Novel Method for Rapidly Constructing Active Power Steady-State Security Regions Incorporating the Equivalent Reactances of TCSCs

: Active power steady-state security regions (APSSRs), which can provide guidance for prevention and control through security checks, is of great importance for the safe operation of power systems in which more and more sustainable energy power generation is integrated. As a mature ﬂexible AC transmission system (FACTS) device, thyristor-controlled series compensators (TCSCs) can carry out series compensation for the transmission line by controlling its equivalent reactance. With the change of the equivalent reactance parameter of a TCSC, the nodal admittance matrix and power ﬂow distribution of the power system also changes. Inevitably, the APSSR will be different. Therefore, it is necessary and important to further incorporate the equivalent reactance parameters of TCSCs in the APSSR expression, which is generally established in the space of node active power injections. In this paper, a rapid construction method of APSSRs incorporating the equivalent reactances of TCSCs is proposed. Firstly, applicability and efﬁciency of the conventional APSSR construction method for power systems with TCSCs are analyzed. Further, with equivalent disconnection of TCSC branches, the effect of TCSC equivalent reactances on the distribution of active power ﬂow through changing the structure parameters is treated as modifying node active power injections. On this basis, explicit expressions of APSSRs with a single TCSC equivalent reactance parameter and double TCSC equivalent reactance parameters are derived, respectively. Moreover, by deducing the general formula of APSSRs with multiple TCSC equivalent reactance parameters, the feasibility of the proposed method for power systems with multiple TCSCs is analyzed. Eventually, via benchmarks with different scales and a different number of TCSCs, validity and superiorities of the proposed method in computational efﬁciency are demonstrated.


Introduction
With the continuous integration of sustainable energy power generation such as wind power and photovoltaics, and the increase of load uncertainty in power markets under competitive mechanisms, the power flow of smart grids is becoming more complex with random variations of power supply output and uncertainty of load [1][2][3][4]. As an important part of security region methodology, steady-state security regions can be used for real-time on-line security analysis and monitoring as well as prevention and control of the power system [5].
The concept of a steady-state security region (SSR) was first proposed by [6] in 1975. It is the set of node power injections that can ensure the safe and stable operation of a power system under a given network structure. An active power SSR (APSSR), which is more concerned with the transmission network, mainly considers the branch active power flow constraint and output active power constraints For the conventional construction method of APSSRs, the nodal admittance matrix is parameterized by considering the equivalent reactance parameter of the TCSC. As there is a high computational burden caused by the inversion of the parameter matrix, the efficiency of the conventional method is not ideal to construct an APSSR incorporating the equivalent reactance of the TCSC.
In this paper, a rapid construction method of APSSRs, which incorporates the equivalent reactance parameter of TCSCs, is proposed. Firstly, the applicability and efficiency of the conventional APSSR construction method for the power system with a TCSC are analyzed. It is pointed out that the efficiency of the existing method will not be ideal due to the heavy computational burden in obtaining the inversion of a high-dimensional parametric matrix caused by the parameterization of the node admittance matrix. To avoid this issue with equivalent disconnection of TCSC branches, the effect of TCSC equivalent reactances on the distribution of active power flow through changing the structure parameters is treated as modifying node active power injections. On this basis, the explicit expressions of APSSRs with a single TCSC equivalent reactance parameter and double TCSC equivalent reactance parameters are deduced, respectively. Moreover, by deducing the general formula of the APSSR with multiple TCSC equivalent reactance parameters, the applicability of the proposed method for a power system with multiple TCSCs is analyzed. For the conventional construction method of APSSRs, the nodal admittance matrix is parameterized by considering the equivalent reactance parameter of the TCSC. As there is a high computational burden caused by the inversion of the parameter matrix, the efficiency of the conventional method is not ideal to construct an APSSR incorporating the equivalent reactance of the TCSC.
In this paper, a rapid construction method of APSSRs, which incorporates the equivalent reactance parameter of TCSCs, is proposed. Firstly, the applicability and efficiency of the conventional APSSR construction method for the power system with a TCSC are analyzed. It is pointed out that the efficiency of the existing method will not be ideal due to the heavy computational burden in obtaining the inversion of a high-dimensional parametric matrix caused by the parameterization of the node admittance matrix. To avoid this issue with equivalent disconnection of TCSC branches, the effect of TCSC equivalent reactances on the distribution of active power flow through changing the structure parameters is treated as modifying node active power injections. On this basis, the explicit expressions of APSSRs with a single TCSC equivalent reactance parameter and double TCSC equivalent reactance parameters are deduced, respectively. Moreover, by deducing the general formula of the APSSR with multiple TCSC equivalent reactance parameters, the applicability of the proposed method for a power system with multiple TCSCs is analyzed.

