Online Estimation of Disturbance Size and Frequency Nadir Prediction in Renewable Energy Integrated Power Systems

The displacement of conventional synchronous generation by Inverter-Based Resources (IBRs) poses critical challenges to the frequency stability of Renewable Energy (RE) integrated power systems. An increased shift towards green energy by integrating RE sources that provide little or no inertia results in a high Rate of Change of Frequency (RoCoF) and deteriorated frequency nadir/zenith following a credible system disturbance. Prediction of frequency metrics, such as frequency nadir and RoCoF, immediately after disturbances will help grid operators to take preventive/corrective actions, such as the deployment of faster frequency control, to ensure secure and stable grid operation. This paper presents an easy-to-implement analytical method to estimate disturbance size in a power system immediately following a contingency which is then used for predicting frequency nadir. The proposed estimation method uses active power measurements from a limited number of monitoring nodes and an adaptive bus admittance matrix of the system for the disturbance size estimation. The estimated disturbance size is then used to predict frequency nadir using a Neural Network (NN) based method. The performance and accuracy of the presented approach are evaluated using a standard IEEE 39 bus system and a real-life Gujarat power system in India through extensive simulations in DIgSILENT PowerFactory, considering various cases of disturbance size, type, and location.

Abstract-The displacement of conventional synchronous generation by Inverter-Based Resources (IBRs) poses critical challenges to the frequency stability of Renewable Energy (RE) integrated power systems.An increased shift towards green energy by integrating RE sources that provide little or no inertia results in a high Rate of Change of Frequency (RoCoF) and deteriorated frequency nadir/zenith following a credible system disturbance.Prediction of frequency metrics, such as frequency nadir and RoCoF, immediately after disturbances will help grid operators to take preventive/corrective actions, such as the deployment of faster frequency control, to ensure secure and stable grid operation.This paper presents an easy-to-implement analytical method to estimate disturbance size in a power system immediately following a contingency which is then used for predicting frequency nadir.The proposed estimation method uses active power measurements from a limited number of monitoring nodes and an adaptive bus admittance matrix of the system for the disturbance size estimation.The estimated disturbance size is then used to predict frequency nadir using a Neural Network (NN) based method.The performance and accuracy of the presented approach are evaluated using a standard IEEE 39 bus system and a real-life Gujarat power system in India through extensive simulations in DIgSILENT PowerFactory, considering various cases of disturbance size, type, and location.Index Terms-Disturbance size, frequency nadir, inertia, rate of change of frequency, synchronizing power coefficient.

