Abstract：In the current studies on stability of weak grid connected permanent magnet synchronous generator(PMSGs) wind farm, single machine equivalence is generally used for wind farm, and it is usually assumed that the wind farm is an ideal system, that is, the parameters, outputs and collector lines of all PMSGs are identical. However, there is still a lack of related theoretical support for the prerequisites of single machine equivalent modeling for PMSGs wind farm. In addition, in the actual wind farm, only the parameters of PMSGs are identical (that is, the same type), the outputs and collector lines of PMSGs are different, which is a nonideal system. Whether the single machine equivalent modeling is effective remains to be studied. In response to the problems, this paper presents the criterions for the PMSGs wind farm to meet the single machine equivalence and proposes the prerequisites of single machine equivalent modeling. It provides a certain theoretical support for the application of single machine equivalence. Firstly, three criterions are proposed to determine whether the PMSGs wind farm can be equivalent to a single machine model. There is at most one pair of critical eigenvalues in the system (i.e. weak damping or negative damping eigenvalues). The grid side state variables must participate in this oscillation mode. The critical eigenvalues of single machine equivalent model and detailed model are equal. Secondly, the oscillation characteristics and single machine equivalent modeling of the ideal system are studied analytically and theoretically. The state space model of weak grid connected PMSGs wind farm is established. The eigenvalues and eigenvectors of the state matrix are analytically analyzed using the matrix similarity transformation theory. It shows that there are one set of in-phase modes and n-1 sets of anti-phase modes in an ideal system with n PMSGs. The in-phase modes can be excited by both machine and grid sides, while the anti-phase modes can be only excited by machine sides. The weakest oscillation mode must be the in-phase mode under normal conditions. The single machine equivalent model of PMSGs wind farm is obtained based on the aggregation method. The theoretically analysis results show that the eigenvalues of equivalent model are equal to the in-phase mode of detailed model. Therefore, the single machine equivalence of PMSGs wind farm meets the three criterions. The single machine equivalent model can accurately represent the oscillation characteristics of detailed wind farms. Furthermore, in nonideal system, the Monte Carlo method is used to analyze the oscillation characteristics and single machine equivalent modeling. A total of 2000 groups of samples are generated randomly, where only the parameters of PMSGs are identical in every sample. The cluster analysis of eigenvalues calculated in each sample shows that the distribution of eigenvalues is similar to that in ideal system. There is at most one pair of critical eigenvalues, which meets the first criterion. The relative participation on in-phase modes and anti-phase modes shows that, the grid side only has a high participation on in-phase modes, while nearly no participation on anti-phase modes, which satisfies the second criterion. The comparison of critical eigenvalues between single machine equivalent model and detailed model shows that the errors are very small. Although they do not strictly meet the third criterion, the two critical eigenvalues are approximately equal. Therefore, the single machine equivalence is still effective. The time-domain simulation results show that the responses of single machine equivalent model are close to that of detailed model. The single machine equivalent model can reflect the oscillation mode, and has high accuracy. The following conclusions can be drawn: 1) The three criterions are proposed to determine whether the PMSGs wind farm can be equivalent to a single machine model. 2) The prerequisites of single machine equivalent modeling for PMSGs wind farm are the parameters, outputs and collector lines of all PMSGs are identical, which are theoretically proved. 3) The single machine equivalence is still effective, where only the parameters of PMSGs are identical, and the outputs and collector lines are different.
李龙源, 付瑞清, 吕晓琴, 王晓茹, 段虎昌. 接入弱电网的同型机直驱风电场单机等值建模[J]. 电工技术学报, 2023, 38(3): 712-725.
Li Longyuan, Fu Ruiqing, Lü Xiaoqin, Wang Xiaoru, Duan Huchang. Single Machine Equivalent Modeling of Weak Grid Connected Wind Farm with Same Type PMSGs. Transactions of China Electrotechnical Society, 2023, 38(3): 712-725.
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