Abstract
In the existing small-signal stability constrained optimal power flow (SSSC-OPF) algorithms, only the rightmost eigenvalue or eigenvalues that do not satisfy a given threshold, e.g., damping ratio threshold and real-part threshold of eigenvalue, are considered in the small-signal stability constraints. The effect of steady-state, i.e., operating point, changes on eigenvalues is not fully taken into account. In this paper, the small-signal stability constraint that can fully reflect the eigenvalue change and system dynamic performance requirement is formed by analyzing the eigenvalue distribution on the complex plane. The small-signal stability constraint is embedded into the standard optimal power flow model for generation rescheduling. The simultaneous solution formula of the SSSC-OPF is established and solved by the quasi-Newton approach, while penalty factors corresponding to the eigenvalue constraints are determined by the stabilization degree of constrained eigenvalues. To improve the computation speed, a hybrid algorithm for eigenvalue computation in the optimization process is proposed, which includes variable selection for eigenvalue estimation and strategy selection for eigenvalue computation. The effectiveness of the proposed algorithm is tested and validated on the New England 10-machine 39-bus system and a modified practical 68-machine 2395-bus system.
Set of eigenvalues to be considered in the unconstrained optimization | |
Set of dominant variable serial numbers for electromechanical oscillation mode k | |
Set of dominant variable serial numbers in the unconstrained optimization | |
Set of serial numbers of all state variables | |
, | Iteration numbers of one-dimensional search in the unconstrained optimization, and |
l | Iteration number of quasi-Newton approach |
B. Parameters | |
Penalty factors for real-part constraints of eigenvalue and damping ratio constraints | |
Convergence tolerance | |
Scaling factor | |
Threshold for determining NR in Algorithm 1 | |
Real-part threshold of eigenvalue, and | |
Damping ratio thresholds, and | |
Time interval | |
Cost coefficients of the generator | |
, | State matrix and identity matrix |
The maximum iteration number in Algorithm 1 | |
Numbers of power flow equations and technical limits | |
Number of considered contingency scenarios | |
Numbers of generators and state variables | |
Vector of ramping limits | |
S | Threshold for selecting eigenvalue computation strategy |
Penalty factors for the technical limit and the small-signal stability constraint | |
C. Variables | |
Real and imaginary parts of the | |
The largest real part of eigenvalues | |
The | |
Eigenvalue corresponding to the minimum damping ratio | |
The minimum damping ratio of all modes | |
Variation of the minimum damping ratio between optimal power flow (OPF) and small-signal stability constrained optimal power flow (SSSC-OPF) solutions | |
Variation of the largest real part of eigenvalues between OPF and SSSC-OPF solutions | |
Variation of augmented Lagrangian function value between OPF and SSSC-OPF solutions | |
Optimal step length for the unconstrained optimization | |
, |
Deviations of parameter and gradient vectors associated with the |
Approximation to inverse of Hessian matrix of augmented Lagrangian function | |
H at operating point | |
, | Total numbers of H restarts and optimal step length search |
, | Total number of calls to QR algorithm and implicitly restarted Arnoldi approach |
Total number of calls to the first-order eigenvalue sensitivity computation subroutine without sensitivity computation for updating H | |
K | Column vector consisting of state variables and Lagrangian multipliers |
L( |
Augmented Lagrangian function value at |
Nλ | Number of modes considered in |
NR | Number of selected dominant state variables |
Active power output of the generator | |
R, Q | Intermediate variables in Algorithm 1 |
Uk, Wk |
Right and left eigenvectors corresponding to the |
Voltage magnitude of the generator | |
Sensitivity of with respect to state variable xi in the unconstrained optimization | |
x, y | Column vectors of state variables and Lagrangian multipliers |
POWER flow distribution affects both the steady-state and stability performance of power systems. For computational reason, the standard optimal power flow (OPF) model usually does not consider stability constraints or simply takes the form of a branch phase angle difference [
The SSSC in the SSSC-OPF model needs not only to adequately describe the stability of the system, but also to consider the computation burden. Therefore, for small-scale power systems, the SSSC includes all or some of the eigenvalues; for large-scale power systems, only several critical eigenvalues can be considered due to the computation burden. In [
At present, the SSSC-OPF problem is mainly solved by gradient-based optimization algorithms [
In this paper, a practical algorithm for the simultaneous solution of the SSSC-OPF problem is proposed. The SSSC is formed here by analyzing the eigenvalue distribution on the complex plane, which is composed of the real part of eigenvalue and the damping ratio of some critical modes. These critical modes include all unstable and underdamped modes as well as those that may become unstable and underdamped as the system operating point changes. The SSSC is integrated with the standard OPF model, and the penalty factors for these constrained eigenvalues are determined by their corresponding stabilization degree. The quasi-Newton approach is employed to simultaneously solve the proposed SSSC-OPF model, and a hybrid algorithm using approximate and accurate computation is proposed to compute eigenvalues in the optimization process.