I. INTRODUCTION
T HE growing concern about environmental challenges and global heating due to fossil fuel depletion has resulted in the transition of power systems into cleaner, greener, integrated energy systems.The rapidly growing integration of renewable energy from solar and wind has replaced centralized synchronous power plants, thus reducing system inertia.The decrease in system inertia poses severe challenges to the frequency stability of the power grids since frequency variations are very likely to breach the security limits during disturbances, such as a generator or load tripping [1].Loss of Generation (LoG) is one example that can result in frequency instability in modern power systems characterized by high IBR penetration levels [2].Improved frequency control schemes are expected to deliver fast-frequency support to low inertia-related issues.Such control techniques require suitable coupling with wide-area monitoring systems (WAMS) [3], [4], [5].Some examples of fast frequency controls are Adaptive Under Frequency Load Shedding (AU-FLS) [6], the use of Energy Storage Systems (ESS) [7], and synthetic/virtual inertia from IBRs [8], [9], [10], [11].
Fast Primary Frequency Response (PFR) is essential for stabilizing the frequency after a significant disturbance.Frequency nadir is an important indicator for PFR's monitoring and control.It can be used to estimate spinning reserve requirement and load shedding settings in the resource scheduling (to provide PFR) and under-frequency load shedding (UFLS) schemes, respectively [12].Therefore, frequency nadir or RoCoF prediction is essential to examine the frequency stability of the system during system disturbances.This also enables the Transmission System Operators (TSO) to react faster by releasing necessary frequency support or deploying preventive measures to retain system frequency within its secure limits, thereby avoiding unnecessary load shedding.However, achieving fast and robust activation of frequency support services is a major challenge.The frequency support should be delivered depending on disturbance size and location by utilizing the minimum resources possible and should minimize the impact of such actions on the system's normal operation (e.g., power flows) [5].There are various modes of activating frequency control actions, such as using proportional response as used by governor systems or responses based on thresholds in cases of under/over frequency load shedding, etc.Such triggering methods are robust but slow, while the RoCoF-based triggering is fast but is prone to maloperation due to the challenges in accurately assessing RoCoF under disturbed conditions.This maloperation can be avoided by the combined use of RoCoF and frequency as used in most AUFLS schemes, which is limited by its slow response [6].
In conventional UFLS schemes, the non-essential load shedding is done sequentially based on priority-oriented assumptions and past experiences [13], whereas adaptive mechanisms focus on minimizing the number of loads to be shed [14].Most of the frequency control schemes that have been proposed so far assume that a precise estimate of the power imbalance is readily available to operators immediately after the occurrence of the disturbance [6], [14], [15].For a significant power imbalance in a system, the frequency deviation will be more, thus making estimation of the disturbance size an important task for system operators of larger power systems to tackle associated frequency stability issues [16].Therefore, power system dynamic modeling, estimation of inertia, disturbance size, and prediction of parameters such as RoCoF, frequency nadir, etc., play a critical role in designing frequency control schemes.Most of the identified methods for inertia estimation available in the literature [17], [18], [19] use frequency measurements subsequent to disturbances in the system (disturbance-based estimation) for inertia estimation.Several challenges in using frequency measurements to estimate inertia, power mismatches, etc. need to be tackled.One such issue is inaccuracy associated with utilizing a frequency signal measured from a single location, which will not accurately represent system frequency during dynamics [20].On the contrary, several methods use calculated RoCoF values using measurements from Phasor Measurement Units (PMU) as a part of a WAMS [21].It is logically possible to follow an approach to obtain frequency and RoCoF values as first and second derivatives of synchro-phasor angle, which can be measured using PMUs.However, such a method is sensitive to noise [22].Also, this approach demands frequency monitoring at all system nodes to effectively accommodate local frequency behavior during transient periods following system events.This is not different from the case where instead of localized values, a Center of Inertia (CoI) based RoCoF is calculated using the RoCoF of each generator from PMU measurements.However, the study presented in [15] reveals that the measurements of RoCoF using PMUs are inconsistent with RoCoF values separately calculated from generator shaft speed for the initial transient period.All these observations conclude that accurate and fast determination of frequency parameters such as RoCoF is challenging for a system experiencing a contingency.
One method to keep possible frequency deviations within the threshold is by predicting frequency parameters such as nadir and RoCoF immediately after the disturbance occurrence.Therefore, this paper proposes an analytical method to estimate the active power imbalance subsequent to a disturbance which is then used to predict the frequency nadir.This is a two-stage method, with the first stage estimating disturbance size using immediate active power changes measured from a limited number of generator terminals combined with the bus admittance matrix of the power grid.In stage 2, the nadir is predicted by assuming a linear relationship between the nadir and RoCoF where the latter is linearly related to the estimated disturbance size and available kinetic energy of the system [23].A Neural Network (NN) based model is trained and employed using available sample data collected from various case studies to predict frequency nadir accurately.The main contributions of this paper are summarized below: 1) An easy-to-implement analytical method is proposed to estimate disturbance size in a power system immediately following a system disturbance.
2) The proposed method significantly improves over the existing synchronizing power coefficient (SPC) method by accommodating the following realistic factors in the estimation of disturbance size: a) System dynamics during and after disturbance, using an adaptive bus admittance matrix (Y bus ), captures the changes in network parameters which is less prone to error than its offline counterpart.b) Active power contribution of frequency and voltagesensitive loads to power mismatch at the onset of the disturbance.c) Inertial contribution from Type 1/ Type 2 wind turbine generators during the system disturbance.d) Contribution from non-synchronous fast power services during the system disturbance.3) A frequency nadir prediction method is proposed for a RES-dominated power system.Subsequently, following a disturbance, system operators can use these predictions for efficient system frequency control.4) Frequency nadir prediction uses the estimated disturbance size immediately following a disturbance.Hence, contributions from loads, type1/type 2 wind turbines, and non-synchronous fast-frequency services are also incorporated.The proposed frequency nadir prediction method is more accurate and outperforms other existing nadir prediction methods.The rest of the paper is organized as follows; the modeling of test systems and an overview of the conventional SPC method are described in detail in Section II.Section III explains the proposed method for the estimation of disturbance size.The case studies and results are discussed in detail in Section IV.Finally, Section V presents the concluding remarks.

