A Study of Dynamic Equivalence Using the Similarity Degree of the Equivalent Power Angle in Doubly Fed Induction Generator Wind Farms

This paper presents a study of dynamic equivalence for doubly fed wind farms (DFWFs) in large power systems. The core idea is to find an attribute representing the critical characteristics of a doubly fed induction generator (DFIG) and use it for clustering in the equivalent modeling process. Using the voltage and flux linkage equations of DFIGs, we derived an electromagnetic power expression. Similar to the power angle concept for synchronous generators, an equivalent power angle (EPA) is proposed and shown to represent the dynamic characteristics of a DFIG in the test system. We introduce the similarity degree of the EPA based on similarity theory and construct a similarity index of the EPA for clustering a DFWF. A practical algorithm was developed for parameter aggregation of the equivalent model. In addition, a simulation of DFWFs in Yunnan, China, was conducted to show the feasibility and high precision of the proposed method.


I. INTRODUCTION
With the rapid developments in wind power in recent years, the doubly fed induction generator (DFIG) has become a popular type of wind turbine (WT) in most wind farms (WFs) and now plays a prominent role in power systems. WFs usually consist of dozens or even hundreds of WTs, and this leads to tremendous challenges in the dynamic analysis of bulk power systems because detailed models of WFs create an enormous computational burden and are time-consuming. An equivalent model of a doubly fed wind farm (DFWF) substantially reduces the scale while exhibiting similar responses and is powerful enough to fulfil the simulation requirements in the above scenario. Such models have received considerable attention.
Studies have shown some achievements regarding the equivalence of WTs [1]- [10]. Similar to coherency-based The associate editor coordinating the review of this manuscript and approving it for publication was Junjian Qi . dynamic equivalents of synchronous generators, there are two crucial processes in building a reasonable equivalent model of a DFIG: rational clusters and parameter aggregation.
As the most crucial and first step, a rational cluster initiates DFIG WT groups with similar characteristics. The physical properties of wind power are generally chosen as pivotal factors during the clustering process. For example, Ref. [1] and Ref. [2] used wind speeds for coherent identification. Ref. [3] considered the wake effect factor and the action of the blade pitch angle as the classification index for the same WF. However, various operating conditions and the complex properties of wind power, especially wind intermittency and uncertainty, create challenges for these methods.
Considering the composition of DFIGs, researchers have also proposed that converter controllers and crowbar protection should both be used for clustering and aggregation because of the close link with the dynamic characteristics of the DFIG. Based on this idea, [4] developed a refined dynamic equivalent model for a DFWF. However, whether this model can meet the simulation requirements for large power grids with a high penetration of wind power, where certain WFs need to be largely reduced, is unclear.
Recently, several artificial algorithms have been applied to clusters of coherent WTs, including the K-means algorithm, the density peak clustering algorithm, and the fuzzy clustering algorithm [5]- [7]. These algorithms provide new insights into the way WTs are processed and clustered. However, the algorithms must be trained with massive datasets, which is time-consuming.
Parameter aggregation is used to identify adequate parameters for an equivalent model and ensure that the accuracy of the dynamic responses is comparable to that of the original system. For instance, the equivalent models of group generators and their optimized parameters can fit aggregated transfer functions within tolerable accuracies, and the dynamics of the active/reactive power and voltage at the point of common coupling (PCC) are consistent with the original system [8]. Ref. [9] proposed an approach in the time domain based on structure preservation, which is more easily implemented than the traditional aggregation method in the frequency domain. Ref. [10] developed a dynamic weighted aggregation equivalent modeling method in which the weighting factors for each generator were calculated according to a Weibull distribution of wind speeds.
Notably, almost all of these studies involved a multiequivalent model of WTs within the same WF only, ignoring the scenario of equivalent models for various WFs [11]. A multi-equivalent model obtained by the above method can work well for a small-scale power system. In contrast, such a model will not be accepted for the dynamic analysis and security assessment of main power grids in which one must extensively reduce and equalize the generating units, networks, and loads at remote terminals. Considering these circumstances, the aim of this paper is to achieve these goals. A reasonable equivalent method with acceptable accuracies and flexibilities for various WFs is significant and can dramatically reduce the scale and time of the simulations.
To overcome these limitations, this paper proposes a novel method of dynamic equivalence for DFWFs. The core idea of this paper is inspired by the slow coherency and aggregation of synchronous generators that successfully use natural or inherent properties to derive reduced models and obtain the parameters [12]- [15]. The process of coherency and aggregation takes advantage of the power angles of generators after disturbances and identifies groups of coherent generators based on similar swing curves of the generator power angles. Then, models for the equivalent generators and optimized parameters are chosen to fit the aggregated transfer functions within tolerable accuracies. For studies of the dynamic equivalence of DFWFs, the core idea of the aforementioned studies is still attractive, but the following issues must be resolved: (1) It is critical to find properties that can properly represent the dynamic characteristics of DFWFs.
(2) The proposed method can effectively cluster WFs into various groups with the above properties.
(3) A proper and practical parametrization for the equivalent model must be identified after clustering the WFs.
Using the voltage and flux linkage equations of WFs, we derive the equivalent power angle (EPA) between the equivalent transient internal voltage (ETIV) and the PCC voltage [16]. The EPA represents the dynamic characteristics of a WF, and use of the EPA in the clustering of DFWFs is novel. However, the EPA varies widely with the operational conditions and disturbances and challenges the conventional slow-coherency criterion. This paper introduces the similarity degree of the EPA based on similarity theory and applies the similarity degree to the EPA for clustering. A practical algorithm based on power weights is used to aggregate the parameters of the equivalent model. In addition, time-domain simulations are carried out to validate the proposed method.

