Preventive control approach for voltage stability improvement using voltage stability constrained optimal power flow based on static line voltage stability indices

Voltage stability improvement is a challenging issue in planning and security assessment of power systems. As modern systems are being operated under heavily stressed conditions with reduced stability margins, incorporation of voltage stability criteria in the operation of power systems began receiving great attention. This study presents a novel voltage stability constrained optimal power flow (VSC-OPF) approach based on static line voltage stability indices to simultaneously improve voltage stability and minimise power system losses under stressed and contingency conditions. The proposed methodology uses a voltage collapse proximity indicator (VCPI) to provide important information about the proximity of the system to voltage instability. The VCPI index is incorporated into the optimal power flow (OPF) formulation in two ways; first it can be added as a new voltage stability constraint in the OPF constraints, or used as a voltage stability objective function. The proposed approach has been evaluated on the standard IEEE 30-bus and 57-bus test systems under different cases and compared with two well proved VSC-OPF approaches based on the bus voltage indicator L - index and the minimum singular value. The simulation results are promising and demonstrate the effectiveness of the proposed VSC-OPF based on the line voltage stability index.


Introduction
During recent years, the planning and the operation of large interconnected power systems while improving system stability and security have become important concerns in the daily operation of modern power networks. Therefore there is a renewed interest in developing optimal power flow (OPF) models that incorporate additional constraints and new objective functions to enhance the security of electricity markets. As several blackouts around the world have been related to voltage phenomena [1,2], much more interest has been devoted by planning engineers to the voltage stability constrained OPF (VSC-OPF) problem. The present paper deals with including the voltage stability issue in the conventional OPF to effectively improve system voltage stability as well as to reduce power losses when subject to unexpected contingencies such as generation outages, tripping of a transmission line in a heavily loaded system or an unpredictable increase in load demand.
In the literature, many methods and techniques have been reported for voltage stability analysis and voltage collapse prediction. Some of these methods are based on PV-QV curves [3], modal analysis [4], singular value decomposition [5], sensitivity method [6], energy function [7], continuous power flow [8] and bifurcation theory [9].
On the other hand, a number of static voltage stability indices have been widely used for evaluating and predicting the proximity of the system to voltage instability. These indices have been grouped into two categories. Bus voltage stability indices have been applied to provide information about the stability condition of the buses and to identify the weakest bus in the system such as L-index [10], voltage collapse prediction index [11], voltage stability index (VSI) [12] and an improved VSI (IVSI) [13] whereas, the line indices-based techniques have been used to determine the most critical line in an interconnected system, for example, fast VSI (FVSI) [14], line stability index (Lmn) [15], voltage collapse proximity indicator (VCPI) [16] and the line voltage factor (LQP) [17].
Several works have been investigating the most efficient and accurate manner to incorporate these different approaches as voltage stability criteria into the OPF formulation. An OPF for maximising voltage security through the use of the minimum singular value (MSV) of the power flow Jacobian was proposed in [18] and a voltage security constrained OPF based on the loading level parameter was formulated in [19]. In these works, the authors tried to minimise the operating costs and the losses whereas maximising the distance to the voltage collapse. The voltage stability margin (VSM) obtained by the continuation power flow (CPF) was adopted as a static voltage constraint in the multicontingency VSC-OPF model [20]. In [21], a VSC-OPF formulation incorporating a linear margin enhancement constraint (MEC) is proposed to enhance the interface flow margin. In [22], an optimal dispatch with voltage stability constraint using the bifurcation technique was applied to improve the voltage stability in deregulated power systems.
In addition, many researchers have introduced bus voltage indices in the OPF problem. In [23,24] Huang et al. and Kim et al. presented an OPF with voltage stability constraint based on the L-index to investigate the voltage constraint effect on the fuel cost and the system voltage stability. In [25], the L-index was adopted as the voltage stability constraint to compute the load curtailment evaluation. The L-index has also been used as an objective function for voltage stability enhancement in both nominal situations [26][27][28][29] and contingency conditions [30,31]. In [13], the total sum of an improved bus voltage index (IVSI) for all the system buses is used as an objective function for optimising the settings of the compensation devices.
Even though the above works have made important contributions and improvements in the VSC-OPF problem, the use of the line VSIs in the OPF model has not been investigated extensively. Kargarian et al. [32] studied the minimisation of the market payment for the reactive power and the system energy losses, and simultaneously maximising the voltage stability by reducing the largest line stability index (Lmn) of the network. The FVSI and the LQP are used as objective functions in the optimal location of the FACTS devices for minimising the power system losses and the voltage stability improvement [33,34].
Moreover, the good characteristics of the line VSIs motivated the purpose of a VSC-OPF based on the line voltage indices. These indices involve both power and voltage variation and hence provide reliable information about the proximity to voltage instability. In addition, the choice of the VCPI index is inspired by its good properties (accuracy and robustness) in predicting voltage collapse. Comparison studies of the performance of several line VSIs [35,36] have shown an agreement between the different line stability indices which were found coherent with their theoretical background. However, the VCPI has the best accuracy and robustness in predicting voltage collapse [37]. Also, the VCPI index has some strong points such as simplicity, fast numerical calculation, flexibility for simulating any type of topological and load modifications in the network and its application for real time simulation [38]. Furthermore, a VSC-OPF study has revealed that incorporating the VCPI index in the OPF is more efficient for voltage stability enhancement and power losses reduction.
This work is focused on including the VCPI index in the classical OPF to effectively improve the voltage stability as well as minimise the power losses. The OPF problem is formulated as a non-linear optimisation problem with equality and inequality constraints. The objective functions are the minimisation of the fuel cost and the improvement of the voltage stability in both stressed and contingency conditions. The paper is organised as follows. The description of the VCPI is given in Section 2. In Section 3, the mathematical formulation of the OPF problem with voltage stability consideration is presented. Section 4 explains the VSC-OPF approach for preventive control, whereas Section 5 details the concept implementation of the proposed approach. Section 6 presents and discusses the numerical results. Section 7 summarises a comparative study. Finally, conclusions and future works are outlined in Section 8.

