Abstract
The deployment of dynamic reactive power source can effectively improve the voltage performance after a disturbance for a power system with increasing wind power penetration level and ubiquitous induction loads. To improve the voltage stability of the power system, this paper proposes an adaptive many-objective robust optimization model to deal with the deployment issue of dynamic reactive power sources. Firstly, two metrics are adopted to assess the voltage stability of the system at two different stages, and one metric is proposed to assess the tie-line reactive power flow. Then, a robustness index is developed to assess the sensitivity of a solution when subjected to operational uncertainties, using the estimation of acceptable sensitivity region (ASR) and D-vine Copula. Five objectives are optimized simultaneously
① total equipment investment; ② adaptive short-term voltage stability evaluation; ③ tie-line power flow evaluation; ④ prioritized steady-state voltage stability evaluation; and ⑤ robustness evaluation. Finally, an angle-based adaptive many-objective evolutionary algorithm (MaOEA) is developed with two improvements designed for the application in a practical engineering problem: ① adaptive mutation rate; and ② elimination procedure without a requirement for a threshold value. The proposed model is verified on a modified Nordic 74-bus system and a real-world power system. Numerical results demonstrate the effectiveness and efficiency of the proposed model.
GROWING integration of renewable energy reduces the inertia of a power system and makes it more vulnerable to disturbance caused by power equipment failure or severe weather conditions [
Dynamic reactive power compensation devices such as static synchronous compensator (STATCOM) are better candidates to alleviate the voltage stability issue because of their ability to provide dynamic and continuous reactive power in a short time (usually in the millisecond scale) [
As for the uncertainties, since extreme weather condition is major direct or indirect causes for some of the large scale blackout events [
For a reactive power source planning problem considering various and conflicting objectives such as planning and operational cost, voltage stability, optimal reactive power flow, and power loss, a multi-objective (3 or fewer objectives) optimization model [
In this paper, an adaptive many-objective robust optimization model of dynamic reactive power sources is proposed to enhance the voltage stability of the power system with a high wind penetration level. The major contributions are as follows.
1) Tie-line power flow index (TPFI) is proposed to assess the reactive power transmission for the minimization of cross-region reactive power transmission.
2) Based on the estimation of acceptable sensitivity region (ASR), a robustness index (RI) is developed to assess the robust optimality of the solutions quantitatively when subjected to correlated uncertainties. There is no requirement for probability distributions of any operational uncertainty considered in the robust optimality analysis. D-vine Copula is adopted to represent the correlation between operational uncertainties and identify the low-probability correlated scenarios that might lead to a conservative estimation of ASR.
3) An index that can effectively differentiate the common voltage deviations and the delayed voltage recovery is proposed to evaluate the short-term voltage stability. This index is with a self-adaptive strategy to assign weighting factors to different buses, enabling a high-efficient reactive power compensation.
One other minor contribution of this study is to improve the applicability of a many-objective evolutionary algorithm (MaOEA) for a practical reactive power source planning problem in the following aspects: ① an adaptive mutation rate to increase the diversity; and ② an angle-based elimination procedure to avoid unrealistic parameter tuning for a complicated practical engineering problem.
Voltage stability is defined as the ability of a power system to maintain steady voltages at all buses in the system after being subjected to a disturbance from a given initial operating condition [
In this study, two indices are employed to assess the voltage stability of a wind-penetrated power system. An improved voltage collapse proximity index (VCPI) considering the priorities of power lines is used to assess the voltage stability in the pre-contingency stage. An adaptive transient voltage severity index (TVSI) is used to evaluate voltage stability after a contingency. Tie-line reactive power flow is assessed using TPFI.
TVSI proposed in [
(1) |
(2) |
With the growing integration of wind turbines in the power system and the inherent weather-sensitive nature, the extent of the voltage deviation and the time used to restore to an acceptable voltage level, which are not fully revealed using TVSI, should be considered in the evaluation of voltage stability. Therefore, an adaptive index , which is based on the original TVSI, is used to describe the immediate voltage behavior of buses, as proposed in our previous work [
(3) |
An illustration for the calculation of the index is shown in

Fig. 1 Illustration for calculation of .
