Abstract
With the rapid integration of renewable energy, wide-band oscillations caused by interactions between power electronic equipment and grids have emerged as one of the most critical stability issues. Existing methods are usually studied for local power systems with around one hundred nodes. However, for a large-scale power system with tens of thousands of nodes, the dimension of transfer function matrix or the order of characteristic equation is much higher. In this case, the existing methods such as eigenvalue analysis method and impedance-based method have difficulty in computation and are thus hard to utilize in practice. To fill this gap, this paper proposes a novel method named the smallest eigenvalues based logarithmic derivative (SELD) method. It obtains the dominant oscillation modes by the logarithmic derivative of the k-smallest eigenvalue curves of the sparse extended nodal admittance matrix (NAM). An oscillatory stability analysis tool is further developed based on this method. The effectiveness of the method and the tool is validated through a local power system as well as a large-scale power system.
OSILLATION issues are of great concern in modern power systems with massive integration of renewable energy [
The time-domain methods mainly include the eigenvalue analysis and electromagnetic transient (EMT) simulation [
The widely recognized frequency-domain methods are based on the impedance concept, for instance, the source-load method [
To solve the problem of oscillatory stability analysis in practical large-scale power systems, this paper proposes a novel method named the smallest eigenvalues based logarithmic derivative (SELD) method. It has the following features:
1) For the modeling of large-scale power systems, the frequency-coupled admittance network model (FCANM) represented with sparse extended NAM is used to improve the performance in storage.
2) The implicit restarted Arnoldi method (IRAM) is applied to calculate the k-smallest eigenvalues of the sparse matrix with the great improvement in time complexity of computation, thus making the proposed method applicable to extremely large-scale power systems.
3) A segmentation function based logarithmic derivative criterion is proposed to identify the dominant oscillation modes based on the curves of the k-smallest eigenvalues.
An oscillatory stability analysis tool is further developed to implement the proposed method for an actual large-scale power system. The effectiveness of the proposed method and tool is verified in a local power system and a large-scale power system, respectively.
The rest of the paper is organized as follows: Section II introduces the problem of oscillation analysis in a practical large-scale power system. Section III proposes the SELD method for analyzing the oscillatory stability of large-scale power systems and develops an oscillation stability analysis tool. Section IV validates the effectiveness of the method and tool in two practical systems with different sizes. Section V gives a brief conclusion.
The oscillatory stability of a practical large-scale power system in China Southern Power Grid is analyzed. The diagram of the large-scale power system is shown in