Line VSI
The VCPI was proposed by Moghavvemi and Faruque [16] that has rigorously demonstrated its accuracy and reliability on the standard IEEE test systems with different load nodes. This indicator is adopted in our study to investigate the stability of each line of the system by determination of the critical line referred to a bus. The VCPI index is based on the concept of maximum power transferred through the lines of the network as presented in Fig. 1 and it is defined as follows The numerator is the real or reactive power transferred to the receiving end and it is obtained from the power flow calculations. The denominator is the maximum active or reactive power that can be transferred through a line. It can be calculated by the following equations where V S is the sending end voltage; Z S is the line impedance; θ is the line impedance angle and f = tan −1 (Q r /P r ) is the phase angle of the load impedance. The values of the VCPI index increase with the increasing of the power flow transferred by the transmission lines and vary from 0 (no load condition) to 1 (voltage collapse). Hence, the critical line will be the line with the highest VCPI value and the load bus connected to the line will be the vulnerable bus in the system. This indicator is simple and fast to calculate compared with the bus voltage stability indicator L-index [10] which requires additional computing time for the inversion of the complex bus admittance submatrices. The VCPI index considers the power variation in addition to the voltage variation, this allows us to take into account the power limits of the system equipments (such as generators) which are not considered in the L-index [39]. The index is flexible for simulating any type of system topology and load modifications, and easily online implemented in a realistic system. From the above equations, it is clear that the rise in voltage increases the values of P r(max) and Q r(max) (as they are proportional to V 2 S ), resulting in lower VCPI indicator values therefore improving the voltage stability limit. Hence, this principle is investigated in a VSC-OPF problem by limiting (constraint) and minimising (objective function) the VCPI index to enhance the system voltage stability. As a result of the voltage stability improvement, the line current magnitude is reduced as along with the power losses.

