Abstract
A day-ahead voltage-stability-constrained network topology optimization (DVNTO) problem is proposed to find the day-ahead topology schemes with the minimum number of operations (including line switching and bus-bar splitting) while ensuring the sufficient hourly voltage stability margin and the engineering operation requirement of power systems. The AC continuation power flow and the uncertainty from both renewable energy sources and loads are incorporated into the formulation. The proposed DVNTO problem is a stochastic, large-scale, nonlinear integer programming problem. To solve it tractably, a tailored three-stage solution methodology, including a scenario generation and reduction stage, a dynamic period partition stage, and a topology identification stage, is presented. First, to address the challenges posed by uncertainties, a novel problem-specified scenario reduction process is proposed to obtain the representative scenarios. Then, to obtain the minimum number of necessary operations to alter the network topologies for the next 24-hour horizon, a dynamic period partition strategy is presented to partition the hours into several periods according to the hourly voltage information based on the voltage stability problem. Finally, a topology identification stage is performed to identify the final network topology scheme. The effectiveness and robustness of the proposed three-stage solution methodology under different loading conditions and the effectiveness of the proposed partition strategy are evaluated on the IEEE 118-bus and 3120-bus power systems.
ONE of the distinctive features of modern power systems is the growing penetration of renewable energy sources into power systems, which has the notable impacts on the secure and stable operation of power systems [
Various controls have been suggested and investigated, such as var sources [
To extend the NTO, many researchers have incorporated the NTO into the optimization operations of power systems. For example, a day-ahead scheduling model while considering transmission switching is proposed to restrict short-circuit current and minimize system operation cost [
Besides the cost and the security constraints, the voltage stability constraints have been considered in the NTO problems. The linearized indicators are popularly employed to measure the voltage stability degree in the NTO problem, e.g., the L-indicator [
Moreover, from the operational viewpoint, a practical issue is that too many operations for NTO are undesirable because of the potential risks for the secure operation of power systems and the heavy work for system operators. Hence, a limited/minimal number of operations is crucial and more practical for the online application of the NTO [
This paper focuses on minimizing the number of NTO schemes required while ensuring that the static voltage stability achieves the desired stability level. To this end, a novel formulation called day-ahead voltage-stability-constrained network topology optimization (DVNTO) is proposed in this paper, which considers the inherent uncertainty of renewable energy sources and loads. It is very challenging for the proposed DVNTO problem to find the best NTO scheme from the large number of NTO candidate schemes. Hence, an effective and fast solution methodology is needed. The main contributions of this paper are as follows.
1) The comprehensive NTO schemes (including line switching-in, line switching-out, and bus-bar splitting) are incorporated into the DVNTO problem to demonstrate the effect of line switching and bus-bar splitting on improving the voltage stability. To flexibly mimic the NTO schemes, a general NTO model is presented, which is suitable for fast screening the NTO candidate schemes.
2) A DVNTO problem formulation is proposed to find the day-ahead NTO schemes with a minimal number of operations. The AC continuation power flow constraints, AC power flow constraints, and uncertainties from renewable energy sources and load fluctuations are incorporated in the formulation, instead of the linearized constraints.
3) A problem-specified scenario construction method is presented to address the nonlinear impacts of uncertainties on the voltage stability margin.
4) A three-stage methodology is developed to solve the proposed DVNTO problem, which involves the scenario generation and reduction stage, dynamic period partition stage, and topology identification stage.
The remainder of this paper is organized as follows. The modeling of topology optimization and uncertainty are described in Section II. The DVNTO problem formulation and the solution methodology are presented in Section III and Section IV, respectively. Numerical studies on the IEEE 118-bus and 3120-bus power systems are extensively analyzed in Section V. Section VI concludes this paper and the possible future work.
Both line switching and bus-bar splitting are considered as the possible controls for the NTO in this paper, which can be illustrated by

Fig. 1 Illustration of NTO. (a) Topolgy before NTO. (b) Line switching-out. (c) Bus-bus splitting.
Four line branches are connected on the bus-bars and . The possible controls include line switching (line is switched into service in
How to flexibly mimic all the possible schemes and identify the optimal NTO scheme quickly from a lot of candidates is a very challenging task. To flexibly mimic the changes of topologies, a general NTO model is proposed, inspired by Ward equivalent theory in this paper. Taking the network shown in

