Issue 
Sci. Tech. Energ. Transition
Volume 79, 2024
Decarbonizing Energy Systems: Smart Grid and Renewable Technologies



Article Number  47  
Number of page(s)  13  
DOI  https://doi.org/10.2516/stet/2024041  
Published online  05 August 2024 
Regular Article
Reduced order calculation method of subsynchronous oscillation mode of PMSG based wind farm connected with power grid based on Arnoldi method
School of Electrical Engineering, Shenyang University of Technology, Shenyang, Liaoning 110870, China
^{*} Corresponding author: 996545558@qq.com
Received:
28
February
2024
Accepted:
4
June
2024
Large scale wind turbine connected to the power grid is easy to induce subsynchronous oscillation. “Curse of dimensionality” will be encountered when the subsynchronous oscillation mode is analysed and evaluated based on the fullorder model of the wind farm. It is urgent to research the fast and effective analysis method of subsynchronous oscillation mode when the wind farm is connected to the power grid. Therefore, a reduced order calculation method of the subsynchronous oscillation mode of PMSG based wind farm connected with a power grid based on the Arnoldi method is proposed. This method can quickly obtain the eigenvalues corresponding to the subsynchronous oscillation mode through iterative approximation when PMSGbased wind farm connected to the power system. This can be used to analyze the subsynchronous oscillation of largescale wind farms connected to the power grid and provide an effective means for the evaluation of it. By changing system stability, collector network structure, and wind farm interconnection scale, the accuracy of the eigenvalues distribution between the fullorder system and the reduced order system are compared. Moreover, the reduced order and computation time are considered. The results present that the proposed method can quickly and accurately obtain the subsynchronous oscillation mode in the system.
Key words: PMSG / Subsynchronous oscillation / Eigenvalue analysis / Model order reduction / Arnoldi method
© The Author(s), published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Developing and utilizing wind power is an important step to help achieve decarbonization goals in the energy sector. The cumulative installed capacity of wind power in China has reached 330 million kilowatts by the end of 2021, meanwhile, it is expected to reach 2.4 billion kilowatts by the end of 2050, indicating a rapid development trend in wind power generation [1–3]. However, as the scale of wind power grid connections increases, sub synchronous oscillation accidents in wind power grid connections occur frequently [4–6]. In 2015, an SSO accident occurred at a direct drive wind farm in Hami, Xinjiang, when it was connected to the Xinjiang power grid via a longdistance nonseries compensation line, posing a serious threat to the safe operation of the power system [7–9].
The eigenvalue method is an effective method for analyzing the SSO problem of wind farm interconnection. However, due to the characteristics of wind farms such as a large number of units, diverse operating conditions, and high model orders, conducting eigenvalue analysis on the fullorder model of wind farms often encounters the problem of “curse of dimensionality”. Therefore, it is an inevitable trend to reduce the fullorder model reasonably. Common simplified methods include aggregation modeling and model reduction techniques.
Aggregation modeling aims to reduce the order of wind farm models by establishing single or multimachine equivalent models. The single machine equivalent model refers to equating the entire wind farm into one unit, with the research goal of obtaining equivalent unit parameters that can improve model accuracy. Common methods include the capacity weighting method [10–12] and the parameter identification method [13, 14]; however, when there are significant differences in the operating conditions of the fans onsite, the accuracy of the model is difficult to ensure. Multimachine equivalence refers to dividing the wind farm units into several clusters based on certain indicators and equating them into one unit [15–17], and characterizing the entire wind farm into the form of parallel connection of each equivalent unit; the focus of the study is on the determination of clustering indicators [18, 19]. Multimachine equivalence has good applicability for scenarios with complex working conditions, but as for large wind farms, obtaining clustering indicators for each unit and forming an aggregation model after changing the operating point will consume a lot of additional computing resources, even exceeding the feature value calculation of the detailed model.
