Decoupling and dimension reduction method for distribution system security region

The application of the security region methodology in a practical distribution system with large scale normally requires large computer memory and high computation time. To overcome this problem, this article proposes a decoupling and dimension reduction method, which can significantly accelerate the calculation of distribution system security region (DSSR) and is important for the application of the DSSR theory in large ‐ scale distribution systems. First, the definition of DSSR dimension reflecting the size of so-lution space and the time complexity is proposed. And the solution algorithm for DSSR dimension is also given. Second, a decoupling and dimension reduction method suitable for the analysis of DSSR is proposed. Following the method, an incidence matrix can be obtained from the DSSR expressions, which can be further divided into multiple block matrices. According to the feeder combinations of the block matrices, the distribution system can be decoupled into multiple sub ‐ networks for more efficient analysis. Finally, a 10kV distribution network is used in case study to validate the proposed method. The results for a time ‐ consuming calculation, that is, TSC curve calculation, show that the proposed method can reduce the computation time significantly, making the time ‐ consuming calculation suitable for the analysis of large ‐ scale cases.

number of loops and so on [19].Evaluation indicators and analysis methods are proposed to reflect the sufficiency level of the scale of networks [20].For distribution system, a complete evaluation indicator system and comprehensive model of distribution network scale are established based on the principle of grey relational analysis and analytic hierarchy process [21].The scale of DSSR is also a fundamental issue in the study of the DSSR.
The scale of an object or system can be described as dimension [22].Dimension can be represented by the number of parameters required to describe an object [23].Two forms of DSSR dimension are proposed to describe the scale of DSSR [24].One is named binary dimension: it is defined with two variables, the number of security boundaries (Q b ) and the number of variables (Q v ) in them.The other is named unary dimension: the number of equivalent feeders is introduced, denoted as N e , and the DSSR dimension is donated as D u , which is expressed in the most commonly used form of N e � N e .
The research of the DSSR dimension has a certain foundation, but the large scale of the real distribution system leads to a very high dimension of its security region.To address this challenge, dimension reduction methods for DSSR are required.Existing dimension reduction methods in DSSR research are primarily employed for visualising DSSR.In these methods, the power injections at two or three important nodes in the distribution system are selected for visualisation while keeping the power injections at other nodes constant.The authors in refs.[10,25], and [26] visualised the static security region of the distribution network in 2-dimensional cross sections.Wan et al. [11] study the high-dimensional boundary by mapping the maximum uncertainty boundary of distributed generators in active distribution networks to a 2-dimensional plane.Effective decoupling methods for DSSR have not been proposed in existing studies.Therefore, this paper proposes a decoupling and dimension reduction method for DSSR.The superiority of the proposed method is in the dimension reduction of the expression of DSSR, which can significantly improve the calculation speed of DSSR-associated calculations.The main contributions of this paper include: (1) A more effective definition of DSSR dimension comparing to the existing relevant study is proposed for dimension reduction of DSSR, which can further contribute to the decoupling of distribution systems.(2) A DSSR dimension calculation method based on DSSR expressions and incidence matrix is proposed, which has demonstrated its effectiveness in reducing the dimension of DSSR.(3) A decoupling and dimension reduction method for DSSR is proposed to enable rapid security region-based analysis and accelerate the security analysis of the real-world distribution systems.The effectiveness of the proposed method has been validated by calculating Total Supply Capability (TSC) curve.Compared to the existing method for calculating TSC curve, the proposed method can significantly increase the calculation speed while reducing the memory requirement.