Fig. 2 Illustration of proposed general NTO model. (a) Before NTO. (b) After NTO.
Before the NTO, the voltage equation of the power system can be described as:
(1) |
where , , and are the self-admittance matrices of the internal buses, boundary buses, and external buses, respectively; , , , and are the mutual admittance matrices; , , and are the matrices of the bus voltages; and , , and are the matrices of injection currents. Eliminating the first line in (1) by the Gaussian elimination method, the internal subsystem can be represented as an equivalent network (i.e., the mesh network in
(2a) |
(2b) |
where is the admittance matrix of the equivalent internal subsystem.
After the NTO, the topology of the internal subsystem is changed, whereas the boundary buses and the external subsystem remain unchanged. Hence, the voltage equation of the power system after the NTO can be described as:
(3) |
where the superscript “+” denotes the elements changed due to the NTO. Similarly, the internal subsystem can be represented by a new mesh network shown in
(4a) |
(4b) |
Hence, the difference between network topologies before and after the NTO can be equivalent to the admittance change of two mesh networks and , i.e.,
(5) |
The proposed model is suitable for any switching action such as the NTO and shunt switching, and supports the fast-screening task to screen out those invalid NTO schemes by a linearized method. The detailed steps are summarized in Section IV-B.
In this paper, the uncertainties are modeled by scenario-based methods. R-vine Copula method is used to fit the correlations according to the historical data to model the spatial correlations among renewable energy sources. Assume that an available prediction technique can provide day-ahead predicted loads and renewable energy sources. The hourly prediction errors of load demands and the outputs of renewable energy sources can be described by the normally-distributed random variables. Assume that the standard deviations are and for the renewable energy source on load buses i and j at the hour, respectively, and the (predicted) mean values are and , the outputs of renewable energy sources and load demands on bus i can be represented as scenario sets () and ( ), respectively. According to the previous studies in [
(6) |
where and are the elements of the sets and , respectively. Then, all the day-ahead scenarios of renewable energy sources and loads can be composed, for example, the scenario can be composed by:
(7a) |
(7b) |
where is the total number of buses of the power system.
The proposed problem aims to determine the day-ahead optimized network topologies altered by the minimal number of necessary actions to ensure enough hourly voltage stability margin up to a desired value and the engineering operation requirements of power systems. Hence, the proposed problem formulation can be described as:
(8) |
s.t.
(9) |
(10) |
(11) |
(12a) |
(12b) |
where and are the network topologies at the and hour, respectively; maps the difference of network topologies, i.e., the needed number of line switching and bus-bar splitting to alter the network topology from to ; and denote the AC continuation power flow equation and the AC power flow equation in the scenario and the topology , respectively; and are the voltage vector and the load margin in the scenario , respectively; is the voltage magnitude of bus in the scenario ; and are the lower and upper limits of the voltage magnitude of bus in the scenario , respectively; is the set of buses; and are the apparent power of line i-j in the scenario and its limit, respectively; and is the power injection variation in the scenario , specified by the rescheduling active and reactive power generations, the load demand, and their uncertainties.
The objective function (8) maps the minimal number of required operations to change the network topologies for the day-ahead power system. Constraints (9) and (10) are the AC nonlinear continuation power flow equations and the AC power flow equations of the day-ahead power system, respectively. Constraint (11) is the hourly load margin requirement. Constraints (12a) and (12b) represent the operational engineering constraints, including the voltage magnitude limit constraint (12a) and the thermal limit constraint (12b).
The nonlinear impact of the power injection variations on the load margin to static voltage stability limit has been studied in [

Fig. 3 Illustration of variation of active power injection.
A three-stage methodology is presented to solve the DVNTO problem, whose framework is shown in