Brief Illustrations of the Existing APSSR Construction Method
According to the literature [7,8], brief illustrations of the existing APSSR construction method can be summarized as follows.
For a high voltage transmission network, by ignoring the effect of reactive power, the following assumptions can be established with the DC power flow model: (1) The resistance of transmission lines is much smaller than the reactance, thus G ij ≈ 0. G ij is the element of the real part of the nodal admittance matrix. (2) Voltage phase angle difference θ ij of branch i-j is very small, therefore sinθ ij ≈ θ ij , cosθ ij ≈ 1.
Under these assumptions, the active power flow equation of the power system can be simplified as follows: where P Gi and P Di are generation output active power and load active power of node i, respectively. b ij is the susceptance of branch i-j. Further, Equation (1) can be transformed into the following Equation (2): where P is the vector of node active power injection. θ is the vector of node voltage phase. X = B −1 , B is the imaginary part of nodal admittance matrix Y. Thus, the active power flow of branch i-j can be expressed as a function of node active power injections as Equation (3).
where K l = (k d1 , k d2 , . . . , k dm , . . . , k dn ), n is the total number of nodes without the slack node. k dm = −b ij (X im − X jm ), X im is the element of X in row i and column m. Therefore, the APSSR of branch i-j can be represented as: where k dm is the hyper-plane coefficient (HPC) of P m . From Equations (1)-(4), it can be found that: When the system contains a TCSC, Y will be changed with the variation of the equivalent reactance of the TCSC. Then, X = B −1 is about to change and further affect the APSSR via essentially influencing k dm . Therefore, it is necessary to further incorporate the equivalent reactance parameter of the TCSC in the existing APSSR expression, which is established in the space of node active power injections.

Applicability Analysis for Power System with TCSC
For the grid whose structure parameters are fixed values, and Y, B and X are both constant matrices, the APSSR can be constructed conveniently with the existing method as Equations (1)-(4).
However, if there is a TCSC whose equivalent reactance is denoted as X TCSC , Y will be parameterized as Y(X TCSC ). Correspondingly, B and X will be parameter matrices as B(X TCSC ) and X(X TCSC ) = B(X TCSC ) −1 as well. And, it can be seen that: (1) When using Equations (1)-(4) to construct an APSSR of power grid with a TCSC, inversion of the n-order parameter matrix B(X TCSC ) is needed. (2) Moreover, the inversion of a parameter matrix is much more time-consuming than the inversion of a numerical matrix with the same dimension.
Therefore, the efficiency of the existing method is not ideal to construct an APSSR incorporating the equivalent reactance of a TCSC, due to the high computational burden caused by the inversion of parameter matrix B(X TCSC ).
In other words, to realize rapid construction of the APSSR expression with X TCSC , the key issue that needs to be solved is to avoid or overcome the inversion of the n-order parameter matrix B(X TCSC ) caused by the parameterization of Y.

Derivations of APSSRs with a Single TCSC and Double TCSCs
To avoid parameterization of the X = B −1 calculation, this paper proposes to equivalently disconnect the TCSC branch. The active power flowing through the original TCSC branch is equivalent to two X TCSC related node active power injections. Then: (1) Via the equivalent disconnection of the TCSC branch, the network structure parameter no longer contains the variable X TCSC , and Y and B will be constant matrices. That is, X = B −1 is also the constant matrix, which can be obtained conveniently. (2) The effect of X TCSC on the APSSR can be analyzed by influencing node active power injections rather than parameterization of X = B −1 .
With the above analysis, the explicit expressions of APSSR with a single TCSC and double TCSCs are derived, respectively.