II. MODELLING OF TEST SYSTEMS AND SYNCHRONIZING POWER COEFFICIENT METHOD
Two different test systems are considered in this study.Test system-1 is the standard IEEE 39 bus system shown in Fig. 1.It is a 10-machine system with a total demand of around 6 GW.It is important to mention that in this study, test system-1 is not modified by adding any RE-based generators, and all loads are modeled as constant impedance loads (CIL) unless mentioned otherwise.Test system-2 is the real-life Gujarat State power grid with a total installed capacity of 41 GW.Gujarat is a RE-rich state in India, with almost 38% of total installed capacity (as of Jan 2022) being contributed by RE sources, majorly solar PV and wind energy.The system parameters used for modeling the Gujarat state grid are taken from [24].Both test systems are simulated in DIgSILENT PowerFactory software.
As explained in Section I, estimation of disturbance size is essential to deliver a response that will aid in limiting the frequency deviation and ensure stable and secure power system operation subsequent to a severe contingency.A Synchronizing Power Coefficient (SPC) based method is introduced in [26] for the detection and estimation of the disturbance size and its location.It is briefly discussed in the following subsection.

A. Estimation of Contingency Response From Synchronous Generators Using Synchronizing Power Coefficient Method
The conventional SPC method proposed by Negar shams et al. is a two-step approach in which the disturbance is detected in the first step, followed by the assessment of its size and location in the second step.This method uses active power measurements from generator terminals measured through PMUs and the system impedance matrix.When subjected to a disturbance, the power mismatch created in the system is immediately compensated for by the kinetic energy of the rotating machines, predominantly the synchronous generators present in the system.The immediate distribution of power mismatch observed at each generator terminal depends on its electrical distance to the disturbance location as well as the total disturbance size [14].Moreover, the disturbance location has a unique pattern that can define the power distribution at generator terminals following the disturbance.The SPCs are calculated between synchronous generators and the disturbance location, which is then used to calculate total power deviation using the SPC method.
In an N-bus power system, with synchronous generators modeled as a constant voltage source behind a reactance (classical power system model) [13], electrical power output, P e,i from the ith synchronous machine is represented using the following equation: ) where E i and V i represents the voltages of generator terminal and bus respectively (note E i equals to V i for generator buses), δ ij is the angle between E i and V j , G ij and B ij are real and reactive part of off-diagonal elements of bus admittance matrix (Y bus ).
Synchronizing power coefficient (K ij ) is defined as an electrical power change of ith synchronous generator (SG i ) due to a change in angle between SG i and SG j , assuming all other angles in the system are staying constant, shortly following a disturbance [13].Equation ( 2) represents the calculation of K ij : where δ ij0 represents δ ij value at steady state conditions.For an N bus power system with M number of synchronous generators, synchronizing power coefficient matrix (K M×N ) is defined using (2).Considering the occurrence of a system disturbance, active power change observed at terminals of an SG is given by: where K iu is SPC between SG i and bus u (disturbance location), P u d represents size of disturbance that occurred at time t d , P e,i (t d ) and P e,i (0) represents post and pre-disturbance active power output of SG i .From (3) it is seen that the total power mismatch the system is subjected to is distributed among all synchronous generators according to the SPC values, which depend on the disturbance location [13].Therefore, by calculating SPC values corresponding to system disturbances and the power changes monitored at the generator terminals, the total power change in the system can be estimated accurately.With the help of several different test cases, the robustness and accuracy of the conventional SPC method have been examined in [26].

B. Limitations of the Conventional SPC Method
The conventional SPC method described in Section II-A has some limitations, as highlighted below, that restrict its application in practice.a) Magnitude of the internal voltage of the SGs is assumed to be constant shortly after an active power disturbance in a system, whereas the voltage angle between generator terminal and bus (δ ij ) varies.But in practice, this assumption is invalid.The disturbance is analyzed only by considering initial active power changes at different SGs.However, the reactive power changes at the SG terminals are not negligible; hence, the voltage magnitude does not remain constant.Disturbances such as an increase or decrease of load by small size may not impact the internal voltage magnitudes much, unlike significant events such as generator outages, load tripping, etc.Therefore, it is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
essential to consider the changes in the internal voltage magnitude of generators and their angle following system events.b) The SPC method assumes that the network consists of only constant power loads that do not change during and after the event.Thus, the change in total active power observed by the system is equal to the size of disturbance.However, a real power system consists of static and dynamic loads with its significant share being voltage and frequency sensitive, highly influences the total power change, especially in the first 1-2 s following the disturbance.There are voltage-dependent loads with dynamics attributed to voltage variations and frequency-dependent loads and governors, which are influenced by the system's frequency response.Therefore, system dynamics play a key role in total active power mismatch and should be considered in the respective estimation approach.c) Another assumption in the conventional SPC method is that subsequent to a disturbance, the total power mismatch is compensated only by the kinetic energy of rotating machines of the system, primarily the SGs, through their inertial response.However, there are other components in the grid that contribute to the power imbalance along with the generators.In practice, the active power imbalance is shared among SGs, voltage and frequency-dependent loads, Type-1 and Type-2 WTGs if available in the system, the non-synchronous generators provided with fast power reserves, and other frequency and voltage-sensitive elements.The initial contribution from these components during the disturbance is significant and should be considered while estimating the total disturbance size.