| Background of DSSR dimension
The cardinality of a finite set represents the size of the set, which is just the number of elements in the set.And if a vector space V has a basis of cardinality K, we say that V is Kdimensional [23].Similarly, the DSSR dimension describes the size of DSSR space.Higher dimension requires more parameters to describe a DSSR, which indicates that the DSSR problem has a larger scale.Two forms of DSSR dimension are proposed to describe the scale of DSSR [24].One is defined with two parameters, the number of security boundaries and the number of variables in them.Although this definition can represent the scale of the security region, there are shortcomings: firstly, the two-parameter approach is not normal for DSSR users to construct the problem space; secondly, comparing the scale of two DSSRs is difficult when the comparison results of these two parameters are not consistent.Thus, another form of DSSR dimension is defined with one parameter, that is, the number of equivalent feeders, which is denoted as N e .For the distribution network with N feeders, after considering the feeder link, it is equivalent to the full contact distribution system with N e feeders, and the dimension is N e � N e .This dimension cannot completely distinguish feeders with links.For example, the real dimension of a distribution network with n single-links is 2, but the result of unary dimension is 4n ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 9n−5 p [24].In ref. [24], the researchers clarify that a distribution network with n single-links is equivalent to a virtual full-link network with 4n  ffi ffi ffi ffi ffi ffi ffi ffi 9n−5 p feeders.The unary dimension is the square of the number of the feeders in the equivalent full-link network.The details on how to obtain the number of the feeders of equivalent virtual full-link network can refer to [24].The definition of DSSR dimension in ref. [24] is not convenient to use since it is computationally complex and hard to be obtained from the incidence matrix of the DSSR expressions.

| A new definition of DSSR dimension
In order to overcome the shortcomings of large computational cost and difficult transformation of incidence matrix and to conform to the traditional definition of dimension as well, an integer responding to the number of parameters, a new DSSR dimension is defined in this paper.The number of parameters is the number of linked feeders, the maximum of which determines the dimension of DSSR space.
For a distribution network with N feeders, the DSSR dimension D m is denoted as k is the number of groups of feeder combinations.A feeder combination is the combination of feeders linked with each other through the tie switches [24].m i is the number of feeders in the ith feeder combination.Physically, the DSSR dimension D m in this paper denotes the maximum number of feeders which link with each other in a distribution system.This dimension partly reflects the interdependence of the feeders in the distribution system, which can be used for decoupling the distribution system.

| Relationship between DSSR dimension and time complexity of DSSR calculation algorithms
Assuming that n is the size or number of input data, T(n) is the execution times of a given code.Algorithms generally use a simplified estimate O(f(n)) of T(n) to measure the execution speed of the code, and this simplified estimate O(f(n)) is called the time complexity [27].The time complexity O(f(n)) is the extent to which the solution time increases when the input variable n increases.
The DSSR dimension expresses the scale of DSSR problem, thus determining the complexity of constructing DSSR or searching optimal solutions in DSSR space by using a specified algorithm.The DSSR dimension and the time complexity of DSSR calculation algorithms have one-to-one correspondence.For example, for the calculation of TSC curve [28], if the dimension of the power grid is D m , then the time complexity of the conventional TSC curve algorithm is O(n Dm ).

| DSSR DIMENSION CALCULATION METHOD
The steps of the method for calculating the DSSR dimension are as follows.For clarity, each step is explained with a fivefeeder case network as shown Figure 1.

| DSSR expression
Step 1: Obtain the DSSR expression Ω DSSR based on detailed feeder interconnection relationship and component parameters (including the transformer and feeder capacities) of the distribution system [7].DSSR is defined as the set of all operating points that satisfy N-1 security constraints of the distribution system.Therefore, the feeder load and transformer load should satisfy N-1 security constraints including feeder N-1 security constraints and transformer N-1 security constraints.
1-1: Feeder N-1 security constraints indicate that the power load at the feeder out of service can be transferred to other connected feeders through tie switches, which do not incur any overloading problem.Assuming feeder m is out of operation after N-1 contingency, the power load at feeder m can be transferred to feeder n via the tie switch, then the N-1 constraint for feeder m can be expressed as follows: where S F;x (x = m, n) is the power load supplied by feeder x; c F;n is the capacity of feeder n.Equation (2) guarantees that after load transfer from feeder m to feeder n, no overload problems occur.N-1 security constraints for other feeders can follow the same principle as in Equation ( 2).Take the N-1 security constraint for feeder 1 of the case in Figure 1 as an example, when feeder 1 is out of operation after N-1 contingency, the power loads S F;1 at feeder 1 can be transferred to feeder 2 via the tie switch.Then the N-1 constraint for feeder 1 can be expressed as follows: In Figure 1, there is also a two-supply-one-backup connection mode containing feeder F3, F4, F5.For brevity, feeder F3 is used as a backup for feeder F4 and F5.
1-2: Transformer N-1 security constraints express the security after N-1 contingency at each transformer, in a similar way as feeder N-1 security constraints.When transformer i is out of operation after N-1 contingency, all its feeder loads are transferred to other transformers via the tie switches.The security constraint regarding transformer i should guarantee that no overloading problems occur in all the transformers after N-1 contingency at transformer i, which is expressed in Equation (4), where S i;j T ;tr is the load transferred from transformer i to transformer j; c T;j is the rated capacity of transformer i.
Take the N-1 constraint for transformer T 1 of the case in Figure 1 as an example, when transformer T 1 is out of F I G U R E 1 A five-feeder case network for illustrating the calculation steps of distribution system security region (DSSR) dimension.
XIAO ET AL. operation after N-1 contingency, its power load S F,1 at feeder F1 should be transferred to transformer 2 via the tie switch, and the power load S F,3 at feeder F3 should be transferred to transformer T 3 .The N-1 constraint for transformer T 1 can be expressed as follows: In summary, the DSSR expression Ω DSSR consisting of the above constraints can be formulated as For the case in Figure 1, the DSSR expression Ω DSSR can be formulated as