APSSR Incorporating a Single TCSC
As in the DC power flow model shown in Figure 2, assume that TCSC1 is located near node l on branch l-m. Also, the original active power injection of each node is defined as (P 10 , P 20 , . . . , P m0 , . . . , P n0 ). And, it can be seen that: (1) When using Equations (1)-(4) to construct an APSSR of power grid with a TCSC, inversion of the n-order parameter matrix B(XTCSC) is needed. (2) Moreover, the inversion of a parameter matrix is much more time-consuming than the inversion of a numerical matrix with the same dimension.
Therefore, the efficiency of the existing method is not ideal to construct an APSSR incorporating the equivalent reactance of a TCSC, due to the high computational burden caused by the inversion of parameter matrix B(XTCSC).
In other words, to realize rapid construction of the APSSR expression with XTCSC, the key issue that needs to be solved is to avoid or overcome the inversion of the n-order parameter matrix B(XTCSC) caused by the parameterization of Y.

Derivations of APSSRs with a Single TCSC and Double TCSCs
To avoid parameterization of the X = B −1 calculation, this paper proposes to equivalently disconnect the TCSC branch. The active power flowing through the original TCSC branch is equivalent to two XTCSC related node active power injections. Then: (1) Via the equivalent disconnection of the TCSC branch, the network structure parameter no longer contains the variable XTCSC, and Y and B will be constant matrices. That is, X = B −1 is also the constant matrix, which can be obtained conveniently. (2) The effect of XTCSC on the APSSR can be analyzed by influencing node active power injections rather than parameterization of X = B −1 .
With the above analysis, the explicit expressions of APSSR with a single TCSC and double TCSCs are derived, respectively.

APSSR Incorporating a Single TCSC
As in the DC power flow model shown in Figure 2, assume that TCSC1 is located near node l on branch l-m. Also, the original active power injection of each node is defined as (P10, P20, …, Pm0, …, Pn0). Active power flows out from node l and injects into node p through the TCSC branch is as Equation (5).
As in the equivalent schematic diagram shown in Figure 3, by the substitution theorem, the TCSC branch can be disconnected. Also, Plp can be equivalent to two active power injection net increments ΔPT1 and ΔPT2 at node l and node p, respectively. 11 12 TCSC1 Active power flows out from node l and injects into node p through the TCSC branch is as Equation (5).
As in the equivalent schematic diagram shown in Figure 3, by the substitution theorem, the TCSC branch can be disconnected. Also, P lp can be equivalent to two active power injection net increments ∆P T1 and ∆P T2 at node l and node p, respectively. (1) According to Equation (6), via the above equivalent disconnection of TCSC branch l-p, active power injections of the two endpoints l and p can be modified as the following Equation (7): (2) With the disconnection of the TCSC branch, the structure parameters of the system will no longer contain the variable XTCSC1. That is, Y, B and X are constant matrices. Therefore, according to Equations (4) and (6), the APSSR of branch i-j can be expressed as follows: (3) In Equation (8), it can be seen that there still exists an intermediate variable θlp, which shall be eliminated and be written as an expression composed of (P10, P20, …, Pm0, …, Pn0) and XTCSC1. Referring to Equation (2), the following equation can be derived: Therefore, θlp can be obtained as follows: (4) Substituting Equation (10) into Equation (8), the APSSR of branch i-j for the system with a single TCSC can be derived as Ω1.
From Equation (11), the hyper-plane coefficient (HPC) of Pm0 for Ω1 can be obtained as λm1. (1) According to Equation (6), via the above equivalent disconnection of TCSC branch l-p, active power injections of the two endpoints l and p can be modified as the following Equation (7): (2) With the disconnection of the TCSC branch, the structure parameters of the system will no longer contain the variable X TCSC1 . That is, Y, B and X are constant matrices. Therefore, according to Equations (4) and (6), the APSSR of branch i-j can be expressed as follows: (3) In Equation (8), it can be seen that there still exists an intermediate variable θ lp , which shall be eliminated and be written as an expression composed of (P 10 , P 20 , . . . , P m0 , . . . , P n0 ) and X TCSC1 . Referring to Equation (2), the following equation can be derived: Therefore, θ lp can be obtained as follows: (4) Substituting Equation (10) into Equation (8), the APSSR of branch i-j for the system with a single TCSC can be derived as Ω 1 .

