Improved multi objective particle swarm optimization based reactive power optimization for ensuring voltage security of power systems

In this study an improved multi objective particle swarm optimization (IMOPSO) algorithm is proposed for power system reactive power optimization with the objective of ensuring voltage security. The multi objective particle swarm optimization (MOPSO) is improved by introducing an adapted binary crossover (ABX) to the new positions obtained by the basic particle swarm optimization (PSO) algorithm. Additionally, diversity maintenance strategy is added to the algorithm by employing crowding distance (CD) calculation. The developed algorithm is tested and compared with standard MOPSO and non dominated sorting genetic algorithm (NASGA II). The comparison is made based on the degree of closeness to the true pareto front, as measured by the inverted generational distance (IGD), and based on diversity, as measured by the CDs . The test is made using ZDT1, ZDT2, and ZDT3 test functions. The IMOPSO showed improved performance over MOPSO and NASGA II algorithms in terms of convergence to the true pareto front (PF) and in terms of the speed of convergence as well as in maintaining diversity. The algorithm is then implemented to reactive power optimization of IEEE 14 bus test system. For the implementation purpose, the voltage stability and voltage deviation components of voltage security are formulated as a multi objective functions. The implementation has resulted diverse options of optimal settings of reactive power controlling parameters. The optimal settings proved to produce an improved voltage security as measured in terms of voltage deviation and voltage stability.