Fig. 1 Diagram of a large-scale power system.
The order of the characteristic equation becomes extremely high due to the large number of electrical equipment. The total numbers of nodes and branches are 14647 and 20309, respectively. There are approximately 6872 transmission lines, 11804 transformers, and 1459 thermal units or synchronous condensers in this system. Besides, there are around 86 wind farms or photovoltaic power plants in the system. The total capacity of renewable energy sources is about 18000 MVA. The transmission lines are considered as a -equivalence model, which is a three-order circuit. The transformers can be equated to the inductors. The thermal units or synchronous condensers can be modeled as an 18-order state space model. For simplification, each wind farm is aggregated as the wind turbine generators (WTGs) in parallel connection. The order of the WTG model is 8 at least [
For the eigenvalue analysis method, according to the estimated order of the characteristic equation, the dimension of matrix A of the state space equation is approximately 1
Conventional impedance-based methods analyze the stability by the closed-loop transfer function matrix. There are mainly two schemes. The first scheme uses the stability theory of the multiple-input multiple-output (MIMO) system [
(1) |
(2) |
where i is the node number chosen to aggregate; and are the voltage and current with coupled frequencies of node i, respectively; and subscript ii means the row and the column of the matrix by 2×2 partition.
However, both schemes have difficulties in computation. For the first one, the algorithm of determinant calculation is normally based on the LU decomposition. The determinant is the production of diagonal elements of L and U, where L and U are the decomposed lower and upper triangular matrices, respectively. The LU decomposition is based on Gaussian elimination. The transformation of the original matrix into an upper triangular matrix via the Gaussian elimination needs O(
Therefore, it is necessary to develop a method with high computational efficiency to analyze oscillatory stability for extremely large-scale power systems.
According to the impedance-based method, each component in a power system can be modeled as an FCAM represented by a 2×2 matrix [
(3) |
where , , , and are the small-signal voltage and current phasors of two coupled frequencies fp and 2f1-fp, respectively, and f1 and fp are the fundamental and non-fundamental frequencies, respectively; superscript means the conjugation of phasor; is the angular frequency of fp; and Yij is the element of FCAM.
The FCANM can be described by the extended NAM . Then, the network is expressed as:
(4) |
where is the nodal voltage vector; and is the nodal current vector.
The FCANM can be represented by:
(5) |
(6) |
where is the diagonal matrix with each diagonal element representing the FCAM; is the extended node-branch incidence matrix; I2 is a two-dimensional identity matrix; denotes the Kronecker product; and D is the node-branch incidence matrix generated by the system topology.
The huge number of AC nodes in a large-scale power system would result in problems in storage and calculation of FCANM. The cost of the matrix multiplication in (5) is FLOPs, where is the number of nodes; b is the number of branches; and nf is the number of frequency points. It is extremely time-consuming for a large-scale power system.
The sparse matrix technique is very effective in addressing this issue. Similar to the power flow calculation, there are very few non-zero elements in (e.g., 0.1% non-zeros elements for a practical system). Besides, is a diagonal matrix. Therefore, both the matrices are sparse.
In (5), the operation of the zero elements in the matrix is not necessary. Meanwhile, the storage of these zero elements is also redundant. The sparse matrix uses three vectors u, v, w to store and operate the non-zero elements in the matrix only, where u, v, and w are the vectors of horizontal coordinates, vertical coordinates, and values of non-zero elements, respectively. As a result, the massive storage and calculation of zero elements are eliminated. Using sparse matrix can accelerate the calculation by tens or even hundreds of times. The memory requirement can also be greatly reduced. The larger the scale of the power system, the more significant the benefits. Therefore, the computational efficiency in constructing FCANM is highly improved by the sparse extended NAM, which is the extended NAM with sparse matrix representation.
To improve the computational efficiency, the k-smallest eigenvalues of the matrix rather than the determinant or the inverse of the matrix is used, where is at any . The k-smallest eigenvalues are the k eigenvalues with the smallest modulus. By using them, the computational efficiency of oscillatory stability analysis is improved from O(
The SVD of is given by:
(7) |
(8) |
where and T are the orthogonal matrices, and the rows of and the columns of T are the eigenvectors; and is the eigenvalue.
For on the diagonal of , these exists:
(9) |
where xi is the eigenvector of .
The modal impedance is defined as . Each peak value of the modal impedance is usually corresponding to an oscillatory mode [
The IRAM is applied to calculate the k-smallest eigenvalues of the n-dimensional matrix A. The key idea of IRAM is to decrease the dimension. The specific process is introduced as follows.
Define the Krylov subspace of the n-dimensional matrix A as:
(10) |
where x is a n-dimensional vector; m is an integer; and is the set of all possible vectors with linear combinations of the vectors.
The Arnoldi process is an efficient way to reduce the dimension of matrix A, thereby improving the computational efficiency. It transfers the n-dimensional matrix A to a m-dimensional matrix Hm by Krylov subspace, where .
The first step of the Arnoldi process is to find the orthonormal basis (11) of the Krylov subspace (10). The Gram-Schmidt process is used in this step, i.e.,
(11) |
(12) |
(13) |
where hj,i is the Gram-Schmidt coefficient.
(14) |
The Gram-Schmidt coefficients form the upper Hessenberg matrix .
(15) |
(16) |
where .
The transformation of Arnoldi process can be written as:
(17) |
where is without the last row; and .
(18) |
(19) |
According to (18), the eigenvalues and eigenvectors of are calculated to approximate the eigenvalues and eigenvectors of A. This approximation holds when the term is small or:
(20) |
where is the tolerance; and is the
In the Arnoldi process, the restriction (20) does not usually hold. Therefore, the restart process is used to iterate the initial value of the algorithm until (20) holds. The IRAM improves the Arnoldi process by adding the restart process and using the implicit QR decomposition.
The focus is on the k-smallest eigenvalues, while leaving eigenvalues ignored. The Arnoldi process begins with a unit vector v and computes the m step Arnoldi decomposition (13). Then, the eigenvalues and eigenvectors of are calculated by the reduction to the upper Hessenberg form. The eigenvalues are sorted from the largest to the smallest by the magnitude. The largest eigenvalues are used as a shift to enhance the calculation of the required k-smallest eigenvalues. The restart is chosen by the filtering polynomial pr(Hm) in (21). The pr(Hm) shifts away from the undesirable m-k eigenvalues, as opposed to shifts closer to the desirable k eigenvalues.
(21) |
where I is the m-dimensional identity matrix.
Apply the following implicit QR decomposition to these eigenvalues.
(22) |
where is the orthogonal matrix; and is the upper triangle matrix.
The Hm is iterated to by:
(23) |
(24) |
Then, the restart initial vector is derived as [
(25) |
where is the column of Qm in the previous iteration; is the element at the row and column of ; is the previous initial vector; and is the element at the
We use as the new initial vector and apply the Arnoldi process to obtain the eigenvalues and eigenvectors. If (20) holds for the k-smallest eigenvalues, return the k-smallest eigenvalues. Otherwise, repeat the above restart process.
Since the Arnoldi process uses the Gram-Schmidt process, the approximate number of FLOPs for the algorithm is [
D. Identification of Dominant Oscillation Modes by Segmentation Function Based Logarithmic Derivative Criterion
The dominant oscillation modes of a power system are identified by the logarithmic derivative of the k-smallest eigenvalue curves. Due to the utilization of the small-signal linearized model, the general form of eigenvalue curves takes the following expression:
(26) |
where and are the zeros and poles of the , respectively; N and M are the numbers of zeros and poles, respectively; and K is a constant coefficient.
The
The logarithmic derivative of a function is defined as the derivative of the logarithm of the function (27). Its differential form is given in (28), which is more pragmatic.
(27) |
(28) |
where is the derivative of .
The logarithmic derivative of in (26) is presented as:
(29) |
From (29), it can be observed that is the sum of minus the sum of . and have the same expression, as shown in (30). The real- and imaginary-part curves of are shown in
(30) |