APSSR Incorporating Double TCSCs
Based on section A in this chapter, it is further assumed that branch s-t is also a TCSC branch (namely TCSC2), whose equivalent impedance is denoted as X TCSC2 .
According to Equations (5)- (7), equivalent disconnection is carried out for TCSC2 as well. Thus, active power injections of node s and node t can be modified as the following equation.
(1) With the disconnections of TCSC1 and TCSC2, Y, B and X are constant matrices. Referring to Equations (4) and (8), the APSSR of branch i-j with double TCSCs can be expressed as Equation (14). where (2) In Equation (14), it can be seen that there are two intermediate variables θ st and θ lp , which shall be eliminated and be written as the expressions composed of (P 10 , P 20 , . . . , P m0 , . . . , P n0 ), X TCSC1 , and X TCSC2 . Referring to Equations (2) and (9), the following equations can be established.
For Equation (15), there are two variables and two equations. Therefore, the explicit expressions of θ lh and θ st can be achieved as Equation (16).
(3) Substituting Equation (16) into Equation (14), the specific expression of the APSSR for the system with two TCSCs can be derived as Ω 2 . From Equation (17), the hyper-plane coefficient (HPC) of P m0 for Ω 2 can be obtained as λ m2 .

Feasibility Analysis of the Proposed Method for the System with Multiple TCSCs
Moreover, the feasibility of the proposed method for the system with multiple TCSCs is analyzed from the perspective of the solvability of linear equations.
Based on the derivations in Chapter 3, the following analyses can be illustrated.
(1) If the number of TCSCs in the system is K, with equivalent disconnections of the TCSCs, the general expression of the APSSR of branch i-j can be summarized as Equation (19) according to Equations (9) and (14).
where a Km , b Kx are constants that can be obtained based on X.
(4) Substituting Equation (21) into Equation (19), the explicit general expression of the APSSR with multiple TCSCs can be eventually derived as the following equation.
Energies 2018, 11, 551 8 of 14 (5) From Equation (22), the hyper-plane coefficient (HPC) of P m0 for Ω K can be obtained as λ mK . where, From Equations (19)-(23), the following conclusions can be summarized: (1) The feasibility of the proposed method is theoretically not affected by the number of TCSCs. Of course, with the increase of K, concrete expression of Equation (22) will be gradually complicated. (2) The computational burden of the proposed method mainly depends on the inversion of K-order matrix Ts K×K (X TCSC1 , . . . , X TCSCK ). Also, the number of TCSCs is generally much smaller than the number of nodes of a power grid (i.e., K << n). Therefore, the proposed method has much higher computational efficiency compared with the existing method, by avoiding inversion of the n-order parameter matrix B(X TCSC ).

Cases Studies
The effectiveness and superiorities of the proposed method are verified with different scale benchmarks. Specifically, this chapter will firstly demonstrate the effectiveness of the proposed method via the IEEE 10-generator 39-node system [17] shown in Figure 4.  Further, with the IEEE 3-generator 9-node system [18], IEEE 18-node system [19], and IEEE 10generator 39-node system, the superiorities of the proposed method in computational efficiency are verified by changing system scale and the number of TCSCs.
All data in this chapter are obtained by a desktop computer with i7-4790@3.60GHZ CPU and 8G RAM. The software environment is Windows 10 Professional and MATLAB 2013a.  Further, with the IEEE 3-generator 9-node system [18], IEEE 18-node system [19], and IEEE 10-generator 39-node system, the superiorities of the proposed method in computational efficiency are verified by changing system scale and the number of TCSCs. All data in this chapter are obtained by a desktop computer with i7-4790@3.60GHZ CPU and 8G RAM. The software environment is Windows 10 Professional and MATLAB 2013a.

The Scenario with a Single TCSC
Firstly, the effectiveness of the proposed method is tested by scenario 1 as follows: Scenario 1: Suppose a TCSC, whose equivalent reactance is defined as X TCSC1 , is connected in series with branch 9-39 in Figure 4, and the APSSR of branch 3-4 incorporating X TCSC1 is to be constructed.
After equivalent disconnection of the TCSC branch, via Equations (5)-(10), k dm and e m can be obtained as shown in Table 1. k r1 , k r2 , k dl and k dp can be derived as listed in Table 2. Then, according to Equation (11), the APSSR of branch 3-4 can be obtained as the following equation: On one hand, set X TCSC1 = 0.003, hyper-plane coefficients (HPCs) of Equation (24) can be obtained as the results in the 2nd and 5th columns of Table 3.
On the other hand, the reactance parameter of branch 9-39 is set as 0.025-0.003 (namely 0.022; 0.025 is the original reactance of branch 9-39), then the existing method is used to construct the APSSR. The obtained HPCs are as shown in the 3rd and 6th columns of Table 3. In Table 3, it can be seen that the results in the 2nd and 5th columns are in perfect agreement with the results in the 3rd and 6th columns, respectively. That is, for X TCSC1 = 0.0030, an APSSR of Equation (24), which is constructed by the proposed method, is correct.
For other scenarios with different values of X TCSC1 at different branches, test results are in line with the above conclusion.