Model reduction technique means using rigorously proven mathematical methods to simplify the state space matrix of the target system and obtain a reduced order model that meets the requirements of the problem [20]. Common reduction methods include but are not limited to Krylov subspace method, balanced truncation method, and orthogonal decomposition method. Compared to aggregation modeling, this type of method takes the state space equation of the full order system as the object, and the required indicators can be obtained from the state space matrix itself, eliminating the need for additional operations on wind turbines from the physical level. The balanced truncation method is used to obtain a reduced order wind farm model in [21, 22], which can maintain a lowfrequency response, for the subsequent design and tuning of relevant controllers. A simulation of doubly fed wind farms connected to DC transmission systems is constructed in [23]. The singular value decomposition method, balanced truncation method, Krylov dynamic subspace method, and related improved algorithms are used to reduce the order of wind farms of different scales. The effectiveness of the reduced order model is verified by comparing the retention of key characteristic values, frequency response, and timedomain simulation before and after the reduction. The study in [24] proposes a singular perturbation reduction method for directdrive wind farms based on dominance analysis, with the ultimate goal of reproducing the dynamic characteristics and subsynchronous oscillation characteristics of the fullorder system, to achieve the reduction of the directdrive wind farm system. The complete model of an invertertype distributed power source is divided into multiple time scale features and ignores fast dynamic order reduction [25]. In the field of wind farmconnected SSO, research on the reduction of state space equations is mostly based on the principle of state matrix similarity transformation, and there is little attention paid to the model reduction techniques [26–28]. Although the principle of similarity transformation solves the “curse of dimensionality” problem in eigenvalue analysis, it often assumes that the operating conditions of the units are similar. In addition, the units are all parallel to the outlet busbar, which makes it difficult to fully consider the differences in operating conditions of the units in the field. On the other hand, the differences in the collector network have significant limitations. The model reduction technique is a universal method for linear systems, which has no physical structural requirements for wind farms and is theoretically applicable to various wind farms with collector networks.
In summary, this article proposes a reduced order calculation method for PMSG wind farm connected SSO mode based on the Arnoldi method (subordinate to the Krylov subspace method). The Arnoldi method, as a partial eigenvalue calculation technique, has been used in traditional power systems to reduce the order and solve the key eigenvalues of lowfrequency oscillations and has successfully solved the problem of “curse of dimensionality” in lowfrequency oscillations [29–31]. This article modifies it to quickly calculate the current SSO mode in the scenario of a large PMSG wind farm connected to a weak grid. Firstly, this article establishes a fullorder state space model for wind farms. Secondly, the general process of model reduction technology and the basic principle of the Arnoldi method were elucidated, and their modification was made to address the SSO problem of wind farm networking. Finally, in the Matlab environment, a wind farm interconnection calculation system was constructed. By changing system stability, collector network structure, and wind farm interconnection scale, the accuracy of the eigenvalues distribution between the full order system and the reduced order system are compared. Moreover, the reduced order and computation time are considered. The results verified the effectiveness, applicability, and speed of the method.
2 Linear modeling of PMSG networked system
2.1 PMSG wind farm networking structure
The PMSG wind farm networking system is shown in Figure 1, which includes M parallel branches and a total of N PMSGs; each branch is composed of several units with the same parameters and control strategies, which are connected to the collector network through stepup transformers. Each branch is connected in parallel to the 35 kV busbar at the wind farm outlet and is boosted to 220 kV through a transformer at the wind farm outlet before being connected to the AC power grid through the line.
Figure 1 PMSGsbased wind farms interfaced with AC power grid. 
In Figure 1, L_{ck} (k = 1, 2,..., N) is the equivalent inductance of the collector line connected to the kth unit; L_{t} is the equivalent inductance of the generator transformer, the equivalent inductance of the transformer at the turbine end of each wind turbine unit is the same. L_{T} is the equivalent inductance of the wind farm stepup transformer; L_{g} is the equivalent inductance of the line, L_{sys} is the system equivalent inductance, C_{cp} is the grid connection capacitance, N_{j} is the serial number of the fan at the end of branch j, and N_{0} = 0 is recorded. Defining Short Circuit Ratio (SCR):(1)
where U_{sys} is the effective value of system voltage, ω_{g} is the grid angular frequency, and S_{w} is the installed capacity of the entire wind farm.
2.2 Linearization model of PMSG unit
The topology of PMSG is shown in Figure 2, where the subscripts abc and dq represent the coordinates of abc and dq, respectively. i_{sabc} is the terminal current, u_{sabc} is the terminal voltage, e_{abc} is the gridconnected voltage, i_{gabc} is the gridconnected current.
Figure 2 Model structure of PMSG. 
The permanent magnet wind power generation system consists of a wind turbine, a permanent magnet synchronous generator, a fourquadrant inverter, and a controller. The power generation system adopts maximum power point tracking and outputs the rotor speed reference value ω_{rref} based on the wind speed v_{w}.
