1. Introduction A short-circuit fault in a power system is a complex electromagnetic transient process. When using the symmetrical component method to analyze and calculate asymmetrical short-circuit or asymmetrical phase-loss faults in a three-phase symmetrical power system, the impedance parameters of the three-phase asymmetrical branches at the fault port are always equivalently converted into three independent positive, negative, and zero-sequence symmetrical current sources. This transforms the calculation of the difficult-to-solve three-phase asymmetrical fault into the calculation of three independent and symmetrical three-phase circuit networks that are easier to solve. In practical engineering calculations, each sequence network is considered a linear network. For example, if the subscript (S) represents the sequence and n(S) is the number of independent nodes in each sequence network, when using the symmetrical component method to calculate the fault, it is necessary to solve a large sequence circuit network with n(S) independent nodes, which is a considerable computational workload. In many cases, we may be interested in a local network within a large network, where the number of associated nodes r(S) is much smaller than n(S). Alternatively, to speed up the fault calculation or due to limitations in computer capacity, it is often necessary to reduce the node equations of the large network to equivalent node equations with a predetermined number of nodes. This paper proposes to apply the REI equivalent method [1] to the calculation of short-circuit current in multi-node distribution systems. The application of local node equivalent equations to the extended REI equivalent network is studied, which simplifies the mathematical model for calculating short-circuit current in multi-node distribution systems and makes the short-circuit current calculation program simpler and more practical. [b]2 Local Node Equivalent Equations[/b] The local node equivalent equations divide large network nodes into two categories: retained nodes r(S) and eliminated nodes e(S), e(S) = n(S) - r(S). During the calculation, the nodes to be retained are numbered first in each sequence network, and then the nodes to be eliminated are numbered. Then, the node admittance equation Y(S)U(S)＝I(S) is divided into blocks according to the nodes to be retained and the nodes to be eliminated, and we can get [img=239,48]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/65-1.gif[/img] （1） In the formula, Y(S) is the node admittance matrix; U(S) is the node voltage column vector; I(S) is the node injected current column vector; Yrr(S) includes the self-admittance and mutual admittance of the retained node; Yre(S) and Yer(S) are the mutual admittance between the retained node and the eliminated node; Yee(S) includes the self-admittance and mutual admittance of the eliminated node; Ur(S) is the retained node voltage column vector; Ue(S) is the eliminated node voltage column vector; Ir(S) is the retained node injected current column vector; and Ie(S) is the eliminated node injected current column vector. Since Ur(S) should remain unchanged before and after the equivalent reduction, the equivalent admittance equation of the remaining nodes is obtained by eliminating e(S) nodes using the Gaussian elimination method. The equation is (Yrr(S) - Yre(S)Y-1ee(S)Yer(S))Ur(S) = Ir(S) - Yre(S)Y-1ee(S)Ie(S) (2) It is abbreviated as YNrr(S)Ur(S) = INr(S) (3) where YNrr(S) is the equivalent admittance matrix of the remaining nodes; INr(S) is the equivalent current source [2]. YNrr(S) = Yrr(S) - Yre(S)Y-1ee(S)Yer(S) (4) INr(S) = Ir(S) - Yre(S)Y-1ee(S)Ie(S) (5) From equations (4) and (5), it can be seen that when the node to be eliminated is an active node, that is, when Ie(S) is not equal to zero, then the sub-block Yrr(S) of the retained node corresponding to the original network Y(S) and the power supply Ir(S) of the retained node need to be corrected; if the node to be eliminated is a passive node, that is, when Ie(S) is equal to zero, then there is no need to correct the current source of the retained node. In practical engineering calculations, it is assumed that the system is in a three-phase symmetrical operating state before the fault, so the Ie(2) of the negative sequence network and the Ie(o) of the zero sequence network are both zero. If we apply equations (4) and (5) to modify Yrr(S) into YNrr(S) and Ir(S) into INr(S), two situations will occur when calculating the short-circuit current of a multi-node distribution network: (1) When calculating YNrr(S) and INr(S) according to the local node equivalent equation, the original data of the entire network needs to be input, which is a lot of work; (2) When calculating a distribution network with