Introduction
As power systems get more loaded and loaded than ever, voltage security continues to be the concern of power systems.To cope up with this evolution of power systems, efficient voltage security improvement strategies are demanded by power system operators.
Voltage security can be insured by maintaining voltage stability and by keeping bus voltages within the standard limits [1,2].Voltage stability is the ability to maintain steady state voltages at all system buses, whilst voltage limit maintenance is related to providing quality power to customers.
Reactive power provision improves the voltage stability of a power system by improving the maximum active power transfer capability of a transmission network [3].Additionally, reactive power compensation minimizes the transmission reactive voltage drop, giving improved voltage profile at the load points [4,5].
The two components of voltage security needs, i.e. ensuring voltage stability and voltage profile improvement, through reactive power management poses a two objective optimization problem.
It is customary that reactive power optimization is formulated as a single objective optimization problem, with such single objectives as loss minimization, operating cost minimization, voltage deviation minimization [6][7][8][9].However, the presence of many objectives that need simultaneous reactive power optimization forced the treatment of the subject to employ multi objective optimization.Some researches that focus on reactive power optimization, thought to treat voltage stability from the objective function which is based on voltage profile deviation [10].However, voltage profile deviation cannot always be indicator of voltage instability, for voltage stable systems can have voltage deviations within the set limit [11].This situation demands the inclusion of an objective function based on voltage stability indices.In this regard, the most utilized indices include static voltage stability margin [12,13], fast voltage stability index [14], line loading index [15] and line stability index [16].
The other problem, that is observed in the inclusion of voltage stability to the objective function set, is a double inclusion of voltage stability issue to the set of objective functions, in the form of both transmission losses and stability indices [12,16].However, studies well established transmission active power loss can be used as an indicator of voltage instability [17,18].
In this regard, this paper begins the formulation of the voltage security objective function from the basic components of voltage security, i.e. voltage stability and voltage limit maintenance.These components are sufficient for identifying the voltage security level of the system.
Multi objective reactive power optimization problems, as a component of the general multi objective optimization problem, are solved using two ways.The first approach is converting a multi objective problem to a single objective problem.The single objective is expressed as a weighted sum the individual objectives [13,14].This approach poses a serious problem of weight assignment for each objective.
The second way of solving multi objective optimization problems is based on the pareto optimality principle which comes up with a number of solutions with equal chance to be chosen.The set of the optimal solutions is identified as pareto optimal solution or pareto front (PF).According to pareto optimality principle, in the PF no solution is superior than the other [10,16].The selection of the PF member is done based on the principle of dominance.
A number evolutionary optimization algorithms are employed to solve multi objective reactive power optimization problems.Particle swarm optimization [19], Genetic algorithm family techniques [20] , differential Evolution [16], multi objective immune system [12] are some to mention.
Despite applying the multi objective evolutionary algorithms to handle reactive power optimization problems, investigations to improve the performance of these algorithms, as applied to reactive power optimization are rare.As a general study of multi objective optimization algorithms, there are studies made to improve their performance [21,22].
In this work, MOPSO algorithm is chosen for handling our problem.Particle swarm optimization is known for fast convergence, robust adaptability and relative simplicity for implementation [10] [23,24].
The problem with PSO optimization algorithm is convergence to the false optimal positions.In the case of MOPSO, this problem is seen affecting the quality of the pareto front [25,26].Due to this problem, introducing a disturbance to escape false traps is proposed as a solution by literature [27,28].This improves the quality of the pareto front.
The most prevalent disturbance techniques employed include mutation [29,30] and multi swarm PSO techniques [31,32].The works that employed crossover as a strategy are few.[33] introduced double refinement as a cross over for selecting from a blend of a trial population and previous best positions based on a probabilistic selection.Quadratic interpolation based crossover [34,35], an arithmetic crossover [36] multi parent based cross over [37] are employed in PSO.However, these crossover techniques are designed for single objective PSO.The performance to a multi objective PSO is not explored.This creates a need to study the performance of applying crossover to multi objective PSO.The other gap is crossover enhanced PSO is not enough researched as applied to reactive power optimization.
As a means of improving the quality of the pareto front and improving the convergence speed, in this work, we introduced a disturbance based on adapted binary crossover (ABX), which is adapted from [38,39] to MOPSO.Originally, this crossover technique was employed in genetic algorithms.Hence, applying binary crossover to MOPSO creates a new dimension to utilize the advantage of this crossover technique.
In our algorithm, Improved MOPSO (IMOPSO), positions are, initially, created using normal PSO position estimation technique.Then, ABX technique is applied to these positions to create more potential solution set.The newly searched positions are, then, mixed with the original positions and best positions are selected from this merged group based on dominance and crowding distance.The introduction of crowding distance to MOPSO is used as a diversity maintenance strategy.
The performance of IMOPSO is evaluated using ZDT1, ZDT2 and ZDT3 test functions [40].The method used for performance evaluation is inverted generational distance (IGD) and crowding distance.The IGD shows the quality of the pareto front in terms of closeness to the true pareto front [41][42][43], and the CD measures the diversity of the PF.The result is compared with popular algorithms of standard MOPSO and NASGA II algorithms.The comparison shows an improved performance of IMOPSO over MOPSO and NASGA II algorithms.
Finally, this algorithm is applied to IEEE 14 bus test system reactive power optimization with objective of improving the voltage deviation and voltage stability.The method effectively identified the optimum value of the control variables.With these optimum settings voltage deviation and voltage stability are seen improved.
The unique contributions of this paper are; • An improved algorithm (IMOPSO) that shows superiority to standard MOPSO and NASGA II algorithms.
• Application of the developed algorithm to reactive power optimization of IEEE 14 bus system for improving voltage security.
• An improved formulation of the voltage security based objective functions, in a way avoiding the discrepancies seen in the formulation by previous works.
The rest of the paper is arranged as follows.Section 2 contains the basics of multi objective optimization and the IMOPSO development.Section 3 contains results and discussions.Finally, section 4 presents the conclusion of the study.

Voltage security based multi objective optimization problem formulation 2.1. Multi objective optimization problem
Multi objective optimization problem is characterized by having more than one objective.The objectives can be going along each other or conflicting.
The multi objective problem can be formulated as:

Subject to the equality and inequality constraint;
In our specific problem of reactive power optimization, the objective is formulated as a two-objective problem.This formulation begins from the basic voltage security definition, which contains both voltage stability and voltage limit violation.

Voltage stability index based objective function
In this study, the voltage stability based objective is formulated using load bus based stability index.The index is known by the name active power margin index (PPMI) [44].The index measures the distance of point of collapse from current operating point in terms of normalized active margin index.The PPMI is formulated as; The value of the index varies between 0 and 1.When the value of PPMI closes to 1 the load bus is more voltage stable, while nearing to zero the bus is getting more unstable.Hence, the index in nature entails a maximization problem.In order to avoid contradiction with the second objective function, which is clearly a minimization problem as discussed latter, we converted the objective to a minimization problem by finding the complement of PPMI.
For the objective is to improve the whole system voltage stability, we formulated the objective function as the average of this complement of all load buses as in equation (4).The average is taken not to have large values of the function due to summation effect, in case of large power systems who have many hundreds of load buses.
nlb is the total number of load buses.