The machineside inverter adopts rotor flux orientation control. the rotor angle θ_{r} provides a reference angle for the dq transformation. The error between the rotor speed ω_{r} and its reference value ω_{rref} is output as the stator daxis current reference value i_{sdref} through the PI link. The difference between the stator dq axis current i_{sdq} and their respective reference values are used to form a reference value for the stator port voltage u_{sdq} through the PI link. The grid side converter adopts gridconnected voltage vector control, with a dq transformation reference angle provided by the phaselocked angle θ_{PLL}, and its terminal voltage u_{t} determined by the external circuit. The error between the voltage of DC capacitor u_{dc} and its reference value u_{dcref} is output as the grid side daxis current reference value through the PI link. The grid side converter adopts grid connected voltage vector control, with a dq transformation reference angle provided by the phaselocked angle θ_{PLL}, and its terminal voltage u_{tabc} determined by the external circuit. The error between the voltage of DC capacitor u_{dc} and its reference value u_{dcref} is output as the grid side daxis current reference value i_{gdref} through the PI link.The qaxis current reference value of i_{sqref} is set to 0.
The dq axis current i_{gdq} of the grid side filtering inductor is subtracted from its respective reference values and then passed through the PI link to form a reference value e_{dq} for the internal electromotive force of the inverter. The output voltage reference values of the machine side and grid side inverters are modulated by PWM and output pulse signals to control the inverters on each side.
Linearize the PMSG model at a certain operating point to obtain its state space as shown in equation (2):(2)
In equation (2), the subscript k represents the serial number of the fan. Δu_{k} = [Δu_{tkx} Δu_{tky}] ^{T} is the incremental value of the input variable, which refers to the component of the PMSG terminal voltage disturbance in the common xy coordinate system. Δy_{k} = [Δe_{kx} Δe_{ky}] ^{T} is the incremental output variable, which refers to the xaxis and yaxis components of the potential disturbance inside the grid side converter. Δ x_{k} is a micro increment of the state variable used to characterize the small deviations of rotor speed, capacitor voltage, the inductor current, various filtering stages, and the control stages of the machine/grid side converter from their respective stable operating points, consisting of 20 elements. A_{k}∈R^{20×20}, B_{k}∈R^{20×2}, C_{k}∈R^{2×20}, D_{k}∈R^{2×2}: A single machine PMSG state space matrix, where R is a vector space composed of real vectors.
2.3 Collecting network model
Figure 3 shows the topology diagram of the collector network after being uniformly reduced to 220 kV. In Figure 3, u_{c} represents the highvoltage side voltage of the wind farm outlet transformer, i_{ck} represents the current flowing through L_{ck}, i_{Σ} represents the outlet current of the wind farm. In the xy coordinate system, the voltampere relationship of the network in Figure 3 is shown in equation (3). In the formula, the subscript xy represents the xy coordinate system, h is the cyclic variable, and Ω_{g} is the matrix reflecting the coupling relationship between the xaxis and yaxis. The expression is shown in equation (4) (4)
Figure 3 Circuit topology of collector. 
In the formula ω_{g} is the grid angular frequency. Based on equation (4), i_{kxy} is the state variable, u_{cxy} and e_{kxy} are inputs, and u_{tkxy} and i_{kxy} are outputs, establishing the state space of the collector network as follows:(5)
In equation (5), Δu_{c1} and Δu_{c2} are the input variables, Δy_{c1} and Δy_{c2} are the output variables, Δx_{c} is the state variable; A_{c}∈R^{2N×2N}, B_{c1}∈R^{2N×2N}, B_{c2}∈R^{2N×2}, C_{c}∈R^{4N×2N}, D_{c1}∈R^{4N×2N} and D_{c2}∈R^{4N×2} are the state space matrixes of the collector network.(6)
2.4 Linearization model for wind farms
Rewrite equations (2)–(7) in the following form:(7)
In the formula, Δx_{rk} is the vector composed of 18 other state variables in Δx_{k} except for i_{gkxy}. And C_{k2} is a 0 matrix with 2 rows and 2 columns. Considering that the current i_{gkxy} of the wind turbine is a linear combination of the current i_{kxy} of the collector network. Abbreviate equation (7) as(8)
In equation (7):(9)
Integrate equations (5)–(9) establishing the state space of the entire wind farm:(10)
In equation (10), Δu_{F} is the input variable, Δy_{F} is the output variable, Δx_{F} is the state variable. A_{F}∈R^{20N×20N}, B_{F}∈R^{20N×2}, C_{F}∈R^{2×20N}, D_{F}∈R^{2×2}.(11) (12)
2.5 Linearization model of the whole system
The power grid subsystem consists of L_{g}, L_{sys} and C_{cp}, and its dynamic behavior is described by equation (13):(13)
Equation (13) has a total of 4 orders, where Δu_{g} is the input variable, which refers to the sum of current disturbances at the wind farm grid connection points. Δy_{g} is the output variable of the power grid and the voltage disturbance of C_{cp.} Δx_{g} is the grid state vector composed of C_{cp} xy axis voltage and AC grid reactance xy axis current increment. A_{g}∈R^{4×4}, B_{g}∈R^{4×2}, C_{g}∈R^{2×4} and D_{g}∈R^{2×2} are state space matrixes of the power grid.