Problem formulation
The main purpose of the OPF problem is to determine the optimal control variables for minimising an objective function subject to several equality and inequality constraints. The problem is generally formulated as follows subject to where f is the objective function to be minimised; g is the equality constraints and h is the system operating constraints.
x is the vector of the state variables and u is the vector of the control variables. The control variables are generator active power outputs and bus voltages. The state variables are the voltages and the angles of the load buses. The objective functions, the conventional constraints and the voltage stability constraint are described as follows.

Objective functions
In this paper, two different objective functions are considered.

Fuel cost:
The first objective function is to minimise the total fuel cost (FC) of the system. The generator cost curves are modelled by quadratic functions and can be expressed as where N g is the number of generator buses; FC is the total fuel cost; a i , b i and c i are the fuel cost coefficients of the ith generator and P Gi is the real power output of the ith generator.
3.1.2 Voltage stability improvement: Maintaining the acceptable voltage stability level under normal, stressed and contingency operating conditions is an important concern in power system planning and operation. For this aim, the minimisation of the total VSI (VCPI T ) is proposed as an objective function to enhance the overall voltage stability of the system. The VCPI T is the sum of the voltage stability indices for all the lines of the system and it is mathematically evaluated as where VCPI i is the VCPI for line i and N l is the number of transmission lines in the system.

Conventional constraints
The conventional constraints of the OPF problem are represented by two categories of constraints described by (9)-(13) as follows.
3.2.1 Equality constraints: represent the non-linear power flow equations: 3.2.2 Inequality constraints: include the system operating and the security limits: where N is the total number of buses in the system; P Gi and Q Gi are the active and the reactive power generations at bus i; P Di and Q Di are the active and the reactive power loads of bus i; G ij and B ij are the transfer conductance and the susceptance between buses i and j, respectively; θ ij is the phase angle difference between the voltages at buses i and j.

Voltage stability constraint
Generally, the voltage magnitude limits for each bus are used as the voltage constraints. However, the voltage limits alone are not sufficient to guarantee an acceptable voltage stability level of the system under different operating conditions. In this paper, a voltage stability constraint based on the VCPI is added to the classical OPF. The aim of this new voltage security constraint is to limit the maximum value of the line index and then move the system far from the voltage collapse. The additional voltage stability constraint is formulated as where VCPI limit is a desired threshold value to ensure a certain system security level and VCPI max is the maximum value of the VCPI index defined as and N l is the total number of lines in the system.

VSC-OPF approach for preventive control
The purpose of the VSC-OPF based on the VCPI index is mainly to move the power system operation state far away from the voltage collapse by increasing the system stability margin. The proposed algorithm is incorporated into an automatic security monitoring and control system (ASMCS) as the preventive control scheme as illustrated in Fig. 2. Two approaches could be implemented by the block VSC-OPF to design a preventive control system for the voltage stability improvement and the power losses minimisation. The VSI is embedded in the OPF formulation as the new voltage stability inequality constraint, or as an objective function with the minimisation of the total VSI. Fig. 2 summarises the general view of an ASMCS. The output of the state estimator is used to compute the vulnerability [40] and the line voltage stability [16] indices in order to verify the system security. When the network is overloaded and the voltage instable, the ASMCS is in corrective mode. In this situation, a first threshold (Threshold 1) for the voltage indices is designed to evaluate whether the system is stable or not. According to the definition in the previous section, the VCPI values increase with the increasing of the power flow transferred by the transmission lines and vary from 0 (no load condition) to 1 (voltage collapse). Therefore a threshold of 90% is considered appropriate in this work, if the maximum VCPI index values exceed this threshold it means that the voltage of the critical line is very likely to collapse. Then, fast control actions such as fuzzy-logic-based generation rescheduling approach [40], load shedding, flexible AC transmission systems (FACTS) controllers, high voltage direct current (HVDC) links and corrective switching (line and bus switching) should be initially carried out to move the system into a voltage secure operating point. On the other hand, if the contingency is already screened, a solution exists, and the preventive control is applied automatically to the system.
After applying the corrective control actions, the system restores the voltage stability and the values of the VCPI index will definitely move below Threshold 1. In that case, a contingency analysis is performed to provide a list of the most severe contingencies, in terms of voltage stability margins from a set of credible contingencies. If after a contingency study the voltage stability margin is insufficient, then it is necessary to further perform the preventive control based on the VSC-OPF to move the system operating point away from the critical point (the VCPI max value is limited by Threshold 2) and thus obtaining adequate security margins.