Voltage deviation based objective function
The second objective function is based on the bus voltage deviation.System load buses are expected to operate at a unity per unit value of voltage.System operator strives to maintain this standard.Whenever the voltage deviates from this standard, the operator adjusts the reactive power resources to minimize the deviation from this standard.There are variations among literature in objective function formulation for the voltage deviation.Some approaches include the sum of absolute valued difference [15,16], sum of squired difference in [10], squired normalized difference [45], square root of the sum of squire of deviations [46].For this work this latter option is chosen.The need for absolute valuation or squiring is to avoid the cancellation effect upon the sum of the differences.Our choice avoids cancellation by squaring each difference and reduces magnitude increment, that come with squaring, by taking the square root of the sum.
The first set is the equality constraints, which contains the active power and reactive power transaction equations of each bus.The system operation is governed by these power flow equations.Power flow constraints; The control variables also operate within the limits which constitute the inequality constraint set.These are given as; The equality constraints and the state variable limits are imposed during power flow computation.The control variable limits are imposed on the computation of the algorithm during the formulation of the position and during position updates.

Improved multi objective particle swarm optimization (IMOPSO) algorithm
IMOPSO is established on a MOPSO platform.The main modification made is in creating positions additional to the set created by the standard MOPSO algorithm.This affects the normal trend of PSO position search mechanism.This, in turn, prevents convergence to false optimal positions.
The additional positions are created using adapted binary crossover (ABX).Crossover is a genetic operation common to genetic based evolutionary algorithms.To implement ABX to MOPSO, potential original positions are selected based on dominance and crowding distance.The crowding distance helps for maintaining the diversity of the solution and the dominance principle gears the solution towards the true pareto front.
In the next sub sections, the major components of the algorithm are discussed, and finally the algorithm is put as a a whole.

Updating speed and position
This is new position assignment process of the PSO class algorithms.In our algorithm, also, it is the core position producing step through position and velocity updating equations as; The constants c 1 and c 2 in equation (15) determines the weight of exploitation and exploration in the search space.c 1 is the exploitation constant while c 2 is the exploration constants.h-is the leader selected from the pareto set on a probabilistic selection approach.Equation (16) using the velocity vector and old positions create the new positions.

Principle of dominance
The dominance determination step is the core process of pareto based multi objective optimizations.A given position x 2 is dominant over the other position x 1 when: (ii) At least there exists one objective function for which f x f x .

( ) ( ) <
The positions from equation ( 16) get subjected to dominance determination and directly the pareto front is determined from this set.
In our approach, however, the pareto front is determined at latter stages.But the potential positions, on which ABX is going to be applied, is selected from the result in equation ( 16).This selection is made based on dominance and crowding distance calculation.

Ranking of the population and crowding distance computation
These features are common in genetic algorithm based algorithms [47].In our algorithm ranking and crowding distance are used for the selecting potential positions for ABX operation.Ranking measures the closeness of the solution set to the true pareto front while CD measures the dispersion of the solution.A more dispersed solution is favored, for it contains a more diverse solution, which gives a more varied choice of solutions.The utilization of CD in our algorithm helps to maintain the diversity of the pareto front.
The crowding distance is measured as a distance between consecutive neighbouring solutions normalized to the maximum range of the objective values [47,48].This is given as; Where: f is the fitness function, M the number of objective functions and x is the particle position.
Once the rank and crowding distance of positions is computed potential positions are selected for implementing the crossover operation.