Fig. 2 Real- and imaginary-part curves of . (a) Real-part curve. (b) Imaginary-part curve.
According to (30) and
1) The real part of the logarithmic derivative curve is larger only near the oscillation frequency and decays sharply at the inverse square rate away from . is the extreme point whose value of the real part of the logarithmic derivative is the negative inverse of the real part of zi.
(31) |
where is the divergence rate.
2) The imaginary part of the logarithmic derivative curve is approximately symmetric about the center of the point , and again has a large absolute value only near , while it decays at an inverse rate away from . As the frequency increases, the imaginary part of the logarithmic derivative curve rises, then falls, then rises again and has a negative slope at .
The properties of the real- and imaginary-part curves of the can be obtained by (29) and the properties of and . Suppose there is a dominant zero .
Firstly, since the real part of decays sharply at the inverse square rate away from , the influence of other zero-poles on the real part of can be approximately neglected near the frequency of the dominant zero zi.
Secondly, due to that the real-part curve has an extreme value on , the stationary points of the real part of can be used to identify the dominated zeros.
Thirdly, based on the surge of the imaginary-part curve at , the slope of the imaginary part of the logarithmic derivative curve can be used to distinguish the zeros from the poles. The slope of the imaginary part of the zero (pole) term at the frequency is negative (positive).
Finally, the real part of the logarithmic derivative is solved for the real part of the zero-pole based on the extreme value of the real part of the logarithmic derivative [
According to the properties of the real- and imaginary-part curves of , the logarithmic derivative criterion is given as follows.
If exhibits as a stationary point at the oscillation frequency and the slope of at is negative, there exists a zero , and the real part can be solved by:
(32) |
Examine all the identified zeros. If there is a zero with a positive real part, it corresponds to an unstable oscillation mode. Conversely, if the real parts of all the obtained zeros are less than 0, the system is considered stable.
For those non-differentiable points, both the real- and imaginary-part curves charge abruptly at this point, thus the slope cannot be calculated, and the above criterion can ignore those points.
A platform of data collection and its interfaces are developed. This platform collects the original data of the power system. The original data include both the static and dynamic data. The static data include the parameters of transmission lines, WTGs, thermal units, transformers, etc. The dynamic data include wind speed, system load, operating condition of power system, etc. These data are transformed into standard data files, which are the inputs of the tool.
The process of the oscillatory stability analysis tool is given in