Fig. 4 Framework of proposed methodology.
This stage aims to employ the scenario generation method in Section II-B to model the uncertainties and obtain the RS set.
The critical issue for the scenario-based modeling method is how to reduce the number of scenarios to reduce the computation burden from a large number of scenarios. According to our experience, to obtain RSs and avoid the mis-elimination of the extreme scenarios, the scenarios should be reduced according to the impact of scenarios on the load margin instead of the distances among the scenarios. Therefore, a tailored scenario reduction method is presented in this paper, whose steps are summarized as follows.
Step 1: cluster all the generated joint scenarios in Section II-B into groups according to the distance between scenario pairs and identify the central scenario of each group. The distance between the scenarios and is:
(13) |
Step 2: choose a scenario group and compute the hourly load margin of the central scenario of this group and the hourly non-zero left eigenvector corresponding to the hourly zero eigenvalue of the Jacobian matrix at the bifurcation point.
Step 3: estimate the hourly load margins of the
(14) |
(15) |
where is the load margin change due to the difference of the scenario and the scenario ; denotes the AC continuation power flow equation in the scenario and the topology ; and is the non-zero eigenvalue of the Jacobian matrix of at the bifurcation point. Repeat Step 3 until the hourly load margins of all the scenario groups are estimated.
Step 4: cluster all the scenarios into groups according to the distance of the estimated load margins by (16). The distance of load margins between two scenarios and is:
(16) |
Select the scenario with minimal load margin of each group as the RS to form the RS set , where is the RS and , is the hourly scenario at the hour of , and is the number of RSs.
The aim of this stage is to partition the hours into several periods according to the worst scenario determined by the following problem (17), which is proposed to determine and identify the first hour with the violation of constraint (11). The following period partition will be performed from .
(17a) |
s.t.
(17b) |
where denotes the RS of ; is the load margin of power system in the scenario ; and is an integer variable, and if , , otherwise .
To partition the hours into periods, the distinctive feature of this paper is that the voltage changes between the bifurcation point and the power flow point are used as the clustered index and a cluster validity index is proposed to evaluate partition results. For bus i, the change of voltage magnitude in the worst scenario is:
(18) |
where and are the voltage magnitudes of bus at the operation point and the bifurcation point at the hour in the worst scenario , respectively. We use a P-V curve shown in

Fig. 5 P-V curve in scenario at the hour.
(19) |
To partition the hours into several intervals, the FCM cluster method is employed with the following objective function in this paper [
(20) |
where c is the number of clusters; is a parameter controlling the fuzziness of the clustering procedure; is the membership degree of to the cluster ; is the membership matrix; and is the output of the cluster center.
A vital point of the FCM cluster method is that the number of clusters and the fuzzy clustering coefficient must be specified in advance. Several research studies, e.g., [
(21) |
(22) |
where cmax is the maximum number of clusters; and are the weights assigned to the effectiveness indexes and by the entropy weight method [
(23) |
(24) |
where ; is the overlap degree between clusters; and is the separation measure between two fuzzy clusters (please refer to [
After this stage, the first period [] is identified and sent to the next stage.
This stage is to determine the optimal topology that can increase the hourly load margins of the first period [] up to the desired requirement with the minimized number of operations by line switching and bus-bar splitting. To solve the problem, a “screening and ranking–verification” strategy is proposed to identify the most effective NTO schemes.
Step 1: screening and ranking.
In this step, the effective NTO schemes for the scenario are screened from a large amount of NTO candidate schemes and then ranked by their load margins.
To fast evaluate the effect of each NTO candidate scheme on improving the load margin, a linear sensitivity-based method is employed and the sensitivity of the load margin on NTO scheme is derived to pre-screen all the NTO candidate schemes in the scenario , including line switching-in, line switching-out, and bus-bar splitting. According to the general NTO model proposed in Section II-A, the changes of topologies due to the NTO can be modeled as the changes of admittance matrices of the two mesh networks (5). Hence, the change of load margins due to the NTO in the scenario can be estimated by (25) and (26).
(25) |
(26) |
where is the number of branches of the mesh networks and ; denotes the AC continuation power flow equation in the scenario and the topology ; is the non-zero eigenvalue of the Jacobian matrix of at the bifurcation point; is the admittance matrix of the mesh network; is the admittance change of two mesh networks and (5); and is the power injection variation in the scenario . After this step, all the NTO schemes with the nonnegative value (i.e., ) are considered as the possible candidate schemes to improve the load margin.
To rank the reserved NTO schemes, the look-ahead margin estimation method [
Step 2: verification.
Another feature of the proposed methodology is that multiple NTO schemes may be obtained in the above step. Hence, we can further identify the “best” NTO scheme that can meet the load margin requirement for all RSs with the largest period length. To identify the best one, a continuous power flow method is used to calculate the load margin for all RSs from until the last hour that meets the load margin requirement is reached.
Our numerical studies show that: ① in some cases, the NTO schemes obtained in Step 1 may not meet the load margin requirement (11) for the whole period under all the RSs; ② in some cases, the NTO schemes obtained in Step 1 can also meet the load margin requirements (11) for the subsequent hours of the current period under all the RSs. Hence, in this step, the NTO scheme that can meet the load margin requirement (11) with the longest time period is the “optimal” solution. To this end, a dynamic period adjustment strategy, inspired by [