The remainder of this paper is organized as follows. Section II presents the SSSC. The model and algorithm of the SSSC-OPF are given in Section III. In Section IV, the effectiveness of the proposed algorithm is tested and validated on the New England 10-machine 39-bus system and a modified practical 68-machine 2395-bus system. Finally, Section V outlines the conclusions.
For small-signal stability, the state matrix A is formed by the linearized model of power systems at an equilibrium point [
(1) |
Each pair of the complex eigenvalues corresponds to an oscillatory mode. The damping ratio of an oscillatory mode is given by:
(2) |
For a power system with NG generators, there are usually electromechanical oscillation modes. To ensure the system dynamic performance, the eigenvalues of all electromechanical oscillation modes must have negative real parts, and the damping ratio should not be less than a given threshold , e.g., 5% [
(3) |
For SSSC-OPF problems, if the SSSC1 of (3) is not satisfied, the system operating point will be adjusted to improve these modes. However, as the system operating point changes, some stable modes may become unstable and vice versa, i.e., eigenvalue oscillations may occur during the solution process. This oscillation may lead to poor convergence of the algorithm, or even make the algorithm difficult to converge.
To alleviate the oscillation of constrained eigenvalues during the solution process, the complex plane is divided into three parts in this paper, as shown in

Fig. 1 Distribution range of eigenvalues.
Region 1 indicates the modes that do not satisfy SSSC1. All eigenvalues in region 1 are included in . Region 2 represents the modes with insufficient stability or security margin, i.e., the eigenvalues are close to the imaginary axis or . Since the eigenvalues move during the optimization process, the eigenvalues in region 2 may cross the boundary to region 1. Hence, the eigenvalues in region 2 should be considered in . The modes in region 3 are relatively stable and can be ignored. However, if the eigenvalues in region 3 move to region 2, they should also be considered in . From the above analysis, it can be observed that not only contains multiple critical modes, but also changes dynamically. Mathematically, can be described by:
(4) |
And the proposed SSSC2 can be expressed as:
(5) |
If and , the SSSC2 of (5) becomes the SSSC1. The SSSC2 not only reflects the dynamic performance requirement of the system, but also fully considers the effect of the operating point changes on eigenvalues. It should be noted that the purpose of SSSC is not to require all modes to satisfy the SSSC2, but to prevent the eigenvalues in region 2 from moving to region 1 when the modes in region 1 are improved during the optimization process. To achieve this purpose, different penalty factors are set for eigenvalues in different regions.
SSSC-OPF problems are proposed by adding the SSSC to the standard OPF model [
(6) |
(7) |
(8) |
(9) |
where indicates the objective function; is the vector corresponding to the equality constraints, e.g., power flow equations; is the vector corresponding to the inequality constraints, including technical limits for active power, reactive power, voltage magnitudes, and branch flow; and is the vector corresponding to the compact form of the SSSC, which is the implicit function of x.
