Frequency Nadir Estimation Using the Linear Characteristics of Frequency Control in Power Systems

Abstract

With the increasing integration of inverter-based resources, the contribution of synchronous generators to power grids has decreased, resulting in a reduction in system inertia. Currently, acquiring the reserve power to mitigate the volatility associated with renewable-energy sources are difficult. Hence, evaluating the frequency stability of power systems and formulating operational and planning strategies based on these evaluations are becoming increasingly important. In this paper, we propose the definition and formalization of the frequency nadir index (FNI) by identifying the linear characteristics of inertia, frequency regulating reserve space (FRRS), and governor free (G/F) reserve in relation to frequency nadir, particularly under assumed contingencies such as generator dropout and sudden load fluctuation. Furthermore, we present an effective methodology for evaluating frequency stability using the FNI equation by extracting inertia, FRRS, and G/F reserve information from the power-system data. This approach is straightforward and efficiently computes the frequency nadir based on the size of the generator failure using readily available power-system data within a short timeframe.

1. Introduction

A power system is defined as something that generates and consumes electricity concurrently. Ensuring a balance between the supply and demand of electricity is vital to regulating and sustaining frequency fluctuations within a specified range. Hence, continuous monitoring of power demand is imperative. During periods when industrial development was not as extensive, the power grid was relatively small and less heavily loaded, facilitating the monitoring of power demand. Nonetheless, with the increasing sophistication of industries, particularly in the data industry, electricity demand has had an unprecedented surge. With the increasing share of power generation sourced from inverter-based resources (IBRs), power facilities have evolved into larger and more intricate structures. This transformation has made the dependable and efficient operation of power systems a challenge when relying on conventional methods.

Various countries, including South Korea, are progressively expanding their reliance on renewable energy as a strategic response to climate change and fossil fuel depletion. Expanding the use of renewable energy in power generation to achieve carbon neutrality has become a global trend. The South Korean government has unveiled its 2050 carbon- neutral strategy and associated scenario. According to this plan, a total power generation capacity of 335 GW will be required to fulfill 80% of the electricity demand from renewable-energy sources by 2050 [1,2]. Additionally, the 10th Basic Plan for Long-term Electricity Supply and Demand aims to enhance the contribution of renewable power generation facilities such that they constitute 45% of the overall power generation infrastructure by 2036, projecting an installed renewable-energy capacity of 108 GW. Within this capacity, solar power constitutes the majority at 92%, comprising 65.7 GW of solar energy and 34 GW of wind energy [3].

Unlike the inherent capabilities of synchronous generators (SGs), IBRs present the challenge of effectively maintaining reserves of active and reactive power that can be dynamically controlled in response to fluctuations in the power grid. Furthermore, IBRs, such as those employed in solar and wind energy systems, do not provide inertia or governor support to power grids [4,5]. These components are essential for maintaining the frequency stability of power grids. Because the attributes of IBRs diverge from those exhibited by conventional SGs, the intermittent fluctuations associated with renewables can exert a significant impact on the frequency stability of the power system, particularly as the proportion of renewables within the grid continues to increase [6,7,8]. Therefore, the proficient evaluation and continual monitoring of the frequency stability in real time is crucial. Regarding frequency stability, inertia is a predominant factor influencing the rate of change of frequency (ROCOF). Adequate inertial energy from SGs plays a pivotal role in constraining abrupt frequency fluctuations within a power system, ensuring that the frequency remains within a normal range, particularly during disturbances such as generator failure. The diminished power-grid inertia resulting from renewable-energy generation amplifies the frequency decline attributed to the exacerbated ROCOF during the initial phases of a disturbance [9,10]. A reduced frequency nadir within the power grid can precipitate extensive regional power outages.

Several methods have been proposed for predicting the inertia, ROCOF, and frequency nadir of a power system and for employing these measures to evaluate frequency stability. In [11], a novel model simplified the estimation of the maximum frequency deviations in small isolated power systems during generation or load disturbances, accounting for the spinning reserve regulation speed and streamlining calculations by replacing regulator closed loops with simple ramp gains. Reference [12] proposed evaluating the frequency stability using the AEMO method. This method establishes a minimum threshold for inertia by considering the relationship between the inertia and fast frequency control ancillary services (Fast FCAS). Consequently, it ensures the ability of the power system to return to normal operation under a contingency. In [13], a simplified estimation method was proposed for non-center-of-inertia (COI) frequency scenarios, eliminating the requirement to acquire the COI frequency and ROCOF. The estimation approach is streamlined based on the linearization of the regulation power to enhance the accuracy and simplify the estimation process. In [14], a neural network (NN) method was proposed to predict the frequency nadir using measurements from the early stages after an event. This approach offers superior accuracy to static value-based predictions. Reference [15] proposed an online method to estimate the power-system inertia constants during normal operations using dynamic modeling based on phasor measurement unit (PMU)-measured bus frequency and active power output, extracting inertia constants from the model’s unit step response in the time domain and updating them in real time through a sliding window and exponential smoothing techniques. Reference [16] proposed virtual inertia to extend the concept of inertia in SGs and proposed an inertia evaluation method tailored for power grids with high-penetration electronic devices. Additionally, a minimum inertia estimation method considering a fast frequency response was presented, building on the virtual inertia technology of grid-connected inverters. Previous studies have not utilized the linear characteristics of frequency control to evaluate frequency stability by considering the inertia, frequency regulating reserve space (FRRS), and governor-free (G/F) reserve information.