A. DEFINITION OF THE EPA
The dynamic characteristics of a DFIG closely rely on its composition. As shown in Fig. 1, a DFIG mainly consists of an induction generator and two back-to-back converters. Hence, to find a physical quantity that characterizes the operating state of a DFIG, it is necessary to analyze the mathematical model of the induction generator and the back-to-back converter. The voltage equations of the stator and rotor of the DFIG are shown in (1) for the d-q synchronous reference frame: where u, i and represent the voltage, current, and flux linkage of the DFIG, respectively; R is resistance of the DFIG; the subscripts s and r denote the stator and rotor quantities, respectively; p is a differential operator; s is the slip; and the subscripts d and q indicate the direct-axis and the quadrature-axis, respectively.
The flux equations of the stator and rotor are shown in (2): where L m is the mutual inductance. VOLUME 8, 2020 By combining (1) and (2), the voltage can be written as where X ss is the equivalent reactance, which is calculated as The back-to-back converters involve a rotor-side converter (RSC) and a grid-side converter (GSC). Fig. 2 shows the control block diagram of the RSC and GSC. Using a stator-flux orientation, the RSC provides decoupled control of the active and reactive power, and the GSC is employed to maintain a constant voltage on the DC-link. Fig. 2 shows that the control of the DFIG eventually reflects variables such as the terminal voltage, current, and flux linkage of the DFIG. Thus, we can use these variables to construct E . Let where E represents the voltage under control. Considering X ss R s , the voltage drop on the stator can be ignored, and the ETIV of the DFIG is From (6) we find that the DFIG and synchronous machine have the same form between the transient potential E and terminal voltage U s . Similar to the power angle concept from synchronous generators, δ dfig is defined as the EPA, which is the angle between E and U s [16], [17]. For the sake of analysis, we let E be aligned with the q axis to easily obtain the relationships among the power factor angle ϕ, inner power factor angle θ, and EPA δ dfig , as shown in Fig. 3. We can derive the output power P es from the stator side of the DFIG: The electromagnetic power P e can be calculated as In the electromechanical transient process, the DFIG rotor speed changes slowly, and the slip s has little impact on the flux linkage t . The rotor flux t is decoupled from the mechanical motion of the rotor in the dynamic process. According to (5), the EPA is independent of the physical rotor position but changes in the electromechanical transient process. When a fault occurs, the dynamic characteristics of the DFIG can be triggered, and the EPA may aggravate the consequence caused by the fault [18]. In the next section, we examine an EPA with characteristics that suit the identification of the coherence between DFIGs.