| Incidence matrix
Step 2: Generate the incidence matrix from the DSSR expression.
The incidence matrix [24] is used to store the linking relationship between feeders in a distribution system, which is denoted as A as follows: where a ij is the link between feeder i and feeder j.When feeder i and feeder j are linked, a ij ¼ 1; otherwise a ij ¼ 0 The key to the incidence matrix is the linking relationship between each of the two feeders in a distribution system, which can be identified from the DSSR expressions.The identification process can be summarised in the following two situations.
First, if the variables of power loads at two feeders (e.g.feeder i and feeder j) appear in the same constraint as in Equation ( 4) or Equation ( 6), these two feeders are directly linked with each other and the corresponding matrix element a (i,j) is set as 1.This can be illustrated by the example of the first constraint in Equation ( 7) (i.e. S F;1 þ S F;2 ≤ c F;2 ) Since the constraint contains the power loads S F,1 and S F,2 , their associated matrix element a (1,2) is set as 1, indicating that feeders F 1 and F 2 are directly linked and the power loads under these feeders should not exceed the capacity of the transferred feeder F 2 .
Second, if the variables of power loads at feeder i and feeder j appear in different DSSR constraints, but these constraints are dependent in terms of these two variables, the two feeders are indirectly linked.Under this situation, the matrix element a (i,j) should also be set as 1.This is because if constraints associated with two variables are not independent, a change of one variable will affect the range of the value for the other variable.Therefore, the two variables (or the two feeders) are indirectly linked with each other.Take the fourth and the fifth constraints of (7) (i.e. S F,4 + S F,3 ≤ c F,3 and S F,5 + S F,3 ≤ c F,3 ) as an example.From these constraints, the allowable transferred power loads S F,4 and S F,5 are both influenced by the power load S F,3 , which indicates the indirect linking relationship between feeders F4 and F5.Therefore, their associated matrix element is recorded as 1.
Following the generation process of the incidence matrix above, the whole incidence matrix for the case network in Figure 1 is shown in Figure 2 for reference.

| Block matrix
Step 3: Divide the incidence matrix into block matrices [29].If a matrix is partitioned by sequential partitions of its rows and columns, the resulting partitioned matrix is called a block matrix.
The obtained incidence matrix in Step 2 can be further divided into block matrices by identifying non-zero elements following two ways.

F I G U R E 2
The incidence matrix and the results of block matrices of the five-feeder case network.