Application of ABX to MOPSO algorithm
The disturbance introduced to the MOPSO algorithm is adapted from the crossover operation, simulated binary crossover, which is used in genetic algorithm [38,39].In this operation two potential parents are taken and the genetic information is exchanged to produce two new offspring.In our case, the two parents are two positions nps1 and nps2 selected based on rank and crowding distance.The offspring are identified as intermediate positions nip, in our case.
The new intermediate positions (nip) are computed as; b -is a distribution index and determines degree of exploration and exploitation.b close to 0 the new intermediate position becomes similar with the np's.This means the exploitation is more intensive.b close to 1 the operation explores the search space much more.Hence, this approach adds a more exploration and exploitation effort to the PSO equation (15).b is computed as; Other wise 21 is the distribution index for cross over, u is the random number generated between 0 and 1.
The intermediate positions then get mingled with the original positions to produce larger position size.This increases the originally planned size of positions.To maintain the planed size, next, positions are selected from this large population based on dominance and crowding distance.This gives the new position set with planed size.
The utilization of crowding distance for the selection of resulting positions improves the diversity of the algorithm.Now, the algorithm for the IMOPSO is generalized as follows.
The IMOPSO algorithm 1. Initialize position and velocity.
2. Compute the fitness function values and determine the dominant positions for the initial positions.
3. Select a leader from the dominant set on a probabilistic approach.
5. Perform power flow with the new position and compute the fitness functions.
6. Select potential positions using ranks and crowding distance for crossover.
7. Perform crossover on the selected position and produce intermediate positions.10.Check if the constraints are respected, if not make adjustments for over/under_the_limit quantities.
11.If the maximum iteration is reached or the minimum error is achieved terminate the search.Else go to step 3.

Performance measure of IMOPSO
The performance of the IMOPSO is measured based on two criteria.The first criterion is the quality of the pareto front and the second is the diversity of the solution.In this work, the quality of the pareto front is measured using inverted generational distance (IGD) and the diversity of the solution is measured using crowding distance.
The IGD is the first line of the performance evaluation tool.CD would apply if the pareto fronts are the same or approximately the same.If two pareto fronts are different in terms of IGD, applying CD for comparison leads to erroneous result.The reason is a low quality pareto front may have higher values of CD, due to the position of the PF at high values in the objective space.
The inverted generational distance measures the closeness of the pareto front to the true pareto front [41][42][43]49].The IGD is given as; N-is the number of non-dominated vectors.d i, j ( )is a Euclidean distance between the i th solution of existing pareto front and the j th element of the true/reference pareto front.The lower the value of the IGD indicate the closer the pareto front to the true pareto front.
We utilized the test functions ZDT1, ZDT2 and ZDT3 for assuring the consistency of the results.The pareto fronts are also graphically depicted for elaborating the meaning of the IGDs and CDs.

Implementation of IMOPSO for voltage security ensuring reactive power optimization
Once the algorithm gets tested using standard test functions, next, it is implemented on IEEE 14 bus test system.The IEEE 14 bus system is the simplest test system containing the necessary control parameters for voltage security study.These control parameters are generator voltage outputs, reactive power compensations and tap changing transformers.
The IEEE 14 bus test system data is taken from Matpower simulation software [50].Accordingly, the test system contains five generators, five transformers (this includes two winding representations of three winding transformers), one reactive power compensation as control variables.Additionally, the system contains 9 load buses.
The algorithm is implemented on Matlab.The power flow computations, which are important for the calculation of the objective functions, are performed using Matpower power system analysis software.

Test of IMOPSO for standard test functions
For the test of the algorithm ZDT1, ZDT2 and ZDT3 test functions are used.In the performance evaluation, standard MOPSO and NSGA -II algorithms are used for comparison purposes.Based on figure 1, MOPSO produces the less dominant pareto front of all.NASGA II shows the next less dominant front, while IMOPSO is producing the dominant front of all.For a narrow range NASGA II shows dominance comparable to IMOPSO, but IMOPSO is seen generally dominating.
Further, the IGDs are shown in figure 2 below.The graph of the IGDs reveals two things.The first is the quality of the pareto front in coming closer to the true pareto front.The second thing revealed is the speed of convergence.The method that settles to its best front in earlier iterations has fast convergence.
The IGD graphs show the dominance of IMOPSO over MOPSO and NASGA II algorithms in terms of closeness to the pareto front and speed of convergence.IMOPSO settles to its minimum distance at earlier iterations.
Seeing the magnitude of IGDs, the result from figure 2 consolidates the result in figure 1 above.The IGD value of IMOPSO is smallest as compared to the counter parts.This means the pareto front of IMOPSO is nearest to the true pareto front.NASGA II and MOPSO comes in the next order.Comparing MOPSO and NASGA II, MOPSO shows faster convergence at early iterations than NASGA II.This result go in line with the findings of the literature, which speak about the fast convergence of PSO [24].But, though slower, NASGA II, finally, achieves lower IGD than PSO, which makes it more preferable in terms the quality of the pareto front.
These results show the superiority of IMOPSO over the other methods used for the comparison, in terms of close up to the true pareto front and speed of convergence.
The results from the test function ZDT2 and ZDT3 are shown in figure 3 next.The results show the same improvement made as in the case with ZDT1 test function.