Using a fixed weighting factor for a bus throughout the whole optimization process is not cost-efficient, since the marginal cost is high to increase installed capacity of a reactive power source at the bus with slow recovery speed and poor voltage response at the beginning stage when it has been compensated adequately. For the case of a fixed weight factor, the focus of the optimization process will be distracted because the importance of the buses with poor voltage response initially should not be prioritized at a later stage of the optimization. Therefore, a self-adaptive approach is used to determine the weighting factors as proposed in our previous work [
(4) |
(5) |
For a practical utility grid, it is usually divided into different operating regions to limit the reactive power transmission between regions and make it easier to regulate the voltage. However, in existing studies on the planning problem of reactive power resources, the tie-line reactive power flow is not optimized. Instead, it is only considered as a steady-state constraint. If the reactive power flow is not optimized, it is possible that there is a large reactive power flow, but still within the constraint, through the tie-lines, which is detrimental to the voltage stability of the power system. This is because long-distance or regional transmission of reactive power is unfavorable, particularly large tie-line power flow usually exacerbates the severity of the contingency (at or near tie-lines). So, the implementation of the tie-line reactive power flow as an objective can optimize the reactive power flow to alleviate the situation.
Although not as effective as reactive power source deployment, optimization of tie-line power flow will be an economic option to alleviate the reactive power deficiency problem during a contingency. Therefore, TPFI is proposed in this paper for the purpose of optimization.
(6) |
Line-based VCPI, of which mathematical details can be found in [
(7) |
The proposed model uses electromagnetic transient simulation in Power System Simulator/Engineering (PSS/E) to obtain the post-contingency voltage trajectories for the dynamic analysis and voltage stability evaluation. The electromagnetic transient model cannot be solved by prevalent commercial solvers like CPLEX. Besides, the studied reactive power source deployment problem relates to many planning and operating criteria such as device investment, steady-state and short-term voltage stabilities, and operational costs. These non-linear and sometimes conflicting (such as high stability versus low cost) objectives should be optimized simultaneously instead of being formulated as a single-objective model. Specifically, combining five objectives as one objective through the weighting-sum method has the following issues: ① it is difficult to subjectively determine the weighting factors (for a weighted-sum single-objective model) for each sub-objective; ② the units of each sub-objective might be inherently different; and ③ sensitivities of each sub-objective are different.
Pareto optimality theory [
A compact mathematical model, presented as (8)-(10), is used to explain the concept of sensitivity region for a many-objective optimization problem. Uncertainties incorporated in the robust optimality analysis include outputs of wind turbines and load levels.
(8) |
s.t.
(9) |
(10) |
Since the optimal solution is obtained under a specific operational status, the random variations of the uncertain variables will result in the change of objective values. A robust solution means the solution is less sensitive to variations of operating conditions. To describe this robustness, the concept of ASR [

Fig. 2 Illustration of ASR.
Subscript “0” denotes the original variable and state. Edge of ASR is formulated by the maximum acceptable change of the objective function when a specific uncertain variable changes. An example of two uncertainties, namely and , are shown in
Conventionally, the optimality of a solution is actually evaluated with a group of specific scenarios. However, for an engineering problem, the changes of different uncertainties might deteriorate the optimality to some extent.

Fig. 3 Negative impact of 3 typical operational uncertainties on voltage response after a fault.
Define nominal Pareto front (NPF) as the Pareto front generated with a fixed . Then, RPF, all of which are inferior to NPF, can be illustrated in

Fig. 4 RPF in 3 typical situations.
Besides, unlike other robust analysis methods, probability distribution of the studied variables is not needed in the following robust analysis. Therefore, given the difficulty in narrowing the gap between planned voltage performance and the actual performance of a power system with high wind penetration level and high percentages of motor loads, the implementation of robust optimality is acceptable.
As shown in
In practice, the actual contour of ASR is usually far from symmetric and regular. Therefore, evaluating the robustness of a solution using an accurate calculation of the shadowed area is very difficult and cannot reveal the sensitivity of the solution in the worst scenario. As discussed in
MSD of ASR is defined as the MSD within the ASR, in terms of the change rate of when subjected to n uncertainties.
