Optimal power flow with enhancement of voltage stability and reduction of power loss using ant-lion optimizer

: In this work, the most common problem of the modern power system named optimal power flow (OPF) is optimized using the novel meta-heuristic optimization algorithm ant lion optimizer (ALO). ALO is inspired by the hunting process of ant-lions in the natural environment. ALO has a fast convergence rate due to the use of roulette wheel selection method. For the solution of the optimal power flow problem, standard 30 bus IEEE system is used. ALO is applied to solve the suggested problem. The problems considered in the OPF problem are fuel cost reduction, voltage profile improvement, voltage stability enhancement, minimization of active power losses and minimization of reactive power losses. The results obtained with ALO is compared with other methods like firefly algorithm (FA) and particle swarm optimization (PSO). Results show that ALO gives better optimization values as compared with FA and PSO which verifies the strength of the suggested algorithm.


Introduction
At the present time, the optimal power flow (OPF) is a very significant problem and most focused objective for power system scheduling and operation (Bouchekara, Abido, Chaib, & Mehasni, 2014).The OPF is the elementary tool which permits the utilities to identify the economic operational and many secure states in the system (Duman, Güvenç, Sönmez, & Yörükeren, 2012;Niknam, Narimani, Jabbari, & Malekpour, 2011).The OPF problem is one of the utmost operating desires of the electrical power system (Carpentier, 1962).The prior function of OPF problem is to evaluate the optimum

PUBLIC INTEREST STATEMENT
Today every country load demand is increasing but the generation of electric power is limited due to limited availability of fossil fuels.Power system utility has the responsibility to provide power with continuity, reliability and security.The power system is also responsible for providing power at minimum cost.At present in deregulated power system economy is of main concern and security of the system is the basic need.This perspective article describes for the current situation electrical engineers are facing a leading problem of management of existing power in an effective and efficient way to feed the rapidly increasing demand of customers.
operational state for Bus system by minimizing each objective function within the limits of the operational constraints like equality constraints and inequality constraints (Bouchekara, Abido, & Boucherma, 2014).Hence, the optimal power flow problem can be defined as an extremely nonlinear and non-convex multimodal optimization problem (Abou El Ela & Abido, 2010).
From the past few years too many optimization techniques were used to solve the optimal power flow (OPF) problem.Some traditional techniques are used to solve the proposed problem have been suffered from some limitations like converging at local optima, not suitable for binary or integer problems and also have the assumptions like the convexity, differentiability, and continuity (Bouchekara, 2013).Hence, these techniques are not suitable for the actual OPF situation (AlRashidi & El-Hawary, 2009;Frank, Steponavice, & Rebennack, 2012a).All these limitations are overcome by meta-heuristic optimization methods like genetic algorithm (GA), particle swarm optimization (PSO), ant colony optimization (ACO), differential evolution algorithm (DEA) and harmony search algorithm (HSA) (Frank, Steponavice, & Rebennack, 2012b;Yildiz, 2012).
In this paper, a newly introduced meta-heuristic optimisation technique named ant-lions optimizer (ALO) is implemented to solve the optimal power flow problem.The ALO technique is a biological and sociological inspired algorithm.This technique is follows the hunting process of the antlions.Key steps of hunting ants such as ants random walk, builds traps, trapping of ants, grasping foods, and rebuilding the traps are applied.The capabilities of ALO are finding the global solution, fast convergence rate due to the use of roulette wheel selection, can handle continuous and discrete optimization problems.
According to no free launch theorem state that single Meta-heuristic algorithm is not best for every problem so we considered ant-lions optimizer for continues optimal power flow problem.In this work, the ALO is applied to standard 30 bus IEEE test system (Lee, Park, & Ortiz, 1985) to solve the OPF (Bakirtzis, Biskas, Zoumas, & Petridis, 2002;Belhadj & Abido, 1999;Bouktir, Labdani, & Slimani, 2005;Ongsakul & Tantimaporn, 2006;Soliman & Mantawy, 2012) problem.There are three objective cases considered in this paper that have to be optimize using ant-lion optimizer (ALO) technique are fuel cost reduction, voltage stability improvement, and voltage deviation minimization.The result shows the optimal adjustments of control variables in accordance with their limits.particle swarm optimisation (PSO) and firefly algorithm (FA) are the most popular algorithms in swarm algorithms.So, the results obtained using ALO technique has been compared with particle swarm optimisation (PSO) and firefly algorithm (FA) techniques.The results show that ALO gives better optimization values as compared with different methods which prove the strength of the suggested method.

Ant-lion optimizer technique
The ALO technique reflects the intellectual activities of antlions in catching ants in the environment.The ALO algorithm inspired by the interface of antlions and ants inside the pit.To model such interfaces, ants have to travel over the exploration space, and antlions are permitted to pursuit them and become stronger using traps (Mirjalili, 2015).