hundreds of nodes under the condition of only microcomputer, due to the limitation of computer capacity, the admittance matrix Y(S) of the entire network nodes cannot be formed before the block division, so the block division is not possible and the program cannot proceed. If the REI equivalent method is combined with the local node equivalent method, the problem of short-circuit current calculation of multi-node distribution networks will be further simplified. [b]3 Applying the REI equivalent method to short-circuit current calculation[/b] The REI equivalent method was proposed by P. Dimo ​​et al. and applied to the power flow calculation and security analysis of developed countries [3]. It divides the nodes of the power grid into two groups: nodes that should be retained and nodes that should be eliminated. First, the active nodes among the nodes to be eliminated are grouped into several groups according to their related properties. Each group is replaced by a virtual equivalent active node, which is then connected to it through a lossless virtual network (REI network). The active and reactive power injected into the virtual active node is the algebraic sum of the active and reactive power of the active nodes in that group. After connecting to the REI network and the virtual equivalent node, the original active nodes become passive nodes. Then, all the passive nodes to be eliminated are eliminated using conventional methods. The equivalent transformation process is shown in Figure 1. [img=280,274]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/t65-1.gif[/img] Fig.1 Simplified processing of REI equivalent network The values ​​of each admittance in the REI network are determined by the following method: For each active node to be eliminated, the relationship of its injected current is [img=199,25]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/65-2.gif[/img] （6） In the formula [img=11,22]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/65-3.gif[/img][sub] k [sub] represents the active node injection current to be eliminated; [img=18,24]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/65-4.gif[/img] represents the conjugate value of the active node injection power in the original network; [img=23,24]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/65-5.gif[/img] represents the node complex voltage conjugate value of the basic power flow solution. [img=149,50]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/65-6.gif[/img] （7） Where [img=11,22]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/65-3.gif[/img][sub] R [/sub] is the injected current of the virtual active node. When constructing the REI network, the parameters should keep the injected power of each active node in the original network unchanged. Therefore, [img=264,27]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/65-7.gif[/img] (8) where Yk is the admittance between node G and the node to be eliminated in the REI equivalent network. To satisfy the conditions for lossless meshes, then [img=89,38]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/66-1.gif[/img] （9） [img=220,53]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/66-2.gif[/img] （10） [img=145,27]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/66-3.gif[/img] （11） Where [img=51,24]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/66-4.gif[/img] are the injected power and complex voltage of the virtual active node R, respectively; YR is the admittance between the virtual active node R and node G. In equation (8), UG can be arbitrary. Taking UG = 0, the construction of the REI network becomes unique, thus obtaining [img=269,49]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/66-5.gif[/img] （8）′ [img=265,26]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/66-6.gif[/img] （11）′ After constructing the REI network, we need to eliminate those nodes that are not of interest. Assume that the network admittance matrix after expanding the REI network is Y. The subscript E represents the set of nodes to be eliminated, and the subscript I represents the set of nodes to be retained. Thus, Y can be written as: [img=112,43]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/66-7.gif[/img] (12) Eliminate all nodes in the node set E to obtain a simplified network composed of nodes in the node set I, as shown in Figure 1(c). The node admittance matrix after external equivalence is YIIN＝YII－YIEYEE－1YEI (13) The equivalent current source is INI＝II－YIEYEE－1IE (14) The equivalent admittance equation of the retained node is simply YNIIUI＝INI (15) Applying this theory to the short-circuit current calculation, we see that equations (13)～(15) correspond to equations (3)～(5) respectively. The difference is that after connecting the REI network and the