Based on the frequency stability evaluation conducted using Power System Simulation for Engineering (PSS/E), this paper proposes an approach for the efficient evaluation of frequency stability. Simulating the frequency stability of a nationwide power system using PSS/E presents several challenges. Significant time is required to simulate the frequency stability of a nationwide power system. Additionally, a complex procedure is needed to verify the initialization of power flow data (SAV) and dynamic data (DYR), which contain information about the nationwide power system. These difficulties are addressed by employing a straightforward equation, termed the frequency nadir index (FNI), without relying on PSS/E frequency stability simulations. The proposed method identifies the linear characteristics of key factors such as inertia, FRRS, and G/F reserve in relation to the frequency nadir to facilitate this evaluation. With the FNI, the frequency nadir of the power system can be computed in response to generator drop-outs, thereby enabling the effective management of frequency stability. The frequency nadir estimation becomes feasible through expedient and straightforward calculations of grid data. This estimation can be accomplished by supplying real-time system data to the frequency-stability evaluation system. These improvements can significantly enhance the reliability of power systems, enabling them to operate economically and consistently. This involves preventing large-scale generator cascades and load shedding via under-frequency relay (UFR) and reducing the potential for wide-area blackouts [17].

2. Frequency Response Characteristics

In power systems predominantly reliant on SGs as the primary source of power generation, the imperative tasks of frequency control and the preservation of frequency stability have emerged as critical concerns, not only under normal operating conditions but also in the presence of disturbances. Figure 1 shows the control measures for three distinct time intervals in the event of a disturbance, such as a generator failure, resulting in a frequency decline. (i) Initial inertial response exhibited by the SG, (ii) frequency response resulting from the generator’s droop characteristics and load damping, and (iii) restoration of frequency to its reference value through automatic generation control (AGC). The inertial response of a power system is an intrinsic physical response caused by the SG mass [18].

In Figure 1, the frequency nadir is determined through the composite interplay of inertia, FRRS, and G/F reserve. The primary frequency reserve is influenced by the following two characteristics. The first is the number of SGs engaged in the droop control. When more SGs participate in the droop control, the frequency stability improves. The second is the FRRS of SGs engaged in droop control. During disturbances, such as generator dropouts, a generator with sufficient FRRS will exhibit a significant droop response as the frequency drops, thereby contributing to improved frequency stability.

3. Frequency Maintenance Criteria and Reserve Requirements in South Korea

To maintain grid frequency, the Korea Power Exchange (KPX) issues directives to power generators and instructs them to provide electricity to the grid, which may involve adjusting the active power output of the power generation and energy storage system (ESS). During normal operational conditions, the grid frequency should be maintained at 60 ± 0.2 Hz. In the event of failure in a single generator with maximum capacity, the grid frequency must be maintained at a minimum of 59.7 Hz. In the event of failure of two generators or the activation of a special protection system (SPS), the grid frequency should be maintained at a minimum of 59.2 Hz. In the event of an emergency, the grid frequency should be maintained within the range of 57.5 to 62 Hz. In South Korea, the largest individual capacity generator is a nuclear power plant with a 1.4 GW capacity. In the worst-case scenario, this would involve the failure of the N-2 and failure of the generator. The frequency maintenance criteria for South Korea are presented in Figure 2 and

Table 1. Specifics presented in Figure 2.

	N-1	N-2 or SPS	On Emergency
Maintenance Range	(minimum) 59.7 Hz	(minimum) 59.2 Hz	57.5–62.0 Hz
Recovery	(within 1 min) 59.8 Hz	(within 1 min) 59.5 Hz (within 10 min) 59.8 Hz	-

[19].

Generator reserves must be secured to satisfy South Korea’s frequency maintenance criteria during normal operating conditions and under failure. In the event of a fault, an SPS can be employed to ensure transient stability, voltage, and frequency stability. An SPS comprises protective relays and communication equipment, serving as a system designed to mitigate a variety of extensive failures, such as isolated grid events, generator failures, and transmission-line cascade tripping. When a generator trips with an SPS to ensure the transient stability, a rapid drop in frequency occurs, requiring reserves to be established to adhere to the frequency maintenance criteria shown in Figure 2 and Table 1. The generator reserve requirements for South Korea are listed in

Table 2. Reserve requirements in South Korea.

	Types of Reserves	Method	Response Time	Power Generation Output Time	Reserve Capacity
In normal situation	Frequency control reserve	AGC	(Operate) within 5 min	more than 30 min	more than 700 MW
In emergency	1st reserve	G/F	(Operate) within 10 s	more than 5 min	more than 1000 MW
	2nd reserve	AGC	(Operate) within 10 min	more than 30 min	more than 1400 MW
	3rd reserve	Central Controllable Load Dispatching Generator	(Secure) within 30 min	-	more than 1400 MW

[19].

A centrally controllable load-dispatching generator is defined as a generator with a capacity of 20 MW or more that operates under KPX. In addition to Table 2, hydroelectric power is operated as an additional reserve among the central controllable load-dispatching generators, capable of commencing operation in less than 20 min and sustaining output for over four hours.

Renewable-energy generation to achieve carbon neutrality goals is growing worldwide, and South Korea is following this trend. As renewable-energy generation increases, the proportion of SGs must be reduced to match the power supply. This decrease in inertia highlights the increasing importance of ensuring reserves in maintaining frequency stability during grid faults such as generator failures, which can result in frequency drops.