Operators of ALO algorithm
As ants travel randomly in search space when finding the prey, a random walk is selected for demonstrating ants' effort and it is given by Equation (1) (Mirjalili, 2015): where cumsum computes the cumulative sum, n is the maximum No. of iteration, t is the step of ants random walk (iteration), and r(t) is a stochastics function defined as follows (Mirjalili, 2015): (1) where t is the step of ants random walk and rand represent a random number created by constant circulation in the interval of [0, 1] (Mirjalili, 2015).
The location of ants are kept and used during optimisation in the given matrix (Mirjalili, 2015): where M Ant = the matrix for storing the location of every ants, A ij = the value for j th variable (dimension) of i th ant, n = the No. of ants and d = the total No. of variables.
For calculating individual ant, a fitness function is used in optimisation and subsequent matrix saves the fitness value of each ants (Mirjalili, 2015): where M OA = the matrix for storing the each ant fitness, A ij = the value of j th variable of i th ant, n = the total No. of ants and f = the objective function.
So we suppose that ants, as well as the antlions, are hide anywhere in the search area.So as to store their locations and fitness values, the following matrices are used: where M Antlion = the matrix for storing the location of individual antlion, AL ij = the value of j th variable of i th antlion, n = No. of ants and d = the No. of variables.
where M OAL = the matrix for storing the fitness of individual antlion, AL ij = the value of j th variable of i th antlion, n = No. of ants and f = the objective function.

Random walk of ants
Each of the behaviors is mathematically modeled as (Mirjalili, 2015).
The random walks of ants is calculated by Equation ( 6): (2) where a i = minimal of the random walk of i th variable, b i = the maximum of random walk in i th variable.

Trapping in antlions pits
The trapping in ant-lion's pits is calculated by Equations ( 7) and (8):

Sliding ants towards antlion
The sliding ants towards ant-lion calculated by Equations ( 9) and ( 10): where I = ratio, c t = the minimal of total variables at t th iteration, and d t = the vector containing the maximum of total variables at t th iteration.

Catching prey and re-building the pit
Hunting prey and re-arranging the pits calculated by Equation ( 11): where t = the current iteration, Antlion j t = the location of chosen j th antlion at t th iteration, and Ant i t = the location of i th ant at t th iteration.

Elitism
Elitism of ant-lion calculated using roulette wheel by Equation ( 12): where R t A = the random walk nearby the antlion chose by means of the roulette wheel at t th iteration, R t E = the random walk nearby the elite at t th iteration, Ant t i = the location of i th ant at t th iteration (Mirjalili, 2015).

Optimal power flow problem formulation
As specified before, OPF is Optimized power flow problem which provides the optimal values of control (independent) variables by minimizing a predefined objective function with respect to the operating bounds of the system (Bouchekara, Abido, Chaib, et al., 2014).The OPF problem can be mathematically expressed as a non-linear constrained optimization problem as follows (Bouchekara, Abido, Chaib, et al., 2014): Minimize f (a, b)

Control variables
The control variables should be so manage to fulfill the power flow equations.For the OPF problem, the set of control variables can be formulated as (Bouchekara, Abido, Chaib, et al., 2014;Bouchekara, Abido, & Boucherma, 2014): where P G = real power output on the generator buses excluding on the reference bus, V G = magnitude of voltage on generator buses, Q C = shunt VAR compensation, T = tap settings of the transformer, NGen, NTr, NCom = No. of generating units, No. of tap changing transformers and No. of shunt VAR compensation devices, respectively.

State variables
There is a need for variables for all OPF formulations for the characterization of the Electrical Power Engineering state of the system.So, the state variables can be formulated as (Bouchekara, Abido, Chaib, et al., 2014;Bouchekara, Abido, & Boucherma, 2014): where P G 1 = Real power output on the reference bus, V L = magnitude of voltage on load buses, Q G = reactive power generation of all generators, S l = line loading or power flow, NLB, Nline = No. of PQ buses and the No. of lines, respectively.

Constraints
There are two OPF constraints named inequality and equality constraints.These constraints are explained in the next topic.

Equality constraints
The physical condition of the system is defined by the equality constraints of the OPF.Basically these are the load flow equations which can be explained as follows (Bouchekara, Abido, Chaib, et al., 2014;Bouchekara, Abido, & Boucherma, 2014).

Real power constraints.
The real power constraints can be formulated as follows (Bouchekara, 2013):

Reactive power constraints.
The reactive power constraints can be formulated as follows (Bouchekara, Abido, & Boucherma, 2014): where δ ij = δ iδ j , NB = total No. of buses, P G = real power generation, Q G = reactive power generation, P D = active power load demand, Q D = reactive power load demand, B ij and shows the susceptance and conductance among bus i and bus j, respectively.