(11) |

Fig. 5 Illustrations of MSD for ASR and FSR. (a) MSD for ASR. (b) MSD for FSR.
s.t.
(12) |
where is an acceptable change caused by ; and .
Although is efficient, in terms of computation burden, in estimating the ASR, it might lead to a conservative result if the shape of the ASR, which of course cannot be known in advance, has several points that are much closer to the origin of the coordinates than other points on the edge of ASR. As illustrated in

Fig. 6 Illustration of improved estimation of ASR to avoid conservativeness.
(13) |
where is the set of correlated scenarios that are within the area surrounded by arc determined by and the x-y coordinates; is the set of all the scenarios studied; and is the minimum acceptable ratio of . Instead of using Monte Carlo method or other conventional random sampling methods that consider the operational uncertainty separately, to generate the scenarios for (13), the correlated , considering the dependent uncertainties of outputs of wind turbines and load demands, is generated based on D-vine Copula [
Similar to ASR, FSR is defined as the maximum variations of uncertainties within the constraints. It is used to estimate the feasibility of the solution. An example is shown in
(14) |
s.t.
(15) |
(16) |
(17) |
A 5-objective robust optimization model of dynamic reactive power sources is formulated with following sub-objectives: ① total equipment investment; ② adaptive short-term voltage stability evaluation; ③ tie-line power flow evaluation; ④ prioritized steady-state voltage stability evaluation; and ⑤ robustness evaluation.
A risk-based assessment for contingency analysis is used in this paper. It is carried out in following four steps: ① select the index for assessment; ② select a critical contingency (with probability); ③ evaluate the contingency; and ④ calculate the risk-based evaluation results. Specifically, the
(20) |
Like , a risk-based assessment is also used for the
(21) |
The
(22) |
For a many-objective optimization problem, it is hard to strike a trade-off between the robustness and the voltage performance only based on and , because they just provide a reference for robustness comparison. For instance, with and , the planners and the operators cannot tell whether this installation plan solution is robust. Therefore, an RI is developed as (23) to represent the robust optimality of the planned reactive power source installation decisions when subjected to multiple uncertainties quantitatively. serves as a reference and it can be selected based on an acceptable .
(23) |
Constraint (24) is for power flow balance. The steady-state operational limits are expressed as (25). Constraint (26) is for the rotor angle stability. It is used to make sure that the rotor angle deviation between 2 generators during a period of T () is smaller or equals . Constraint (27) is the time-varying voltage trajectory constraint of low-voltage ride through (LVHT) or high-voltage ride through (HVRT).
(24) |
(25) |
(26) |
(27) |
Generally, there are 5 steps in MaOEAs: initialization, fitness assignment, offspring selection, crossover and mutation, and environmental selection. Fitness assignment is essentially the focus of most of MaOEAs such as non-dominated sorting genetic algorithm III (NSGA-III). However, few algorithms can perform well in convergence and diversity simultaneously. MaOEA with coordinated selection strategy (MaOEA-CSS) is proposed in [
It is favourable to have a smaller mutation rate for the first few population generations since their diversity is relatively good. However, if the diversity becomes poor as the generation number increases, it is more suitable to have a higher mutation rate for the improvement of diversity. An inflexible mutation rate, which is insensitive to the diversity, might generate unsatisfactory offspring.
For environmental selection step, Euclidean distance is used in MaOEA-CSS to eliminate inferior solutions. A threshold value is needed to compare the Euclidean distances, which plays a critical role in convergence and diversity performance of the whole population. However, there is no universal method to determine the threshold for a specific problem. It varies significantly even for the benchmark problems, let alone a practical engineering problem that is much more complicated.
To further improve MaOEA-CSS, an adaptive angle-based MaOEA is developed with 2 improvements designed specifically for the reactive power source deployment problem.
A self-adaptive mutation rate is proposed to address the complexity of a practical engineering problem based on the crowdedness of the individuals and the generation number. First, a crowdedness index (CI), which is exclusively determined by the population crowdedness, is defined as (28).