4. Dynamic Simulation of Frequency Stability Using PSS/E

In South Korea, KPX functions as an independent system operator (ISO), and Korea Electric Power Corporation (KEPCO) serves as the transmission operator (TO) [20]. Both entities engage in frequency stability evaluations using PSS/E. Previous studies introduced dynamic simulations that focused on power-system transients and frequency stability using PSS/E [21,22,23,24,25,26]. Performing a frequency stability analysis using PSS/E requires the availability of the SAV and DYR files. The DYR file includes dynamic models for the generators, exciters, governors, stabilizers, FACTS (SVC, STATCOM and TCSC), LCC HVDC, VSC HVDC, etc. Although dynamic models are generic models supplied by PSS/E, user-defined model (UDM) can be employed as required. The flowchart in Figure 3 shows the procedure for conducting the frequency stability analysis using PSS/E.

For a frequency stability analysis, the SAV and DYR files must be initialized and, in some cases, the UDM file must be verified. The time required to conduct a nationwide power-system frequency-stability simulation using PSS/E is substantial. Currently, KEPCO utilizes data gathered from supervisory control and data acquisition (SCADA) to generate an SAV file for the national power system, which can be simulated using PSS/E. Because real-time power-system data files change continuously, initializing the SAV and DYR files for frequency stability analysis is challenging. Even when they are initialized, a significant amount of simulation time is required using PSS/E. Hence, a more efficient method to evaluate the frequency stability in the event of disruptions, such as generator failure, utilizing power-system data is required.

5. Methodology

In this paper, we examine the primary factors influencing the frequency nadir during disturbances that affect frequency fluctuations. We establish a linear relationship with the frequency nadir by individually controlling each of these factors and introduce an efficient method for calculating the frequency nadir based on this relationship. The controlled parameters include the inertia, FRRS of the generators participating in droop control, and G/F reserve, encompassing three key elements. When a power system experiences an imbalance between supply and demand caused by events such as generator loss or a sudden increase in load, an imbalance occurs between the mechanical and electrical powers of the SGs. However, the occurrence of a generator failure or a sudden load increase causes changes in the kinetic energy of the rotor, ultimately contributing to the preservation of power balance, as shown in the system swing Equation (1) [27]. Additionally, as the value of H increases, the rate of frequency fluctuation during disturbances decreases.

$$2H\frac{d{w}_m}{dt}\approx P_m-P_e$$

(1)

The damping effect of the load is disregarded in Equation (1), where $w_m$ represents the angular speed of the rotor (pu), H denotes the inertia constant (s), and $P_m$ and $P_e$ are the mechanical and electrical powers, respectively (pu). The inertia constant of the ${SG}_i$ is defined as $H_i$, which is calculated as the ratio of the kinetic energy ($E_{KE,i}$) stored in the rotating mass at the rated speed to the machine’s rated power ($S_i$), as shown in (2) [27].

$$H_i=\frac{E_{KE,i}}{S_i}$$

(2)

Additionally, in a multi-machine power system, the equivalent inertia constant of the entire power system is given using (3) [28].

$$H_{sys}=\frac{{\sum }_{i=1}^n{H}_i{S}_i}{{\sum }_{i=1}^n{S}_i}=\frac{{\sum }_{i=1}^n{E}_{KE,i}}{{\sum }_{i=1}^n{S}_i}=\frac{{Inertia}_{sys}}{S_{sys}}$$

(3)

$H_{sys}$ represents the system’s equivalent inertia constant; ${Inertia}_{sys}$ is the total kinetic energy of the SGs; $S_{sys}$ is the rated capacity of the system; and $n$ signifies the total number of generators in service. In this paper, ${Inertia}_{sys}$ is considered the system inertia, and renewable-energy generation is treated as the negative load, which is offset by the positive load. This approach leads to conservative frequency stability results. The frequency stability evaluation using the FNI equation presented in this paper is conservative and assumes that the asynchronous generators have no inertia. The droop is calculated using (4), which is a well-known droop characteristic of a generator [29].

$${\delta }_i=\frac{\frac{∆f}{f_n}}{\frac{{∆P}_i}{P_{max,i}}}\times 100 [\%]$$

(4)

${∆P}_i$ represents the output variation of the generator $i$ caused by the droop control when $∆f$ occurs in the time domain where the recovery frequency appears after a disturbance. $P_{max,i}$ represents the maximum power output that can be generated with the governor of generator $i$ in the DYR. It is determined using the maximum value of the governor. $f_n$ is the nominal system frequency in Hz. The governor-free response quantity (GFRQ) refers to the frequency response capability of the governor of a generator after a disturbance, such as generator loss or a sudden load increase. In South Korea, the GFRQ is considered the G/F reserve, as shown in Table 2. It is determined by the capability of the droop control of SGs to respond to frequency changes of 0.2 Hz in the power system [30]. The GFRQ of generator $i$ is calculated as shown in (5).

$${GFRQ}_i=P_{max,i}\times \frac{∆f\times 100}{f_n\times {\delta }_i}$$

(5)

In (6), ${GFRQ}_{total}$ is the total GFRQ in the entire power system, and $m$ is the total number of the governors in service in the DYR.

$${GFRQ}_{total}=\sum _{i=1}^m{GFRQ}_i$$

(6)

To ensure that the droop control can adequately respond to frequency fluctuations, we require an adequate amount of FRRS for generator output variations; the FRRS for generator $i$ is defined as shown in (7).

$${FRRS}_i=P_{max,i}-P_{gen,i}$$

(7)

$P_{gen,i}$ is the current output of generator $i$ before the disturbances occur. In (8), ${FRRS}_{total}$ is the total FRRS in the entire power system, and $m$ is the total number of governors in service in the DYR, which is identical to Equation (6).

$${FRRS}_{total}=\sum _{i=1}^m{FRRS}_i$$

(8)