Inequality constraints
The boundaries of power system devices together with the bounds created to surety system security are given by inequality constraints of the OPF (Abou 3.2.2.4.Security constraints.These comprise the limits of the magnitude of the voltage on PQ buses and line loadings.Voltage for every PQ bus should be limited by its minimum and maximum operational bounds.Loadings over each line should not exceed its maximum loading limit.So, these limitations can be statistically stated as (Bouchekara, 2013): The control variables are self-constraint.The inequality constrained of state variables comprises a magnitude of load (PQ) bus voltage, active power production at reference bus, reactive power production, and loading on line may be encompassed by an objective function in terms of quadratic penalty terms.In which, the penalty factor is increased by the square of the disrespect value of state variables and is included in the objective function and any impractical result achieved is declined (Bouchekara, 2013).

Penalty function can be mathematically formulated as given below:
where P , V , Q , S = penalty factors, U lim = Boundary price of the state variable U. (20) If U is greater than the maximum value, U lim takings the maximum value, if U is lesser than the minimum value, U lim takings the value of that limit.This can be shown as follows (Bouchekara, 2013):

Application and results
The ALO method implemented for the OPF solution for standard 30-bus IEEE test system and for a number of objectives with dissimilar functions.The used program is written in MATLAB R2014b computing surroundings and used on a 2.60 GHz i5 PC with 4 GB RAM.In this work, the number of search agents or number of ants is selected to be 40.

IEEE 30-bus test system
With the purpose of elucidating the effectiveness of the suggested ALO algorithm, it has been verified on the 30-bus IEEE standard test system as displays in Figure 1.The test system selected in the present work has these equipment (Bouchekara, 2013;Lee et al., 1985): six generating units, four regulating transformers and nine shunt VAR compensators.
In addition, generator cost coefficient numbers, the line numbers, bus numbers, and the upper and lower bounds for the control variables are specified in (Lee et al., 1985).In given test system, five diverse objectives are considered for various purposes and all the acquired outcomes are given in Table 1.The very first column of this table denotes the optimal values of control variables found where: (28) • signifies the power and voltages of generator 1 to 6.
Further, fuel cost ($/h), real power losses (MW), reactive power losses (MVAR), voltage deviation and L max represent the total generation fuel cost of the system, the total real power losses, the total reactive power losses, the load voltages deviation from 1 and the stability index, respectively.Other particulars for these outcomes will be specified in the next topic.
The control parameters for ALO, FA, PSO used in this problem are given in Table 1.

Case 1: Minimization of generation fuel cost
The very common OPF objective that is generation fuel cost reduction is considered in the case 1.Therefore, the objective function Y signifies the total fuel cost of every generators and is calculated by Equation (29) (Bouchekara, Abido, Chaib, et al., 2014): where f i shows the fuel cost of the i th generator.f i , may be formulated as follow: where u i , v i and w i are the cost coefficients of the i th generator.The cost coefficients data are specified in Lee et al. (1985).
The variation of the total fuel cost for different algorithms is presented in Figure 2. It demonstrates that the suggested method has outstanding convergence characteristics.The comparison of fuel cost obtained with diverse techniques is shown in Table 2 which displays that the results obtained by ALO are better than the other methods.The optimal ideals of control variables achieved by various methods for case 1 are specified in Table 3.By means of the same settings i.e. control variables boundaries, initial situations, and system values, the results achieved in case 1 with the ALO (29)  technique are equated to different methods and it displays that the fuel cost is greatly decreased compared to the initial case (Bouchekara, 2013).Quantitatively, it is reduced from 901.951 to 799.155 $/h.

Case 2: Voltage profile improvement
Bus voltage is considered as very essential as well as important security and service excellence indices (Bouchekara, 2013).Here the goal is to increase the voltage profile by reducing the voltage deviation of load buses from the unity 1.0 p.u.
Hence, the objective function may be formulated by Equation (31) (Bouchekara, Abido, & Boucherma, 2014): The variation of voltage deviation with different algorithms over iterations is sketched in Figure 3.It demonstrates that the suggested method has good convergence characteristics.The statistical values of voltage deviation obtained with different methods are display in Table 4 which displays that the outcomes obtained by ALO are enhanced than the other methods.The optimal values of control variables achieved by different algorithms for case 2 are specified in Table 5.By means of the same settings, the results achieved in case 2 with the ALO technique are compared to some other methods and it displays that the voltage deviation is significantly reduced compared to the initial  case (Bouchekara, 2013).It has been made known that the voltage deviation is decreased from 1.1496 to 0.1222 p.u. using ALO technique.