3-1:
Obtain the diagonal block matrices (D i ).For the generation of block matrix D 1 , firstly use a 11 as the first element, then find the largest all-ones square matrix as D 1 .The generation of other block matrices follows the similar way.The difference is the first element for D i (i≠1) uses the first element of the remaining diagonal elements after generation of D 1 -D i-1 .The generation of the block matrices stops when a nn is included.Following the generation process of the block matrices, the scale of D i is d i � d i , and the dimension of D i is d i .
3-2: Obtain the non-diagonal block matrices (E i ), all with non-zero elements.The scale of E i is e i � e j , and the dimension of E i is e i + e j .Due to the symmetry of the incidence matrix, a more efficient way is to obtain the nondiagonal block matrices in the upper-triangular part of the incidence matrix.And the diagonal block matrices in the lower-triangular part is symmetric with those in the uppertriangular part.
The division of the incidence matrix and the results of the block matrices of the case network are shown in Figure 2.
As with Section 2.2, the DSSR dimension is defined as the maximum value of the number of feeders with links in this paper.Since one block matrix represents a group of feeder combination (i.e. the combination of feeders linked with each other through the tie switches), the DSSR dimension in Equation ( 1) is equal to the maximum value of the block matrix dimension as follows: d i is the number of rows and columns of the diagonal block matrix; e i and e j is the number of rows and columns of the non-diagonal block matrix respectively.For the case in Figure 1, at most three feeders (i.e.F1, F2, F3 or F1, F4, F5 or F3, F4, F5) are directly or indirectly linked.Therefore, the dimension of the network is 3, which can be obtained from the diagonal block matrix D 1 with the maximum rows/columns (or the non-diagonal matrix E 1 or E 2 with the maximum sum of rows and columns) in Figure 2.

| DECOUPLING AND DIMENSION REDUCTION METHOD OF DSSR
Based on the DSSR dimension calculation method, this section provides the decoupling and dimension reduction method for DSSR.From the obtained block matrices, the distribution network can be decoupled as follows.The remaining steps are as follows.
Step 5: Extract the feeder combinations corresponding to the diagonal block matrix D i and the non-diagonal block matrix E i .The feeders associated with the row and column elements of a block matrix can form a group of feeder combinations.
Step 6: Divide the distribution network into several subnetworks according to the feeder combinations.The feeders in each group of feeder combinations can form a sub-network.In this regard, feeders without electrical connections after N-1 fault are decoupled.In other words, a large-scale distribution network can be divided into several smallerscale sub-networks.Accordingly, the corresponding highdimensional security region of the network can be divided into several lower-dimensional security regions, which achieves the dimension reduction for the high-dimensional security regions.

| Overview
The 10 kV distribution network case in Figure 3 [24] is used for the case study.This case has 4 substation transformers and 28 feeders.The capacities of transformers and feeders are 80MVA and 10MVA respectively.The calculations of DSSR are carried out on a 64-bit Window 10 with Intel(R) Core(TM) i5-10200H @ 2.40 GHz CPU, 8 GB RAM.

| DSSR expressions
Step 1: Obtain the expression of DSSR for the case network.
The power load should satisfy N-1 security constraints.
The DSSR expression for the 28-feeder case (denoted as Ω DSSR ) is established as below: S F;i�j means the sum of S F;i to S F;j .For instance, S F;1�5 is Ω DSSRx (x = 1,2,3,4) are the N-1 security constraints of the four two-supply-one-backup connection modes in the case.For each two-supply-one-backup connection, one feeder is used as a backup for two other feeders.For clarity, Ω DSSRx (x = 1,2,3,4) are written separately as follows:

| Incidence matrix
Step 2: Generate the incidence matrix from the DSSR expression.The identification process can be summarised in the following two situations.First, if the variables of two power loads at two feeders (e.g.feeder i and feeder j) appear in the same constraints as in Equation (10), the corresponding matrix element a (i,j) is set as 1.For example, S F;1�5 þ S F;15�21 ≤ c T ;3 , power loads such as S F,1 , S F,2 , S F,3 are in this formula, then a (1,2) , a (1,3) and a (2,3) are recorded as 1.
Second, if the variables of power loads at feeder i and feeder j appear in different DSSR constraints, but these constraints are dependent in terms of these two variables, the matrix element a (i,j) should also be set as 1.For example, S F; 13  � � , the a (13,14) is recorded as 1.

| Block matrix
Following the step 3 in Section 3.3, the divided results of block matrices are shown in Figure 4.
From Figure 4, the incidence matrix can be divided into four diagonal block matrices (i.e., D 1 -D 4 marked in yellow) and six non-diagonal block matrices (i.e.E 1 -E 6 marked in blue).D 1 -D 4 and E 1 -E 6 are numbered from the top-left to the bottom-right.
Following the definition of DSSR dimension in Section 3.4, the dimensions of the block matrices for the 28-feeder case are summarised in Table 1.The scales of these block matrices are also shown for reference.

| DSSR dimension
From Table 1, the DSSR dimension is 12, which corresponds to the largest dimension among the block matrices, that is, the dimension of E 1 , E 2 or E 5 .Therefore, the time complexity of DSSR calculation algorithms is O(n 12 ).