Fig. 6 Illustration of two situations in Step 2.
Situation 1: the obtained NTO schemes in Step 1 are used to verify the load margin requirement (11) under each RS from the first hour . The NTO scheme with the longest time period is the solution, and update . is the last hour of this period that the NTO scheme can meet the requirement of load margin (11).
Situation 2: the obtained NTO schemes in Step 1 are used to verify the load margin requirement (11) for the subsequent hours until the last hour without violating (11) is met, and update . is the last hour outside the period that the NTO scheme can meet the requirement of load margin (11).
Based on the above process, the NTO scheme for the period is determined, and sent to the period as the new initial topology to perform the proposed methodology till the NTO schemes for 24 hours are determined.
The effectiveness of the proposed methodology, the proposed partition strategy, and the performance of the comprehensive NTO have been tested on IEEE 118-bus and 3120-bus power systems [
1) Method M1: the method does not employ the proposed verification step.
2) Method M2: the method does not employ the proposed period partition step.
3) Method M3: the method conducts the period partition based on the net loads.
Assume that 3 photovoltaic (PV) power stations are installed at buses 72, 73, and 74, and 3 wind generators are installed at buses 90, 91, and 92. The lower and upper bounds of bus voltage magnitudes are 0.94 p.u. and 1.06 p.u., respectively. The day-ahead active power and reactive power of loads are shown in

Fig. 7 Day-ahead load level and hourly load margins in worst scenario of IEEE 118-bus power system.
The outputs of the proposed methodology are as follows.
1) In stage I, a total of 1000 joint scenarios are generated and 4 RSs are obtained by the proposed scenario reduction method.
2) After stage II, the ineligible hourly load margins in the 4 RSs are listed in
RS 1 | RS 2 | RS 3 | RS 4 | ||||
---|---|---|---|---|---|---|---|
(p.u.) | (p.u.) | (p.u.) | (p.u.) | ||||
02:00 | 1.8412 | 02:00 | 1.8206 | 06:00 | 1.8531 | 02:00 | 1.8732 |
07:00 | 1.8313 | 06:00 | 1.8384 | 07:00 | 1.8623 | 07:00 | 1.8663 |
11:00 | 1.8642 | 17:00 | 1.8576 | 11:00 | 1.8035 | 17:00 | 1.8617 |
17:00 | 1.7785 | 17:00 | 1.8674 | 23:00 | 1.8524 | ||
23:00 | 1.7842 |
3) In stage III, the NTO scheme with switching out the line 114 is determined as the best solution for the first period 01:00-06:00 and it can support the load margin requirement in this period. Hence, the solution for the first period is the topology with switching out the line 114.
The 24 hours are partitioned into 4 time periods by the proposed methodology, as shown in
Time partition | NTO scheme | |||
---|---|---|---|---|
Period | Duration | No. of line switching-in | No. of line switching-out | Bus-bar splitting scheme No. |
1 | 01:00-06:00 | 114 | ||
2 | 07:00-16:00 | 114 | 185 | |
3 | 17:00-22:00 | 185 | 57 | |
4 | 23:00-24:00 | +190 |
Note: “+” represents the execution of bus-bar splitting.
Case | Splitting bus-bar No. | Bus-bar splitting scheme No. | Bus-bar splitting scheme | |
---|---|---|---|---|
No. of line connected on one bus | No. of line connected on another bus | |||
118 | 17 | 65 | 21, 23, 178 | 22, 36, 39, L |
37 | 136 | 47, 51, 52 | 48, 50, 53, S | |
68 | 190 | 107, 183 | 104, 126 | |
75 | 210 | 120, 185, L | 115, 116, 117 | |
94 | 264 | 146, 147, L | 145, 150, 155 | |
3120 | 47 | 104 | 91, 92 | 95, 255 |
1085 | 2840 | 1113, 1114 | 1176, L | |
2576 | 9242 | 2275, 2261 | 2278, L | |
2810 | 9560 | 2660, 3646 | 2693, 2702 | |
3100 | 10425 | 384, 2770 | 2628, 2768, 2780, 3671, L |
Note: “L” represents the load branch on the splitting bus-bar, and “S” represents the shunt branch on the splitting bus-bar.
To evaluate the effectiveness of the proposed methodology, the load margins before and after NTOs in the 4 RSs of 24 hours are calculated by the continuation power flow method, as shown in