III. PROPOSED METHOD FOR DISTURBANCE SIZE ESTIMATION AND FREQUENCY NADIR PREDICTION
The novelty of the proposed estimation method lies in the possibility of addressing the limitations mentioned above, associated with the conventional SPC method, to improve the accuracy of estimation of the disturbance size.The system dynamics that play a significant role in the power system response following a severe disturbance should not be ignored in the estimation methodology.Hence, a dynamic bus admittance matrix that takes in the system dynamics is used to calculate the SPC values of the system.Furthermore, the contribution from loads and renewable energy sources towards the total power imbalance is also included in the estimation method to accommodate the system variations during the disturbance.The proposed method is a two-stage method: Stage 1 -Disturbance size estimation Stage 2 -Frequency nadir prediction This section explains both stages of the proposed method in detail.Sections III-A to D discuss the modifications made to overcome the limitations of the conventional SPC method for improving the accuracy of disturbance size estimation.Section III-E discusses the NN-based method to predict frequency nadir (stage 2) using the estimated disturbance size in stage 1.

A. Dynamic Bus Admittance Matrix
The elements of the bus admittance matrix (Y bus ) are used to calculate synchronizing power coefficients in the conventional SPC method.Y bus represents a linear transmission network model that derives a relationship between bus voltages and respective bus injection currents, as shown in (4).
where v bus is the complex-valued n vector of all bus voltages and i bus is the corresponding complex-valued n vector current injections at all buses.Y bus is usually calculated using offline models of network elements obtained during their design.As mentioned in Section II-B, the power system parameters vary depending on the characteristics of system events.They are seldom captured by offline models, which do not get updated corresponding to the system variations, hence providing erroneous or inaccurate results.To avoid these shortcomings of offline calculation of Y bus , which ignores the system dynamics in the conventional SPC method, a dynamic bus admittance matrix calculation has been proposed in this estimation method which is then used to calculate synchronizing power coefficients of the network.Such a method is preferable since it is more accurate and adaptive to system changes.
In [27], a measurement-based approach using least square estimation for calculating Y bus is explained for a power system using phasor measurements.The voltages and current injections at all nodes of the network are measured for several time intervals and are used to calculate the elements of Y bus .
The voltage and current measurements taken at a time t i at bus j are denoted by v j (t i ) and i j (t i ) respectively, then v(t i ) and i(t i ) gives n vectors of voltages and currents at ith time sample and is represented by the following equations: where T indicates transpose operation.V bus and I bus are defined as n×m voltage and current measurement matrices where n is the number of buses in the network and m is the number of time samples and is given by: The matrices V bus and I bus are obtained from the measurements and, Y bus is calculated using (9) based on the least squares estimation method considering minor measurement errors.

