Ambient-Data-Driven Modal-Identification-Based Approach to Estimate the Inertia of an Interconnected Power System

A novel approach for estimating the inertia of an interconnected power system is presented using the identification of interarea oscillation modes (frequency, damping and mode shape) extracted from ambient data. The proposed method concentrates on estimating the values of the effective inertia of each area rather than the equivalent inertia of the entire system. Based on an equivalent two-machine system (ETmS) obtained by combining small signal stability analysis (SSSA) with the structure of the power grid, we derive a mathematical relationship between the effective inertia of each area and the interarea oscillation modes. Furthermore, the interarea oscillation modes can be extracted from ambient data, and the developed scheme enables an online estimation of the inertia only by using the outputs measured by PMUs. The performance of the proposed methodology is tested via numerical simulation cases and real data.

Inertia directly affects the response of a power system's frequency, power angle and other electrical quantities on the electromechanical time scale; it is also one of the important parameters that is used to measure the immunity of the system [1], [2]. Due to the high penetration of large-scale renewable sources connected to the system through power electronics, the inertia decreases [3], [4], and its distribution also changes [5], [6]. Therefore, monitoring and determining the inertia level and distribution are very important to ensure the security of a power system for a transmission system operator (TSO). VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ In modern power systems, elements such as the frequency dependence of the load and the power electronic interfaces for generation, storage and the load contribute to the effective inertial response of the system, besides conventional synchronous generators (SGs) [7], [8], and this characteristic will become even more prevalent in the future. The inertia is complex and strongly time-varying. Estimating the effective inertia using PMU is considered a viable technology. This study presents an approach for estimating the effective inertia of different areas of an interconnected power system from ambient PMU measurements.
Several techniques have been proposed for the estimation of inertia using PMU measurements. In the early stage, the estimation of the inertia of entire systems was developed and tested using the PMU measurements of frequency events [9]- [15]. Such methods are mainly based on the relationship between the unbalanced active power and the ratio of change of frequency (RoCoF) and depend on data recorded in a large disturbance. Typical research cases including the inertia of the Western Electricity Coordination Council (WECC) system [10] and the Great Britain (GB) power system [15] were estimated using recorded disturbances, and an improved online simulation was subsequently realized. In [16], the relationship between the equivalent inertia and the root of the swing equation is derived, so that the inertia can be estimated by the oscillation signal rather than the frequency deviation caused by sudden power mismatch. In [17], the inertia is estimated by the eigenvalues and eigenvectors extracted from the PMU-measured oscillation power and frequency signals.
However, the inertia estimated based on recorded data for a large disturbance only reflects the inertial response of the system when the disturbance occurs, which makes it difficult to achieve continuous online inertia estimation.
In an actual system, the PMU records a large amount of ambient data in real time. Moreover, previous studies have indicated that the ambient data contain rich electromechanical response information [18], [19]. Some scholars have carried out studies on ambient data-based inertia estimation methods. Based on the Laplace transform of the swing equation, the close-loop frame [20] and ARMAX model [21] were used to estimate the inertia based on ambient data of the active power and frequency deviation. In [22], the relationship between inertia and frequency was established based on a training algorithm, which examined the inertia estimation problem from a statistical point of view. Based on an aggregated power system model, the inertia estimation problem was built and described as a regression equation that can be solved by dynamic regressor extension and mixing [23]. In [24], the variant feature of the system inertia is expressed as a stochastic model according to the random behaviour of the power system, so that the inertia can be estimated by a likelihood function.
At present, most data-driven inertia estimation approaches have been developed on the basis of the relationship between the active power and frequency changes. As one of the most important electromechanical parameters of a power system, the inertia not only affects the frequency response but also influences other electromechanical behaviors of the system, such as electromechanical oscillation. The technical solutions for estimating the inertia based on the oscillation modes (frequency, damping and mode shape) extracted from the PMU have been proven to be feasible. However, the existing techniques using modal information are limited to post-event analysis using an equivalent system model and are unrealistic in terms of obtaining the effective inertia of different areas.
To address the above concerns, this paper develops a methodology for online estimation of the effective inertia of different areas of an interconnected power system by employing the values of interarea oscillation modes, i.e., modal frequencies, damping and mode shape, calculated from online ambient data using PMU. The main contribution of this paper and the presented method are as follows: 1) According to the results of small signal stability analysis (SSSA) and the cut set, an equivalent two-machine system (ETmS) is constructed; then, on the basis of existing research [25], we develop an expression for the ratio of the effective inertia of the different areas using the interarea modal eigenvalue and mode shape; 2) This paper builds an effective inertia estimation strategy that relies solely on ambient data for the interconnected power system; 3) Both simulations and real measured case studies confirm the effectiveness of the proposed effective inertia estimation scheme.
The remainder of this paper is organized as follows. Section 2 discusses the details of the proposed approach. Section 3 reviews recursive stochastic subspace identification (RSSI) and discusses an online application scheme for the proposed effective inertia estimation approach. Sections 4 and 5 present two cases to evaluate the performance of the proposed method. Section 6 presents the conclusions.