B. CHARACTERISTICS OF THE EPA
In a DFIG, the mechanical part and electrical parts can be decoupled. The WTs, RSC, and GSC mainly affect the output power of the grid-connected bus. There are three types of control modes in the RSC: constant voltage control, reactive power control, and power factor control. We constructed a simple test system to examine the characteristics of the EPA. As shown in Fig. 4, the two DFIGs in the test system can adopt identical or different control strategies. The parameters of T1 and T2 are same. The values of Z 1 and Z 2 are equal. The parameters of the system in Fig. 4 are shown in TABLE 1. The control strategy of DFIG1 is constant voltage control, whereas constant reactive power control is applied in DFIG2. We focus on observing the active power and the  EPA after a fault occurs. When t=5 s, a single-phase ground fault occurs, which lasts for 0.1 s before removal. Figs. 5-8 present the curves of the active power and EPA in these scenarios.
In Fig. 5 and Fig. 6, under the same control mode, the active powers of the two DFIGs exhibit a similar tendency and amplitude after the faults. This means that two DFIGs should be clustered into a coherent group when the active power and EPA curves of the two DFIGs are consistent. From Fig. 7 and Fig. 8, we can see that the two DFIGs represent different dynamic characteristics under various control modes. However, the EPA shows more apparent and intuitive differences between the two DFIGs. This result reflects the amplification effect of the EPA after the fault. The simulation indicates that taking the EPA as an observable variable for coherent identification is reasonable.

III. AN INDEX OF THE SIMILARITY DEGREE FOR THE EPA
In the previous section, we discussed the derivation and characteristics of the EPA. For a conventional synchronous generator, the swing of the power angles can act as the criterion for coherent identification under different disturbances, as shown in (9).
This conventional criterion cannot apply directly to the EPA because the EPA varies with the operational conditions and disturbances. In this paper, a novel method that fully VOLUME 8, 2020 utilizes inherent attributes is proposed. Based on similarity theory [21], we employ the Prony algorithm to analyze and extract the properties of the EPA curves, such as the frequency of oscillation and damping ratio, as the Prony algorithm is a commonly used signal processing algorithm [22]. The Prony method uses a linear combination of a set of exponential terms to fit equally spaced sampling data, from which information such as the damping and frequency of the signal can be extracted. We use these properties to construct a similarity index. Finally, we apply the similarity degree to DFIGs and successfully cluster the DFIGs.

A. THE BASIC PRINCIPLE OF SIMILARITY THEORY
Similarity theory can describe the degree of similarity between two objects that are similar in composition or characteristics. Similar elements are units that have common attributes and characteristics between objects with different values. Suppose that object A consists of m elements and that object B has n elements. According to the principle for selection, there are l pairs of similar elements between objects A and B. The degree of similarity between the i th pair of similar elements is q(ui), and the weight of q(ui) is β i . The degree of similarity Q between objects A and B is a function of the above quantities, which can be expressed as: If objects A and B correspond to a curve, the corresponding elements may be the frequency, amplitude, damping, phase, etc., with a total of l. Then, equation (10) can be specified as: Artificially setting a threshold ν when Q (A,B) is equal to or greater than ν, A and B can be considered to be similar. Using similarity theory, we can depict the similarity degree of the EPA, which can be used as the coherent clustering criterion for DFIGs.