B. Inertial Contribution From Load in Response to a System Power Disturbance
In most of the literature, the disturbance size estimation methods assume that the total active power mismatch (ΔP (t)) created in the power system is equal to the size of contingency (ΔP cont ), such as the output of the disconnected generator or Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.load right after the disturbance, represented as follows: (10) This assumption holds only if the estimation takes place at the instant of disturbance so that the change in the system loads and frequency-controlled reserves will not substantially affect the power change.However, in practice, the total active power imbalance the system will experience could deviate significantly from the ΔP cont .This deviation mainly results from the voltage and frequency dependency of loads.To predict the frequency nadir following a disturbance, an accurate measure of the power imbalance at the time of disturbance should be available with the TSO, thus making a precise and quick estimation of the disturbance size essential.Furthermore, if the measurements are taken at a later stage of the event, the estimated disturbance size will be different from the ΔP cont not only due to the mentioned frequency and voltage-dependent loads but also due to responses from generator governors.It is important to mention that in this study, the voltage and frequency dynamics of the loads are included, while the responses from governors and their contribution to power change are not considered for the proposed estimation approach.
The electrical power change (ΔP e (t)) observed in the system is written as: where ΔP L (t) is change in total demand resulted from the disturbance attributed to the frequency and voltage-dependency of the loads.
In [16], authors proposed a method (RV method) to calculate the power deviation caused by load demand changes due to voltage and frequency variations in the power system following a disturbance.The RV method is a combination of the R method that accommodates frequency dynamics of the system loads and the V method, which is based on the argument that voltage dependency of loads also results in power deviation owing to load demand changes after system disturbance.Thus, the difference in total load demand is expressed as: where h 1 (f (t)) represents the power deviation due to frequencydependent loads, h 2 (V (t)) is the change in power from voltagedependent loads.
In steady-state, mechanical power change and frequency changes are related linearly as ΔP m = −RΔf where R is a constant named system gain, ΔP m is mechanical power change, and Δf is the frequency deviation at the system steady-state.However, this linear relation is true only for steady-state conditions and is not valid to express system dynamics.Since this work considers all system dynamics while estimating the size of the disturbance, the gain R is substituted by R(t), a varying time function [13] which will accommodate dynamics associated with Δf (t).By replacing h 1 (f (t)) in ( 12) by R(t)Δf (t) and obtaining these values by calculation and measurements, the contribution from frequency-dependent loads towards the power imbalance is obtained.
In this approach, an aggregate load model is used to analyze the behavior of the total system load.This load model comprises constant power (P), constant current (I), and constant impedance (Z) components [28] and is monitored by data collected from generator buses.The total voltage-dependent load demand at a time t, P LV (t) is given by: where l is the number of loads present in the system, P L0 i and V L0 i are the initial values of active power load and voltage at bus i, V L i is the voltage at load bus i at a time instant t and k z , k i , k p indicate fractions of constant impedance, constant current and constant power loads connected to the system.It is assumed that these fractions are already available with the TSOs.
With the assumption that the data is available only from the generator buses, the total pre-disturbance power production, P prod is used to replace l i=1 P L0 i in (13).Furthermore, the aggravated voltage profile of the connected aggregate load model is approximated by using voltage measurements taken at only generator buses by: where n is total number of generators, E i (t) is the voltage at generator bus i at time t and E i 0 is the pre-disturbance voltage at generator bus i.Thus using ( 13) and ( 14) the change observed in total demand is given by: In the RV method, at first, the V method is applied to approximate the changes in voltage dependent loads through h 2 (V (t)) followed by the R method employed to accommodate frequency-dependent load changes using h 1 (f (t)).While calculating h 2 (V (t)), only change in power from voltage-dependent loads is considered.The frequency dependency of the loads and the governor response is not incorporated.So, the time of application of the V method is chosen to be before 500 ms and as soon as possible immediately following the system event since governors start to respond to disturbances after about 500 ms to 1 s [29].
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C. Inertial Contribution From Type 1 and Type 2 Wind Turbine Generators
The inertial response of a WTG depends on the electrical characteristics of its individual wind turbine technology.Fixedspeed WTGs have different inertial responses compared to conventional synchronous generators.However, they do not intrinsically reduce the inertia of the system due to their electromechanical characteristics.On the contrary, the variable speed WTGs do not inherently exhibit similar inertial responses owing to their rotating mass being decoupled from grid frequency unless controlled for that specific purpose [30].Type 1/Type 2 wind turbines are directly connected to the power system, hence can release kinetic energy stored in their blades, gearbox, and other rotating parts, thus contributing to frequency support.
To quantify this released inertial support from Type 1 and Type 2 WTGs, the torque equation-based induction machine model [29] is considered: where, J tot is the total inertia of the machine in kg-m 2 , T e and T m represents electrical and mechanical torque in N-m, respectively, and ω m is the angular velocity of the rotor in rad/s.The rated mechanical torque of the machine is defined using rated mechanical power P mr , nominal electrical frequency of the network ω n in rad/s, P z represents number of pole pairs, and slip n is the nominal slip.
The mechanical equation is modified by using rated mechanical torque as base: (19) where T ag,tot is the total acceleration time constant in s defined using rated machine parameters, t e and t m indicate electrical and mechanical torque in pu.Assuming a linear relationship between torque and power output of the machine represented by p e and p m (19) is modified into: Using (20) and measuring the change in frequency at the wind turbine bus, the inertial response from Type 1/Type 2 WTGs is calculated and used to obtain the total power imbalance immediately following the disturbance.