Combining equations (11)–(13), obtain the state space of the entire PMSG wind farm interconnection system at order 20N+4:(14) (15)
The formula, Δx_{sys} represents the increment of the entire system state variable. A_{sys}∈R^{(20N+4)×(20N+4)} is the overall system state matrix.
3 Order reduction calculation of SSO mode for PMSG wind farm interconnection
3.1 Principle of model reduction
The general idea of model reduction is shown as follows [30]:
Given an norder linear timeinvariant system with pinput and qoutput(16)
A, B, C, and D are the state space matrixes. x is its state variable, u and y are its input and output variables, respectively. Find the reduced transformation matrix W, V∈R^{n×r}. Make the system:(17)
are the state space matrixes of the reduced order system. R is the target order and r<n. are state variables for reduced order systems. are output variables for reduced order systems. If matrix V is a standard column orthogonal matrix, then W = V. For the reduced order system of equation (17), it is required that the dynamic characteristics of its dominant characteristic frequency band are close to the original system.
The central idea of the SSO mode reduction calculation for PMSG wind farm interconnection is to find the reduction transformation matrices which are W and V, ensuring the error between the SSO modes of the full order matrix A_{sys} and its reduction matrix Ã_{sys} is small enough. In practical application, in order to match the convergence of specific algorithms, corresponding preprocessing is often performed on Asys before order reduction.
3.2 Arnoldi method
The Arnoldi method was developed by American scholar W. E. Arnoldi proposed in reference [31], which can be used to approximately solve largescale subsets of eigenvalues. The following text will introduce the basic principle of this method and modify it to be applicable to the SSO problem of PMSG wind farm interconnection [32–35].
3.2.1 Arnoldi method basic steps
The central idea of the Arnoldi method is to construct a set of standard column orthogonal vectors v_{1}, v_{2}, …, v_{r} in a given Krylov subspace, and use V = [v_{1}, v_{2}, …, v_{r}] and V^{H} as the reduction transformation matrix. The specific steps for order reduction are as follows:
 1.
For the nth order objective matrix A, given any ndimensional nonzero column vector b_{1}, the first column base v_{1} in the reduced order transformation matrix V is obtained by unitizing it.
 2.
Calculate the number α+1 column of standard orthogonal vectors.
 3.
Organize the reduced order transformation matrix V = [v_{1} v_{2} … v_{r}].
 4.
The elements of reduced order system state space matrix Ã are as follows:
λ and x_{λ} represent the eigenvalues and corresponding eigenvectors of matrix A, respectively. I represents the identity matrix. For characteristic equations (A − λI)x_{λ} = 0, the approximate value of x_{λ} can be obtained in Krylov subspace span{v_{1}, v_{2}, …, v_{r}}, which is composed of v_{1}, v_{2}, …, v_{r}. If x_{r} = Vy_{r}, then (V^{H}AV)y_{r}−λ_{r}y_{r} = 0. Obviously, y_{r} is the reduced matrix Ã, corresponding to its characteristic eigenvector of characteristic value λ. By solving λ_{r}, then an approximate subset of the eigenvalues of the fullorder system can be obtained. As the order r increases, λ_{r} will gradually converge to the eigenvalues with the maximum and minimum modulus values in the original system [27].
3.2.2 Reorthogonal process
The Arnoldi method is theoretically correct, but in numerical calculations, the subsequently generated vectors will lose orthogonality with each other due to the accumulation of errors during iteration, causing the algorithm failure. Therefore, it is necessary to perform reorthogonalization on the vector generated by equation (18) to maintain the orthogonality with the previous vector [32]. Perform improved GramSchmidt orthogonalization on the vectors generated in equation (18) [33]:(20)
In the formula, μ is the loop variable. This cycle continues until the ratio of modulus values before and after the operation of b_{α} is less than a given threshold (with a value greater than 1) and ends. Afterwards, standardizing b_{α}, setting as v_{α+1} to proceed to steps 2) to 4). Meanwhile, reference [27] points out that in most cases, reorthogonalization can be effectively achieved by cycling once as shown in equation (20).
3.2.3 Order reduction calculation for SSO mode in PMSG wind farm interconnection
For the PMSG wind farm connected SSO mode, when using the Arnoldi method for order reduction calculation, the value of the reduction order r directly affects the accuracy of the SSO mode and the speed of the calculation process. However, the Arnoldi method does not provide a method for determining r. Therefore, the following text discusses how to determine the value of r and proposes a specific method for applying the Arnoldi method to reduce the order of PMSG wind farm connected SSO mode.
According to the properties of the Arnoldi method in Section 3.2.1, as the value of r increases, the SSO mode of the order reduction matrix Ã_{sys} will approximate the SSO mode of the full order matrix A_{sys}. Therefore, this article proposes an iterative method for determining the value of r, as follows:

Set an initial reduction order r_{0} and calculate the eigenvalues for each order as the step size gradually increases in order of Δr.

The geometric distance d between the SSO mode of the current order system and the SSO mode of the previous order system in the complex plane is used as the accuracy evaluation index, and this index is iteratively calculated.

If d is less than 10^{−3} in a certain iteration, it is considered to meet the accuracy, and the iteration will be stopped. The current r value is the target value, and the SSO mode at this time is the calculated result.
The above iterative method ensures the accuracy of the calculation results, but it is difficult to ensure the speed of the convergence process. According to Section 3.2.1, it is known that the eigenvalues of the reduced state matrix obtained through the Arnoldi method will first converge to the eigenvalues with the maximum and minimum modulus values in the original system, as the order r increases. For a PMSG wind farm connected system, the modulus of the SSO mode is much greater than that of the lowfrequency mechanical mode and much smaller than that of the highfrequency electromagnetic oscillation mode with faster attenuation. Therefore, in order to obtain an SSO pattern with high accuracy, the number of iteration times will increase and the calculation speed will decrease accordingly.
To ensure the speed of the calculation process, before applying the Arnoldi method, perform the following displacement transformation on the original state matrix A_{sys}:(21)
In the formula, is the transformed state matrix, which is a complex square matrix. λ_{s} is a complex number near the SSO mode, and I_{sys} is a unit matrix of the same order as A_{sys}. According to equation (21), it can be seen that the eigenvalues of matrix and the eigenvalues λ_{sys} of matrix A_{sys}, the following relationship exists:(22)
seting up λ_{s} = α_{s}+jω_{s}. If the eigenvalues of A_{sys}’ corresponding to the SSO mode of the full order system are λ_{sso}’, then λ_{sso}’ can be seen as the full order SSO mode moving left in the complex plane α_{s}, moving down ω_{s.} At this point, the SSO mode of A_{sys} becomes the eigenvalues with the minimum modulus, and the overall convergence process has directionality, improving the calculation speed.
After transforming A_{sys} into equation (21), determine r according to the iterative steps mentioned earlier, and obtain the SSO mode. But at this point, d is no longer the geometric distance between SSO patterns before and after each iteration, but the distance between the eigenvalues with the smallest modulus.
The initial process is as follows:

Calculate the average wind speed within the field and establish a single machine equivalent model;

Calculate the working points of the single machine equivalent model and unfold the state space As of the single machine equivalent model A_{s};

Calculate the eigenvalues of A_{s} and obtain its SSO mode λ_{s};

Determine the stable operating points of each unit based on the trend;

Expand the fullorder model at the working point to obtain the fullorder state matrix A_{sys}.
Now determine value of the parameters λ_{s}, r_{0} and Δr.

The value of λ_{s} can be selected from the SSO mode in a single machine equivalent network system formed by using the average wind speed. This value can roughly locate the SSO mode of the detailed model. And the order of the single machine equivalent model is only 24 orders, compared to detailed models with thousands of orders, the additional computational resources consumed can be almost ignored. In addition, in order to further reduce the complexity of operations, the influence of the collector network is ignored when forming a single machine model.

For r_{0} and Δr, considering that the order of the wind farm interconnection system model using single machine equivalence is 24 orders, and the model of a single PMSG unit is 20 orders, r_{0} and Δr can be taken as 24 and 20 respectively. In this way, increasing the order of each iteration is equivalent to increasing the equivalent number of fans.
In summary, the flowchart is shown in Figure 4.
Figure 4 Flow chart of SSO mode calculation method based on Arnoldi method. 
4 Example verification
Under the Windows 10 Professional 64bit operating system, with Intel Core i75500 CPU(2.40 GHz). Based on the MATLAB software platform, a wind farm interconnection calculation system is constructed, and the SSO mode reduction calculation method based on the Arnoldi method proposed is used to reduce the SSO mode of the wind farm interconnection system. By changing factors such as system stability, collector network structure, and wind farm interconnection scale, the accuracy, applicability, and computational speed of the proposed calculation method are verified.
4.1 Example 1: Validation of method effectiveness in stable/unstable states
4.1.1 Eigenvalue distribution of full order systems
Make the number of PMSG wind farms connected to the grid N = 100, and the number of convergence branches M = 10, each of branches include 10 units (as shown in Fig. 5), the unit parameters are shown in Table 1. To ensure that the complexity of the entire system state space matrix can be maximized within the wind speed range where SSO occurs frequently, and to maximize the effectiveness of the two reduction methods, this article selects the unit wind speed and the length of the collector line according to the following principles:

Randomly generate 100 different wind speeds in steps of 0.01 m/s between 4 and 8 m/s;

Randomly generate 100 collection line lengths in steps of 1 m between 400 and 800 m.
Figure 5 Connecting scheme of a 100PMSG wind farm. 
Parameters of PMSG.
The current fullorder system has a total of 2004 orders, and the distribution of all eigenvalues is shown in Figure 6(a); Figure 6(b) is a local magnification of Figure 6(a). At this time, the SSO mode is −1.9964 + j238.2858 and the system is stable. Regions 1–4 in Figure 6(a) represent four regions with concentrated distribution of eigenvalues

Region 1: Areas with real parts less than −3000;

Region 2: The region where the real part ranges from −1000 to −500;

Region 3: The region with real parts ranging from −500 to −100;

Region 4: Areas with real parts greater than −100.