II. PROPOSED METHODOLOGY A. PROBLEM STATEMENT
For the interconnected power grid in Figure 1 (a), the system can be reduced to an ETmS, which is based on the definition of the generator group and the critical cut set (comprising multiple tie-lines), as shown in Figure 1 (b), where ASG1 and ASG2 are the aggregated synchronous generators (ASGs) of generator group 1 and generator group 2, respectively. Then, the dynamics of the ETmS can be described by the swing equation [26], expressed as where Linearizing (1) and (2) at the operation point and assuming P m,e = 0, the homogeneous differential equation with a state variable is formed and can be expressed as Subsequently, the model of the interconnected system, which is hiding in (3), can be obtained by the solution of the differential equation. According to modal analysis theory, the real and imagery parts of the solution correspond to the oscillation frequency and decaying coefficient of the electromechanical modal information, respectively. As a result, the relationship between the system parameters and the modal information of the electromechanical oscillation can be expressed as Although the interarea exchange power P e0 can be measured by PMU equipped at a tie-line, directly determining the equivalent power angle δ e is difficult. Because the determination of the equivalent power angle δ e based on network reduction, which is used in [27], has low adaptability for large systems, the determination of the equivalent power angle δ e based on the PMU measurements is developed in this paper. Figure 1 (c) is an equivalent circuit of the ETmS shown in Figure 1 (b), whereV i = V i θ i ,i = 1, 2 is the voltage vector at buses 1 and 2;İ i = I i α i , i = 1, 2 is the equivalent injection current vector of buses 1 and 2; x l is the equivalent reactance between bus 1 and bus 2; andĖ i = E i δ i , i = 1, 2 is the internal voltage vector of ASG1 and ASG2.
For the area where ASG i is located, it is assumed that there are M boundary buses connected to the critical cut set, and there are N lines connected to the boundary buses inside the area, as shown in Figure 2 (a). Thus, the equivalent injection current of bus i can be expressed as For ASG i, the phase relationship between the internal voltageĖ i , terminal voltageV i and injection currentİ i is shown in Figure 2 (b) [28]. Let E i = 1.0 p.u.; the following equation can be written using the sine rule for the triangle shown in Figure 2 The estimate of δ ASGi is generated by solving (1) at every time step. Considering that the ambient data used in this paper have a certain randomness, to reduce the estimation error, this paper uses the statistical mean in a fixed time period as the basis for the subsequent calculation.
Based on the determined power P e0 and angle δ e0 , the equivalent inertia H e can be estimated by extracting the electromechanical oscillation modal frequency and damping from PMU measurements. This idea is basically the same as that proposed in [27]; however, the method proposed in [27] mainly depends on the trajectory after a large disturbance to estimate the equivalent inertia and the effective inertia of the entire system.