In general, the SSSC-OPF problem is a nonlinear programming problem in mathematics. There are many approaches to deal with this problem [
(10) |
(11) |
(12) |
Penalty factors for SSSCs are given as:
(13) |
(14) |
If a mode is located in region 1 ( or ), the system is underdamped or unstable under small disturbances. To improve this mode, and need to be set relatively large. However, the penalty factors for the modes in region 2 ( or ) should be relatively small. In other words, is less than 1. This is because the penalty factors for the modes in region 2 are not to improve the modes, but to prevent the eigenvalues in region 2 from moving to region 1. It is effective to select the penalty factors by starting with a low value and then increasing it during the optimization process [
The Karush-Kuhn-Tucker (KKT) conditions for the SSSC-OPF problem of (10) can be given as follows. and are gradient vectors of the Lagrangian with respect to state variables and Lagrangian multipliers, respectively.
(15) |
(16) |
Since the second-order eigenvalue sensitivity is very time-consuming [
(17) |
where ; and can be obtained by using the one-dimensional search [
(18) |
(19) |
After each iteration, the inequality constraints (8) and (9) are checked and is changed. Near the optimal solution of , the variation of is relatively small. However, due to the highly nonlinear relationship between eigenvalues and operating parameters [
During the simultaneous solution of the SSSC-OPF using the quasi-Newton approach, is obtained by the one-dimensional search in the unconstrained optimization. In the process of searching for , it is necessary to repeatedly compare the value of the augmented Lagrangian function L(K). This requires repeatedly forming the state matrix and calculating eigenvalues, which is time-consuming. For small-scale power systems, an accurate eigenvalue computation, i.e., QR algorithm, IRA approach, can be performed for each iteration. For large-scale power systems, the computation burden of eigenvalues is much greater than that of solving the equilibrium point. To reduce the computation cost, the eigenvalues considered in can also be computed by a hybrid algorithm using accurate eigenvalue computation and eigenvalue estimation.
1) Variable Selection for Eigenvalue Estimation
Since the number of state variables in large-scale power systems may be large, it is also time-consuming to compute the sensitivities of multiple eigenvalues with respect to all state variables in x. To reduce the sensitivity computation time, it is necessary to reduce the number of variables in x for eigenvalue estimation. This means that only dominant variables are used for eigenvalue estimation.
The small-signal stability index is utilized to explain how to select the dominant variables. For an electromechanical oscillation mode k, the first-order Taylor series expansion of at an operating point is given in (20), which can be rewritten as (21).
(20) |
(21) |
In [
To reduce the computation time, if in (21) has a relatively small effect on , it can be ignored. However, all are unknown beforehand, because the purpose of variable selection is to select that needs to be computed. Fortunately, all have been computed in each unconstrained optimization, and can be used for variable selection. For a mode k, the absolute values of all are sorted in decreasing order, i.e.,
(22) |
where represents the absolute value.
Assuming that the first NR in (22) have a relatively large effect on , and the first NR corresponding to the state variables are considered as the dominant variables and will be selected. The value of NR is determined by
Algorithm 1 : determine value of NR |
---|
1: Initialization: set , , , , and ; choose a threshold 2: while do 3: set , , 4: if 5: Output NR and stop 6: end if 7: set 8: end while |
For the mode k, is given by:
(23) |
Multiple modes are included in , and dominant variables for different modes are usually inconsistent. Therefore, for multiple modes, the dominant variables are selected as:
(24) |
Note that the computation complexity of the proposed variable selection approach is very low. To reduce the impact of variable selection on the optimization process, dominant variables can be reselected when needs to be computed in each iteration.
2) Strategy Selection for Eigenvalue Computation
The hybrid algorithm can be used for eigenvalue computation in the one-dimensional search process, but it is still a problem to choose an appropriate eigenvalue computation strategy (accurate eigenvalue computation or eigenvalue estimation) in each iteration. An accurate eigenvalue computation can be done after several iterations, and the eigenvalue estimation will be used for the rest of the iterations. However, this may lead to poor convergence or even non-convergence of the algorithm because the variation of state variables may be large. Therefore, the following condition, as given in (25), is utilized to choose an appropriate eigenvalue computation strategy.