For the sake of discussion, the following text will focus on the behavior of eigenvalues within region 4. Reducing k_{p5} to 0.09 while keeping other parameters unchanged to obtain the distribution map of eigenvalues in region 4 as shown in Figure 7. The SSO mode in Figure 7 is 4.0281+j237.5815, indicating that the system is instable.
Figure 6 Calculation results of 100PMSGssystem (stable). 
Figure 7 Eigenvalue distribution of 100PMSGssystem (unstable, area 4). 
4.1.2 Reduced order solving SSO mode
Firstly, solving the SSO mode in a steady state descending order. The SSO mode of the single machine equivalent model is −1.6874+j239.9005, and the iteration process is shown in Table 2; The comparison of eigenvalues in the full order and reduced order eigenvalue regions 4 is shown in Figure 8.
Figure 8 Comparison of calculation results of full order system and reduced order system (100 PMSGs, stable, area 4). 
Iterative process (100 PMSGs, stable).
According to Table 2, after 5 iterations, the SSO mode converges, and A_{sys} is of the 124th order. From Figure 8, it can be seen that the SSO mode is well maintained before and after order reduction.
Similarly, the reduction calculation process and results in unstable states can be obtained. At this time, the SSO mode of the single machine equivalent model is 4.3897 + j239.1500, and the iteration process is shown in Table 3. The comparison of eigenvalues in region 4 of the full order and reduced order systems is shown in Figure 9. According to the calculation results, the SSO mode that meets the accuracy requirements was still obtained at r = 124 in the unstable state. The results demonstrate the effectiveness of the proposed method in stable and unstable states.
Figure 9 Comparison of calculation results of full order system and reduced order system (100 PMSGs, unstable, area 4). 
Iterative process (100 PMSGs, unstable).
4.2 Example 2: Validation of method applicability under different collecting network structures
Section 4.1 has provided a topology structure for the collector network, which is referred to as Structure 1 in this article; This section will further set up the following two topological structures to verify the applicability of the method for wind farms with different structures:
Structure 2: Make the number of PMSG wind farms connected to the grid N = 100, and the number of convergence branches M = 5, each of which includes 20 units;
Structure 3: Ensure that the number of PMSG wind farms is connected to the grid is N = 100, and each unit is connected in parallel to the generator end busbar through a collection line.
Figure 10 shows the internal wiring diagrams of two types of wind farms with different structures; The operating conditions of the unit and the length of the collector line remain the same as in Section 4.1.
Figure 10 Circuit topology of collector. 
Due to the increasing impedance between the fan near the end of the branch and the grid connection point in structure 2, its SSO mode tends to be more unstable compared with structure 1, and the oscillation frequency decreases by 0.7634 Hz to −1.7153 + j233.7053. The distribution of eigenvalues in region 4 before and after the descending order of this structure is shown in Figure 11. As shown in Figure 11, the reduced order SSO mode has good computational accuracy.
Figure 11 Comparison of calculation results of full order system and reduced order system (Structure 2, area 4). 
The SSO mode of Structure 3 is −2.1014 + j239.9501, which is more stable than Structure 1 and Structure 2. The distribution of characteristic values in region 4 is shown in Figure 12; The reduced order and full order SSO modes in Figure 12 are basically consistent. In summary, the applicability of the method proposed in this article to wind farms with different topology structures has been verified.
Figure 12 Comparison of calculation results of full order system and reduced order system (Structure 3, area 4). 
4.3 Example 3: Rapid validation of methods under different wind farm scales
In this section, under the condition of the chain collector network structure, unit and network parameters as shown in Table 1, N is gradually expanded from 100 units to 600 units with a step size of 100 units, and each branch is maintained to contain 10 units; Comparing the computation time of full order and reduced order to verify the speed of the proposed method.
Correspondingly, generating the wind speed of the unit and the length of the collector line according to the method in Section 4.1. This method can generate a total of 400 different wind speeds; Therefore, when the number of fans is less than 400, the wind speed of each unit should be taken within the above range and ensured to be different from each other. When the number of fans is equal, or greater than 400, these 400 wind speeds will be allocated to 400 units, and the wind speeds of other units will be randomly generated in steps of 0.01 m/s between 4 and 8 m/s. This principle also applies to the generation of collector line length.
The time required for full order and reduced order calculations is shown in Table 4. According to Table 4, as the scale of the wind farm expands from 100 units to 600 units, the proportion of reduced order calculation time relative to full order time monotonically decreases from 60.18% to 12.76%. This result indicates that the order reduction method proposed in this article effectively saves computational costs; The larger the scale of the wind farm, the more obvious the speed of the reduction calculation method.
Reduced order and computation time under different wind farm scales.
In Table 4, the calculation time unit for the full order system is seconds, which is acceptable for general analysis. However, the above results only represent the computation time in the MATLAB computing environment and do not mean that there will be the same computation speed in other environments. Nevertheless, the quantitative relationship between the time consumption of full order and reduced order calculations in Table 4 has certain reference significance for practical engineering calculations and the development of new functions in other power engineering calculation software in the future.
5 Conclusion
Based on the fullorder wind farm networking model, SSO will face the problem of “dimensionality disaster”. This article proposes a reduced order calculation method for PMSG wind farm connected in SSO mode based on the Arnoldi method, which provides a fast and effective evaluation method for wind farm connected in SSO analysis. The main conclusions are as follows:

The SSO mode reduction calculation method proposed for wind farm networking systems can take into account the influence of factors such as the number of wind turbines, the collection network, and unit operating conditions, and has strong applicability.