B. APPLICATION OF SIMILARITY THEORY TO THE EPA
Taking two EPA curves of DFIGs under faults as A and B mentioned above, we use the Prony algorithm to extract the elements of the curves and obtain the elements of the curve in a given order. These elements usually contain many redundant parts with amplitudes close to 0. Therefore, the redundant orders can be eliminated by setting the energy threshold e, which decreases similar elements between the DFIGs. The elimination works as follows: the energies of all elements are ordered from high to low and are summed until the sum is equal to or greater than the product of the total energy E i and threshold e.
The contributing orders of the curves for the EPA are confirmed, which are also similar elements between DFIGs A and B. From the simulation in section II, we can conclude that the frequency and damping are the primary factors affecting the tendency of the curves. We think it is more suitable to define the frequency and damping as the frequency of oscillation and the damping ratio, respectively. The damping ratio represents the rate of decay of the amplitude of the EPA curve. Therefore, the i th order similar elements of the frequency and damping between A and B are shown in equations (13) and (14): where f i and α i are the frequency and damping of the i th element of the EPA curves, respectively, and A and B represent the EPA values of DFIGs A and B, respectively. Considering the similar weight of the frequency and damping, the similarity of the i th similar element is β i1 and β i2 are the weights of the frequency and damping, respectively, which can be set according to the actual requirements. Both the damping and frequency represent key characteristics and trends of the EPA curve, so the weights of the two variables are equally important. The parameters can also be determined by trying different values here. According to the practical problem in this paper, the weights of the frequency and damping were set to be equal. Hence, equation (15) can be written as The ratio of the energy of each order to the total energy is defined as follows: Considering the weight, the similarity of all similar elements can be written as The process of calculating the similarity index under a single fault is given above. Note that the similarity index needs to consider the similarity of the dynamic characteristics of the EPA between the DFIGs under different fault types.
There are k types of faults at the output end of each DFIG. For fault type i, the similarity between the DFIGs can be calculated as Q f (i) , and the similarity index of the coherent identification of different faults is defined as follows: (19) Through calculation of the similarity index between the DFIGs, the coherent DFIGs in the WF are determined.

IV. AGGREGATION OF THE COHERENT DFIGS
After clustering the WFs, we introduce a practical algorithm based on the weighting factors of the DFIG active powers. Considering that WFs in the same cluster are parallel to each other at the aggregated bus, parameters of the equivalent WT are easily obtained from this topology.

A. AGGREGATION OF THE COHERENT BUS
There are N DFIGs in the coherent group, and the ETIV is (20) where E equ is the ETIV of the equivalent WT; P l is the active power of the i th DFIG; and E l is the ETIV of the l th DFIG. E l is connected to E equ through a complex variable ratio phase-shifting transformer (shown in Fig. 9 (b)) with a ratio of According to equation (22), the equivalent bus of the DFIG can be determined by the ETIV and the equivalent reactance (shown in Fig. 9 (c)). The equivalent reactance is where X ssN is the equivalent reactance of the N th DFIG. Because large-capacity WTs in WFs have a greater influence on the dynamic characteristics of the system than other WTs, the dynamic characteristics of WFs can be better maintained if the aggregation takes the active power output of the WT as the weight rather than using the mean value method.

B. PARAMETER AGGREGATION OF AN EQUIVALENT WIND TURBINE
The rated capacity and output characteristics of the coherent groups should remain consistent with those of the original system. Therefore, the parameters of the equivalent WT are where P is the active power, and Q is the reactive power. The subscripts eq represent the parameters of the equivalent WT. For calculation of the shafting parameters of the equivalent WT, a method similar to that for solving the ETIV is adopted. Therefore, the shafting parameters of the equivalent WT are where H gi , H ti , and H si represent the inertia time constant, the inertia time constant of the rotor of the WT, and the stiffness coefficient of the shafting of the i th generator, respectively.