Fig. 3 Process of oscillatory stability analysis tool.
The algorithm of the proposed method in the oscillatory stability analysis tool is given in
Algorithm 1 : proposed SELD method |
---|
Input: data collected by the platform. |
1. Initialize the array of “branch” and “node” by the data. The parameters of the equipment xj (), initial value of power flow, and node-branch incidence matrix D are acquired. |
2. The nodal Ui, δi, Pi, Qi () are calculated by the power flow analysis. |
3. The FCAMs Yj of each branch are obtained by the parameters xj and the nodal Ui, δi, Pi, Qi. |
4. The sparse-matrix-based FCANM is constructed by (5). |
5. The k-smallest eigenvalue curves λl(ω) are obtained by IRAM, where . |
6. The dominant oscillation modes are identified by DL(λl(ω)) (28), (32). |
Output: the dominant oscillation modes of the system. |
A demonstrative example of two parallel RLC circuits, as shown in

Fig. 4 Demonstrative example of two parallel RLC circuits.
The NAM is derived as (33). The circuit parameters are: , H, F, , H, F, . The resonance modes are: and .
(33) |
The eigenvalue of is calculated. The two modal impedance curves 1 and 2 are obtained by calculating the inverse of absolute value of eigenvalues. The extreme points of modal impedance curves indicate the oscillation modes. The curve of the largest modal impedance is selected to contain the extreme points. The modes 1 and 2 are clearly identified by the peak value of the curve of the largest modal impedance in

Fig. 5 Modal impedance curves.
The smallest eigenvalue curve has the expression of (26). The logarithmic derivative criterion is adopted to the smallest eigenvalue curve to obtain the oscillation modes. The logarithmic derivative curves obtained by the proposed method are shown in

Fig. 6 Logarithmic derivative curves obtained by proposed method.
Two practical systems are used to validate the effectiveness of the proposed method and the developed tool. A local power system is used to verify the correctness. The large-scale power system in
The simplified diagram of a real offshore wind-integrated local power system is shown in

Fig. 7 Simplified diagram of a real offshore wind-integrated local power system.
The WTGs in these wind farms are all the same type of direct-drive permanent magnet synchronous generator (PMSG), with a total capacity of 2734.2 MVA. The local load in this area is very small. The power from QZ, YY, and PT wind farms is collected to the 500 kV PL station via 220 kV transmission lines, and then to the EH2 station via a 500 kV transmission line (EH2-PL). The power from QZ6 wind farm is collected to the 500 kV EH1 station via several transmission lines. The EH1 station and EH2 station are connected to the external network equivalence via 500 kV transmission lines. This local power system is equipped with five thermal units, all of which have a rated capacity of 1112 MVA. The BH1-3 thermal power plant is equipped with three thermal units. The BH4 thermal power plant and the YX thermal power plant are each equipped with one thermal unit. The numbers of nodes and branches are 64 and 93, respectively, in this local power system.
The FCAMs of components in this power system are established based on the collected parameters. The FCAMs of the PMSGs are derived in [
To verify the effectiveness of the proposed method, eigenvalue analysis, zero-crossing stability criterion [
With the derived FCAMs of components, the s-domain FCANM is constructed by (5) and next transferred into state-space equations. Then, the eigenvalue analysis can be carried out. It is observed that there is an eigenvalue with a positive real part, . This indicates an unstable oscillation mode at the frequency of 46.23 Hz.
Apply the zero-crossing stability criterion to the extended NAM . Its real and imaginary parts with the frequency around 46 Hz are plotted in