The calculation results of the stable/unstable scenarios of SSO connected to 100 wind farms and the topology structure of series/parallel collector networks show that the geometric distance between the feature roots of the SSO mode calculated by the proposed reduction method and the full order model SSO mode is less than 10^{−4}, indicating the accuracy of the proposed reduction calculation method. The scale of the wind farm has been expanded from 100 units to 600 units. Compared with the full order calculation time, the reduced order calculation time has been reduced from 60.18% to 12.76%, indicating that the larger the wind farm scale, the more obvious the speed of the proposed reduced order calculation method.
References
 Erdiwansyah, Mahidin, Husin H., et al. (2021) A critical review of the integration of renewable energy sources with various technologies, Prot. Control Mod. Power Syst. 6, 1, 3. [CrossRef] [Google Scholar]
 http://www.nea.gov.cn/202201/26/c_1310441589.htm. [Google Scholar]
 National Development and Reform Commission. (2015) China 2050 China 2050 high renewable energy penetration scenario and roadmap study, National Development and Reform Commission. [Google Scholar]
 Xu Y., Geng Y., Yang B. (2021) Study on the mechanism and influencing factors of subsynchronous oscillations induced by wind farms in a fossil fuel power plant, Power Syst. Prot. Control 49, 18, 1–9. [Google Scholar]
 Xiang W., Ban L., Zhou P. (2021) Online prediction and optimal control method for subsynchronous oscillation of wind power based on an interpretable surrogate model for machine learning, Power Syst. Prot. Control 49, 16, 67–75. [Google Scholar]
 Zhang T., Wang H. (2021) Research methods for subsynchronous oscillation induced by wind power under weak AC system: a review, Power Syst. Prot. Control 49, 16, 177–187. [Google Scholar]
 Wang W., Zhang C., He G., et al. (2017) Overview of research on subsynchronous oscillations in largescale wind farm integrated system, Power Syst. Technol. 41, 4, 1050–1060. [Google Scholar]
 Xie X., Liu H., He J., et al. (2016) Mechanism and characteristics of subsynchronous oscillation caused by the interaction between fullconverter wind turbines and AC systems, Proc. CSEE 36, 9, 2366–2372. [Google Scholar]
 Wang J., Jia Q., Liu K., et al. (2021) Analysis of subsynchronous torsional mode of windthermal bundled system transmitted via HVDC based on a signal injection method, Power Syst. Prot. Control 49, 17, 12. [Google Scholar]
 Huang M., Wan H. (2009) Simplification of wind farm model for dynamic simulation, Trans. China Electrotech. Soc. 24, 9, 147–152. [Google Scholar]
 Xia Y., Li Z., Cai X., et al. (2014) Dynamic modeling of wind farm composed of directdriven permanent magnet synchronous generators, Power Syst. Technol. 38, 6, 1439–1445. [Google Scholar]
 Mi Z., Su X., Yu Y., et al. (2010) Study on dynamic equivalence model of wind farms with wind turbine driven doubly fed induction generator, Autom. Electr. Power Syst. 34, 17, 72–77. [Google Scholar]
 Luo K., Shi W., Qu J. (2017) Multimachine equivalent model parameter identification method for doublyfed induction generator (DFIG)based wind power plant based on measurement data, J. Eng. 13, 1550–1554. [Google Scholar]
 Zhang J., He Y. (2020) Parameters identification of equivalent model of permanent magnet synchronous generator (PMSG) wind farm based on analysis of trajectory sensitivity, Trans. China Electrotech. Soc. 35, 15, 3303–3313. [Google Scholar]
 Kunjumuhammed L.P., Pal B.C., Oates C., et al. (2017) The adequacy of the present practice in dynamic aggregated modeling of wind farm systems, IEEE Trans. Sustain. Energy 8, 1, 23–32. [CrossRef] [Google Scholar]
 Yang X., Yue C., Xie H. (2011) An aggregation method of permanent magnet synchronous generators wind farm model, Power Syst. Technol. 35, 2, 115–120. [Google Scholar]
 Wu H., He Y., Zhao B., et al. (2018) Research on dynamic equivalent of wind farm based on improved Kmeans clustering algorithm, Acta Energ. Sol. Sin. 39, 11, 3232–3238. [Google Scholar]
 Yan X., Li J.. (2020) Grouping method of direct drive wind farm based on principal component analysis, Power Syst. Prot. Control 48, 5, 127–133. [Google Scholar]