(25) |
Assume that an accurate eigenvalue computation is performed at and the value of S is given. If (25) is not satisfied at operating point
3) Eigenvalue Estimation with Sensitivity Approach
If (25) is satisfied at , it is assumed that the modes considered in are the same as those considered in . Based on the eigenvalue sensitivity approach, the eigenvalues at
(26) |
(27) |
(28) |
The derivative of A with respect to state variable xi can be obtained using plug-in modeling technology [
For the sake of simplicity, a computation flowchart of the proposed algorithm for the SSSC-OPF problem is given in

Fig. 2 Flowchart of proposed algorithm.
The extension of the proposed algorithm considering both the normal and contingency operating conditions will be discussed below. The OPF model considering SSSCs under the normal and contingency operating conditions of the power system can be formulated as follows. The subscripts 0 and represent the normal operating condition and contingency scenario, respectively.
(29) |
s.t.
(30) |
(31) |
(32) |
(33) |
(34) |
is the compact form of constraints (8) and (9).
The problem in (29)-(34) can be solved by several algorithms, e.g., Benders decomposition utilized in this paper. Using the Benders decomposition, the problem can be decomposed into a normal operating master problem and a set of contingency sub-problems. The whole problem is iteratively solved between the master problem and sub-problems. The proposed algorithm in this paper can be utilized to solve both the master problem and sub-problems. A detailed description of the Benders decomposition used in optimization problems can be found in [
In this section, the proposed algorithm is applied to the New England 10-machine 39-bus system and a modified practical 68-machine 2395-bus system to illustrate the effectiveness. For all systems, the loads are modeled as constant impedance. Except for the slack generator, PGi and VGi of all generators are included in the state variables. The objective function is to minimize the cost of power generation, which can be expressed as:
(35) |
Double precision computation is typically used in small-signal stability analysis, because solving eigenpairs is computationally intensive and introduces relatively large round-off errors. The scaling factor is set to be 1
The New England 10-machine 39-bus system is often used for stability analysis. Details of network parameters, nodal power, and dynamic parameters can be found in [
1) Effectiveness of Proposed Algorithm
To analyze the effectiveness of the proposed algorithm, the following four cases are taken into account. All cases are solved by the proposed algorithm, where all eigenvalues are computed by the QR algorithm.
1) Case 1: OPF without SSSC.
2) Case 2: OPF with SSSC1 (OPF-SSSC1), without considering H restart.
3) Case 3: OPF-SSSC1, considering H restart.
4) Case 4: OPF with SSSC2 (OPF-SSSC2), considering H restart.
Simulation results and electromechanical oscillation modes in four cases for New England 10-machine 39-bus are shown in
Case | L | ζm (%) | l | ||
---|---|---|---|---|---|
Case 1 | 40308.18 | 1.07 | 9 | 106 | |
Case 2 | 43310.49 | 2.96 | 9 | 120 | |
Case 3 | 43210.34 | 2.99 | 14 | 187 | 4 |
Case 4 | 42836.08 | 3.01 | 9 | 117 | 1 |

Fig. 3 Electromechanical oscillation modes in four cases for New England 10-machine 39-bus system.