| Feeder combination
Corresponding to the diagonal and non-diagonal block matrices in Table 1, 10 feeder combinations are obtained in Table 2.
Take E 1 as an example: E 1 contains the elements at the intersection of rows 1-5 and columns 15-21, so the feeders associated with 1-5 rows and 15-21 columns elements can form a group of feeder combination, that is, (F1-F5, F15-F21).

| Decoupling and dimension reduction
The obtained feeder combinations can decouple the distribution network into ten sub-networks with no more than 12 feeders.
The 4 sub-networks from the feeder combinations of diagonal block matrices are shown in Figure 5, while the 6 sub-networks obtained from the feeder combinations of non-diagonal block matrices are shown in Figure 6.Accordingly, the corresponding 28-dimensional security region of the network is divided into 10 lower-dimensional security regions, which achieves the dimension reduction for the highdimensional security region of the original distribution network.

| Application: TSC curve calculation
TSC refers to the maximum load supply capability of a distribution network when it meets the N-1 security criterion [30].Since the load information is contained in the boundary points of the security region of the distribution network, the TSC can be obtained by summing the loads at the boundary points [15].The TSC curve can be obtained by arranging the TSC values corresponding to the respective boundary points in order.Compared with TSC, the TSC curve fully describes the extreme load-carrying capability of a distribution network [15].
The proposed decoupling and dimension reduction method in this paper is applied to accelerate the calculation of TSC curve [28], which is a typical DSSR-associated calculation.To verify the effectiveness of the proposed method, the result is compared with that calculated by using the existing algorithm in ref. [28].It should be mentioned that the algorithm in ref. [28] is also used to calculate the TSC curve after using the proposed method to decouple the case network into multiple sub-networks.
T A B L E 2 Block matrices and its corresponding feeder combinations.

Block matrix
Feeder combination F I G U R E 5 Sub-networks obtained from the feeder combinations of diagonal block matrices.

F I G U R E 6
Sub-networks obtained from the feeder combinations of non-diagonal block matrices.

T A B L E 1
The scale and dimension of block matrices of the 28-feeder case.Since the existing algorithm in ref. [28] is based on sampling the state space of security region, this DSSR-associated calculation is very time-consuming when applying to largescale distribution networks.By using the method proposed in this paper, the dimension of DSSR is reduced from 28 to 12. Therefore, the time complexity of DSSR calculation algorithm is also reduced from O(n 28 ) to O(n 12 ), which reduces more than 50%.The sampling step s when sampling the strict boundary is 10MVA.The number of sampling points is denoted as