Fig. 8 Hourly load margins before and after NTOs in 4 RSs.
To show the effectiveness of the verification step proposed in stage III, this example is also tested by method M1. The time partition and NTO scheme are summarized in
Time partition | NTO scheme | |||
---|---|---|---|---|
Period | Duration | No. of line switching-in | No. of line switching-out | Bus-bar splitting scheme No. |
1 | 01:00-04:00 | 114 | ||
2 | 05:00-09:00 | 114 | 128 | |
3 | 10:00-14:00 | 128 | 26 | |
4 | 15:00-16:00 | 26 | 90 | |
5 | 17:00-21:00 | |||
6 | 22:00-24:00 | 90 | +190 |
Note: “+” represents the execution of bus-bar splitting.
Assume that there are 6 PV stations at buses 1983, 1984, 1993,, 2100, 2144, and 2146 and 6 wind generators at buses 2274, 2300, 2306, 2310, 2345, and 2349. The 55 load buses 430-440, 1100-1110, and 2800-2840 are divided into three different types of loads that vary according to the predicted load demand and 7 generators (at the buses 2791, 2794, 2797, 2803, 2814, 2823, and 2828) are scheduled to supply the increased load demand.
The hourly load demands and load margins are shown in

Fig. 9 Day-ahead load level and hourly load margins in worst scenario of 3120-bus power system.
After the proposed methodology is adopted, the 24 hours are partitioned into 4 time periods, as shown in
Time partition | NTO scheme | |||
---|---|---|---|---|
Period | Duration | No. of line switching-in | No. of line switching-out | Bus-bar splitting scheme No. |
1 | 01:00-09:00 | 3654 | ||
2 | 10:00-18:00 | 3654 | 2960 | |
3 | 19:00-22:00 | 2960 | 3060 | |
4 | 23:00-24:00 | 3060 | +10425 |
Note: “+” represents the execution of bus-bar splitting.

Fig. 10 Hourly load margins before and after NTOs in 3 RSs.
To demonstrate the effectiveness of the proposed partition strategy, the simulation comparison is conducted between the proposed methodology and method M2. The results obtained by the two methods are summarized in Tables
Time partition | NTO scheme | |||
---|---|---|---|---|
Period | Duration | No. of line switching-in | No. of line switching-out | Bus-bar splitting scheme No. |
1 | 01:00-04:00 | 64 | ||
2 | 05:00-09:00 | 64 | 3410 | |
3 | 10:00-15:00 | 3410 | 546 | |
4 | 16:00-18:00 | 546 | 3541 | |
5 | 19:00-22:00 | 3541 | 135 | |
6 | 23:00-24:00 | 135 | +10425 |
Note: “+” represents the execution of bus-bar splitting.
1) For the number of periods, 4 periods are identified by the proposed methodology, yet 6 periods are obtained by method M2, which are more than those of the proposed methodology.
2) For the number of operations, the network topology needs to be changed at 01:00, 10:00, 19:00, and 23:00 with the total number of operations of 7 by the proposed methodology. By method M2, the network topology needs to be changed at 01:00, 05:00, 10:00, 16:00, 19:00, and 23:00 with the total number of operations of 11, which is much more than that of the proposed methodology.
For comparison, the example on the IEEE 118-bus power system is tested by the proposed partition strategy based on the voltage changes in (18), (19) and the method M3. The day-ahead curves of wind power stations WT1-WT3 and PV power stations PV1-PV3 are shown in

Fig. 11 Day-ahead curves of renewable energy sources and net load. (a) Wind power station. (b) PV power station. (c) Net load.