V. CASE STUDY
We take a grid integrated with WFs in Yunnan province, China, as a case study. This small part of the Yunnan power grid with concentrated wind power is selected to verify the equivalent effect in DFWFs. There are 17 DFWFs in this system, as shown in Fig. 10. To avoid excessive equivalent lines, preliminary groups are necessary. These clustering groups usually depend on the connected buses and electrical distance. Thus, the preliminary results of clustering for these WFs can be obtained as WF1 = [YISG1, YISG2, FETUG1, From the simulations of the two types of faults and Figs. 11-15, we find that the DFIGs in WF2, WF3, and WF5 are coherent due to their identical dynamics. However, modifications should be made to WF1 and WF4. In WF1,     FETUG1, FETUG2, and DLKG1 are similar and should be coherent in one group, whereas YISG1 and YISG2 and DLKG2 and DLKG3 consist of the second and third groups, respectively. Similar modifications should be made for WF4; MPTG and LFAG should be grouped, and LJSG and DZSG are coherent. The final clustering can be confirmed by the similarity degree.

B. COHERENCY IDENTIFICATION BASED ON THE SIMILARITY INDEX
Through observation of the EPA results, some curves do not quantitatively identify the coherency. Thus, the EPA curve under contingencies is used to extract a series of data. The data are used to calculate the similarity index between the DFIGs. If the similarity index of the EPA Q EPA is 0.85 or greater, the two DFIGs are coherent. The results of the calculation are shown in Tables 2 -4.  From the results of Tables 2 -4, we can obtain the final clustering with the similarity index. The pro-fault power curves are shown to verify the validity of the clustering results. Figs. 16-19 show the active power output of the WT under contingencies. Fig. 16 shows that the coherence between FETUG1 and FETUG2 is better than that of the FETUG1 and DLKG1 group, which is in accordance with the results in Table 2. These WFs were divided into 8 coherent groups, meaning that the number of WFs decreased by 53% in this case. Applying this method for the entire power grid in Yunnan province, we would obtain similar reduction effects.    The simulation further illustrates that methods based on the similarity degree of the EPA are feasible and very precise.

C. PARAMETER AGGREGATIONS OF THE COHERENT GROUP
After coherent identifications, we carried out the following parameter aggregations. We compared the proposed algorithm   in this paper with a traditional algorithm, the average algorithm, as shown in Fig. 20. We aggregated two DFIGs in WF2 with different capacities. The main parameters of the DFIGs in the test system are shown in Table 5. Fig. 20 compares the proposed algorithm with the average algorithm through the active power of line APU_W22-LF22. The curve of the power aggregation algorithm is closer to that of the original system, which indicates the superiority of our algorithm.

D. EQUIVALENT EFFECT VERIFICATION
After equivalences, the number of WFs decreased by 53% in the equivalent system compared with the original. Figs. 21-23 show the results of the clustering and aggregation algorithm   proposed in this paper. Comparisons of the active power, reactive power, and voltage between the original system and the equivalent system demonstrate that the systems are almost identical. This result also indicates that the proposed method can maintain the dynamic characteristics of the original system. Therefore, the clustering and aggregation method proposed in this paper is reasonable and has a high precision.
There are approximately 132 WFs in the Yunnan power grid. Using the proposed method, the scale of the power grid can be reduced while retaining high fidelity. The equivalent system  will save time in the modeling and simulation of dynamic analysis.

VI. CONCLUSION
To reduce the main power grid for dynamic analysis and security assessments, this paper presents a study of the dynamic equivalence for DFWFs. The EPA, the angle between the equivalent transient internal voltage (ETIV) and the PCC voltage, can represent the critical characteristics of a DFIG. Thus, we can introduce the similarity degree of the EPA based on similarity theory to cluster DFWFs. A similarity index of the EPA is used as the criterion for coherent identification. The power weight algorithm can be used to obtain aggregate parameters for the equivalent model. The case study verifies the reduction effect and precision of the method compared with the original system. The simulation results for DFWFs in Yunnan province, China, show the feasibility and high precision of the proposed method.