D. Contribution From Non-Synchronous Fast Frequency Reserves to System Disturbance
Electrically decoupled wind turbines such as Type 3/Type 4 WTGs and large-scale solar PV systems with no inherent inertial response can also participate in frequency support either by virtual inertia emulation or active power control.An emulated inertial response can be supplied from variable speed WTGs, either through overproduction or deloading scheme, whereas emulating fast frequency response from LSPV systems is mainly Fig. 2. Droop based frequency support scheme [26].done using hybrid PV-Energy Storage Systems.Likewise, in active power control, these wind turbines and solar PV can curtail their output power to provide power reserve at the expense of loss of profits.
In this study, to consider the contribution from nonsynchronous power sources, a droop-based frequency control scheme, as shown in Fig. 2, is considered to regulate the active power control output from a WTG/LSPV proportional to the frequency variations [32].In Fig. 2. R is the droop constant, Δf is the measured frequency deviation, and ΔP IR is the power contribution from the non-synchronous sources towards the total power imbalance following the system disturbance.
This droop controller adequately improves the frequency recovery process after the disturbance where the active power is adjusted in accordance with its linear characteristics.
A flow chart of the proposed method to estimate the size of disturbance is shown in Fig. 3, indicating that the total disturbance size is obtained considering the individual contributions from synchronous generators, frequency and voltage-dependent loads, and renewable energy-based sources integrated into the power system.
The disturbance size estimation method is executed in a system through the following steps: Step 1: Collect the required measurements from the selected PMU nodes/generator terminals immediately after the disturbance.
Step 2: From the measurements, obtain the Y bus matrix and calculate the SPC values.
Step 3: Using the SPC values calculate the distribution of active power imbalance at all SG terminals.Step 4: Calculate the active power changes from remaining sources as explained in Sections III-B to D.
Step 5: Combine the calculated individual active power contributions to estimate the total power imbalance.

E. Frequency Nadir Prediction Using the Estimated Power Disturbance
Having determined the size of the perturbation, the final step of the proposed method is to predict the frequency nadir immediately following the disturbance.Frequency nadir prediction has been discussed in the literature [33], [34].However, use of such methods for RE-dominated low-inertia grids is not well explored.In the proposed method, a neural network (NN) is employed to predict the frequency nadir using the estimated size of the disturbance.NN is used because it provides more  accurate and reliable estimates than the analytical approaches for solving non-linear power system problems.
Fig. 4 represents the structure of the proposed NN-based frequency nadir prediction model.It takes two inputs: (a) the disturbance size estimated using stage 1 of the proposed method and (b) the kinetic energy of the system, which is assumed to be available with the TSOs.There is one hidden layer with ten neurons, which introduces nonlinear computation.The Sigmoid function is used as the activation function for the hidden layer, whereas output layer uses a linear function.
One of the concerns for training an NN-based nadir prediction model is lack of sufficient data from real power system disturbances.However, using the detailed simulation models of the power systems available with the TSOs, the dataset can be generated under various operating conditions for several possible system contingencies.This dataset can then be used to train the NN model.Once the model is trained offline, it can be used for further applications in the power grid.In this study, extensive simulations are performed on both test systems to generate the data, which is then used to train the proposed NN model using nntool by MATLAB.

IV. RESULTS AND DISCUSSIONS
The effectiveness of the proposed method is investigated in this section using five test cases: in the IEEE 39 bus test system and the Gujarat state grid.Two scenarios named "Load Change event" and "Generator Outage event" are used for the study.The former scenario represents a load change of 40 MW at a selected bus in the IEEE 39 bus system, and the latter refers to different generator outages in both test systems.In all test cases, power measurements are made using a limited number of PMUs placed at selected generator buses.The performance metric used for evaluating the proposed estimation method is percentage error (E % ) which is defined as follows: where P imb act. is the actual power imbalance observed in the system and P imb est. is the estimated power imbalance using the proposed method.
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A. Estimation of Disturbance Size in IEEE 39 Bus System
In this subsection, three case studies that are carried out for both scenarios of the load change event and the generator outage event are explained.
1) Case Study A.1: In this example, a load change event is created for the study.A step load increase of 40 MW is applied at bus 12, bus 15, and bus 38.PMUs are placed at various generator buses to measure power changes following the scenario.Table I shows the of the proposed estimation method applied for the load change event at bus 12.The values of SPC and Generator Disturbance Distribution Matrix (GDDM) [26] used in the calculation of active power imbalance and the estimated values of active power imbalance are given in Table I.The actual power imbalance observed in the system during the studied scenario is 40.23 MW.In contrast, the estimated value of active power imbalance using PMU measurements taken from all ten generators is 39.74 MW with E % = 1.21.
The summary of case A.1 results is given in Table II, which compares the E % metric for all three instances of simulated load change events at buses 12,15, and 38 for both the conventional SPC method from the literature [26] and the proposed modified SPC method using dynamic Y bus .In this case study, all ten generators are assumed to be equipped with PMUs.The power change measurements at all generators are used to calculate the size of disturbance.Load contribution towards the power imbalance is assumed to be negligible in this case study.
2) Case Study A.2: Another set of case studies is carried out to assess the performance of the proposed method over the conventional SPC method for different generator outage event scenarios.When the power system experiences a generation outage, the power deficit created in the system, equal to the power output of the tripped generator, is distributed among the remaining generators.However, under such a scenario, the system topology and SPC values are different before and after the disturbance.
In the conventional SPC method, change in SPC values due to the system dynamics is not accommodated.Instead, a slight alteration is introduced in the estimation method.This alteration uses observations from initial simulations, which showed that the use of pre-disturbance SPC values results in the calculated size of disturbance being almost double the active power output of the disconnected generator.Hence for a generator outage event, the estimated power using the conventional SPC method has to be divided by two to obtain the actual size of the power disturbance that occurred in the system [33].But this alteration did not provide a better disturbance size estimation under generator outage events.However, in the proposed modified estimation method, the values of SPC are calculated using dynamic Y bus , which incorporates system dynamics such as topology changes during the disturbances.These SPC values are then used to calculate the disturbance size of the system using the proposed method.Different generator outage events are simulated in IEEE 39 bus system, and the disturbance size is calculated using the conventional SPC and the modified SPC methods using data from three PMUs.The observations from Fig. 5 indicate that the modified SPC method gives a far better and more accurate estimation of disturbance size than its conventional counterpart.
3) Case Study A.3: In this case study, the effectiveness of the proposed estimation method for different numbers of PMU monitoring nodes is evaluated.Loss of generation of G4 with a power dispatch of 632 MW is simulated, and voltage and current at all nodes are measured for calculating Y bus which in turn is used to calculate SPC values.The total power mismatch in the system is then estimated by the proposed modified SPC method using power change measurements from PMUs placed at generator terminals.The E % metric calculated for various combinations of monitored PMUs in the estimation method is shown in Table III.The results of case study A.3 show that PMU placement for power output measurement at generator terminals influences the accuracy of the size estimation using the modified SPC method.By monitoring generators that are in closer proximity to the disturbance location, estimation error can be reduced using a limited number of monitoring nodes.On the contrary, using PMUs at all generator terminals provides the best accuracy estimate of disturbance size, as seen in Table III.