From
The eigenvalue changes for Case 3 and Case 4 are provided by
Case 3 | Case 4 | ||||
---|---|---|---|---|---|
λm | ζm (%) | λm | ζm (%) | ||
1.03 | 1 | 1.03 | 2 | ||
1.03 | 1 | 2.56 | 2 | ||
2.24 | 1 | 2.49 | 3 | ||
2.24 | 1 | 2.99 | 3 | ||
2.24 | 1 | 2.99 | 3 | ||
2.76 | 2 | 3.00 | 3 | ||
2.97 | 2 | 3.00 | 3 | ||
2.98 | 2 | 3.01 | 3 | ||
2.97 | 1 | 3.01 | 3 | ||
3.01 | 0 | ||||
2.99 | 1 | ||||
2.99 | 1 | ||||
2.99 | 1 | ||||
2.99 | 1 |
2) Feasibility of Hybrid Algorithm Without Variable Selection
Since the New England 10-machine 39-bus system is relatively small, there is no need to reduce the number of state variables in x. Taking Case 4 as an example, the SSSC-OPF results under different threshold S are provided in
S | L | ζm (%) | l | Computation time (s) | ||||
---|---|---|---|---|---|---|---|---|
42836.08 | 3.01 | 9 | 117 | 1 | 117 | 0 | 44.73 | |
0.01 | 42590.68 | 3.01 | 12 | 170 | 1 | 67 | 4 | 31.14 |
0.05 | 42720.13 | 3.01 | 12 | 143 | 1 | 30 | 4 | 22.12 |
0.15 | 42780.83 | 3.00 | 12 | 153 | 1 | 24 | 4 | 21.13 |
0.25 | 42826.50 | 3.01 | 12 | 165 | 1 | 16 | 3 | 18.96 |
3) Comparison with Existing Approaches
In [
The results of different algorithms for New England 10-machine 39-bus system are provided in
Algorithm | σr | ζm (%) | Δσr | Δζm (%) | ΔL (%) | Computation time (s) |
---|---|---|---|---|---|---|
Proposed | 0.19 | 3.01 | 0.12 | 1.94 | 5.66 | 31.14 |
SQP-GS [ | 0.20 | 0.16 | 3.17 | 40.60 | ||
SA [ | 0.32 | 5.00 | 0.24 | 4.22 | 4.46 | 86.39 |
Algorithm | Machine | Runtime environment |
---|---|---|
Proposed |
Dell Precision T7920 with a 2.4 GHz CPU and 16 GB RAM | Visual Fortran 11.0 |
SQP-GS [ |
Dell Precision T5810 with a 3.5 GHz CPU and 64 GB RAM | CPLEX 12.60 |
SA [ |
1.7 GHz Intel Xeon CPU and 32 GB RAM | MATLAB 9.2 |
A modified practical 68-machine 2395-bus system shown in

Fig. 4 A modified practical 68-machine 2395-bus system.
The system consists of 66 synchronous machines and 2 synchronous compensators, where 57 generators are rescheduled to improve the small-signal stability. There are 2395 buses, 5488 transmission lines, 1496 transformers, and 458 constant impedance loads. The generators are described by the sixth-order model. Except for the equivalent generator, all the generators are equipped with excitation system, turbine governor, and power system stabilizer [
1) Comparison of OPF-SSSC1 and OPF-SSSC2
Results of OPF-SSSC1 and OPF-SSSC2 are provided by
OPF-SSSC | L | ζm (%) | l | Computation time (s) | ||
---|---|---|---|---|---|---|
OPF-SSSC1 | 115677.92 | 3.98 | 23 | 233 | 7 | 8821.81 |
OPF-SSSC2 | 116633.21 | 4.17 | 10 | 106 | 1 | 4137.84 |
2) Variable Selection for Eigenvalue Estimation
Since the modified practical 68-machine 2395-bus system is relatively large, it is necessary to reduce the number of state variables in x. The thresholds are set to be and . Taking the second iteration of the OPF-SSSC2 as an example, the logic of variable selection can be explained. According to the results of the second iteration, there are four modes with damping ratios smaller than . The eigenvalues and damping ratios of the four modes, , , , and , are shown in
No. | λ | ζ (%) | Dominant variable serial number | |
---|---|---|---|---|
Each mode | All considered mode | |||
1107, 1108 | j3.4817 | 4.11 | 40, 84, 52, 38, 28, 18 | 40, 84, 52, 88, 62, 60, 28, 18, 38, 2, 68, 10 |
1003, 1004 | j5.4111 | 5.11 | 84, 38, 52, 60, 40, 10, 28, 18 | |
1010, 1011 | j4.7853 | 4.66 | 40, 84, 52, 88, 62, 60, 28, 18, 38, 2, 68, 10 | |
975, 976 | j7.5428 | 5.23 | 40, 60, 84, 88, 28, 52 |
In the proposed algorithm, all state variables of the modified practical 68-machine 2395-bus system are and the corresponding serial numbers are [].