B. QUANTIFICATION OF THE EFFECTIVE INERTIA OF DIFFERENT AREAS
It should be noted that the estimated inertia H e = H ASG1 H ASG2 (H ASG1 + H ASG2 ) using (4) and (5) is the equivalent inertia of the ETmS rather than the effective inertia of the area. H ASG1 and H ASG2 can be determined based on the quantitative relationship between them.
In [25], it was proven that the inertia can be estimated using the modal eigenvalue and its corresponding mode shape as In theory, the inertia can be estimated using (7) once the modal eigenvalues and mode shape are identified. However, it should be emphasized that in (7), the mode shape must be the element of the right eigenvector corresponding to the power angle. In practice, measuring the power angle is a difficult task, especially for the ETmS and ambient data used in this paper. Therefore, equation (7) is mainly employed to determine the quantitative relationship between H ASG1 VOLUME 8, 2020 and H ASG2 . Ignoring the equivalent damping of the ASG and assuming that the interarea oscillation modal eigenvalue and corresponding eigenvectors extracted by the average frequency of the two areas are λ and ϕ, respectively, it is known that K s,12 = K s, 21 for the ETmS according to the definition of the synchronous power coefficient. Thus, we can obtain the following: At this point, the effective inertia of ASG1 and ASG2 can be simultaneously estimated while extracting the modal information and initial power angle from the ambient data.

III. AMBIENT-DATA-DRIVEN INERTIA ESTIMATION SCHEME
The scheme is the basis for realizing an effective inertia online estimation to accurately extract the interarea electromechanical oscillation modes (frequency, damping and mode shape). In this section, the RSSI algorithm is briefly reviewed first, and then the ambient data-driven effective inertia estimation scheme is presented.

A. EXTRACTION OF OSCILLATION MODES BASED ON RSSI
As a mature system identification method, the stochastic subspace identification (SSI) algorithm has been successfully applied to ambient-data-driven electromechanical oscillation mode extraction. The covariance-driven stochastic subspace identification (SSI-COV) algorithm constructs the covariance matrix based primarily on observation data and then estimates the state matrix A and observation matrix C in the state space model shown in (9) on the basis of a singular value decomposition of the covariance matrix.
where x k is the system state variable, y k is the measurement, w k is process noise and v k is measurement noise. The oscillation frequency, damping and mode shape can be obtained by analyzing the eigenvalue of the following continuous time matrix [29]: where t is the sampling period of the measurement data. The traditional SSI algorithm is time consuming in single calculations, which is not suitable for an online application. Therefore, Prof. Vaithianathan Venkatasubramanian proposed the recursive adaptive subspace identification (RASSI) algorithm [30], which increases computational efficiency and realizes online tracking of the oscillation modes.  number of PMUs will be installed in future power grids. In the presented scheme, ambient data of the voltage, current and frequency measured by the PMUs are used to obtain the effective inertia.
Before performing the scheme for the effective inertia estimation based on ambient data, it is necessary to construct the ETmS according to the results of an SSSA of the operational structure and power flow and determine the required measurements and the area where the effective inertia can be estimated. Then, the relative effective inertia of different areas can be estimated using the following steps.
1) Use the measured ambient data of the voltage and current of the boundary buses connected to the critical cut set to estimate the equivalent power angle δ ASGi and extract the modes from the measured ambient data of the frequency of each area. This provides the key intermediate parameters for the inertia estimation scheme.

2) Determine the quantitative relationship between H ASG1
and H ASG2 based on the extracted modal information. 3) Estimate the equivalent inertia H e using the equivalent power angle and modal frequency and damping extracted in step 1.

4) Calculate the area inertias H ASG1 and H ASG2 using the results of steps 2 and 3 combined with the equation
This concept is illustrated in Figure 3 for the estimation of the effective inertia of the relative area for any ETmS.