If the ${FRRS}_i$ is insufficient when the power system experiences a disturbance causing a frequency drop, the increase in power generation is limited by $P_{max,i}$. The G/F reserve of generator $i$ is given using (9).

$${G/F}_{reserve,i}=\mathrm{m}\mathrm{i}\mathrm{n} \{{FRRS}_i,{GFRQ}_i\}$$

(9)

The ${G/F}_{reserve,i}$ is determined by selecting a smaller value between generators ${FRRS}_i$ and ${GFRQ}_i$. When each generator is dispatched to guarantee the adequacy of the ${FRRS}_i$, the ${G/F}_{reserve,i}$ equals ${GFRQ}_i$. Consequently, in practical applications, the total G/F reserve is similar to the total GFRQ. The total G/F reserve from all operating generators with governors in service is the G/F reserve of the entire grid, which closely approximates the GFRQ of the entire grid, as shown in (10).

$${G/F}_{reserve,total}=\sum _{i=1}^m{G/F}_{reserve,i}\approx {GFRQ}_{total}$$

(10)

In this paper, we treat the G/F reserve and GFRQ as synonymous and intentionally exclude the influence of the ESS on frequency stability during disturbances. Our focus is solely on examining the linear characteristics of the three aforementioned factors—inertia, FRRS, and GFRQ—and their impact on the frequency nadir. A list of abbreviations for the terminology is shown in the Appendix A.

5.1. Response Characteristics of the Frequency Nadir as Inertia Changes

The inertia of a power system plays a dominant role in determining the ROCOF and frequency nadir. Greater power-system inertia results in a lower ROCOF when the power system experiences a frequency drop, such as due to a generator failure or a sudden load increase. Furthermore, owing to the presence of load damping, which changes with frequency, the power system converges to a single frequency drop point when a contingency occurs. A linear relationship between inertia and the frequency nadir is observed in the system swing Equation (1). By assuming a linear relationship between the ROCOF and the frequency nadir, inertia and the frequency nadir are also assumed to have a linear relationship. These assumptions were also employed in the linear regression model of the ENTSO-E report ‘Future System Inertia 2’ [31]. In this paper, based on the PSS/E simulation results, we observed that the relationship between inertia and frequency nadir is nearly linear. When the power generation, FRRS, and GFRQ of the power system are maintained at constant values while altering only system inertia, a linear characteristic emerges with the frequency nadir within the practical operational range of the power system.

In this study, frequency stability in the South Korean power system was analyzed using PSS/E simulations. While varying the inertia as an independent variable, the inertia of each generator was uniformly scaled to maintain constant power generation, FRRS, and GFRQ. In practical scenarios, inertia changes as generators are added to or removed from the grid. Nevertheless, artificially manipulating the H constant of each generator can provide valuable insights into the connection between the inertia and frequency nadir. Thus, when the FRRS and GFRQ are held constant while the inertia is independently increased, a linear relationship between the inertia and the frequency nadir is observed, as shown in Figure 4. This linear relationship is represented using (11).

$$f_{Nadir}\propto {Inertia}_{sys}$$

(11)

Nonetheless, note that for the validity of the relationship expressed in (11), the FRRS must exceed the amount of generator failure. This process is described in detail in the following section.

5.2. Response Characteristics of the Frequency Nadir as FRRS Changes

The FRRS represents the space required to adjust the power generation through droop control for each generator in response to a frequency decrease caused by disturbances and which is defined in (7). When the inertia and GFRQ are constant, while FRRS is independently increased, the relationship between FRRS and the frequency nadir is shown in Figure 5. The smaller the FRRS relative to the amount of generator failure, the lower the frequency nadir, as shown with the sum of the exponential functions in (12), which are defined using coefficients a, b, c, d.

$$f_{Nadir}=a·{exp}^{b·FRRS}+c·{exp}^{d·FRRS} (\mathrm{i}\mathrm{f} FRRS<F2)$$

(12)

However, if the FRRS is large relative to the amount of generator failure, the frequency nadir changes only minimally, as shown in (13).

$$f_{Nadir}\approx constant (\mathrm{i}\mathrm{f} FRRS\ge F2)$$

(13)

Within the practical operational range of the power system in South Korea, the FRRS exceeds the value of the N-2 generator failure. Consequently, in the simulation conducted in this study, because the FRRS was larger than the amount of the N-2 generator failure, it did not affect the frequency nadir. In South Korea, a 2.8 GW generator failure would constitute the worst-case scenario, and the reserve operation of the South Korean power system maintains at least 3.1 GW of the FRRS (the sum of the frequency control, primary, and secondary reserves) to ensure frequency stability.

5.3. Response Characteristics of the Frequency Nadir as GFRQ Changes

In the event of a disturbance within the power system that induces a frequency decline, an increased value of GFRQ, as indicated in (6), results in a greater participation of generators in the droop response. Depending on the generator failure scenario, ensuring an appropriate GFRQ is essential. In power-system operation in South Korea, a minimum G/F reserve of 1 GW should be maintained, as shown in Table 2. The 1 GW represents the GFRQ value, which encompasses the sum of changes in generator output determined using the droop characteristics of SGs in service within the power system when the power-system frequency undergoes a 0.2 Hz change.

KPX manages DYR files containing droop characteristics supplied by power generation companies. These files are shared with KEPCO to facilitate the frequency-stability simulations. In this study, we validated the droop characteristics provided by power generation companies by calculating and verifying them using the frequency change and output change of each generator during the time interval in which the recovery frequency was observed following N-2 generator failure.

