Static model of DFIG wind farms considering control strategies

The static equivalent models of DFIG grid-connected currents vary under different control strategies. Traditional static equivalent algorithms fail to consider these differences, posing risks in static security analysis. In this study, based on the internal structure of DFIG, grid-connected current models for DFIG under different control strategies are established. The appropriate model is selected during the current calculation process based on the coupling relationship between active power (P) and reactive power (Q) of DFIG. Subsequently, generator nodes to be retained are selected according to the reactive power sensitivity of external network generators based on the internal network, and the node admittance matrix is modified to perform static equivalence. Results show that the proposed model in this study has higher accuracy compared to the traditional Ward equivalent model.


Introduction
The large-scale grid integration of wind power has made the power grid structure increasingly complex.Traditional static equivalent models such as the Ward equivalent method are unable to capture the control strategies of new energy units and the internal states of the units.Consequently, static security analysis based on these models results in significant errors, hindering the safe and stable operation of the system [1-  3]   .In [4], a consistent static equivalent model for wind farms, considering both power flow and sensitivity, was proposed to maintain consistency in power flow and sensitivity before and after equivalence.This approach ensures the adaptability and accuracy of the wind farm's static equivalent, but it overlooks the internal structure of wind turbines.When faults occur, they fail to effectively reflect the internal safety of the turbines.In [5], an external network static equivalent model is introduced considering line loss sensitivity consistency, which is advantageous for analyzing line losses.However, it does not address how wind turbine equipment provides support within the internal network under transient conditions.In [6], an improved Ward equivalent method is presented, where Ward equivalence is carried out based on updated data from the external network, ensuring that the equivalent model can respond to changes in the external network.However, this method involves extensive computations and does not investigate the impact of new energy integration on equivalent models [7-10] .
DFIG exhibits different power flow solutions and static equivalent models under various control strategies [11] .Existing literature has overlooked this issue.To bridge this gap, this paper establishes the static equivalent model of DFIG grid-connected systems while considering different control strategies and retaining their internal structures.The challenges encountered in establishing the static equivalent model include the following: (1) DFIG has different grid integration current flow models under different control strategies, thus

DFIG power flow models under different control strategies
In power flow calculations, system nodes are typically categorized into PQ nodes, PV nodes, and slack nodes.
Active power constraint equation of PQ node and PV node is as follows: Reactive power constraint equation of PQ node is: The Newton-Raphson method is used to iteratively calculate Equations ( 1) and (2).
where Psys and Qsys represent matrices of active and reactive power imbalances in the system, respectively; H and N represent partial derivative matrices of active power constraints concerning node voltage phase angles and magnitudes, respectively; J and L represent partial derivative matrices of reactive power constraints concerning node voltage phase angles and magnitudes, respectively; sys and Usys represent matrices for modifying node voltage phase angles and magnitudes, respectively.
Figure 1 illustrates the structure of the DFIG, where WT represents the wind turbine, AC and DC represent the alternating current and direct current components respectively, PCC denotes the Point of Common Coupling, and Subscripts s, m, r, and g respectively represent the stator, excitation, rotor, and grid-side converter (GSC) components.
The above model represents a fixed reactive power model.When the DFIG participates in scheduling, the reference value for reactive power, denoted as QDFIG,set, needs to be calculated based on the dispatch instructions from the power system.However, dispatch departments often specify power factors, and DFIG operates based on a fixed power factor.In this case, when the DFIG is grid-connected with a fixed power factor angle  and QDFIG,set is a non-fixed value, additional constraints should be added.

Voltage control strategy for DFIG power flow model
When the DFIG operates under the constant voltage control strategy, it is required to maintain the terminal voltage magnitude Us at a constant value.In this case, the reactive power constraints for DFIG can be removed, and Equation ( 2) can be modified as follows: The Jacobian matrix is modified as follows: (1) We remove the reactive power constraints represented by Variables Qs,sys/ and Qs,sys/U from matrix Jsys.
(2) When the wind turbine voltage magnitude Us is constant, the constraint equations with partial derivatives concerning Us are removed from Matrices Jsys and JDFIG,sys.
(3) With the changes in the internal reactive power constraint equations of DFIG, we correspondingly add the partial derivatives concerning system node variables in matrix JDFIG,sys.