Diversity of the pareto fronts
In this work, the diversity of the pareto fronts is measured in terms of the crowding distance (CD).Larger value of CD means a more diverse pareto front which is highly favoured.Smaller value of CD indicates a less diverse pareto front.The crowding distance of the pareto fronts from the three algorithms for the three test functions is shown in table 1 below.
From the table MOPSO seems to produce a more diverse front as the mean value of CDs is the highest.However, it is important to note that the PF from MOPSO is situated in the less dominant solution space as discussed and depicted by the previous section.This means MOPSO fails the first line of performance check when compared with IMOPSO and NASGA II algorithms.
In this analysis, despite IMOPSO produces a more dominant PF than NASGA II, this difference doesn't result in a larger impact on the CD computation.Hence, the computation takes this under consideration.
Table 1 contains the mean value of the CDs and the distribution of CDs around the mean as measured in terms of the standard deviation.Larger values of mean indicate higher crowding distance values.Larger value of the standard deviation means a large deviation from the mean.We need larger mean value and smaller standard deviation, for the PF to be diverse.Larger mean and smaller standard deviation indicate individual CDs are at higher values and are close to the mean.Hence, comparing NASGA II and IMOPSO, IMOPSO owns higher mean values and smaller standard deviation for the three functions.This indicates the CDs are at higher values which mean a more diverse position is obtained using IMOPSO.