Implementation of the VSC-OPF algorithm
The proposed VSC-OPF defined in the previous section is solved by using the fmincon function provided by the standard optimisation toolbox of MATLAB [41]. In this optimisation algorithm, an approximation of the Hessian matrix of the Lagrangian function is calculated at each iteration and the problem is solved by using a line search procedure.
The following steps describe the computational procedure for solving the VSC-OPF presented in the diagram of Fig. 3.
(1) Read the system data.
(3) Calculate the VCPI for all the transmission lines by using (1). Find the critical line with the highest VCPI index value.
(4) If the objective of the optimisation study is to restrict the value of the VCPI index within a range of 0 to VCPI limit to achieve a required voltage stability level and to study the effect of the security voltage on the generation cost, then the approach with the VCPI as the voltage constraint is completed as follows: † Formulate the cost objective function (7).
† Construct the conventional constraints given by (9)- (13). † Construct the voltage stability constraint given by (14). (5) If the aim of the problem is the improvement of the overall voltage stability of the system and the minimisation of the power losses, then the VCPI index is used as an objective function and the OPF is executed as follows: † Formulate the voltage stability objective function (8).
† Calculate the conventional constraints given by (9)- (13). † Calculate the derivatives of the conventional constraints. † Solve the optimisation problem by the line search method. (6) The voltage stability condition has been achieved when the objective function and the control variables converge. At that time, the execution of the algorithm stops, if not, it is repeated from Step 3.

Simulation results and discussions
To verify and investigate the effectiveness and the performance of the proposed approach, the standard IEEE 30-bus and 57-bus depicted in Fig. 4  The results are carried out by using MATLAB 7.11 environment with AMD Phenom(tm) II processor, 2.79 GHz and 3 GB RAM.
In this simulation study, the VCPI index is incorporated in the optimisation problem in two ways. First, it can be added to the OPF constraints as the new voltage stability constraint. On the other hand, the VCPI index can be minimised by formulating the index as the objective function of the optimisation problem. The two approaches are applied for the voltage stability enhancement and the power losses minimisation under the stressed and the contingency conditions in the system.
Since the main purpose of this paper is the performance evaluation of the proposed VSC-OPF approach based on the line VSI (VCPI), the OPF problem has been solved for different cases described as follows: † Case 1: Cost function. In this first case, the minimisation of the fuel cost of generation is considered as an objective function. The generator cost coefficients for the IEEE 30-bus system [44] and the IEEE 57-bus system [43]ar edefined in Tables 6 and 7, respectively. In this case, the VCPI is included as the voltage stability constraint in the OPF model. This will allow limiting the VCPI index of the most critical line in the system and then increasing the voltage stability margin of the system. Consequently, the optimal solution can satisfy the economic and the system security requirements simultaneously. † Case 3: Total sum of the VSI. From the system security point of view, an objective function which incorporates an improvement of the system voltage stability is found to be more efficient. As the VCPI index value indicates the proximity of the system to the voltage collapse, the minimisation of the sum of the voltage stability indices (VCPI T ) is selected as an objective function, such that the overall system voltage stability is improved. Thus, the lower the value of the VCPI T , the better the voltage stability.
In this study, the MSV of the reduced power flow Jacobian matrix is used to validate the improvement of the system voltage stability. It is well known that the MSV of the power flow Jacobian is a measure of the voltage stability and an accurate indicator of the system proximity to the voltage collapse point [5]. Therefore the higher MSV value (lower VCPI T value) indicates an improvement in the voltage stability [26,47].