Fig. 12 Period partition results and load margins of method M3.
Comparing
To explore and show the effectiveness of the comprehensive NTO (including line switching and bus-bar splitting) on improving the voltage stability, two examples in the IEEE 118-bus power system and 3120-bus power system are conducted by the comprehensive NTO, line switching, and bus-bar splitting, respectively. The data of the two examples are the same as those in Section V-A and Section V-B. The simulation results of the two examples by line switching and bus-bar splitting are summarized in
Duration | Line switching | Bus-bar splitting scheme No. | |
---|---|---|---|
No. of line switching-in | No. of line switching-out | ||
01:00-06:00 | 114 | +136 | |
07:00-16:00 | 114 | 185 | -136, +65 |
17:00-22:00 | 185 | 57 | -65, +264 |
23:00-24:00 | 57 | 134, 65 | -264, +190 |
Total number of operations | 8 | 7 |
Note: “+” represents the execution of bus-bar splitting, and “-” represents the restoration of bus-bar splitting scheme.
Duration of line switching | Line switching | Duration of bus-bar splitting | Bus-bar splitting scheme No. | |
---|---|---|---|---|
No. of line switching-in | No. of line switching-out | |||
01:00-09:00 | 3654 | 01:00-09:00 | +2840 | |
10:00-18:00 | 3654 | 2960 | 10:00-16:00 | -2840, +9560 |
19:00-22:00 | 2960 | 3060 | 17:00-19:00 | -9560, +104 |
23:00-24:00 | 3060 | 65, 423 | 20:00-22:00 | -104, +9242 |
22:00-24:00 | -9242, +10425 | |||
Total number of operations | 8 | 9 |
Note: “+” represents the execution of bus-bar splitting, and “-” represents the restoration of bus-bar splitting scheme.
It can be concluded that:
1) For the example of the IEEE 118-bus power system, 4 time periods are identified. For the line switching, the network topology needs to be altered at 01:00, 07:00, 17:00, and 23:00, and a total of 8 operations are needed to alter the network topology. For bus-bar splitting, the topology needs to be optimized according to the bus-bar splitting scheme Nos. 136, 65, 264, and 190, and a total of 7 operations are needed. However, from the results in
2) For the example of the 3120-bus power system, 8 operations are needed by the line switching, and 9 operations are needed by the bus-splitting. However, from the results in
An example of the IEEE 118-bus power system is used to test the effectiveness of the proposed methodology under different loading conditions. In this test, 3 PV power stations are installed at buses 53, 54, and 55 and 3 wind generators are installed at buses 103, 104, and 105. The day-ahead active and reactive power loads and hourly load margins are shown in

Fig. 13 Day-ahead load level and hourly load margins.
RS 1 | RS 2 | RS 3 | RS 4 | ||||
---|---|---|---|---|---|---|---|
(p.u.) | (p.u.) | (p.u.) | (p.u.) | ||||
02:00 | 0.9978 | 04:00 | 0.9592 | 02:00 | 0.9968 | 02:00 | 0.9824 |
09:00 | 0.9918 | 14:00 | 0.9812 | 04:00 | 0.9280 | 04:00 | 0.9491 |
15:00 | 0.9561 | 15:00 | 0.7074 | 09:00 | 0.9883 | 09:00 | 0.9598 |
19:00 | 0.9359 | 23:00 | 0.9789 | 15:00 | 0.9483 | 15:00 | 0.9638 |
23:00 | 0.9366 | 19:00 | 0.9855 | 19:00 | 0.9849 | ||
23:00 | 0.9645 |
The results obtained by the proposed methodology are summarized in
Time partition | NTO scheme | |||
---|---|---|---|---|
Period | Duration | No. of line switching-in | No. of line switching-out | Bus-bar splitting scheme No. |
1 | 01:00-08:00 | 185 | ||
2 | 09:00-13:00 | 185 | 82 | |
3 | 14:00-18:00 | 82 | 30 | |
4 | 19:00-24:00 | 30 | +210 |
Note: “+” represents the execution of bus-bar splitting.

Fig. 14 P-V curves before and after NTO in worst scenario.
This paper develops a DVNTO problem to improve the static day-ahead voltage stability of power systems by the comprehensive NTO with a minimum number of operations. A three-stage solution methodology is proposed to solve the problem. To flexibly mimic the change of topology by the NTO, a general NTO model is proposed and tailored for the screening task to quickly screen out those ineffective NTO schemes by a linear sensitivity-based method. To model the uncertainty, a scenario construction method is developed, specified for the static voltage stability problem, which can avoid the mis-elimination of the extreme scenario. To reduce the number of operations for NTO, a dynamic period partition strategy is presented. Extensive testing results on the IEEE 118-bus power system and 3120-bus power system demonstrated that the proposed methodology can effectively solve the proposed problem under different loading conditions with promising results.
Some of the future work is to further optimize the reactive power resources with the NTO to improve the voltage stability of power systems. Another relevant work is to improve the proposed problem by considering the other important constraints such as static stability and frequency stability constraints.
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