Application of IMOPSO to IEEE 14 bus system
In this implementation, the generator voltages are allowed to vary between the range minimum 0.9 pu and maximum 1.1 pu.The transformer tap positions are allowed in the range 0.95 and 1.05.The reactive power compensation is allowed to vary between the minimum value of 0 MVAR to the maximum value of 20 MVAR.The iteration is run for 200 iterations.After, the final iteration the pareto front, in figure 4, is obtained.
For comparison purpose the result from NASGA II and standard MOPSO algorithms are also shown.These depictions consolidate the result from the previous section 3.1.In this specific application, for lower f PPMI and higher f VD IMOPSO and NASGA II show similar performance.But, moving towards lower f VD IMOPSO continues to dominate both NASGA II and standard MOPSO.
The discussions afterwards, are based on the pareto front from IMOPSO.The pareto front of IMOPSO in figure 4 contains 200 possible operating points meeting the objectives and the equality and inequality constraints.The voltage deviation objective function (f VD ) goes from a minimum of 0.0188 to the maximum of 0.3059 while the voltage stability index objective function goes from a minimum of 0.6131 to the maximum of 0.7281.
Interpreting these functions in terms of voltage deviation and individual bus index, each bus will have average minimum voltage deviation of 0.0457 and average maximum voltage deviation of 0.1844, in this optimal front.The deviation is measured from the nominal voltage of 1pu.Based on the standard, i.e.IEC 60038, load buses are allowed to vary 5%  (0.5 pu value) from the nominal voltage setting.Hence, the variations resulted from IMOPSO well respect this standard.
In terms of the stability index PPMI, using the relation in equation (4), the index varies from average minimum PPMI value of 0.9191 to the average maximum PPMI index of 0.9319.Based on equation (3) these values indicate large gap between current operating power (P opr ) and the maximum possible active power transfer (P max ).This means the system is far enough from voltage instability under these optimal operating points.From the set of optimum values, the selection depends on the specific need of the system operator.Whether to incline to voltage deviation or voltage stability, within this optimum set, is up to the operator's specific preference.Below, the value of the control variables for the 200 optimal positions, in the pareto front, are revealed in figure 5.The control variables are generator output voltages, transformer tap positions and the value of the reactive power compensation.For more elaboration of the discussion, table 2 shows important features of the data set.
Discussing the generator output voltages based on figure 5 and the table 2, the highest generator output voltage hits are the same for all generators.The minimum generator output is recorded for generator 3. The highest among minimum generator output voltage is made by generator 1.This attributes to the stabilizing capacity of this generator and its service as a slack bus.The highest mean value goes to generator 1.The standard deviation from the central value is minimum for this generator too.That means generator 1 is operated at higher voltage values almost for all pareto front points, with minimum deviation.
Continuing speaking generator output voltage, the lowest mean value is owned by generator 3.This means for the majority of the pareto points generator 3 is operated at relatively lower voltage output.The standard deviation for this generation is also the largest.That means generator 3 owns a large set of operating voltage output values.
The maximum limit of the generators, i.e. 1.1 pu, is hit by all the generators but the minimum limit 0.9 pu is hit by only one generator.This infers optimum operating points favour high generator output voltages.
Coming to the transformer tap changer position, the minimum tap changer position 0.95 is hit by all the transformers.Whilst, the maximum tap changer position is hit by 2 transformers out of 5 transformers.This indicates the optimum operating points favour the lower tap changer position.
In order not to congest the space and for the results goes similarly with the selected transformers, figure 5 contains only the positions of 3 transformers out of five.
Transformer 5-6 shows a minimum variation in values around the mean for the standard deviation is the smallest.The mean is also the smallest for this transformer.Transformer 5-6, Transformer 4-7 and Transformer 4-9 are operated at lower tap changer positions, as revealed by lower maximum and average values of the tap positions.
Coming to the reactive power compensation at bus 9, the compensation goes from the minimum 1MVAR to the maximum 20 MVAR.However, the mean value 7.03 MVAR, which is less than the median 11 MVAR, shows the reactive power compensation favours lower amount of compensation.

Conclusion
In this study, an improved multi objective particle swarm optimization (IMOPSO) is proposed for reactive power optimization to meet the objective of ensuring voltage security.IMOPSO employed an adapted binary crossover to the standard MOPSO algorithm.Additionally, diversity maintenance strategy using crowding distance calculation is incorporated to the algorithm.The comparative performance of the algorithm is then evaluated using inverted generational distance and crowding distances on standard test functions ZDT1, ZDT2 and ZDT3, as compared with standard MOPSO and NASGA II algorithms.IMOPSO resulted a pareto front nearest to the true pareto front with better convergence speed and a more diverse pareto front.The algorithm is then applied to IEEE 14 bus test system for voltage security ensuring reactive power optimization.For this purpose, objectives, relevant to voltage security only, are defined, and system constraints are imposed.The algorithm identified the value of control variables, which give the optimal voltage secure operation.The implementation provided considerable alternatives of optimum operating points, which give the operators increased choice.The optimization achieved low voltage deviation and improved voltage stability as measured

8 . 9 .
Merge intermediate positions to the population from step 4. Select the new position set based on dominance and CD respectively, with intended size of positions.

3. 1 . 1 .
Quality of the pareto front In order to understand the results from IGD, first the pareto fronts are visualized on the solution space graphically in figure 1.Then, the IGDs of the pareto fronts are calculated and they are shown in figure 2. For discussion of the results ZDT1 is chosen.Not to repeat the same discussion, only the results from ZDT2 and ZDT3 are shown in figure 3.

Figure 2 .
Figure 2. Inverted generational distance of the Algorithms.

Figure 4 .
Figure 4. Pareto front of voltage security determining objectives.

Figure 5 .
Figure 5.The control variable values at the optimum operation of IEEE 14 bus system.

Table 1 .
Crowding distance computation of the pareto fronts from the three algorithms.

Table 2 .
The variation of the control variables the voltage stability index.This shows the promising application of IMOPSO for voltage security improvement. by