virtual equivalent node, the original active node becomes a passive node. If the virtual equivalent node R is used as the retained node, then IE is zero. Equation (14) becomes INI＝II (16) As can be seen from equations (8) and (11), the parameters of the REI network are related to the network's operating parameters UK. In power flow calculation and security analysis, the relationship of mutual equivalence with the original network is only satisfied under the basic operating mode. When the system operation deviates from the basic operating mode, if the REI network parameters are kept unchanged, errors will occur. However, in the short-circuit current calculation, due to the adoption of a practical algorithm, UK is taken as the rated voltage before the short circuit at point K, and its per-unit value is 1. Then, equations (6) to (11) can be changed to equations (17) to (24): [img=200,25]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/66-8.gif[/img] （17） [img=86,36]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/66-9.gif[/img] （18） [img=67,24]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/66-10.gif[/img] (19) [img=87,37]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/66-11.gif[/img] (20) [img=53,24]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/66-12.gif[/img] (21) [img=61,26]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/66-13.gif[/img]（22） [img=74,25]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/66-14.gif[/img] (23) [img=65,24]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/66-15.gif[/img] (24) The REI equivalent network for short-circuit current calculation using equations (18) to (24) is easy to implement. After expanding the REI network, the local node equivalent method is used, so there is no need to modify the current source II of the retained node. The modification of the equivalent admittance matrix YII of the retained node can be carried out in a simple way according to its characteristics. [b]4 Analysis and modification of the equivalent admittance matrix of the retained node[/b] The equivalent admittance matrix of the retained node is modified according to equation (13). For multi-node distribution networks, the two situations mentioned in Section 2 will occur. Analyzing the differences between the retained node admittance matrices YII and YNII before and after the correction reveals the following: ① The two matrices have the same order; ② If Yij represents the mutual admittance in YII and YNij represents the mutual admittance in YNII, then YNij = Yij (i ≠ j), meaning the impedance parameters between the retained nodes remain unchanged; ③ The self-admittance YNii = Yii (i ≠ m), indicating that none of the other nodes connected to node i have been eliminated; ④ The self-admittance YNmm ≠ Ymm (m ≠ i), indicating that at least one of the nodes connected to node m has been eliminated. Analysis shows that for YNII, only the YNmm and Ymm corresponding to node m, which are associated with the eliminated nodes, are not equal. The number of nodes in YII that possess the properties of node m is extremely limited, because in practical power distribution networks, the network can often be divided into retained and eliminated parts at a certain bus. Therefore, to correct YII to YNII, it is only necessary to calculate YNm1m1～YNmnmn one by one according to the definition based on the network structure. [b]5 Example[/b] The power supply system of Shanxi Tianji Coal Chemical Group has two 110 kV incoming lines; two self-provided generators of 30 MW and 16 MW respectively; more than a dozen 2 MW asynchronous motors; ten large substations; and more than 180 electrical devices that require short-circuit current calculation. After simplification to the maximum extent possible, there are still more than 170 nodes and more than 200 branches that need short-circuit current calculation to meet the requirements of safe and reliable power supply management, such as relay protection setting calculation and reasonable selection of electrical equipment. However, the computer used for power supply management in this unit is limited by capacity and cannot calculate networks with more than about 100 nodes under Windows 95 or 98 environment. Using the method proposed in this paper, we developed a short-circuit current calculation software for this system, which runs well on the existing computer and meets the power supply management requirements of this unit. 5.1 Algorithm steps (1) Decompose the network shown in Figure 2 into two (or more) parts at appropriate busbars, with a network of about 100 nodes as the internal system and the rest as the external system. The nodes of the internal system are reserved nodes. When numbering nodes, first number the passive nodes in the retained