When the inertia and FRRS are held constant, while the GFRQ is independently increased, a linear relationship between the GFRQ and frequency nadir is observed, as shown in Figure 6. This linear relationship is represented using (14).

$$f_{Nadir}\propto {GFRQ}_{total}$$

(14)

This step is a precursor to deriving the FNI equation. The formalization of the FNI equation, to be used as a basis for frequency-stability evaluation, utilizes the light-load planning data (SAV and DYR) for the worst-case scenario. To ensure that generators can sufficiently increase power generation through the droop control as the frequency decreases, the FRRS must have a margin of at least 10% higher than the amount of the generator failure. This requirement is followed for practical power-system operations in South Korea. However, with the increase in renewable energy, the proportion of synchronous generators operating during light-load condition is lower than usual, making it challenging to secure an over GFRQ of 1 GW.

5.4. Frequency Nadir Estimation Using the FNI

In the previous section, we discuss the effect of inertia, FRRS, and GFRQ on the frequency nadir during disturbances such as generator failure. In Section 5.1, we observed a linear correlation between the frequency nadir and the changes in inertia. In Section 5.2, we showed that when the FRRS is higher than the amount of generator dropout, the frequency nadir remains relatively constant and unaffected. In Section 5.3, we confirmed a linear correlation between the GFRQ and frequency nadir. In this step, we converted the frequency nadir to FNI from the data obtained in the previous step. Subsequently, we conducted linear regression on the GFRQ and FNI datasets to formalize the FNI equation. This equation serves as the basis for evaluating frequency stability, and the results are conservative because the light-load planning data for the worst-case scenario is used. By extracting the inertia and GFRQ values from other power-system data and substituting them into the FNI equation, both FNI and frequency nadir can be calculated through simple linear calculations. The frequency nadir achieved through GFRQ incrementation can be used to calculate the corresponding FNI value at a specific GFRQ value, as shown in (15).

$$FNI=\frac{Generator failure amount\times 0.1}{60-f_{Nadir}}[\mathrm{M}\mathrm{W}/0.1 \mathrm{H}\mathrm{z}]$$

(15)

The linear relationship between the inertia, GFRQ, and the frequency nadir continues even when the FRRS exceeds the number of generator failures. In practical scenarios, the FRRS is expected to exceed the number of generator failures. As discussed in Section 5.2, the FRRS in South Korea includes a total of at least 3.1 GW of the sum of the frequency control, primary, and secondary reserves. Thus, inertia and the GFRQ can be used to formulate the FNI equation, enabling the evaluation of frequency stability. To align the dimensions of G/F reserve with inertia, we apply the $\rho$ coefficient, as shown in (16), defining it as the virtual inertia.

$${Inertia}_{G/F}=\rho ·{G/F}_{reserve}$$

(16)

The synthetic grid inertia is defined as the sum of the inertia represented using the constant H of the SG and the virtual inertia of the G/F reserve, as shown in (17).

$${Inertia}_{total}={Inertia}_{sys}+{Inertia}_{G/F}$$

(17)

The frequency nadir can be determined by incorporating the synthetic grid inertia and a scaling factor ($A$) for standardizing dimensions in the FNI equation, as shown in (18).

$$\begin{array}{ll}FNI& =\frac{{Inertia}_{sys}}{A}+\frac{\rho ·{G/F}_{reserve}}{A}\\ & =\frac{{Inertia}_{sys}}{A}+\frac{{Inertia}_{G/F}}{A}\\ & =\frac{{Inertia}_{total}}{A}\left[\mathrm{M}\mathrm{W}/0.1 \mathrm{H}\mathrm{z}\right]\end{array}$$

(18)

After the linear regression analysis on the GFRQ and FNI datasets using (15), the parameters $A$ and $\rho$ can be derived. The unit of FNI is MW/0.1 Hz, enabling the grid operator to intuitively evaluate how many generator failures may affect a 0.1 Hz drop in grid frequency. With the FNI equation, the frequency nadir can be computed swiftly by considering the amount of generator dropout, as shown in (19).

$$f_{Nadir}=60-\frac{Generator failure amount}{FNI\times 10}$$

(19)

The SAV file of the power system provides a list of the generators in service, whereas the pre-existing DYR file contains essential details, including the generator’s H constant, installed capacity (MVA), and droop characteristics. These parameters play a crucial role in determining the power-system inertia and GFRQ. In a subsequent step, we can derive the FNI equation for the system. In conclusion, the FNI equation has the significant advantage of evaluating the frequency stability without requiring to simulate the frequency stability using PSS/E, as shown in Figure 7.

Figure 8 shows the flow chart of the frequency-stability evaluation using the FNI equation proposed in this paper. The frequency-stability evaluation can be evaluated by applying the data including the inertia and GFRQ values from SAV files to the appropriate FNI equation. In the final step of the flowchart, simulation results are included for the specific case where different lists of governors in service are used. However, they operate at the same load size, differing only in GFRQ, as shown in Figure 7. Additionally, simulation results for other load conditions were also examined in Section 6.

In the next section, we used the proposed method to evaluate the frequency stability of power-system data under light-load conditions in South Korea. This choice was based on the consideration that the worst-case scenario for frequency stability occurs when an SG fails during a light-load operation. In this scenario, synchronous generators constitute the smallest proportion among all generators supplying the load, resulting in the most significant frequency drop in the event of a generator failure.

