Nowadays, the changes of economic, environment, and regulations are forcing the electric utilities to operate systems at maximum capacity. Therefore, the operation and control of power system to improve the system stability has been receiving a great deal of attention. This paper presents an approach for enhancing the static voltage stability margin and reducing the power losses of the system with voltage securityconstrained optimal power flow (VSCOPF) that is based on static line voltage stability indices. The control approaches incorporate the voltage stability criteria into the conventional OPF. The minimization of the summation of fast voltage stability index (FVSI), line stability index (
Voltage instability is recognized as the major sources of power system insecurity, for which several partial or full power interruptions have been related to this problem. Nowadays, electric power systems have been operating closer to their stability limit. Hence, this problem is a great challenge for power system stability and control [
In the literature, the voltage stability criterion is incorporated into the OPF formulation with different approaches. These can be classified into two categories, i.e., voltage stability objective function and constraint. For the objective function to improve stability margin, several studies have introduced bus voltage indices into the OPF problem, which has been an efficient method for enhancing the voltage stability. One example is the objective function to minimize the sum of the squares of Lindex, which can provide the optimal setting of various control devices against voltage collapse [
With respect to incorporation of the voltage stability constraint, many other studies have reported that it can control the system in a secure manner. The reactive reservebased contingency constrained optimal power flow (RCCOPF) concept was formulated as the reactive reserve constraint to improve the voltage stability margin of postcontingency states [
Few studies have used the line voltage stability indices in the OPF problem. The VSCOPF approach based on the voltage collapse proximity indicator (VCPI) has been presented and included in OPF formulation in two ways, the proposed objective function and constraint [
As previously mentioned, incorporation of the static line voltage stability indices into the OPF problem has not extensively been investigated. Moreover, these indices provide accurate information about the proximity to voltage collapse. Both voltage variation and power are incorporated into the indices [
This study is focused on incorporating the static line voltage stability indices into the conventional OPF formulation to improve the static voltage stability margin and power losses of the system. The fast voltage stability index (FVSI), line stability index (
Static line voltage stability indices are formulated based on the power transmission concept of a twobus system. A single line of the twobus system is illustrated in Figure
Twobus system.
Musirin [
An FVSI less than 1.00 indicates a stable condition. If the FVSI is close to 1.00, the particular line is close to instability. If the FVSI goes beyond 1.00, the bus voltage will suddenly drop, resulting in system collapse.
The line stability index (
It is defined
Naishan et al. [
The LVSI is used to identify the system conditions that are similar to the FVSI and
All static line voltage stability indices use a combination of system variables and elements of the admittance matrix, which is simple and requires less computation. In addition, the limits of the system elements, such as the VAR limits of the generators, can be considered in the analysis. Therefore, these indices are flexible to simulate any type of load pattern and system network. One disadvantage of such indices is the limitation in the estimation accuracy. They only express the critical lines. However, based on equations (
The optimal power flow (OPF) aims to identify the optimal solution for an objective function that is subject to several equality and inequality constraints, such as the power flow constraints, system operating limit constraints, and limits on equipment. The general OPF formulations are determined as follows [
This objective function aims to minimize the total fuel cost of active power injection. It is a simple summation of the individual polynomial cost functions of active power generation for each generator. And the power generation is the control variable for this function. It is defined as follows:
The static line voltage stability indices are incorporated into the conventional OPF problem as the objective functions. The purpose of objective functions is to minimize the summation of stability indices. The control variables involve active power generation except at slack bus and generator bus voltages. Three objective functions are proposed as follows:
Voltage stability margin enhancement based on the FVSI:
Voltage stability margin enhancement based on
Voltage stability margin enhancement based on the LVSI:
There are three types of constraints of the OPF problem worth considering. Equations (
These equality constraints are the set of the real and reactive power balance equations:
The integration of inequality of dependent variables into the penalized objective function should be maintained within their limits to refuse infeasible solutions. These variables are including slack bus active power generation, load bus voltage magnitudes, reactive power generations, and apparent power flows. The penalty function can be expressed as follows [
The computational procedure of the VSCOPF problem is presented in the flowchart of Figure