nodes, then number the power supply nodes in the retained nodes, then number the boundary nodes, and finally number the nodes to be eliminated, as shown in Figure 2. The reactance parameters in the figure are per-unit values ​​calculated from the actual components. For the sake of illustration, the resistance parameters that should be calculated for the low-voltage network are omitted. Nodes 13 and 14 are boundary nodes. (2) Group the active nodes with relevant properties (such as power supply, large asynchronous motor, etc.) in the external system to be eliminated into a group, as shown in nodes 15-17 and 18-20 in Figure 2. Using the above method, calculate YR and YK for each group, and construct the REI network as shown in Figure 3. [img=320,172]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/t67-1.gif[/img] Fig.2 Distribution network sample [img=250,132]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/t67-2.gif[/img] Fig.3 Make up supply mains nodels of outside part system into REI network SR1 = S15 + S16 + S17, connected to boundary node 13 via impedance X13 ~ R1, resulting in nodes 15 to 17 becoming passive nodes and can be included in the elimination node set E. The same method can be used to correct active nodes 18 to 20. Nodes 13 and 14 with power supply are called boundary nodes and are assigned to the reserved node set I. Their equivalent network is shown in Figure 4. [img=310,164]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/t67-3.gif[/img] Fig.4 Reservation nodes equivalent network (3) Input the original data according to the equivalent network in Figure 4, and the short-circuit current is calculated to be accurate enough. Under this condition, the admittance matrix only corrects the external system power supply, but does not correct the external system admittance, which is represented by YN1II. The difference between YN1II and YNII is the same as analyzed in Section 4, only the self-admittance of the boundary points is not equal. If a more accurate short-circuit current value is required, YN1II needs to be corrected to YNII, that is, the self-admittance of the boundary node YN1mm needs to be corrected to YNmm. In the example, the admittance at boundary point 13 is YN11313. The correction method is to first calculate Y1313 according to Figure 2, and then calculate Y[img=168,32]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/67-3.gif[/img][img=115,34]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/67-4.gif[/img]. Using this method, Y [sup]N[/sup][sub]mm[/sub] is calculated one by one, and Y [sup]N[/sup][sub]II[/sub] is used as the retained node admittance matrix applied in the calculation process. The short-circuit current of any branch of the power distribution network, under any operating mode, at any point and of any type, can be calculated using the above method. 5.2 Accuracy Analysis Taking the three-phase short circuit at point 13 in the example as an example to illustrate the calculation accuracy, the fault at this point is most affected by the short-circuit current supplied by the asynchronous motor. (1) Calculate using the conventional short-circuit current calculation method [img=297,58]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/67-5.gif[/img] (2) Calculate using the short-circuit current calculation method proposed in this paper [img=316,49]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/67-6.gif[/img] (3) When the nodal admittance matrix YN1II is corrected to YNII, Id＝2.63 5.3 Flowchart [img=270,347]http://zszl.cepee.com/cepee_kjlw_pic/files/wx/zgdjgcxb/zgdj2000/0004/image4/t88-1.gif[/img] [b]6 Conclusions[/b] (1) Multi-node power distribution systems can be decomposed into internal and external systems at appropriate busbars using the method proposed in this paper, and each part can be treated as an internal system to complete the short-circuit current calculation. (2) The role of the REI equivalent network is to connect the power sources in the external system to the boundary nodes, and the boundary nodes are assigned to the reserved nodes. The original active nodes become passive nodes, which significantly improves the accuracy of short-circuit current calculation. [b]References:[/b] ［1］ Zhu Junwei. Power System Analysis (Volume 1)［M］. Beijing: China Water Resources and Electric Power Press, 1995, 11: 119-124. ［2］ Mi Linshu, Jiang Shifang. Computer-Aided Analysis of Power Grid Relay Protection［M］. Beijing: China Water Resources and Electric Power Press, 1995, 11: 25-44. ［3］ Wu FF, Narasimhamurthi N. Necessary Condition for REI Reduction to be Exact［C］. IEEE PES Winter Meeting, 1979: A79 065-4. ［4］ Chen Yamin. Power System Calculation Program and Its Implementation［M］. Beijing: China Water Resources and Electric Power Press, 1995, 11: 11-146.