Block matrix Scale Dimension
is the number of points that can be sampled on each boundary and N is the number of feeders [28].
The calculation process and computation time of a 36feeder case are given in the Appendix A.
The results of TSC curve of the 28-case network before and after using the proposed method in this paper are compared, which are shown in Table 3.
After sampling the strict boundary, TSC curve is plotted by taking the sampling number as abscissa and the total feeder load as ordinate [28].The TSC curves of sub-networks were calculated; then the TSC curve "summary after decoupling" was obtained, the detailed data and process is in Appendix B. After summary, the TSC curve is exactly the same as that before decoupling, which verifies the effectiveness of the proposed method.The reason why the TSC curve with the use of decoupling is the same as the one without decoupling is as follows: Firstly, the upper and lower limits of the TSC curve before and after decoupling are the same.There are no overlap feeders in which there are tie switches connected between different sub-networks after decoupling, which ensures independent analysis of the sub-networks.In this regard, the range of the feeder load that satisfy N-1 security constraints stays the same after decoupling.Secondly, since the sampling step is constant, the sampling values of the feeder loads before and after decoupling are the same.These two factors result in the fact that the range of the total load of each boundary point of DSSR (i.e. the TSC value) remains unchanged.
Secondly, the shape of the TSC curve stays the same after the decoupling of the distribution network.The shape of the TSC curve is determined by the proportion of different TSC values on the TSC curve.Since the ranges of different feeder load and the sampling values are unchanged after decoupling, the distribution and proportion of feeder loads in each subnetwork are the same as those before decoupling.The summary of TSC curve after decoupling aims at obtaining the Cartesian product of the vectors of the points on the security boundary, and each element of the Cartesian product is composed of the feeder variables selected from each vector of the points on the security boundary in order.Therefore, the distribution and proportion of the feeder loads after summation are the same as those before decoupling.Since the proportions of different TSC values on TSC curve before and after decoupling are the same, the shape of the TSC curve keeps the same.
In summary, the upper and lower limits and the shape of the TSC curve before and after decoupling are the same, hence the TSC curve with the use of decoupling is the same as the one without decoupling.
The computation time and memory are compared in Table 4.
The total time of calculating TSC curve after decoupling is 7.842 ms, which comprises 6.348 ms for the establishment and decoupling of the incidence matrix and 1.494 ms for the TSC curve calculation.The total time for calculating TSC curve using the proposed method in the paper is only approximately 0.05% of the time before decoupling.This shows that the proposed method can significantly accelerate the calculation of TSC curve especially in large-scale distribution networks.With a fixed sampling step, the number of sampling points for calculating the TSC curve grows exponentially with the number of feeder lines.By using the method proposed in this paper to split the incidence matrix of the security region into blocks, the distribution network can be decoupled into multiple subnetworks with fewer feeder lines, thereby significantly reducing the number of sampling points and improving the performance of TSC curve calculation.The condition of the acceleration is that the distribution network can be decoupled and the DSSR dimension can be reduced.The only one exception is that the network is fully linked between feeders, which is explained in Appendix C. Since a real distribution network contains many substations, of which the feeders are normally not fully linked as the case in Appendix C. The methodology in this study can be applicable in most cases.
The computation time and memory are compared in Table 4.The memory during the calculation process is mainly used for storage of the vectors of sampling points on the security boundary.(i.e. the feeder load information contained in the points on the TSC curve).Considering we create int8 variable in MATLAB that takes up 1 byte, the memory usage can be expressed as x*N ÷ 1024(kb), where x represents the number of sampling points on the security boundary and N represents the number of feeders.As an illustration, consider the first row of Table 4, which corresponds to the network prior to decoupling.With 28 feeders, a total of 2048 TSC curve points are obtained, yielding a memory usage of 56(kb).
The maximum requested memory of the sub-networks is 0.1875kB with only approximately 0.33% of the requested memory before decoupling.This shows that the proposed method can significantly reduce the computation memory of TSC curve.The main memory is used for storing the vectors of the sampling points on the security boundary.Since the vectors of the sampling points on the security boundary is constantly needed during the calculation of the TSC curve, the memory cannot be freed during the calculation.

| CONCLUSION
For a large-scale distribution system, the application of the security region methodology normally requires large computer memory and high computation time due to the high dimension of DSSR.To overcome this problem, this paper provides a decoupling and dimension reduction method for the practical application of DSSR, which has significant implications for applying security region analysis methods to large-scale distribution networks in practice.The main contributions are as follows: (1) DSSR dimension is defined as the maximum number of feeders with links, which is more suitable for dimension reduction compared to the existing definition of DSSR dimension.This definition is able to overcome the shortcomings of large computational cost and difficult transformation of incidence matrix and conform to the traditional definition of dimension as well, an integer responding to the number of parameters.(2) A DSSR dimension calculation method is proposed, which involves formulating the DSSR expressions associated with the target distribution system, extracting the incidence matrix from these expressions, and generating block matrices based on the obtained incidence matrix.From the case study, the method is easy to be implemented.(3) A decoupling and dimension reduction method for DSSR based on the dimension calculation method is proposed for the first time.By utilising the method, a distribution system can be decoupled into multiple smaller sub-systems with fewer feeders.Accordingly, this lowers the dimension of the distribution system, facilitating more efficient security analysis of the distribution system.
A 28-feeder case and a 36-feeder case were used to verify the effectiveness of the proposed method.For the calculation of TSC curve (which is proved to be time-consuming in previous studies), the decoupling and dimension reduction method performs more efficiently than the previous studies.For the two cases, the time complexity is reduced by more than 50%, increasing the computation speed by more than a thousand times.Other cases also work well but the specific speedup depends on the specific case.The proposed method is of great significance for applying DSSR-associated calculations to largescale distribution networks.The decoupling method can not only be used in DSSRassociated calculation, but also be used to divide the largescale distribution system into smaller-scale sub-systems, which is worthy of further study in distribution system analysis.In addition, distributed generators, grid-connected energy storage, and other components in smart distribution network will also be considered in the future.