Fig. 8 Real and imaginary parts of extended NAM.
Next, the proposed method is employed and the smallest eigenvalue is calculated by the IRAM. The real and imaginary parts of the logarithmic derivative within the frequency range of 45.5-47 Hz are plotted in

Fig. 9 Real and imaginary parts of logarithmic derivative within frequency range of 45.5-47 Hz.
There are several stationary points within the colored intervals of the real-part curve. Since the slope of the imaginary-part curve is positive in the blue interval, the stationary point corresponds to a pole of the transfer function. The stationary points in the green intervals are the non-differentiable points of the smallest eigenvalue and cannot be identified by the criterion in Section III-D. Since the slope of the imaginary-part curve is negative in the red interval, the stationary point corresponds to a dominant oscillation mode. By calculating the real part of this mode (32), an unstable mode is identified, and the oscillation frequency is 46.31 Hz.
Comparing the results obtained from the proposed method with the zero-crossing stability criterion, the relative error of the latter in the frequency is 0.15%. Such small relative errors demonstrate the accuracy of the proposed method, as the results closely align with those obtained from the zero-crossing stability criterion.
The proposed and existing methods were also validated in other scenarios, and the results verify the effectiveness of the proposed method as well.
The accuracy of the proposed method is further validated by comparing its results with commercial EMT simulation. When the output reactive power of the WTGs is increased to 0.3 p.u., sub-/super-synchronous oscillation is excited. The output active power of the WTGs is plotted in

Fig. 10 Output active power of WTGs.
The frequency spectrum of the output current in the sub-/super-synchronous band are presented in

Fig. 11 Frequency spectrum of output current in sub-/super-synchronous band.
The large-scale power system in
Firstly, the original data are collected using the dynamic simulation program (DSP) developed by Electric Power Research Institute of China Southern Power Grid. The data are converted into the standard format by our developed tool.
Secondly, the FCAMs are constructed by the data. Due to the large number and type of new energy units in the actual power system, it is difficult to obtain all the new energy unit models from the manufacturers. For simplicity, we use typical models of new energy units in the analysis. The direct-driven PMSGs and photovoltaic units are equivalent by the admittance model derived in [
Then, the extended node-branch incidence matrix is obtained by the system topology and stored by the sparse matrix. The diagonal matrix is obtained by the FCAMs of the branches and stored by the sparse matrix.
Finally, the FCANM is constructed by (5), and the k-smallest eigenvalue curves are obtained by IRAM .
The k-smallest eigenvalue curves are used to identify the dominant zeros by the improved logarithmic derivative criterion.

Fig. 12 Logarithmic derivative curves of the
The above results indicate that the oscillation analysis in the extremely large-scale power system is achievable. The total time consumed for the proposed method is 410.57 s.
As a comparison, the existing methods are applied to this large-scale power system. According to the analysis in Section II-B, the computational efficiency of both the eigenvalue analysis method and conventional impedance-based method is very low. The number of FLOPs of the eigenvalue analysis method is at least, where 1
The aforementioned computation was performed on an Intel Core i7-10700 CPU with a base frequency of 2.90 GHz, 16.0 GB RAM, and Windows 10 operating system.
This paper proposes an SELD method to analyze the oscillatory stability of the large-scale power systems. Compared with the existing methods, the proposed method has following advantages.
1) The sparse extended NAM based modeling method has better performance in storage than the conventional method in the modeling of large-scale power systems.
2) The k-smallest eigenvalues of the sparse extended NAM are used to represent the dynamics of the power system. The IRAM significantly improves the time complexity of computation from O(
3) The segmentation function based logarithmic derivative criterion is proposed to quickly and accurately identify the dominant oscillation modes based on the k-smallest eigenvalues.
The accuracy and computational efficiency of the proposed method and the developed oscillatory stability analysis tool are demonstrated through case studies on two practical systems. In the case of a local power system, the results of the proposed method are equivalent to those of the impedance-based method and EMT simulations. In the case of a large-scale power system with 14000 nodes, the total time consumed for the proposed method is 410.57 s. The proposed method has better computational efficiency over the eigenvalue analysis method and the conventional impedance-based method.
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