Fig. 5 All w
3) Effectiveness of Hybrid Algorithm with Variable Selection
The performance comparison of the hybrid algorithm with variable selection is shown in
Variable | S | L | ζm (%) | l | Computation time (s) | ||||
---|---|---|---|---|---|---|---|---|---|
Dominant variables | 0.15 | 116813.51 | 4.25 | 11 | 124 | 1 | 34 | 5 | 3115.23 |
0.25 | 116847.25 | 4.33 | 10 | 116 | 1 | 27 | 5 | 2720.37 | |
0.35 | 116473.37 | 4.24 | 13 | 142 | 1 | 26 | 3 | 3101.72 | |
0.45 | 116473.37 | 4.24 | 13 | 142 | 1 | 26 | 3 | 3039.26 | |
All variables | 0.15 | 116724.37 | 4.24 | 11 | 106 | 1 | 36 | 7 | 3592.10 |
4) Comparison with Partial Eigenvalue Approach
In the proposed algorithm, the eigenvalues are computed using the hybrid algorithm with variable selection that combines accurate eigenvalue computation and eigenvalue estimation. In addition to the QR algorithm, partial eigenvalue approaches, e.g., Krylov-Schur approach [
S | L | ζm (%) | l | Computation time (s) | ||||
---|---|---|---|---|---|---|---|---|
0.15 | 116812.71 | 4.25 | 11 | 124 | 1 | 34 | 5 | 2837.57 |
0.45 | 116473.97 | 4.24 | 13 | 142 | 1 | 26 | 3 | 2826.66 |
From Tables
Taking the New England 10-machine 39-bus system as an example, the OPF-SSSC2 under the normal operating condition is extended to the security constrained OPF with SSSC2 (SCOPF-SSSC2). The following 14 scenarios are taken into account. 1) Scenario 1: base case. 2) Scenario 2: lines 3-18 and 25-26 are out of service. 3) Scenario 3: lines 16-17 and 4-14 are out of service. 4) Scenario 4: line 6-11 is out of service. 5) Scenario 5: 360 MW load increase. 6) Scenario 6: lines 16-17, 4-14, and 25-26 are out of service. 7) Scenario 7: lines 16-17, 4-14, 25-26, and 1-39 are out of service. 8) Scenario 8: line 21-22 is out of service. 9) Scenario 9: lines 9-39 is out of service. 10) Scenario 10: loading. 11) Scenario 11: loading. 12) Scenario 12: loading. 13) Scenario 13: loading. 14) Scenario 14: loading at buses 16 and 21 and lines 21-22 are out of service. Ramping limits p.u./min [
Scenario | Critical mode | |
---|---|---|
OPF-SSSC2 | SCOPF-SSSC2 | |
1 | ||
2 | ||
3 | ||
4 | ||
5 | ||
6 | ||
7 | ||
8 | ||
9 | ||
10 | ||
11 | ||
12 | ||
13 | ||
14 |
The critical mode for Scenario 1 in column 2 of
A practical algorithm for the SSSC-OPF problem is proposed in this paper. The effectiveness of the proposed algorithm is validated on the New England 10-machine 39-bus system and a modified practical system. The unique features of the proposed algorithm are provided as follows.
1) The SSSC adopted in this paper not only reflects the dynamic performance requirement of the system, but also fully considers the effect of steady-state changes on eigenvalues. The simultaneous solution formula of the SSSC-OPF model is also established and solved by the quasi-Newton approach.
2) Different penalty factors are set for SSSCs according to the stabilization degree of constrained eigenvalues, which effectively alleviates the eigenvalue oscillations when simultaneously solving the SSSC-OPF problem.
3) The solution time of the SSSC-OPF problem is significantly reduced by the proposed algorithm, which includes variable selection for eigenvalue estimation and strategy selection for eigenvalue computation.