APPENDIX A Decoupling and dimension reduction calculation for TSC cur ve of 36-feeder case
For the 36-feeder case, in the computation environment described in 5.1 of this paper, the TSC curve cannot be calculated due to the lack of memory by using the method in ref. [28].After using the proposed method, the time for TSC curve calculation is 4.946 ms and the maximum requested memory is 1kB.
The computation process is given below.

A.1 | Over view
This case in Figure A1 has 4 substation transformers and 36 feeders.The detailed capacities of the transformers and feeders can refer to Section 5.1.

A.3 | Incidence matrix blocking
The divided results of block matrices are shown in Figure A2.

A.4 | Feeder combination
The feeder combinations are obtained in Table A1.
T A B L E A1 Block matrices and its corresponding feeder combinations.

Block matrix Feeder combination
- A. 5 | Decoupling and dimension reduction The 4 sub-networks obtained from the diagonal block matrices are shown in Figure A3.The six sub-networks obtained from the non-diagonal block matrices are shown in Figure A4.

A.6 | TSC cur ve
The TSC curve points of each sub-network after decoupling are summarised to obtain the TSC curve results of the whole distribution network, as shown in Figure A5.
The computation time is shown in Table A2.

APPENDIX B
Process to obtain the "summar y after decoupling" TSC cur ve of 28-feeder case The TSC curves of the sub-networks after decoupling are shown in Table B1, which are obtained by the method in [28].
The sub-networks whose TSC curves are not horizontally straight, E 1 and E 5 , are treated first.Considering the feeders in which there are tie switches connected, E 1 contains feeders F1-F5 and F15-F19.The distribution of the supply capability of (F1-F5, F15-F19) obtained by the method in [28] is shown in Table B2.
Summarising Table B2 with Table B3, the results are shown in Table B4.
Arrange Table B4 in the order of supply capability from smallest to largest.The results are shown in Table B5.
The whole distribution network has a total of 28 feeders (F1-F28), and some feeders are still missing in Table B5.Then look for the missing feeders from other sub-networks.
D 1 contains feeders F6-F9.The distribution of the supply capability of (F6-F9) obtained by the method in [28] is shown in Table B6.
Summarising Table B5 with Table B6, the results are shown in Table B7.
D 3 contains feeders F20-F23.The distribution of the supply capability of (F20-F23) obtained by the method in [28] is shown in Table B8.Summarising Table B7 with Table B8, the results are shown in Table B9.This is the end.Table B9 has contained all 28 feeders.Using the first 2 columns of Table B9, the TSC curve can be obtained by a plot software (Microsoft Office Excel used in this paper).

APPENDIX C
Full-link case network that cannot be decoupled This full-link case network in Figure C1 has 2 substation transformers and 10 feeders.
The corresponding DSSR expression is shown in formula C1.
The corresponding incidence matrix is shown in Figure C2.The incidence matrix of the full-link case is an all-one matrix, which contains only one block matrix, that is, the incidence matrix itself.Therefore, the full-link case network cannot be decoupled.
For real distribution networks with many substations, they are normally not fully linked and can be decoupled even if the structure of each substation is the same as in Figure C1.For example, the two substations in Figure C3 follow the same structure as in Figure C1, but there are no links between some feeders (e.g.no link between F1 and F20).Therefore, the case in Figure C3 can be decoupled.

T A B L E 3
Total Supply Capability (TSC) curves before and after decoupling.

F I G U R E A 3
Sub-networks obtained from diagonal block matrices.F I G U R E A 4 Sub-networks obtained from non-diagonal block matrices.F I G U R E A 5 Total Supply Capability (TSC) curve of 36-feeder case.

T 3 E 4 E 5 E 6 T
A B L E B1 TSC curves of the sub-networks after decoupling.A B L E B2 Sequential distribution of the supply capability of (F1-F5, F15-F19).Sequential distribution of the supply capability of (F10-F14, F24-F28).

F I G U R E C 3
The case after replication of substations in FigureC1.XIAO ET AL.

T A B L E A2 Decoupling
and dimension reduction to calculate Total Supply Capability (TSC) curve time.