B. Estimation of Disturbance Size in Gujarat Power Grid
To prove the robustness and efficiency of the proposed method in a practical power system, we use the model of a real-life Gujarat grid in India, a large-scale real power system with a total installed capacity of 41 GW.Similar to the previous case studies, both scenarios of the load change and generator outage events are carried out, and the size of disturbance is estimated using the proposed method.It is essential to mention that the load contribution and Type 1/Type 2 WTG inertial contribution towards the disturbance size estimation described in Sections II-I-B and C are considered in case studies using the Gujarat grid.
1) Case Study B.1: This case study compares the error in estimation of disturbance size obtained for load change scenarios simulated at different locations with different disturbance sizes for the proposed estimation method with a varying number of PMU measurement nodes.Table IV gives a summary of obtained results.
The accuracy and precision of disturbance power size estimation improve with an increased number of PMUs for measuring the power variations at generator terminals.In the cases where a limited number of PMUs are used, the monitoring points are selected so that it is distributed evenly across the area of the power system, i.e., the cases with the number of PMUs as 10 and 25.It can be observed from the results that by selecting evenly spread generators, the performance of the proposed estimation method is improved significantly.

2) Case Study B.2:
In this case study, the proposed estimation method is evaluated considering the outage of two different generators.Loss of generation of CGPL Mundra, the biggest generator in the Gujarat grid with a power dispatch of 2500 MW, and Wanakbori Thermal Power Station (TPS), with a power output of 800 MW located almost at the center of the Gujarat grid, are simulated for the analysis.Table V represents the obtained results of disturbance size estimation.
The RV method explained in Section III-B has been used to determine changes in the loads due to frequency and voltage dependency following a generation loss.The inertial response from Type-1 and Type-2 WTG systems is calculated using the Moment of Inertia (MOI) relation explained in Section III-C.Also, the contribution from non-synchronous fast frequency reserves to the system disturbance is evaluated as discussed in Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Section III-D.Furthermore, the SPC method modified using a dynamic admittance matrix estimates power change contribution from synchronous generators.Combined, all these calculated power changes give an accurate estimate of the active power disturbance size due to the system event.The proposed estimation method significantly reduces the error in the calculation of disturbance size, as evident from the results of various case studies.

C. Results and Discussion on Prediction of Frequency Nadir
This section discusses the implementation of the proposed frequency nadir prediction method explained in Section III-E.A dataset generated from extensive simulations on the IEEE 39 bus system and the Gujarat power system is used to train the network.The total input and target data are normalized and randomly divided into 70%, 15%, and 15% for training, validation, and test purpose, respectively.The training of NN is carried out using backpropagation, which uses the Levenberg Marquardt optimization method [36].The stopping criteria used for the model training are (i) minimum performance gradient: 10-7, (ii) maximum training epochs: 1000, and (iii) maximum validation failures: 6.The performance measure used here is the Mean-Square Error (MSE).
The NN model is trained offline and can be made available online for immediate use following system events.The time it takes to predict the frequency nadir is negligible provided the inputs are immediately available.As this proposed method uses the measurements within a short time interval immediately after the event occurrence, the prediction accuracy is better than the prediction of nadir using only steady-state values.Furthermore, a comparative analysis is carried out to check the performance of the proposed method as opposed to a frequency nadir prediction method presented in [33].The proposed method gives more accurate predictions, as evident from Table VI.