(28) |
(29) |
A self-adaptive mutation rate is defined as (30) and is the current generation.
(30) |
For the first few generations, is relatively small. As the algorithm proceeds, is increasing with the generation number. Instead of a fixed mutation rate or a mutation rate straightforward in proportion to the generation number, in this paper will be more adaptive and practical since is only determined by the population crowdedness. In this case, both crowdedness of individuals and the generation number are considered in the adaptive mutation rate.
Instead of using the threshold value, which is very difficult to be determined for a practical engineering problem, an angle-based elimination procedure is employed to replace the original elimination procedure in [
For the
Algorithm 1 : elimination procedure |
---|
Input: S (combination of parents and offspring) |
Output: new population |
1. Assign to |
2. Remove m extreme solutions from S and add them to |
3. while do |
4. Find the individual pair () which has the smallest angle in S |
5. Find the worst individual and remove it from S |
6. end while |
7. Assign to |
8. Return |
The details of each step are described as follows and the illustration of computation flowchart is also given in

Fig. 7 Illustration of computation flowchart.
Step 1: initialization. First, algorithm parameters are configured. Then, based on the methodology proposed in our previous work [
Step 2: evaluation of steady-state voltage stability and calculation of tie-line power flow before contingencies. Steady-state voltage stability evaluation is conducted based on . Tie-line power flow before contingencies is also calculated in this stage. The base case evaluation result is stored and will be used for the robust optimality analysis in later stages.
Step 3: evaluation of short-term voltage stability after contingency and tie-line power flow evaluation. Short-term voltage stability of the system is evaluated based on the post-contingency voltage trajectories obtained from time-domain simulations under a set of critical contingencies. Tie-line power flow after contingencies is also calculated in this stage. The base case evaluation result is stored and will be used for the robust optimality analysis in a later stage.
Step 4: robust optimality analysis. As an estimation of ASR, RI is calculated according to (11) and (14) for each solution, with the consideration of correlated uncertainties.
Step 5: calculation of objectives. Firstly, total equipment investment, , , TPFI, and RI are calculated for each solution under each contingency. Then, five objectives are calculated accordingly based on (18), (20)-(23).
Step 6: update of offspring. The best offspring of the current generation is identified and updated. This procedure includes individual comparison, elimination of inferior solutions (using the proposed angle-based elimination procedure) and selection of the best Pareto solution. Then the selected solution is regarded as the best offspring in the generation.
Step 7: termination. If any of the following two criteria is satisfied, terminate the whole procedure and generate an RPF: ① the iteration number reaches the pre-determined maximum number; or ② there is a smaller change in fitness function compared with the minimum tolerance.
Step 8: update of mutation rate and generation of offspring. Update the mutation rate based on the self-adaptive strategy proposed in Section IV-B if the termination does not occur, followed by generation of offspring. Then, go back to Step 2.
Step 9: judgement of stop criteria. All the evaluations and cost calculations of this solution will be stopped if there is an unsatisfied constraint during the time-domain simulation, and the corresponding objective functions are computed with a high penalty factor. Then, go back to Step 4.
A Nordic 74-bus test system [

Fig. 8 Nordic 74-bus test system.
Candidate buses are marked in red circles. Doubly-fed induction generators (DFIGs) are located at buses 4021, 4042, and 4062. Time-varying load dynamics are represented by a complex load model [
Parameter | Value | Parameter | Value |
---|---|---|---|
($) | 1500000 | 2.0 | |
($) | 50000 | (%) | 1 |
0.15 | (%) | 1 | |
ω | LM (%) | 25 | |
δ | 0.1 | SM (%) | 15 |
0.5 | DL (%) | 10 | |
(s) | 0.5 | TS (%) | 10 |
(p.u.) | 0.95 | CP (%) | 10 |
(p.u.) | 1.05 | Kp | 2 |
2.0 |

Fig. 9 RPF consisting of 80 robust optimal solutions.
