023. "Transient Frequency Estimation Methods for the Fast Frequency Response in Converter Control" *Electronics* 12, no. 8: 1862.
https://doi.org/10.3390/electronics12081862