The Scenario with Multiple TCSCs
Moreover, the effectiveness of the proposed method is tested for scenario 2 with multiple TCSCs. Scenario 2: Suppose three TCSCs, whose equivalent reactances are defined as X TCSC1 , X TCSC2 , and X TCSC3 , are connected in series with branch 2-25, branch 3-4, and branch 4-5 in Figure 4, and the APSSR of branch 14-4 incorporating X TCSC1 , X TCSC2, and X TCSC3 is to be constructed.
On one hand, via Equations (19)-(22), the APSSR of branch 14-4 can be obtained as the following Equation (25): Set X TCSC1 , X TCSC2 , and X TCSC3 all equal to −0.002, the HPCs can be obtained by Equation (25) as shown in the 2nd column and 5th column of Table 4.
On  Table 4. In Table 4, it can be seen that the results in the 2nd and 5th columns are in perfect agreement with the results in the 3rd and 6th columns, respectively. That is, for X TCSC1 = −0.002, X TCSC2 = −0.002, and X TCSC3 = −0.002, the APSSR of Equation (25) constructed by the proposed method is correct.
For other scenarios with multiple TCSCs, test results are in line with the above conclusion. Therefore, from the above comparisons and analyses, the following conclusions can be obtained: (1) The proposed method can correctly construct the explicit expression of the APSSR, which consists of the equivalent reactance parameters of TCSCs and node active power injections. (2) The effectiveness of the proposed method is not influenced by the number of TCSCs and the values of TCSC equivalent reactances.

Effects of System Scale and the Number of TCSCs on the Computational Efficiency of the Proposed Method
In the above section, the effectiveness of the proposed method is verified. Further, the effects of system scale and the number of TCSCs on the computational efficiency of the proposed method are tested.
For illustration, the following definitions are given: (1) Method 1: The existing method via inversion of parameter matrix B(X TCSC ) (2) Method 2: The proposed method Detailed tests are illustrated as follows.

The Effect of System Scale
On one hand, the following tests are conducted to investigate the impact of system size/number of nodes on computational efficiency. Under the premise of only one TCSC, Method 1 and Method 2 are used to construct the APPSR incorporating the TCSC equivalent reactance parameter for the three benchmarks in different scale. Eventually, the results for computing time listed in Table 5 below can be obtained. From Table 5, the following conclusions can be obtained: (1) When node number increases, the computing times of both Method 1 and Method 2 increase.
(2) Compared with the proposed Method 2, the computing time of Method 1 increases much faster.
For quantitive comparison, an index of computing time ratio is defined as r ct as follows: where t m1 and t m2 are the computing times of Method 1 and Method 2, respectively. From Table 5, the change of r ct with the increase of the number of nodes can be obtained as shown in Figure 5.

The Effect of System Scale
On one hand, the following tests are conducted to investigate the impact of system size/number of nodes on computational efficiency. Under the premise of only one TCSC, Method 1 and Method 2 are used to construct the APPSR incorporating the TCSC equivalent reactance parameter for the three benchmarks in different scale. Eventually, the results for computing time listed in Table 5 below can be obtained. From Table 5, the following conclusions can be obtained: (1) When node number increases, the computing times of both Method 1 and Method 2 increase.
(2) Compared with the proposed Method 2, the computing time of Method 1 increases much faster.
For quantitive comparison, an index of computing time ratio is defined as rct as follows: where tm1 and tm2 are the computing times of Method 1 and Method 2, respectively. From Table 5, the change of rct with the increase of the number of nodes can be obtained as shown in Figure 5. From Figure 5, it can be seen that rct shows a trend of decrease with the increase of system scale. When the number of nodes is 9, 18, and 39, rct is 2.88%, 0.47%, and 0.21%, respectively.
To sum up, it shows that the superiority of the proposed method in computational efficiency will be more prominent with the increase of benchmark scale. From Figure 5, it can be seen that r ct shows a trend of decrease with the increase of system scale. When the number of nodes is 9, 18, and 39, r ct is 2.88%, 0.47%, and 0.21%, respectively.
To sum up, it shows that the superiority of the proposed method in computational efficiency will be more prominent with the increase of benchmark scale.