IEEE 30-bus system: stressed conditions
To analyse the system under stressed conditions, the active and the reactive loads of each bus are increased to 140% of the base load conditions. In this first situation, the VCPI index value is below Threshold 1 indicating a system voltage secure operation point. However, the obtained voltage stability level is small and assumed not sufficient. Therefore the proposed preventive control based on the VSC-OPF is carried out to move the system away from the voltage collapse and to obtain a new operation point with an adequate voltage stability margin.
To assess the effectiveness of the proposed approach, a comparison between three different cases is performed. The results of this comparison are given in Table 1. The results show that the real power loss (P loss ) is reduced from 13.896 to 8.616 MW, with a percentage reduction of 38%, and the reactive power generation (Q gen ) is decreased from 177.18 to 158.76 MVAR, with a percentage reduction of 10.4%. These positive performances are a good indication about the system relieving from the stressed conditions to a more secure level.  In addition, the highest value of the VCPI index (VCPI max ) corresponding to the critical line is reduced to 0.2600 (improved by 29.17%) in comparison of 0.3671 in case of the OPF without the voltage constraint. The sum of the VCPI index values (VCPI T ) is minimised from 4.6634 to 4.1663 which represents a gain in the voltage stability margin of 10.66%. The MSV is also increased from 0.4704 to 0.4790 indicating an improvement of the voltage stability level of the system as illustrated in Fig. 6.
The voltage stability constrained OPF based on the VCPI index for the stressed conditions results in a reduction of 38% in real power loss and 29.17% reduction in VSI, but the generation fuel cost has increased by 11.3% which is acceptable considering voltage stability enhancement and power losses minimisation and the stressed conditions of the system. Note that the fuel cost decreases when the system is less stressed.

VCPI as an objective function:
Here, the results obtained with the minimisation of the sum of the VCPI index values (Case 3) are compared with those obtained by the addition of the voltage stability constraint to the fuel cost function (Case 2). As indicated from Table 1 and Fig. 5, the real power loss (P loss ) is reduced to 4.080 MW in comparison to 8.616 MW in Case 2, a reduction of about 52.65%. The total VSI (VCPI T ) is reduced from 4.1663 in Case 2 to 3.6212 in Case 3, a reduction of about 13.08%. Finally, the MSV is increased from 0.4790 to 0.4880 demonstrating a good improvement of the system voltage stability as shown in Fig. 6.
From the above results analysis, it is observed that Case 3 has the best performance in the voltage stability improvement and also has the minimum power system losses confirming the advantage of using the VCPI index as an objective function in the VSC-OPF problem. However, this case causes an increase in the fuel cost compared with Case 2. This increase represents the additional cost to improve the voltage security of the system.

IEEE 30-bus system: line outage contingency
Line outage contingency generally causes undesirable operating conditions and has a significant effect on altering the system security that could lead to the voltage collapse. Therefore to maintain the system security against the voltage collapse, it is important to estimate the effect of the contingency conditions on the voltage stability.
In this case study, the line with the highest VCPI index is identified as the most critical line, and therefore selected as a candidate for outage. According to the index values obtained from the stressed conditions presented in Fig. 7,i t is found that line 5 (connecting buses 2 and 5) has the maximum index value means that this line is the critical line of the system. The results have also shown that line 5 has the greatest real power loss 2.90 MW.
To analyse the performance of the proposed VSC-OPF based on the VCPI index under system disturbance, the outage of line 5 (2)(3)(4)(5) is considered at 1.20 times base load conditions. This outage results in a reduced voltage stability level with a maximum value of the VSI VCPI max =0.3291 and highest value of the real power loss P loss = 18.043 MW. The considered contingency is ranked among the five most critical contingencies for this test power system [45,46]. For this specified system scenario, the system voltage stability is concluded to be insufficient. Therefore the preventive control based on the VSC-OPF is executed to improve the voltage stability and to minimise the power losses.
The comparative values of the various variables of the system such as reactive power generation (Q gen ), real power loss (P loss ), the maximum value of the VCPI index (VCPI max ) and the sum of the VCPI index values (VCPI T ) for Cases 1-3 are summarised in Table 2.   Table 2 and Fig. 8). The real power loss (P loss ) and the reactive power generation (Q gen ) are reduced by 8.525 MW and 27.62 MVAR, respectively, corresponding to the 47.25 and 17.07% reduction. Furthermore, it can be observed from Table 2 that the maximum value of the VCPI index (VCPI max )i s significantly reduced from 0.3291 to 0.2220 (32.54% reduction), the sum of the VCPI index values (VCPI T )i s decreased by 14.15% (from 4.3906 to 3.7693) and the MSV is increased from 0.4832 to 0.4898, thus indicating an enhancement in the overall voltage stability of the system as shown in Fig. 9.
For the line outage contingency case, it is obvious that both the voltage stability improvement and the real power loss minimisation are satisfied when adding the voltage stability constraint, with 47.25% reduction in the real power loss and 32.54% reduction in the VSI. However, this case has caused an increase of the fuel cost by 10.03%.