6. Simulation Results

The South Korean power system is shown in Figure 9. In South Korea, most large-scale power generators are located along the coastline. During the peak load periods, approximately 45% of the total load was concentrated in the metropolitan area. Power generated by generators in non-metropolitan areas is transmitted to the metropolitan area through several transmission lines [32]. Owing to insufficient AC transmission lines for transporting power generated in non-metropolitan areas to metropolitan areas, a large amounts of non-metropolitan power generation can result in problems with transient stability, voltage stability, and frequency stability. In South Korea, the most critical contingency scenario among transmission line failures involves a 765 kV #1.2 T/L fault connecting the Yeongdong region to the Seoul metropolitan area. When these lines trip, the generators located in the Yeongdong region accelerate, which results in transient stability problems. The SPS for generator tripping is employed to address this [33,34]. However, even if transient stability is achieved through generators tripping, extensive generator tripping can result in supply and demand imbalances, resulting in frequency-stability problems. To address this problem, we implemented operational measures that simultaneously ensure both transient and frequency stability. These measures involve prior constraints on power generation in the Yeongdong region. Instead of reducing power generation in the Yeongdong region, the output of costly combined-cycle generators in the metropolitan area must be increased to satisfy the supply and demand requirements, although this may have adverse effects on the economic power supply. Furthermore, the significance of frequency stability is increasing in the Honam region because of the extensive tripping of renewable-energy sources that lack low-voltage ride-through (LVRT) capabilities during faults. With the increase in renewable energy, frequency stability is becoming a challenging task; thus, the FNI equation can be applied to efficiently evaluate frequency stability under contingencies.

In this study, PSS/E simulations were conducted using SAV and DYR files based on the 10th Basic Plan for Long-term Electricity Supply and Demand. To account for the worst-case scenario, we employed light-load planning data for 2024. The basic parameters are listed in

Table 3. Parameters of the base-case SAV file.

Number of SGs	SGs [GW]	Net Load [GW]	Load [GW]	IBR [GW]	Proportion of IBR [%]	${\mathbf{I}\mathbf{n}\mathbf{e}\mathbf{r}\mathbf{t}\mathbf{i}\mathbf{a}}_{\mathbf{s}\mathbf{y}\mathbf{s}}$ [GWs]	FRRS [GW]	GFRQ [GW]
88	49	48	56.3	−8.3	14.5	280	3.1	1

.

The total generation of the SG was 49 GW, and the net load, which reflected the renewable generation allocated as a negative load, was 48 GW. If Shin-Gori Nos. 3 and 4 drop out, the SGs in service can provide a system inertia of 280 GWs. As shown in Table 2, the FRRS totaled 3.1 GW (the sum of the frequency control, primary and secondary reserves), considering the 2.8 GW failure of Shin-Gori Nos. 3 and 4. GFRQ represented a 1 GW G/F reserve, which was the primary reserve. To simulate the frequency stability, we used a load-frequency model with the ZIP model coefficients in the Electricity Market Operation Rules [30]. The parameters are listed in

Table 4. Parameters of the ZIP model.

PL [%]	QL [%]	IP [%]	IQ [%]	YP [%]	YQ [%]
41.3	20.3	19.2	30.8	39.5	48.9

.

A damping constant of 1.6611% was applied to the frequency-stability simulations. These values, including the ZIP model coefficients and the damping constant parameters, were consistent with the parameters used by KEPCO and KPX in their frequency-stability simulations. Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14 show the procedure for evaluating the frequency stability through the FNI applied to the 2024 light-load planning data for South Korea.

Figure 10 shows Step 1 of the procedure for frequency nadir estimation using the FNI in the South Korean power system. We display the frequency trend after a 2.8 GW failure of Shin-Gori Nos. 3 and 4 while maintaining the SG generation, FRRS, and GFRQ at constant values and varying the inertia value. To facilitate adjustment of the inertia, FRRS, and GFRQ values, we modified the inertia value for each generator by applying a uniform ratio. We observed that the frequency nadir increased linearly with increasing inertia.

Figure 11 shows Step 2 of the procedure for frequency nadir estimation using the FNI in the South Korean power system. We illustrate the frequency trend after an N-2 failure while maintaining constant inertia and GFRQ values and varying the FRRS value. The frequency nadir did not exhibit a linear increase until the FRRS was increased by the amount of the N-2 generator dropout. Instead, it followed a nonlinear pattern, as described in (12). From approximately 3.1 GW with a 10% margin of N-2 generator failure, we observed that the FRRS was sufficiently secured and it did not significantly impact the frequency nadir.

Figure 12 shows Step 3 of the procedure for frequency nadir estimation using the FNI in the South Korean power system. We illustrate the frequency trend after an N-2 failure while maintaining constant inertia and FRRS values and varying the GFRQ value. Both the frequency nadir and recovery frequency increased with an increase in GFRQ. A linear relationship was observed between the increase in GFRQ and frequency nadir. In the operation of the South Korean power system, Steps 1 to 3 assumed considerable importance as they secured 3.1 GW FRRS, which was greater than the amount of N-2 generator failure. An increase in both inertia and GFRQ independently led to a linear increase in the frequency nadir.

Figure 13 shows Step 4-1 of the procedure for frequency nadir estimation using the FNI in the South Korean power system. After a linear regression analysis on the GFRQ and FNI datasets is performed using (15), the parameters A and ρ that correspond to South Korea’s 2024 light-load planning data can be derived. With A and ρ, the FNI equation can be graphically represented with a red dashed line. When maintaining consistent inertia conditions, variations in the GFRQ result in the corresponding FNI values aligned along a linear trajectory.

The data presented in

Table 5. Calculation of $A$ and $\rho$ of the FNI equation through a linear regression on the 7 SAV files in Step 4-1.