The Effect of the Number of TCSCs
On the one hand, the following tests are conducted to investigate the impact of the number of TCSCs on computational efficiency. For the IEEE 10-generator 39-node benchmark, Method 1 and Method 2 are used to construct the APSSR for the three scenarios with three TCSCs, four TCSCs, and five TCSCs, respectively. Eventually, the results for computing time listed in Table 6 below can be obtained. From Table 6, the following conclusions can be obtained: (1) When the number of TCSCs increases, the computing times of both Method 1 and Method 2 increase. (2) Compared with the proposed Method 2, the computing time of the proposed Method 1 increases much faster.
From Table 6, the change of r ct with the increase of the number of TCSCs can be obtained as shown in Figure 6.  Table 6, the following conclusions can be obtained: (1) When the number of TCSCs increases, the computing times of both Method 1 and Method 2 increase. (2) Compared with the proposed Method 2, the computing time of the proposed Method 1 increases much faster.
From Table 6, the change of rct with the increase of the number of TCSCs can be obtained as shown in Figure 6. From Figure 6, it can be seen that rct shows a trend of decrease with the increase of the number of TCSCs. When the number of TCSCs is 9, 18, and 39, rct is 1.09%, 0.29%, and 0.09%, respectively.
To sum up, it shows that the superiority of the proposed method in computational efficiency will be more prominent with the increase of the number of TCSCs.

Conclusions
In this paper, a fast construction method of active power steady-state security regions (APSSRs) incorporating equivalent reactance parameters of TCSCs is proposed. Concretely, with equivalent disconnection of TCSC branches, the effect of TCSC equivalent reactances on the distribution of active power flow through changing the structure parameters is treated as modifying node active power injections. On this basis, the explicit expressions of APSSRs with a single TCSC equivalent reactance parameter and double TCSC equivalent reactance parameters are deduced, respectively. Furthermore, the general formula of an APSSR with multiple TCSC equivalent reactance parameters is generalized. Eventually, via benchmarks with different scales and different numbers of TCSCs, validity and superiorities of the proposed method are demonstrated as follows: (1) It can correctly construct an explicit expression of the APSSR, which consists of the equivalent reactance parameters of TCSCs and node active power injections. (2) It is suitable for a system with a single TCSC, two TCSCs, and multiple TCSCs. That is, its feasibility is not influenced by the number of TCSCs. From Figure 6, it can be seen that r ct shows a trend of decrease with the increase of the number of TCSCs. When the number of TCSCs is 9, 18, and 39, r ct is 1.09%, 0.29%, and 0.09%, respectively.
To sum up, it shows that the superiority of the proposed method in computational efficiency will be more prominent with the increase of the number of TCSCs.

Conclusions
In this paper, a fast construction method of active power steady-state security regions (APSSRs) incorporating equivalent reactance parameters of TCSCs is proposed. Concretely, with equivalent disconnection of TCSC branches, the effect of TCSC equivalent reactances on the distribution of active power flow through changing the structure parameters is treated as modifying node active power injections. On this basis, the explicit expressions of APSSRs with a single TCSC equivalent reactance parameter and double TCSC equivalent reactance parameters are deduced, respectively. Furthermore, the general formula of an APSSR with multiple TCSC equivalent reactance parameters is generalized. Eventually, via benchmarks with different scales and different numbers of TCSCs, validity and superiorities of the proposed method are demonstrated as follows: (1) It can correctly construct an explicit expression of the APSSR, which consists of the equivalent reactance parameters of TCSCs and node active power injections. (2) It is suitable for a system with a single TCSC, two TCSCs, and multiple TCSCs. That is, its feasibility is not influenced by the number of TCSCs. (3) Compared with the conventional method, it shows much higher efficiency in constructing explicit expressions of APSSRs incorporating the equivalent reactance parameters of TCSCs. Moreover, the larger the system scale or the greater the number of TCSCs, the more significant the superiority in computational efficiency the proposed method is.
In future, the construction method of APSSRs for the system with other FACTS devices, for example, a unified power flow controller (UPFC), will be further studied.