The SSSC-OPF model and algorithm for power systems with high penetration of renewable energy will be developed in future work.
References
H. W. Dommel and W. F. Tinney, “Optimal power flow solutions,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-87, no. 10, pp. 1866-1876, Oct. 1968. [Baidu Scholar]
R. D. Zimmerman, C. E. Murillo-Sánchez, and R. J. Thomas, “Matpower: steady-state operations, planning, and analysis tools for power systems research and education,” IEEE Transactions on Power Systems, vol. 26, no. 1, pp. 12-19, Feb. 2011. [Baidu Scholar]
E. Vaahedi, Y. Mansour, J. Tamby et al., “Large scale voltage stability constrained optimal var planning and voltage stability applications using existing OPF/optimal var planning tools,” IEEE Transactions on Power Systems, vol. 14, no. 1, pp. 65-74, Feb. 1999. [Baidu Scholar]
Y. Fu, X. Zhang, L. Chen et al., “Analytical representation of data-driven transient stability constraint and its application in preventive control,” Journal of Modern Power Systems and Clean Energy, vol. 10, no. 5, pp. 1085-1097, Sept. 2022. [Baidu Scholar]
Y. Lin, X. Zhang, J. Wang et al., “Voltage stability constrained optimal power flow for unbalanced distribution system based on semidefinite programming,” Journal of Modern Power Systems and Clean Energy, vol. 10, no. 6, pp. 1614-1624, Nov. 2022. [Baidu Scholar]
Y. Yu, Y. Liu, C. Qin et al., “Theory and method of power system integrated security region irrelevant to operation states: an introduction,” Engineering, vol. 6, no. 7, pp. 754-777, Jul. 2020. [Baidu Scholar]
C. Chung, L. Wang, F. Howell et al., “Generation rescheduling methods to improve power transfer capability constrained by small-signal stability,” IEEE Transactions on Power Systems, vol. 19, no. 1, pp. 524-530, Feb. 2004. [Baidu Scholar]
C. Li, H. D. Chiang, and Z. Du, “Network-preserving sensitivity-based generation rescheduling for suppressing power system oscillations,” IEEE Transactions on Power Systems, vol. 32, no. 5, pp. 3824-3832, Sept. 2017. [Baidu Scholar]
P. Li, H. Wei, B. Li et al., “Eigenvalue-optimisation-based optimal power flow with small-signal stability constraints,” IET Generation, Transmission & Distribution, vol. 7, no. 5, pp. 440-450, May 2013. [Baidu Scholar]
P. Li, J. Qi, J. Wang et al., “An SQP method combined with gradient sampling for small-signal stability constrained OPF,” IEEE Transactions on Power Systems, vol. 32, no. 3, pp. 2372-2381, May 2017. [Baidu Scholar]
H. A. Seyed, R. Abdorreza, and T. B. Samad, “Optimal re-dispatch of generating units ensuring small signal stability,” IET Generation, Transmission & Distribution, vol. 14, no. 18, pp. 3692-3701, Aug. 2020. [Baidu Scholar]
J. Condren and T. Gedra, “Expected-security-cost optimal power flow with small-signal stability constraints,” IEEE Transactions on Power Systems, vol. 21, no. 4, pp. 1736-1743, Nov. 2006. [Baidu Scholar]
R. Zarate-Minano, F. Milano, and A. Conejo, “An OPF methodology to ensure small-signal stability,” IEEE Transactions on Power Systems, vol. 26, no. 3, pp. 1050-1061, Aug. 2011. [Baidu Scholar]
S. Kim, A. Yokoyama, Y. Takaguchi et al., “Small-signal stability-constrained optimal power flow analysis of multiterminal VSC-HVDC systems with large-scale wind farms,” IEEJ Transactions on Electrical and Electronic Engineering, vol. 14, no. 7, pp. 1033-1046, Jul. 2019. [Baidu Scholar]