The schematic of a few functions performed in the energy control center (ECC).
The dragonfly algorithm is a metaheuristic algorithm. It was inspired by the static and dynamic swarming behaviors of dragonflies in nature [
Separation, alignment, cohesion, attraction to a food source, and distraction of an enemy represent the behavior of dragonflies. The described and formulated of these behaviors are as follows:
Separation: it is the avoidance of the static crashing of individuals into other individuals in the neighborhood, which is formulated in the following equation:
Alignment: it represents the velocity matching of individuals to the velocity of others in the neighborhood. It is able to compute by the following equation:
Cohesion: it is the proclivity of individuals to the center of mass of the neighborhood. It can be calculated by the following equation:
Attraction to a food source: it should be the main objective of any swarm to survive. And it is computed by the following equation:
Distraction of an enemy: it is another survival objective of the swarm, which is formulated in the following equation:
The movement of artificial dragonflies and updation of their positions are simulated by considering the step vector (Δ
The position of the artificial dragonflies is able to be updated by the following equation:
In case the search space is not able to find a neighboring solution, stochastic behavior need to be improved by the moving of the artificial dragonflies around the search space with the application of random walk (Lévy flight). For this case, the position of the dragonflies can be calculated as follows:
Particle swarm optimization is a populationbased stochastic global optimization technique which was first introduced in [
Each particle
The Hybrid DAPSO algorithm combines the prominent points of the DA and PSO algorithms. At the beginning, the global solution is explored with the initialization of the dragonflies in DA. After obtaining the best position of DA, it will be substituted as the global best position in the PSO equation (equation (
The application of the DAPSO algorithm for solving the OPF problem can be described as follows [
Step 1. Initialize the system data including the parameters of DA and PSO, the number of dragonflies and particles, the number of iterations, and the archive size.
Step 2. Generate the initial population of dragonflies and particles.
Step 3. Convert the constrained multiobjective problem to an unconstrained one by using equation (
Step 4. Perform the power flow and calculate the objective functions for the initial population of dragonflies.
Step 5. Find the nondominated solutions and save them to the initial archive.
Step 6. Set the fitness value of the initial population as the food source.
Step 7. Calculate the parameters of DA (
Step 8. Update the food source and enemy of DA.
Step 9. Evaluate the
Step 10. Check if a dragonfly has at least one neighboring dragonfly, then update step vector (Δ
Step 11. If any component of each population breaks its limit, then Δ
Step 12. Set the best position obtained from DA as the global best of PSO (
Step 13. Update the velocity of the particle (
Step 14. If any component of each population breaks its limit, then
Step 15. Calculate the objective functions of the new produced population.
Step 16. Employ the Pareto front method to save the nondominated solutions to the archive and update the archive.
Step 17. If the maximum number of iterations is reached, the algorithm is stopped; otherwise, go to Step 7.
The VSCOPF algorithm is proposed in the previous section. This algorithm is incorporated into the energy control center (ECC) as the preventive control scheme [
This study is focused on incorporating the static line voltage stability indices into the conventional OPF formulation with the goal of enhancing static voltage stability margin and reducing power losses of the system. The IEEE 30bus, 57bus, and 118bus test systems are used to assess and verify the performance and effectiveness of the proposed control approaches. The detailed data are as follows:
The IEEE 30bus test system includes six generators installed at buses 1, 2, 5, 8, 11, and 13. There are four transformers at lines 6–9, 6–10, 4–12, and 27–28 and 41 transmission lines. The total loads are 283.4 MW and 126.6 MVar. The network data are given in [
The IEEE 57bus test system consists of seven generators located at buses 1, 2, 3, 6, 8, 9, and 12; 15 transformers; 80 transmission lines; and 42 loads totaling 1250.8 MW and 336.4 MVar, respectively. The detailed data are taken from [
The IEEE 118bus test system has 54 generators, 9 transformers, and 186 transmission lines. The active and reactive power load demand of the system is 4242 MW and 1439 MVar, respectively. The complete data of these networks are given [
The results of this study are generated using a program developed in MATLAB [
Case 1: the minimization of the generation cost is the objective function as shown in equation (
Case 2: the minimization of the total sum of FVSI in equation (
Case 3: the minimization of the total sum of
Case 4: the minimization of the total sum of LVSI in equation (
This section is to investigate the system performance of proposed approaches. For this study, the continuation power flow is used to determine the maximum loadability of the power system. The nose point of the PV curve represents the maximum loadability; when the system reaches this point, any further increase in the active power transfer will lead to voltage collapse. The system performance of the IEEE 30bus test system is shown in Table
System performance of the IEEE 30bus test system.
Objective functions  