 Zhou P., Li G., Sun H., et al. (2020) Equivalent modeling method of PMSG wind farm based on frequency domain impedance analysis, Proc. CSEE 40, S1, 84–90. [Google Scholar]
 Schilders W., Rommes J., Vorst H.A. (2008) Model order reduction: theory, research aspects and applications, Springer, Berlin, Heidelberg. [CrossRef] [MathSciNet] [Google Scholar]
 Veloso S., Gil E., PulgarPainemal H. (2019) Application of model order reduction to a DFIGbased wind farm in the Chilean system, in: 2019 IEEE Power & Energy Society General Meeting (PESGM), IEEE. [Google Scholar]
 Ghosh S., Senroy N., Kamalasadan S. (2014) Reduced order modeling of wind farms for inclusion in large power system simulations for primary frequency response application, North American Power Symposium, IEEE. [Google Scholar]
 Ali H., Pal B.C., Kunjumuhammed L.P., et al. (2019) Model order reduction of wind farms: linear approach, in: 2019 IEEE Power & Energy Society General Meeting (PESGM), IEEE. [Google Scholar]
 Gao B., Wang G., Shao B., et al. Singular perturbation approximation method based on the dominant degree analysis for direct drive wind farm, Proc. CSEE 42, 7, 2449–2462. [Google Scholar]
 Meng X., Zhou N., Wang Q. Multitime scale model order reduction and stability consistency of IIDG system, Proc. CSEE 38, 13, 3813–3825+4022. [Google Scholar]
 Su T., Du W., Wang H. (2019) A reduced order design method for subsynchronous damping controller of multiPMSGs parallel wind farm, Trans. China Electrotech. Soc. 34, 1, 116–127. [Google Scholar]
 Dong W., Du W., Wang H. (2021) Reducedorder modal computation method for smallsignal stability examination of a wind farm, Trans. China Electrotech. Soc. 36, 7, 1468–1479. [Google Scholar]
 Shao B., Zhao S., Gao B. (2020) Simplified model for studying the subsynchronous oscillation of directdrive wind farms via VSCHVDC system based on similar transformation theory, Proc. CSEE 40, 15, 4780–4791. [Google Scholar]
 Lei W., Semlyen A. (1990) Application of sparse eigenvalue techniques to the small signal stability analysis of large power systems, IEEE Trans. Power Syst. 5, 2, 635–642. [CrossRef] [Google Scholar]
 Zheng W. (2007) Application of the refined Arnoldi method to the calculation of critical eigenvalues in small signal stability analysis, Zhengzhou University. [Google Scholar]
 Tian P., Liu C., Yun F., et al. (2015) An improved IRA algorithm and its application in computation of critical eigenvalues in small signal stability analysis, Proc. CSEE 35, 17, 4318–4325. [Google Scholar]
 Wang X.L., Jiang Y. (2011) Model order reduction technique, Comput. Math. Appl. 62, 8, 3241–3250. [CrossRef] [MathSciNet] [Google Scholar]
 Arnoldi W.E. (1951) The principle of minimized iterations in the solution of the matrix eigenvalue problem, Quart. Appl. Math. 9, 1, 6. [Google Scholar]
 Saad Y. (1980) Variations on Arnoldi’s method for computing eigenelements of large unsymmetric matrices, Linear Algeb. Appl. 34, 1, 269–295. [CrossRef] [Google Scholar]
 Daniel J.W., Gragg W.B., Kaufman L., et al. (1976) Reorthogonalization and stable algorithms for updating the GramSchmidt QR factorization, Math. Comput. 30, 136, 772. [Google Scholar]
All Tables
All Figures
Figure 1 PMSGsbased wind farms interfaced with AC power grid. 

In the text 
Figure 2 Model structure of PMSG. 

In the text 
Figure 3 Circuit topology of collector. 

In the text 
Figure 4 Flow chart of SSO mode calculation method based on Arnoldi method. 

In the text 
Figure 5 Connecting scheme of a 100PMSG wind farm. 

In the text 
Figure 6 Calculation results of 100PMSGssystem (stable). 

In the text 
Figure 7 Eigenvalue distribution of 100PMSGssystem (unstable, area 4). 

In the text 
Figure 8 Comparison of calculation results of full order system and reduced order system (100 PMSGs, stable, area 4). 

In the text 
Figure 9 Comparison of calculation results of full order system and reduced order system (100 PMSGs, unstable, area 4). 

In the text 
Figure 10 Circuit topology of collector. 

In the text 
Figure 11 Comparison of calculation results of full order system and reduced order system (Structure 2, area 4). 

In the text 
Figure 12 Comparison of calculation results of full order system and reduced order system (Structure 3, area 4). 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.