IV. NUMERICAL SIMULATIONS
In this section, the IEEE 16-generator system is considered to demonstrate the accuracy of the proposed scheme. The IEEE 16-generator system is a typical interconnected power system, and the system consists of five areas, as shown in Figure 4. Among these areas, area 3, area 4 and area 5 To simulate the ambient response, the Power System Toolbox (PST) is used to model the test system. In the established system, the generator is adopted the complex model, and the two typical exciters, i.e., IEEE ST1A and IEEE DC1, and power system stabilizers (PSS) are also considered. More detailed parameters and information on the configuration of the test system can be found in [31].

A. OSCILLATION MODE EXTRACTION AND INTERAREA EQUIVALENTS
The modal analysis based on the SSSA for the system with basic operating conditions (Opc) is carried out. Four electromechanical oscillation modes with different oscillation frequencies and damping ratios are obtained, as shown in Table 1. Correspondingly, the mode shapes calculated by SSSA are shown in Figure 5 (a). As seen from the mode shapes, for mode 1, the generators in area 1 and area 2 oscillate against the generators in area 3, area 4 and area 5, and for mode 2, the generators in area 1 mainly oscillate against the generators in area 2.
The model analysis for the IEEE 16-generator system network revealed that two critical cut sets are formed by the tie-line in the system. The first set is Cut set No. In the case of basic Opc, we use PST to obtain a continuous 3 min set of ambient data under the condition that the load in the system randomly fluctuates by 5% of the base value. The average frequency of the relative area is calculated by employing the weighted average method proposed in [21].
Using the per-unit values of the average frequency of the relative area as input signals, RASSI is applied to extract the interarea oscillation modes. The mode extraction results are shown in Table 1. It can be observed that the number of electromechanical modes extracted by RASSI from the ambient data is consistent with that of the model analysis. Moreover, the statistical index, i.e., the mean value and standard deviation, shows that the extracted modes concentrate around the theoretical value within a small range. Correspondingly, VOLUME 8, 2020 the interarea oscillation mode shape extracted by the RASSI algorithm is shown in Figure 5(b), from which it can be seen that the oscillation between the areas in each mode is consistent with the theoretical mode shape.

B. EFFECTIVE INERTIA ESTIMATION
On the basis of accurately extracting the electromechanical oscillation modes, the effective inertia estimation method presented in Section III. B is used to estimate H 1 , H 2 and H 345 . To demonstrate the accuracy of the estimation results, the sum of the inertia of each component is calculated, which is regarded as the actual value [22]. The estimation results are shown in Figure 6. In this paper, when using RASSI to extract the electromechanical oscillation modes, continuous data with a length of 30 s are used as the input window length, so the inertia cannot be estimated in the first 0.5 min. It can be observed from Figure 6 that the estimation results show random fluctuation characteristics. However, the fluctuation is around the actual value, marked as the dotted line, within a small range.
Base on the estimation results, the statistical analysis is carried out, as shown in Table 2. The statistical analysis shows that the mean values of the effective inertias H 1 and H 2 of area 1 and area 2 are 32.31 and 63.92, respectively, which are close to the theoretical values (30.67 and 65.48), and the standard deviations are small. The mean value of the estimated result of H 345 deviates from the theoretical value (108.95) by 15.57, which is larger than the deviation between the actual value and the estimated value of area 1 and area 2. The main reason is that area 3, area 4 and area 5 are equivalent  systems themselves, and the electrical distance between the three equivalent areas (area 3, area 4 and area 5) is large; it is difficult to aggregate the three equivalent areas (area 3, area 4 and area 5) into an equivalent generator.
To verify the adaptability and robustness of the proposed approach to the operation conditions and electromechanical oscillation modes, two operation conditions (Opc 2 and Opc 3) are constructed by changing the exchange power between the areas.
Furthermore, we obtain two sets of new modes. The electromechanical oscillation modal frequency and damping corresponding to the three operation conditions are shown in Figure 7. From the spider plot shown in Figure 7, it can be clearly seen that the electromechanical oscillation modes of the system are obviously different under the three operation conditions considered in this paper. Table 2 also lists the mean and standard deviation of the effective inertia estimation results corresponding to Opc2 and Opc3. Similar to the estimation results of the basic Opc, the estimation results of area 1 and area 2 under Opc2 and Opc3 are close to the actual value. It can be observed from Table 2 that for area 1 and area 2, the deviation between the mean and the theoretical values is within 8% of the theoretical value. The results show that the proposed method has a small deviation in the effective inertia estimation of the area in which the electrical connection is tight, and the proposed method has a strong robustness to the operating condition and electromechanical oscillation modes.
Limited by the randomness of the ambient data, there are a small number of values in the effective inertia estimation results that greatly deviate from the theoretical value in the continuous time period. However, this part of the value does not represent the actual estimation results. In practical applications, the statistical mean of the estimation results in a continuous time period can be taken as the expected effective inertia.