Case 1  Case 2  Case 3  Case 4  

292.83  290.34  289.87  288.12 

87.46  83.33  82.63  60.81 
Loss (MW)  9.4290  6.9408  6.4713  4.7236 
Cost ($/h)  802.21  829.11  831.09  971.55 
Maximum FVSI  0.2471  0.0722  0.0672  0.4212 
Maximum 
0.2483  0.0723  0.0672  0.4221 
Maximum LVSI  0.9019  0.8439  0.8631  0.7219 
Sum FVSI  1.4519  0.6321  0.6273  2.4624 
Sum 
1.4694  0.6396  0.6349  2.4988 
Sum LVSI  6.9357  6.6840  6.5508  5.2089 
Maximum loadability (MW)  856.92  895.64  872.74  912.59 
Variable comparisons for the IEEE 30bus test system.
The maximum and summation values of stability indices of Cases 2, 3, and 4 are also improved from Case 1. These results follow the maximum loadability of the system. Case 4 provides the highest improvement in terms of the maximum loadability, which was 6.50% higher than Case 1. The maximum loadability values of Cases 2 and 3 are also increased from 856.92 MW (base case) to 895.64 MW and 872.74 MW, respectively. Total power generations are similar for all cases to meet the demand. All the proposed cases cost more than the base case. The total cost is increased from the base case by 3.35% for Case 2, 3.60% for Case 2, and 21.11% for Case 3. From the results, it can be observed that the proposed approach to incorporate LVSI index into objective function provides the best results because of its reduction of real power generation and loss, and its increase of system maximum loadability. But the high cost is the drawback of this case. The increment of fuel cost is imposed for the security cost.
The system performance of the IEEE 57bus test system is indicated in Table
System performance of the IEEE 57bus test system.
Objective functions  

Case 1  Case 2  Case 3  Case 4  

1,268.30  1,265.17  1,264.58  1,265.62 

237.70  195.08  196.73  205.57 
Loss (MW)  16.63  14.17  13.28  13.55 
Cost ($/h)  41,828.39  42,122.71  42,078.60  44,054.27 
Maximum FVSI  0.3408  0.2809  0.2411  0.3467 
Maximum 
0.3436  0.2830  0.2425  0.3494 
Maximum LVSI  0.8731  0.8558  0.8572  0.8072 
Sum FVSI  4.4305  2.8404  2.8147  4.7031 
Sum 
4.5054  2.8698  2.8463  4.7656 
Sum LVSI  13.8230  12.7749  12.8361  12.5250 
Maximum loadability (MW)  2,395.84  2,441.80  2,416.11  2,444.43 
Variable comparisons for the IEEE 57bus test system.
The variable comparisons of the system performance for the IEEE 118bus test system are shown in Table
System performance of the IEEE 118bus test system.
Objective functions  