Bus No. | Capacity (Mvar) | Bus No. | Capacity (Mvar) |
---|---|---|---|
41 | 38.0 | g11 | 37.5 |
42 | 105.9 | g14 | 62.9 |
46 | 108.4 | g17 | 45.6 |
2031 | 48.2 |
Five additional models are compared to show the advantages of the proposed model (Model A), as shown in
Model | Objective | Solution algorithm | Selection strategy (from Pareto front) |
---|---|---|---|
A | , , , , | Proposed |
Fuzzy membership [ |
B | , , , , | Proposed | Fuzzy membership |
C | , , , , | Proposed | Best results |
D | , , , , |
NSGA-III [ | Fuzzy membership |
E | , , |
NSGA-II [ | Fuzzy membership |
F | , , , , and RI calculated without (13) and (17) | Proposed | Fuzzy membership |
1) The comparison between Model A and Model B aims to demonstrate the superiority of the evaluation metrics ( and ).
2) The comparison between Model A and Model C also aims to demonstrate the superiority of the evaluation metrics ( and ), even with a more aggressive selection strategy.
3) The comparison between Model A and Model D aims to demonstrate the superiority of the proposed solution algorithm.
4) The comparison between Model A and Model E aims to demonstrate the effectiveness of the implementation of proposed TPFI and RI as objectives.
5) The comparison between Model A and Model F aims to demonstrate the effectiveness of constraints (13) and (17). The constraints are introduced to avoid a conservative estimation of ASR and FSR, respectively.
The simulation results are summarized in
Model | f1 | f2 | f3 | f4 | f5 | Computation time (s) |
---|---|---|---|---|---|---|
A | 32.83 | 0.3024 | 0.6068 | 31.475 | 1.3598 | 249580 |
B | 32.68 | 0.4300 | 0.6213 | 31.553 | 1.3759 | 266076 |
C | 35.14 | 0.3553 | 0.6161 | 31.504 | 1.4027 | 266076 |
D | 32.86 | 0.3199 | 0.6131 | 31.544 | 1.4457 | 272863 |
E | 31.20 | 0.3561 | 0.6267 | 31.479 | 1.4682 | 267459 |
F | 32.96 | 0.4199 | 0.6144 | 31.540 | 1.4569 | 251114 |
The comparison of numerical results between Model A and Model B demonstrates that the proposed voltage stability indices make the investment more cost-efficient: a slight investment increase (0.46%) can make a significant performance improvement in all five objectives. A significant improvement (29.6%) is observed in the evaluation result of the short-term voltage stability ().
Furthermore, even with 7.0% more investment, the voltage performance of Model C (based on original and ) is still much worse than that of the Model A. It should be noted that the decision-making is solely based on results.

Fig. 10 Superiority of compared with original TVSI.
For the original TVSI only considering the voltage violation in the post-contingency stage, the delayed voltage recovery after 6 s has very little impact on the final evaluation results. Based on the definition of original TVSI, which does not differentiate the common voltage deviations (within the required time window) and the delayed voltage recovery, the evaluation result (0.302 after normalized) for short-term voltage stability is calculated by integrating all the voltage deviations. Nonetheless, the inferior voltage recovery process will put wind turbines in danger of automatically triggering and frequent voltage violations. Alternatively, can penalize and reflect the prolonged voltage restoration. For the case of
As shown in

Fig. 11 Sensitivity analysis of and TVSI subjected to various capacities.
Since the proposed TPFI is implemented in the optimization model as an objective (), Model A has the smallest reactive power flow. This is particularly beneficial for a scenario when a major voltage disturbance or fault occurs on the tie-lines, since a fault on a heavy-loaded tie-line is usually a trigger for cascading contingencies. For Model E, of which tie-line power flow is not optimized, there is a large reactive power flow compared with all other models. Since reactive power source deployment in a real-world power system is usually carried out in an area-based way instead of taking the whole system as one entity, the implementation of the proposed TPFI in the optimization model is helpful in minimizing the reactive power flow, compared with considering the reactive power flow as a steady-state constraint.
Among all the models with RI as an objective, Model F (using a conservative way to calculate RI) has the worst robustness. It is exactly the conservative case illustrated in

Fig. 12 Pareto front of solutions from Model D.