J. Xing, C. Chen, and P. Wu, “Optimal active power dispatch with small-signal stability constraints,” Electric Power Components and Systems, vol. 38, no. 9, pp. 1097-1110, Sept. 2010. [Baidu Scholar]
Y. Li, G. Geng, Q. Jiang et al., “A sequential approach for small signal stability enhancement with optimizing generation cost,” IEEE Transactions on Power Systems, vol. 34, no. 6, pp. 4828-4836, Nov. 2019. [Baidu Scholar]
J. Liu, Z. Yang, J. Zhao et al., “Explicit data-driven small-signal stability constrained optimal power flow,” IEEE Transactions on Power Systems, vol. 37, no. 5, pp. 3726-3737, Nov. 2022. [Baidu Scholar]
P. Pareek and H. D. Nguyen, “A convexification approach for small-signal stability constrained optimal power flow,” IEEE Transactions on Control of Network Systems, vol. 8, no. 4, pp. 1930-1941, Nov. 2021. [Baidu Scholar]
D. Pullaguram, R. Madani, T. Altun et al., “Small-signal stability-constrained optimal power flow for inverter dominant autonomous microgrids,” IEEE Transactions on Industrial Electronics, vol. 69, no. 7, pp. 7318-7328, Jul. 2022. [Baidu Scholar]
L. Shi, C. Wang, L. Yao et al., “Optimal power flow solution incorporating wind power,” IEEE Systems Journal, vol. 6, no. 2, pp. 233-240, Jun. 2012. [Baidu Scholar]
Y. Yang, Y. Luo, and L. Yang, “Small-signal stability-constrained optimal power flow model based on BP neural network algorithm,” Sustainability, vol. 14, no. 20, p. 13386, Nov. 2022. [Baidu Scholar]
X. Wang, Y. Song, and M. Irving, Modern Power Systems Analysis. New York: Springer, 2010. [Baidu Scholar]
S. Frank, I. Steponavice, and S. Rebennack, “Optimal power flow: a bibliographic survey II,” Energy Systems, vol. 3, pp. 259-289, Mar. 2012. [Baidu Scholar]
A. M. H. Rashed and D. H. Kelly, “Optimal load flow solution using Lagrangian multipliers and the Hessian matrix,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-93, no. 5, pp. 1292-1297, Sept. 1974. [Baidu Scholar]
K. W. Wang, C. Y. Chung, C. T. Tse et al., “Multimachine eigenvalue sensitivities of power system parameters,” IEEE Transactions on Power Systems, vol. 15, no. 2, pp. 741-747, May 2000. [Baidu Scholar]
J. Nocedal and S. J. Wright, Numerical Optimization. New York: Springer, 2006. [Baidu Scholar]
W. M. Lebow, R. Rouhani, R. Nadira et al., “A hierarchical approach to reactive volt ampere (VAR) optimization in system planning,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-104, no. 8, pp. 2051-2057, Aug. 1985. [Baidu Scholar]
A. Monticelli, M. V. F. Pereira, and S. Granville, “Security-constrained optimal power flow with post-contingency corrective rescheduling,” IEEE Transactions on Power Systems, vol. 2, no. 1, pp. 175-180, Feb. 1987. [Baidu Scholar]
A. M. Sasson, F. Viloria, and F. Aboytes, “Optimal load flow solution using the hessian matrix,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-92, no. 1, pp. 31-41, Jan. 1973. [Baidu Scholar]
M. A. Pai, Energy Function Analysis for Power System Stability. New York: Springer, 1989. [Baidu Scholar]
Z. Wu and X. Zhou, “Power system analysis software package (PSASP) - an integrated power system analysis tool,” in Proceedings of International Conference on Power System Technology, Beijing, China, Aug. 1998, pp. 7-11. [Baidu Scholar]
Y. Li, G. Geng, and Q. Jiang, “An efficient parallel Krylov-Schur method for eigen-analysis of large-scale power systems,” IEEE Transactions on Power Systems, vol. 31, no. 2, pp. 920-930, Mar. 2016. [Baidu Scholar]
R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. Philadelphia: SIAM, 1998. [Baidu Scholar]