V. CASE STUDIES BASED ON REAL MEASUREMENTS
This section considers real measurements of a power grid in North China as an example to verify the validity and adaptability of the proposed method.
The simplified schematic of the system is shown in Figure 8. The DB power grid is relatively large and consists of three closely connected small grids. The NM power grid is an energy base that consists of four large thermal power plants and delivers power to the DB power grid through four 500 kV transmission lines. The transmission distance of 4 lines exceeds 800 km.
The SSSA results show that there is an interarea oscillation mode with a frequency of 0.45 Hz. This mode is mainly expressed in that all generators of the NM power grid oscillate against most generators of the DB power grid. Therefore, the system can be equivalent to an ETmS by using the above four 500 kV tie-lines as a critical cut set.
Based on the ambient data recorded using PMUs during a period of time, the effective inertia of the NM power grid and the DB power grid are estimated. During this time period, two generators with a rated capacity of 667 MVA, three generators with a rated capacity of 733 MVA and one generator with a rated capacity of 556 MVA are connected to the NM power grid. The frequency and active power measured by the PMU are shown in Figure 9. During normal operation, the power system continues to suffer from ambient excitation caused by the load behavior or other stochastic processes.  Thus, the measured frequency and power randomly fluctuate around the operation point.
Next, the effective inertia of the two power grids is estimated by the method proposed in this paper. The statistical VOLUME 8, 2020 results of the estimated values for 5 min are shown in Table 3. The statistical results show that the mean of the estimated effective inertia of the NM power grid is close to the theoretical inertia based on the synchronous generator in the area, while the deviation of the DB power grid is larger. The main reasons are as follows: 1) some generators in the DB power grid do not directly participate in the electromechanical oscillation mode, which is used to build the ETmS, and all the generators in the NM power grid are involved; 2) the motor-based industrial load in the NM power grid accounts for a very low proportion of the overall load. However, the DB power grid is the load center containing a large number of motors, which can also contribute to the effective inertia of the power grid.
The results of the estimation of the effective inertia of the NM power grid verify the effectiveness and feasibility of the proposed algorithm for practical systems and real measurements.

VI. CONCLUSIONS
Based on an in-depth study of the coupling relationship between inertia and electromechanical oscillation modes, an effective inertia estimation method based on ambient data is proposed in this paper. The scheme proposed in this paper relies only on the measured output of PMUs.
Test cases of the IEEE 16-generator system show that the proposed effective inertia estimation method can provide reliable estimation results. In contrast to most existing algorithms, the RoCoF of the center of inertia and the total active power deficit, which are very difficult to obtain in an actual system, are not required in the proposed scheme. An archived case with real measurement data shows that the proposed method can feasibly estimate the effective inertia of each area from ambient data. The focus of future research may investigate the effective inertias of systems with invasive power electronic devices and propose a targeted estimation method.
GUOWEI CAI received the B.S. and M.S. degrees in electrical engineering from Northeast Electric Power University, China, in 1990 and 1993, respectively, and the Ph.D. degree in electrical engineering from the Harbin Institute of Technology, Harbin, China, in 1999. He is currently a Professor of electrical engineering with Northeast Electric Power University. His research interests include power system stability analysis and control, and smart grid with renewable power generation.