Compared with
Due to the highly non-linear and non-convex features, as well as the time-consuming time-domain simulations, the reactive power source deployment problem considering short-term voltage stability issues is naturally with high computational complexity. The total execution time T can be estimated based on the parameters listed in (31).
(31) |
where represents the tolerance degree for constraint (12) or (16). In practice, reactive power source deployment of a large-scale power system is usually studied in an area-based method rather than in a way that takes the whole system as one entity. By employing the area-based method, the self-sufficiency of each area can be maximized and the reactive power transmission between different areas can be minimized [
In the proposed planning model, with the help of proposed TPFI implemented as an objective in the optimization model, the computation complexity is alleviated by dividing the studied system into several smaller ones. When the size of the power system increases, the corresponding increase in computation time is relatively moderate since the number of candidate installation sites in a larger system is not necessary to increase proportionally.
A modified real-world power system in China, with 446 buses and 27 power plants (including wind farms), is adopted to show the efficiency and effectiveness of the proposed model. To better reflect the impact of the increasing penetration level of wind generation, seven thermal power plants in the original power system are replaced with wind farms. As shown in
Test system | Total computation time (s) |
---|---|
Nordic 74-bus (74 buses and 7 candidate buses) | 249580 |
Real-world power system (446 buses and 14 candidate buses) | 386017 |
So, the proposed model is applicable for reactive power source deployment of a real-world transmission network as long as the network can be divided into areas, which is also the operational practice for most electric utility corporations. However, among the factors that affect the computation cost as listed in (31), due to the expensive time-domain simulation (a single run costs a few seconds for a large-scale power system), is the dominant factor in the total computation cost of the proposed model. Therefore, for a modern power system with over thousands of buses, a distributed computing platform [
In this paper, an adaptive 5-objective robust optimization model of dynamic reactive power sources is proposed to enhance the voltage stability of a wind-penetrated power system and improve the robust optimality of the planned STATCOMs simultaneously. The key conclusions are listed as follows.
1) Compared with other models, robust optimal solutions from the RPF are robust to the changes of operational uncertainties. And a balance between the robustness optimality and voltage performance of a solution is achieved. Specifically, the proposed RI not only considers the correlation between uncertainties, but also effectively avoids a conservative estimation of the robustness without a compromise in voltage stability performance.
2) The proposed TPFI implemented in the optimization model as an objective can effectively minimize the reactive power flow, compared with a steady-state constraint approach.
3) The proposed short-term voltage stability index can effectively differentiate the common voltage deviations (within the required time window) and the delayed voltage recovery. Furthermore, this index can correctly reflect the marginal cost of improving the voltage performance of a bus with adequate reactive power support by adaptively changing the priorities of buses.
4) Compared with the original MaOEA-CSS, the proposed adaptive angle-based MaOEA does not need the parameter tuning for the threshold value. The feature is favorable for reactive power source deployment since it is usually difficult to determine such parameters for a practical engineering problem.