Number	${\mathit{I}\mathit{n}\mathit{e}\mathit{r}\mathit{t}\mathit{i}\mathit{a}}_{\mathit{s}\mathit{y}\mathit{s}}$ [GWs]	GFRQ [GW]	F Nadir [Hz]	FNI [MW/0.1 Hz]	Result
1	280	1	59.35	432	A = 1.18 ρ = 230
2		1.097	59.38	449
3		1.188	59.40	468
4		1.295	59.43	493
5		1.394	59.45	510
6		1.491	59.47	525
7		1.585	59.49	547

displays the results obtained from the PSS/E frequency-stability simulations using seven SAV files. Each SAV file was characterized by an inertia of 280 GWs and a varying GFRQ ranging from 1 to 1.585 GW.

The formalized FNI equation for South Korea’s 2024 light-load planning data, employing the parameters $A$(1.18) and $\rho \left(230\right)$, is presented in Equation (20).

$$\begin{array}{ll}FNI& =\frac{{Inertia}_{sys}}{1.18}+\frac{230·{G/F}_{reserve}}{1.18}\\ & =\frac{{Inertia}_{sys}}{1.18}+\frac{{Inertia}_{G/F}}{1.18}\\ & =\frac{{Inertia}_{total}}{1.18}\left[\mathrm{M}\mathrm{W}/0.1 \mathrm{H}\mathrm{z}\right]\end{array}$$

(20)

The G/F reserve is represented using the GFRQ value in GW, and the inertia is in GWs. Although they have different dimensional units, we can obtain the virtual inertia by incorporating the value of $\rho$(230) into the G/F reserve. Moreover, the synthetic grid inertia, which is the sum of the system and virtual inertia, can be obtained. Because they share the same dimensions, grid operators can manage the virtual and synthetic grid inertia more intuitively with the G/F reserve. The advantage of the proposed frequency-stability evaluation method is that the FNI equation is based on the worst-case scenario. Therefore, when the inertia and GFRQ values are known from other SAV files, we can conservatively determine the FNI [MW/0.1 Hz] and the frequency nadir associated with generator failure without requiring a PSS/E frequency-stability simulation.

Table 6. Parameters of the SAV file applied to the FNI equation in Step 4-1.

Number of SGs	SGs [GW]	Net Load [GW]	Load [GW]	IBRs [GW]	Proportion of IBRs [%]	${\mathit{I}\mathit{n}\mathit{e}\mathit{r}\mathit{t}\mathit{i}\mathit{a}}_{\mathit{s}\mathit{y}\mathit{s}}$ [GWs]	FRRS [GW]	GFRQ [GW]
88	49	48	56.3	−8.3	14.5	280	3.1	2

lists the parameters of another SAV file under the light-load planning data for 2024. However, note that in this case, the GFRQ increased to 2 GW.

Table 7. Comparison of the results of applying the SAV file from Table 6 to the FNI equation and the PSS/E simulation in Step 4-1.

${\mathit{I}\mathit{n}\mathit{e}\mathit{r}\mathit{t}\mathit{i}\mathit{a}}_{\mathit{G}/\mathit{F}}$ [GWs]	${\mathit{I}\mathit{n}\mathit{e}\mathit{r}\mathit{t}\mathit{i}\mathit{a}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$ [GWs]	FNI [MW]/0.1 Hz]	FNI	PSS/E	Error [Hz]
			(N-2) F Nadir [Hz]	(N-2) F Nadir [Hz]
460	740	628	59.55	59.54	0.01

presents the results obtained by applying the data in Table 6 to Equation (20). At 280 GWs of inertia and 2 GWs of GFRQ, the FNI value was 628 MW/0.1 Hz. The virtual inertia, considering the G/F reserve, was 460 GWs, whereas the synthetic grid inertia, comprising 280 GWs of system inertia with generator H values, totaled 740 GWs. Under a 2.8 GW failure of Shin-Gori Nos. 3 and 4, the frequency nadir obtained from the FNI equation was 59.55 Hz. This indicated an error of 0.01 Hz with a frequency nadir of 59.54 Hz obtained from the PSS/E simulation, as listed in Table 7. Extracting GFRQ data from SAV files with consistent inertia and then applying them to the FNI equation yielded notably accurate results.

In a previous analysis, we examined a case in which only the GFRQ was varied while maintaining a generation of 49 GW from SGs with 280 GWs of inertia. As the load increased, the number of operating generators that were required to be in service also increased, resulting in an increase in system inertia. To evaluate the accuracy of the FNI and frequency nadir values, we compared the results obtained from the PSS/E frequency-stability simulation with the outcomes derived from the SAV files with varying inertia in conjunction with the FNI equation obtained from the SAV file in the worst-case scenario. Grid data of 59 GW and 69 GW (with 335 GWs and 387 GWs of inertia) were applied to the FNI equation obtained from 49 GW (280 GWs of inertia) and further analyzed. The parameters of the SAV files are listed in

Table 8. Parameters of the SAV files applied to the FNI equation in Step 4-2.

Case	Number of SGs	SG [GW]	Net Load [GW]	Load [GW]	IBRs [GW]	Proportion of IBRs [%]	${\mathit{I}\mathit{n}\mathit{e}\mathit{r}\mathit{t}\mathit{i}\mathit{a}}_{\mathit{s}\mathit{y}\mathit{s}}$ [GWs]	FRRS [GW]	GFRQ [GW]
1	109	59	58	66.3	−8.3	12.3	335	3.1	1
2									1.097
3									1.188
4									1.295
5									1.394
6									1.491
7									1.585
8	167	69	68	76.3	−8.3	10.7	387	3.1	1
9									1.097
10									1.188
11									1.295
12									1.394
13									1.491
14									1.585

.