Case 1  Case 2  Case 3  Case 4  

4,317.40  4,312.46  4,317.68  4,310.57 

249.98  167.77  232.22  183.09 
Loss (MW)  67.88  58.94  64.35  65.16 
Cost ($/h)  144,225.06  150,129.63  155,450.07  149,882.80 
Maximum FVSI  0.3913  0.2600  0.2750  0.3068 
Maximum 
0.3980  0.2670  0.2861  0.3514 
Maximum LVSI  1.0000  0.9878  0.9834  0.9982 
Sum FVSI  11.3469  10.6039  8.7766  9.2529 
Sum 
11.5817  10.8010  8.9422  9.4803 
Sum LVSI  58.6067  54.7991  54.6839  58.3085 
Maximum loadability (MW)  20,906.84  23,101.25  21,964.12  21,139.85 
Variable comparisons for the IEEE 118bus test system.
The PV curves at the weakest bus are also used to verify the effectiveness of the proposed control approach. The weakest bus is identified using the tangent vector. The analyzed tool for this section is similar to that in the previous section. The continuation power flow is employed to obtain the PV curves and tangent vector. Based on the simulation results, bus number 30 is the weakest bus of the IEEE 30bus test system. The PV curves at the weakest bus for all case studies are shown in Figure
PV curves at the weakest bus of the IEEE 30bus test system.
The PV curves at the weakest bus of the IEEE 57bus test system are shown in Figure
PV curves at the weakest bus of the IEEE 57bus test system.
Bus number 95 is the weakest bus of the IEEE 118bus test system and shown in Figure
PV curves at the weakest bus of the IEEE 118bus test system.
It can be observed that the voltage at the weakest bus of the IEEE 30bus and 57bus test systems is improved for all the proposed cases compared with base case. For the IEEE 30bus test systems, the highest increase of voltage is Case 4. And the simulated result of Case 2 can provide the most improvement of voltage for the IEEE 57bus test systems. Only Case 2 can increase the voltage level of weakest bus of the IEEE 118bus test systems.
Line outage contingencies generally change the system configuration, resulting in more stressful conditions. The power system may become less secure. Hence, the assessment of the contingency effect is important to maintain the system security. For this study, the maximum value of line stability indices is used to determine the contingency ranking. The line with highest value is identified as the most critical line and therefore selected to assess the effect of the contingency condition. The chosen buses for the IEEE 30bus, 57bus, and 118bus test systems are line 3027 (connecting buses 30 and 27), line 89, and line 3865, respectively.
All proposed cases performed well in the line outage conditions. Table
System performance for line outage contingencies.
Test system  Variable  Objective functions  

Case 1  Case 2  Case 3  Case 4  
IEEE 30bus test system 

91.98  83.99  82.90  59.58 
Loss (MW)  9.86  7.26  6.72  5.21  
Maximum loadability (MW)  593.44  608.28  593.71  611.49  
Cost ($/h)  803.81  826.89  846.04  972.72  


IEEE 57bus test system 

282.05  262.91  265.46  276.98 
Loss (MW)  21.48  20.80  18.97  17.63  
Maximum loadability (MW)  2,374.60  2,379.13  2,387.12  2,448.90  
Cost ($/h)  43,258.85  45,882.94  46,344.09  46,415.52  