Nomenclature
Symbol | —— | Definition |
---|---|---|
A. | —— | Indices |
, | —— | Indices for generators |
, | —— | Indices for buses, objectives, or constraints |
—— | Index for contingency | |
—— | Index for lines | |
—— | Index for uncertainties or generations | |
—— | Index for an original state | |
—— | Index for Pareto solutions | |
—— | Index for time | |
B. | —— | Sets |
—— | Set of all buses | |
—— | Set of buses of static synchronous compensators (STATCOMs) | |
—— | Set of wind turbine buses | |
—— | Set of all generators | |
—— | Set of synchronous generators | |
—— | Set of lines | |
—— | Set of tie-lines | |
—— | Set of contingencies | |
—— | Set of time | |
C. | —— | Parameters |
, | —— | Lower and upper penalty parameters for voltage violation |
—— | Tolerance for an acceptable deviation | |
—— | Tolerance for constraint gi | |
—— | Rotor deviation under an extreme condition | |
—— | A threshold for the importance of buses | |
—— | A threshold for voltage deviation | |
—— | Installation cost of STATCOM | |
—— | Unit cost of STATCOM | |
—— | Percentage of constant power loads in complex load model | |
, | —— | The minimum and maximum crowdedness of the |
—— | Percentage of discharging lighting loads in complex load model | |
—— | Adaptive transient voltage severity index (TVSI) result of contingency k | |
—— | Tie-line power flow index (TPFI) result of contingency k | |
—— | Prioritized voltage collapse proximity index (VCPI) result of contingency k | |
—— | Time-varying voltage trajectory constraints for high-voltage ride through (HVRT) | |
—— | Parameter for voltage-dependent real power load in complex load model | |
—— | Percentage of large motor loads in complex load model | |
—— | Time-varying voltage trajectory constraints for low-voltage ride through (LVRT) | |
, | —— | The minimum and maximum apparent power flows on line l |
—— | Number of total buses | |
—— | Number of contingencies considered in time-domain simulations | |
, , | —— | Number of generations, population size, and number of decision variables for non-dominated sorting genetic algorithm II (NSGA-II) |
—— | Number of objective functions | |
—— | Number of equality constraints | |
—— | Number of inequality constraints | |
—— | Number of individual (in a generation) | |
—— | Number of total power lines | |
—— | Number of uncertainties | |
—— | The maximum value of generations | |
—— | Parameter for Lp-norm | |
, , | —— | Adaptive weighting factors |
—— | Mutation rate of the first generation | |
, | —— | The minimum and maximum active power outputs of generator g |
—— | Probability of contingency k | |
—— | Priority of line l | |
, | —— | The minimum and maximum reactive power outputs of generator g |
—— | Reference robust radius | |
—— | Percentage of small motor loads in complex load model | |
—— | Fault clearing time | |
—— | The maximum voltage deviation time (lower) | |
—— | The maximum voltage deviation time (upper) | |
—— | End of time-domain simulation time | |
—— | Fault time | |
, | —— | End of voltage deviation time within and beyond (lower) |
, | —— | End of voltage deviation time within and beyond (upper) |
—— | Computation time for a single time-domain simulation run | |
—— | Computation time for a single optimal power flow run | |
—— | Percentage of transformer saturation loads in complex load model | |
, | —— | Lower and upper voltage thresholds |
, | —— | The minimum and maximum voltages of bus i |
D. | —— | Variables |
—— | Uncertainties | |
—— | Variations of objective function i | |
—— | Acceptable deviation of objective function i | |
—— | Variation of uncertainty | |
—— | Active power output of wind turbine w | |
—— | Reactive power output of wind turbine w | |
—— | Rotor angle deviation between two generators during a period of T | |
—— | Voltage angle difference between buses i and j | |
—— | Adaptive weighting factor for bus i | |
—— | Susceptance between buses i and j | |
—— | Reactive power source capacities at bus i | |
, | —— | Normalized and un-normalized crowdedness of the |
—— | Conductance between buses i and j | |
—— | Apparent power flow on line l | |
—— | Probability of contingency k | |
—— | Adaptive mutation rate | |
—— | Active power output of generator g | |
—— | Active power transferred to the receiving end through line l | |
—— | Reactive power output of generator g | |
—— | Reactive power transferred to the receiving end through line l | |
—— | Reactive power in tie-line l | |
—— | Reactive power output of STATCOM | |
—— | Radius of estimated feasibility sensitivity region (FSR) | |
—— | Radius of estimated acceptable sensitivity region (ASR) | |
—— | Sensitive and insensitive directions | |
—— | Adaptive TVSI | |
u | —— | Decision variables |
—— | Value of fitness function of individual d in the | |
, , | —— | Average, the maximum, and the minimum values of fitness functions in the |
—— | Voltage magnitude | |
—— | Voltage magnitude of bus i at time t | |
—— | Voltage magnitude of bus i at original state | |
—— | Prioritized VCPI | |
—— | VCPI of line l | |
—— | Power system states | |
E. | —— | Functions |
—— | Total investment cost | |
—— | Sub-objective function | |
—— | Overall objective function | |
—— | Equality constraint i | |
—— | Equality constraints | |
—— | Inequality constraint i | |
—— | Inequality constraints |
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