For the 14 grid datasets (with 335 GWs and 387 GWs of inertia), the GFRQ varied from 1 to 1.585 GW. Figure 14 shows Step 4-2 of the procedure for frequency nadir estimation using the FNI for the South Korean power system. As shown in Figure 14a, when applying the FNI equation derived from the 49 GW worst-case scenario to the 59 and 69 GW grid data, the slope remained constant owing to the fixed values of constants $A$ and $\rho$. However, the y-intercept underwent alterations because of the changing inertia within the system. Figure 14b shows a comparative analysis between the frequency nadir calculated using the FNI equation and the results obtained from the PSS/E frequency-stability simulation, specifically for the case involving the 2.8 GW failure of Shin-Gori Nos. 3 and 4. By comparing the data derived from the FNI equation with the frequency-stability simulation data obtained using PSS/E, we observed that the FNI values associated with each GFRQ were consistently smaller. This observation reflected a more conservative outcome, which corresponded with the prudent approach in the context of the power-system planning and operations. The error in the two datasets was estimated to be due to the load characteristics, specifically the frequency variations induced by generator failures.

Table 9. Comparison of the results of applying the SAV files from Table 8 to the FNI equation and the PSS/E simulation in Step 4-2.

Number	${\mathit{I}\mathit{n}\mathit{e}\mathit{r}\mathit{t}\mathit{i}\mathit{a}}_{\mathit{G}/\mathit{F}}$ [GWs]	${\mathit{I}\mathit{n}\mathit{e}\mathit{r}\mathit{t}\mathit{i}\mathit{a}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$ [GWs]	FNI [MW/0.1 Hz]	FNI	PSS/E	Error [Hz]
				(N-2) F Nadir [Hz]	(N-2) F Nadir [Hz]
1	230	565	480	59.42	59.43	0.01
2	252	587	499	59.44	59.45	0.01
3	273	608	517	59.46	59.47	0.01
4	298	633	538	59.48	59.50	0.02
5	321	656	557	59.50	59.52	0.02
6	343	678	576	59.51	59.53	0.02
7	365	700	594	59.53	59.55	0.02
8	230	617	524	59.47	59.49	0.02
9	252	639	543	59.48	59.52	0.04
10	273	660	561	59.50	59.54	0.04
11	298	685	582	59.52	59.56	0.04
12	321	708	601	59.53	59.58	0.05
13	343	730	620	59.55	59.59	0.04
14	365	752	638	59.56	59.61	0.05

presents the results obtained by applying the data in Table 8 to Equation (20). In each case, the virtual inertia, synthetic grid inertia, and FNI were observable. These results allow grid operators to intuitively evaluate the operational status of the power system as well as frequency stability. Additionally, the maximum error between the frequency nadir obtained from the FNI equation and PSS/E did not exceed 0.05 Hz. As shown in the previous results, the accuracy of the frequency nadir for N-2 generator failure was remarkably high.

Furthermore, because only a linear calculation was required on the pre-formalized formula, the frequency stability could be efficiently evaluated within a considerably shorter time compared with conducting PSS/E frequency-stability simulations. Currently, KEPCO gathers power flow data from power plants, substations, and customers using SCADA. These data errors are corrected using the state estimation (SE) process, and subsequently our system generates SAV files for the South Korean power system every 5 min. Furthermore, the SAV files are verified for convergence through a power flow solution. In the future, KEPCO plans to establish a real-time frequency-stability evaluation system by applying the generated SAV files to the proposed method in this paper, as shown in Figure 15.

7. Discussion

The study aimed to identify the deterministic factors that affect the frequency nadir and establish its linear characteristics to formalize the FNI equation for effective frequency-stability evaluation. We applied the proposed methodology to South Korea’s planning data for 2024 and verified the formalized FNI equation.

This study had some limitations. The ESS was disabled to eliminate its influence on the frequency nadir. The PSS/E provides renewable-energy-model guidelines, including the REGC, REEC, and REPC. However, renewable-energy generation was treated as a negative load, and the impact of renewable-energy generators on the frequency stability was not considered. Currently, in Korea, frequency stability is evaluated by modeling renewable generators as negative loads, resulting in conservative frequency-stability results. Additionally, we acquired the $A$ and $\rho$ parameters and formalized the FNI equation based on worst-case scenario light-load planning data. Under heavy load conditions and various GFRQ scenarios, potential errors may occur when evaluating frequency stability. Variations in load characteristics owing to frequency fluctuations at different load levels may require adjustments to the $A$ and $\rho$ parameters in the FNI equation. Future research should also focus on this aspect.

8. Conclusions

In this paper, we identify the linear characteristics of inertia, FRRS, and G/F reserve (represented as GFRQ) in relation to the frequency nadir and formulate the FNI equation using linear regression. We then propose an efficient method to evaluate the frequency stability based on this equation. We performed simulations and validated our methodology using 2024 planning data points for the South Korean power system. With the rapid expansion of renewable-energy generation, the evaluation of frequency stability is becoming increasingly crucial. However, evaluating the frequency stability using PSS/E not only requires checking the initial state of the SAV and DYR files for the power system but also consumes a significant amount of simulation time. In contrast, our proposed methodology enables the grid operator to evaluate the frequency stability within a short time by utilizing inertia, FRRS, and GFRQ from the grid data without requiring a PSS/E simulation. In the future, we anticipate that the results of this study will establish a system capable of the efficient and real-time evaluation of frequency stability based on grid data.