VCPI as an objective function:
To verify the effectiveness of using the VCPI index as an objective function (Case 3) under the contingency conditions, the computed results for this case are compared with those of Case 2 (VCPI index as the voltage stability constraint).
As shown in Table 2 and Fig. 8, it is clear that the performance obtained in Case 3 is better than Case 2. The real power loss (P loss ) is 2.923 MW less by 69.29% compared with 9.518 MW obtained in Case 2. The total VSI (VCPI T ) is reduced from 3.7693 to 2.8975, a reduction of about 23.13% and the MSV is improved from 0.4898 to 0.4970 resulting in the voltage stability improvement as shown in Fig. 9.

IEEE 57-bus system: line outage contingency
To evaluate the effectiveness of the proposed VSC-OPF approach in the larger power system, the standard IEEE 57-bus system is considered under the contingency conditions. Based on the contingency analysis, the outage of the critical line (8-9) was identified as a severe case with a VCPI max value of 0.3980 and large real power loss P loss of 30.362 MW. Table 3 summarises the system performance for the different case studies.
6.3.1 VCPI as the voltage stability constraint: From the results given in Table 3 and Fig. 10, it is observed that Case 2 (with the voltage stability constraint) obtains better results than Case 1 (without the voltage stability constraint). The real power loss (P loss ) is reduced from 30.362 to 24.164 MW (20.41% reduction) and the reactive power generation (Q gen ) is reduced from 316.67 to 294.74 MVAR (6.93% reduction). Moreover, the maximum value of the VCPI index (VCPI max ) is significantly reduced from 0.3980 to 0.2800 (29.65% reduction) and the total VSI (VCPI T )i s decreased from 9.4363 to 8.8148 (6.59% reduction). Finally, the MSV is improved from 0.2379 to 0.2393 showing an enhancement of the system voltage stability. On the other hand, the fuel cost has increased by a small margin (0.44%).

VCPI as an objective function:
The results obtained in Table 3 and Fig. 10 indicate that using the VCPI index as an objective function (Case 3) achieves the best reactive power generation, the best power losses and  the best voltage stability margin. Consequently, the total real power loss P loss = 15.989 MW is on average 50% less than the losses obtained in the other cases, the total VSI (VCPI T )i s decreased to a smallest value of 7.3521 and MSV of the reduced power flow Jacobian is improved to 0.2401, thus indicating an enhancement in the voltage stability. From all the previous cases, the VSC-OPF based on the VCPI index approach gives good results regarding the voltage stability improvement and the power losses minimisation. However, using the VCPI index as an objective function in the VSC-OPF problem has the best performance in all respects. In addition, the proposed method has shown an increase in the fuel cost. This increase indicates the extra cost to enhance the voltage stability margin in the stressed and the contingency conditions. From the above statement, it is clear that the proposed algorithm is able to simultaneously satisfy the voltage stability and the power loss objectives. Nevertheless, for achieving the best cost it is necessary to combine both the fuel cost objective function and the voltage stability enhancement objective function by solving a multi-objective OPF problem with an efficient optimisation algorithm [28,48].