IEEE 118bus test system 

302.22  251.41  295.41  258.13 
Loss (MW)  84.54  49.65  71.91  68.57  
Maximum loadability (MW)  16,588.36  23,092.41  21,671.22  17,929.17  
Cost ($/h)  144,668.60  156,898.64  165,175.54  152,214.65 
Second, the transmission losses for line outage contingencies were assessed. Compared to the previous section, the transmission losses are increased due to the line outage contingencies. The loss improvement of the proposed control approaches is greater than the base case. For the IEEE 30bus test system, Case 4 gives the best result with 47.15% reduction compared with base case. The reduction levels of Cases 2 and 3 are 26.39% and 31.88%, respectively. Transmission loss of the IEEE 57bus test system is alleviated for all proposed cases. Case 4 achieves the best reduction with 17.63 MW (17.94% reduction) of loss. However, the losses of Cases 2 and 3 are 20.80 MW and 18.97 MW, respectively. Loss of the IEEE 118bus test system is significantly improved in Case 2 with a reduction of about 41.27% (49.65 MW). Cases 3 and 4 can reduce to 71.91 MW and 68.57 MW, respectively, corresponding to the 14.94% and 18.90% reduction.
Third, the maximum loadability enhancement for the selected case studies is expressed. The percentage increase in the maximum loadability is compared to the postcontingency data. All the proposed cases significantly increased the margin compared to the base case. For the IEEE 30bus test system, the simulation results indicated substantial improvement in the voltage stability margin for all the proposed control approaches. The maximum loadability of the system is enhanced to 608.28 MW for Case 2, 593.71 MW for Case 3, and 611.49 MW for Case 4, whereas the base case provides 593.44 MW. For the IEEE 57bus test system, all the proposed control approaches enhanced the voltage stability margin. Based on the simulated results, the maximum loadability of the base case is 2,374.60 MW. Case 2 can increase loadability to 2,379.13 MW. Cases 3 and 4 also provide the greater level of loadability at 46,344.09 MW and 46,415.52 MW, respectively. Furthermore, for the IEEE 118bus test system, Case 2 is able to raise the highest loadability with 39.21% increase from the base case. The loadability levels of Cases 3 and 4 are 21,671.22 MW (30.64% enhancement) and 17,929.17 MW (8.08% enhancement), respectively. As a result, the proposed control approaches are able to mitigate the voltage collapse, especially when the system is relieved from contingency conditions to a more secure level.
Fourth, the increase in the cost for all the proposed approaches is similar to the postcontingency cost, which is imposed for the security cost. The highest generation cost of the IEEE 30bus test system is in Case 4 with 21.01% increase from base case. Cases 3 and 4 cost 2.87% and 5.25% higher than Case 1. For the IEEE 57bus test system, costs of Cases 2, 3, and 4 are 6.07%, 7.13%, and 7.30% increase from the base case, respectively. The results of the IEEE 118bus test system indicate that Case 3 imposes the highest cost with 14.18% greater than base case. Cases 2 and 4 give 8.45% and 5.22% increase in the generation cost.
Indeed, the incorporation of LVSI into objective function (Case 4) achieves the best reactive power generation, transmission loss, and maximum loadability for the IEEE 30bus test system. Case 4 also provides the great results for the IEEE 57bus test system. The use of FVSI summation as objective function (Case 2) is shown to be more efficient for the larger bus system as indicated in the IEEE 118bus test system. Obviously, all proposed approaches demonstrate the effective improvement in both system stability and losses in the event of line outage contingencies. Therefore, all the proposed control approaches are potential countermeasures for relieving the stressful conditions.
In this paper, the voltage securityconstrained optimal power flow (VSCOPF) based on static line voltage stability indices is presented. The proposed control approaches are formulated into the conventional optimal power flow to enhance the static voltage stability margin and reduce the system losses. Minimization of the sums of fast voltage stability index (FVSI), line stability index (
The performance and effectiveness of the proposed control approaches are investigated on the IEEE 30bus, 57bus, and 118bus test systems. The analysis is carried out with different cases including minimization of the generation cost. The simulation results clearly indicate that all proposed control approaches increase the static voltage stability margin and minimize the system losses under normal and contingency conditions. For line outage contingencies, all the proposed control approaches significantly improved the static voltage stability margin and losses after contingencies. Therefore, the proposed control approaches can be considered efficient preventive control measures for contingency conditions, preventing voltage collapse. However, these approaches need to be judiciously chosen because the performance of the approaches will depend on the situation and size of the system.
All proposed control approaches are based on the simple concept that the complexity of the problem can be reduced. These approaches would be applicable for practical power system operation in today’s competitive market. The multiobjective optimization problem approach that incorporates the static line voltage stability indices could be used in future studies on voltage stability improvement.
Research data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.