D. Application of the Proposed Method in a Real Power System: Using Actual PMU Data From the Gujarat Power Grid
The proposed method is validated using real-life PMU measurements along with SCADA data from the Gujarat state TSOs.The data of ten severe contingencies that occurred in the state between 1st April 2021 and 26th March 2022 has been used for this study.
The proposed method is validated and tested against the following two significant events.1) A line tripping event very close to an 800 MW capacity power plant on 27th April 2021: 800 MW generation loss 2) A 132 kV substation tripping event on 17th July 2021: 338 MW generation loss The frequency, RoCoF, voltage, active power, and reactive power data from all the online PMUs during the selected contingencies were obtained and used to validate the proposed method.First, the disturbance size is estimated using field measurements.For this the step-by-step procedure described in Section III is followed.The estimated size is then used to predict the frequency nadir (FN) using the trained NN model.It is essential to mention that the NN model is trained based on the extensive simulations carried out for the Gujarat grid model and not using real-time data.The predicted frequency nadir is compared with the measured frequency nadir, and the results are shown in Table VII.
The difference in predicted and actual frequency nadir is slightly higher compared to the better and more accurate prediction observed in the simulation-based case studies (Table VI).However, this could be attributed to the lack of adequate required real-time power measurements from the RE-based sources in the Gujarat grid that affected the disturbance size estimation.
Limitations of the proposed method and future works: r The proposed estimation method depends on active power measurements from the power system (PMUs at various monitoring nodes).Hence, any measurement errors will affect the accuracy of the disturbance size estimation and Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.therefore, the prediction of frequency nadir.Nevertheless, the proposed method results in smaller errors in the estimation with a limited number of monitoring nodes than the traditional swing equation based or SPC method with more monitoring nodes.
r The PMU locations for measurements play an important role in estimating the disturbance size.It is observed from the case studies and the results that selecting monitoring nodes (generators) that are evenly spread across the power system gives a more accurate estimate.Therefore, selecting suitable PMU locations will improve the proposal's performance and is a matter for future work.

V. CONCLUSION
This paper proposes an approach for fast and accurate online disturbance size estimation based on a limited set of PMU measurements.The modifications are made in the conventional SPC method to improve the accuracy of the estimate.At the outset, the method is applied to estimate disturbance size on IEEE 39 bus system with only synchronous generations.Subsequently, various RE sources, such as wind turbines and large-scale PV systems, are integrated into the relatively larger and real Gujarat State power grid to perform further estimation studies.The comparison of results obtained using the proposed modified SPC method using dynamic Y bus with the conventional SPC method showed that the proposed method provides a more reliable and accurate online estimation of disturbance size.Furthermore, the estimated disturbance size is used for predicting frequency nadir using an NN-based method.The overall process of online estimation of the size of disturbance with the prediction of maximum frequency deviation for any credible contingency can be used for designing advanced frequency control as well as UFLS schemes for a low inertia power system.

Fig. 4 .
Fig. 4. Structure of NN used with two inputs and 10 hidden layers.

TABLE I RESULTS
FOR CASE STUDY A.1: INCREASE OF LOAD AT BUS 12 BY 40 MW TABLE II RESULTS FOR CASE STUDY A.1: COMPARISON OF E % FOR LOAD CHANGE EVENTS AT DIFFERENT BUSES

TABLE III RESULTS
FOR CASE STUDY A.3: COMPARISON OF E % FOR VARIOUS SET OF PMUS

TABLE IV RESULTS
FOR CASE STUDY B.1: COMPARISON OF E % FOR VARIOUS DISTURBANCES WITH VARIED PMU SETS

TABLE V RESULTS
FOR CASE STUDY B.2: COMPARISON OF E % FOR VARIED DISTURBANCES Table VI shows the results of frequency nadir prediction for various test cases.The studies are performed in both the test systems for load change and generation outage events.

TABLE VI RESULTS
FOR FREQUENCY NADIR PREDICTION: COMPARISON OF ERROR IN PREDICTION OF FREQUENCY NADIR

TABLE VII VALIDATION
STUDY RESULTS: COMPARISON OF FREQUENCY NADIR VALUES IN GUJARAT GRID (F N act.: actual FN, FN pre.: Predicted FN)