Ward-PV equivalent method considering the internal structure of DFIG
The Ward equivalent method divides the power network into external network (E), internal network (I), and boundary network (B).This paper selects external system generator nodes that need to be retained by considering the reactive power sensitivity of power flow calculations under different control strategies, highlighting the reactive support role of generator nodes within the internal system.Generator nodes are converted into PQ nodes, and the original power flow calculation results are considered as their reactive power output.Equation ( 3) is modified as follows: The term Usys/Qsys contains the reactive power sensitivities of internal system nodes to external generator nodes.By calculating the importance weight of internal system nodes comprehensively, generator nodes can be ranked.Based on the ranking results, external generator nodes are selected.
The external nodes E are further divided into E= {E1, EPV1, and internal nodes are divided into I= {I1, where PV1 represents the generator node that needs to be retained, D1 and D2 represent the external and internal wind turbine nodes, respectively, and E1 represents the original external nodes excluding the retained generator and wind turbine nodes.Additionally, I1 represents the original internal nodes excluding the internal wind turbine nodes.The external system to ground capacitance is represented in the form of boundary equivalent injected power, i.e., we eliminate the admittance of each branch to ground from the external system admittance matrix.We can obtain:

Wind power grid connection power flow considering different control strategies
Figure 3 shows the voltage magnitudes of the system under different conditions.Except for PV nodes, the voltage magnitudes under constant power factor control are higher than in the other two cases.Figure 4 presents the voltage phase angle variations under different conditions.Compared to magnitudes, the absolute values of phase angles at hybrid control nodes are higher than those in the other two cases.
Although the phase angle variations at these nodes are relatively small, they exceed the convergence accuracy significantly and cannot be ignored.Table 1 presents the DFIG terminal voltage at Nodes 40 and 41, as well as the voltages at Nodes m, r, and g under the three control strategies.It is observed that the voltage magnitudes at Nodes m, r, and g vary significantly.Particularly, the voltage magnitude at D1 differs greatly between constant power control and constant voltage control.Hence, the differences in control strategies cannot be ignored.Otherwise, during transient processes, the reactive power support of DFIG to the internal network cannot be reflected accurately, affecting the accuracy of static equivalence.

Internal system's sensitivity to reactive power at external generator nodes
This paper utilizes an improved PageRank algorithm to rank the importance of nodes in the case study under three control strategies, selecting non-generator important nodes within the internal system.The importance-ranked nodes are then subjected to a normalization of their weights [12] .Table 2 presents the sensitivity of the voltage of these nodes to the reactive power at external generator nodes.The data in Table 2 shows varying sensitivities under different control strategies.However, the sensitivity trends of the same nodes towards the generator nodes remain consistent.Hence, the importance of the external system's generator nodes remains consistent under different control strategies.Considering these factors, it is possible to choose to retain generator nodes 31 and 32.

The Ward-PV equivalents for power flow based on different control strategies
After identifying the generator nodes to be kept, matrices representing the electrical properties of these nodes are created.These matrices include information about external, internal, and boundary nodes.To form these matrices, we introduce new nodes labeled as m, r, and g, along with their associated connections.When calculating the Ward-PV equivalent, it is crucial to calculate the electrical parameters between the remaining external generator nodes and the boundary nodes.Additionally, we need to determine the power injected at these boundary nodes.
The case study involved simulations carried out under two specific conditions: (1) There was an increase of 10% in the internal network load; (2) Lines 3-18 and 18-17 were disconnected.The results presented in Table 3 demonstrate that the method proposed in this paper shows significantly reduced relative errors in voltage magnitude, active power, and reactive power of the lines when compared to conventional Ward equivalents during simulation.Notably, among the three internal variations outlined in Table 3, there is a significant improvement in relative errors for reactive power.Table 4 illustrates the alterations within the internal system of the original network when the Doubly Induction Generator (DFIG) operates under constant power factor and mixed control scenarios.A static equivalent model, controlled by maintaining a constant terminal voltage of the DFIG, is employed to evaluate the maximum relative errors in voltage magnitude, line active power, and reactive power.Currently, the accuracy of this equivalent model is relatively low.Consequently, in the context of widespread DFIG integration into the grid, it is crucial to establish static equivalent models tailored to various control scenarios for accurate power flow analysis.

Conclusion
This paper conducted power flow analysis for DFIG under various grid integration control strategies.The selection of retained generator nodes was based on reactive power sensitivity, taking into account DFIG control strategies and internal structures to establish static equivalent models.The following conclusions were drawn: (1) Under different control strategies, the variation in the DFIG system's power flow results in fluctuations in voltage amplitude that surpass those in phase angle.Specifically, under constant power factor control, the system's voltage amplitude exceeds the level maintained at a constant terminal voltage.
(2) The static equivalent method proposed in this study demonstrates superior accuracy, particularly in the context of reactive power, when compared to the traditional Ward equivalent method.
(3) The selection of grid connection control strategies is crucial.Using mismatched equivalent models can lead to significant errors, impacting the accuracy of static security analysis.

Figure 3 .Figure 4 .
Figure 3.The bus voltages with different control strategies.
A is the swept area, ci is the c9 factor, λi is an intermediate variable, λ is the tip-speed ratio, β is the pitch angle,  is the radius of the wind turbine, S is the apparent power, ω is the rotational speed, and Subscript B indicates grid reference values.

Table 1 .
Voltages of the DFIG with different control strategies.

Table 2 .
Reactive power sensitivities of internal nodes and external generators with different strategies.

Table 4 .
Equivalent discrepancy with different control strategies.