Comparative study
In this section, the performance of the proposed VSC-OPF based on the VCPI index (minVCPI T ) is further validated by comparing its results with the performance of a VSC-OPF using the L-index as an objective function. As mentioned in the Introduction, the minimisation of the maximum L-index value (minL max ) has been widely used as an objective function for the voltage stability enhancement in both the normal and the contingency conditions. In addition, the MSV of the modified power flow Jacobian matrix is used as a voltage stability indicator to verify the improvement of the voltage stability margin. Therefore the two VSIs namely the L-index and the MSV are considered for the comparison and the validation of the obtained results.
On observing the results given in Table 4, the VSC-OPF based on the VCPI index is shown to be more efficient in reducing the reactive power generation (Q gen ) for all the case studies. However, for the IEEE 30-bus system the VSC-OPF based on the L-index is little better in minimising the real power losses (P loss ) under the stressed and the contingency conditions. Nevertheless, for the larger system (IEEE 57-bus), the VSC-OPF based on the VCPI index achieves the best power loss minimisation. In addition, the two approaches give approximately the same MSV with a difference of about 0.1%, which indicates a similar voltage stability improvement margin. Obviously, in the two operation conditions (three tests), the VSC-OPF based on the VCPI index is the fastest algorithm since it converges in 1.86, 2.14 and 6.82 s compared with 4.64, 4.92 and 25.32 s, respectively, for the VSC-OPF with the L-index. As we know, the L-index calculation (see the Appendix) requires complex matrix inversion of the bus admittance submatrices, thus increasing the burden of the calculation [39]. In terms of the fuel cost, it appears that the proposed VSC-OPF approach has the best generation fuel cost.
Finally, the conclusion drawn from the comparison study is that the two approaches realise nearly the same performance with an advantage of the VSC-OPF using the VCPI index which is computationally efficient and less expensive than the approach based on the L-index. The same conclusion can also be drawn when the proposed approach has been compared with a VSC-OPF based on the maximising of the MSV (or minimal eigenvalue) of the Jacobian matrix [49,50]. The results of the comparison of these different VSC-OPF techniques (minL max , maxMSV and minVCPI T ) are presented in Table 5.
These encouraging results make the proposed VSC-OPF based on the VCPI index a promising candidate for the power system optimisation problems considering the

Conclusions
In this paper, a novel VSC-OPF approach for the voltage stability preventive control has been presented. The proposed approach is addressed by incorporating the VCPI into the classical OPF problem. The VCPI index is first used as the voltage stability constraint, this requires only one additional voltage constraint added to the conventional OPF constraints. Hence, the dimension of the resulting optimisation problem is similar to that of a conventional OPF, and this feature help in reducing the complexity and improving the numerical solution feasibility of the problem. On the other hand, this index is formulated as the OPF objective function and minimised to improve the voltage stability of the system. The effectiveness and the robustness of the proposed VSC-OPF based on the VCPI index are tested and demonstrated on both the IEEE 30-bus and the IEEE 57-bus systems. The simulation results obtained under the stressed and the line outage contingency conditions are promising and clearly show the potential of the proposed approach to enhance the power system voltage security by improving the system voltage stability and minimising the power losses. It is also found that the approach based on the VCPI index achieves a performance comparable with the other reported approaches (minL max and maxMSV) with a superiority in terms of the computational efficiency and the solution cost.
The proposed technique is based on a simple concept and can be practically applicable for the online voltage security assessment. In future works, this technique could be easily combined with other stability constraints such as the transient stability or the small signal stability and could be implemented in multi-objective optimisation problems.

L-index calculation
The L-index [10] is a quantitative measure to evaluate the voltage stability at each bus of the system and to estimate the proximity to the voltage collapse. The value of the L-index of load bus j is calculated by where V i and V j are the voltage magnitudes at generator bus i and load bus j, respectively. δ i and δ j are the phase angles at bus i and bus j, respectively, and θ ji is the phase angle of term F ji . N g is the number of generators and N the total number of buses. The values of F ji are obtained from the submatrix F LG calculated as follows where Y LL and Y LG are the submatrices of the Y bus matrix in (18).
I L , I G and V L , V G represent the currents and the voltages at the load buses and the generator buses. The indicator value varies in the range between 0 (no load) and 1 (voltage collapse). The bus with the highest L-index value will be the most vulnerable bus in the system.

Generator cost coefficients
See Tables 6 and 7.