Case 3: Voltage stability enhancement
Presently, the transmission networks are enforced to work nearby their safety bounds, because of cost-effective and environmental causes.One of the significant characteristics of the network is its capability to retain continuously tolerable bus voltages to each point beneath standard operational environments, next to the rise in load, as soon as the network is being affected by disruption.The  unoptimized control variables may cause increasing and unmanageable voltage drop causing a tremendous voltage collapse (Bouchekara, Abido, & Boucherma, 2014).Hence, voltage stability is inviting ever more attention.By using various techniques to evaluate the margin of voltage stability, Glitch and Kessel have introduced a voltage stability index called L-index depends on the viability of load flow equations for every node (Kessel & Glavitsch, 1986).The L-index of a bus shows the probability of voltage breakdown circumstance for that particular bus.It differs between 0 and 1 equivalent to zero loads and voltage breakdown, respectively.
For the given network with NB, NGen and NLB buses signifying the total No. of buses, the total No. of generator buses and the total No. of load buses, respectively.The buses can be distinct as PV buses at the top and PQ buses at the bottom as follows (Bouchekara, Abido, & Boucherma, 2014): where Y LL , Y LG , Y GL andY GG are co-matrix of Y bus .The subsequent hybrid network of equations can be expressed as: (32)  The matrix H is given by: Hence, the L-index denoted by L j of bus j is denoted as follows: (34) Hence, the stability of the whole system is described by a global indicator L max which is presented by (Bouchekara, 2013): The system is more stable as the value of L max is lower.
The voltage stability can be enhanced by reducing the value of voltage stability indicator L-index at every bus of the system (Bouchekara, 2013).
Thus, the objective function may be given as below equation: Y voltage_stability_enhancement = L max The variation of the L max index with different algorithms over iterations is presented in Figure 4.The statistical results obtained with different methods are shown in Table 6 which displays that ALO gives improved results than the various techniques.The optimal values of control variables obtained by various methods for case 3 are display in Table 7.After implementing the ALO approach, it seems from Table 7 that the value of L max is considerably decreased in this case compared to initial (Bouchekara, 2013) from 0.1723 to 0.1140.Thus, the distance from breakdown point is improved.

Case 4: Minimization of active power transmission losses
In the case 4 the Optimal Power Flow goal is to reduce the real power transmission losses, that can be represented by power balance Equation (38) (Bouchekara, 2013):    8 which made sense that the results obtained by ALO give better values than the other methods.The optimal values of control variables obtained by different algorithms for case 4 are displayed in Table 9.By means of the same settings the results achieved in case 4 with the ALO technique are compared to some other methods and it displays that the real power transmission losses are greatly reduced compared to the initial case (Bouchekara, 2013)

Case 5: Minimization of reactive power transmission losses
The accessibility of reactive power is the main point for static system voltage stability margin to provision the transmission of active power from the source to sinks (Bouchekara, 2013).
Thus, the minimization of VAR losses are given by the following expression: It is notable that the reactive power losses may not essentially positive.The variation of reactive power losses with different methods shown in Figure 6.It demonstrates that the suggested method has good convergence characteristics.The statistical values of reactive power losses obtained with different methods are shown in Table 10 which displays that the results obtained by ALO are better than the other methods.The optimal values of control variables obtained by different algorithms for case 5 are given in Table 11.It is shown that the reactive power losses are greatly reduced compared to the initial case (Bouchekara, 2013) from −4.6066 to −25.076 using ALO technique.

Robustness test
In order to check the robustness of the ant-lion optimizer for solving continues optimal power flow problems, 10 times trials with various search agents for cases 1-5.Tables 2-11 presents the statistical results achieved by the PSO, FA and ALO algorithms for OPF problems for various cases.From these tables, it is clear that the optimum objective function values obtained by ant-lion optimizer are near to every trial and minimum compare to PSO and FA algorithms.Its proofs the robustness of ant-lion optimizer (ALO) to solve OPF problem.

Conclusion
In this work, ant-lion optimizer, firefly algorithm, and particle swarm optimization algorithm are successfully applied to standard 30-bus IEEE systems to solve the optimal power flow problem for the various types of cases: fuel cost, active power loss, reactive power loss, voltage deviation and voltage stability index.The obtained results give the optimum sets of control variables with ALO, PSO and FA Algorithms which demonstrate the effectiveness of the different techniques.The solutions obtained from the ALO approach has good convergence characteristics and gives the better optimum results compared to FA and PSO techniques which confirm the strength of recommended algorithm.Further, we can improve the algorithms efficiency using different types of penalty handling approaches: adaptive, Deb, MQM, static methods etc. and different randomization techniques: adaptive and levy flight approaches for better exploration and exploitation.You are free to: Share -copy and redistribute the material in any medium or format Adapt -remix, transform, and build upon the material for any purpose, even commercially.The licensor cannot revoke these freedoms as long as you follow the license terms.
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Cogent Engineering
Subject to s(a, b) = 0 where a = vector of state variables, b = vector of control variables, f(a, b) = objective function, s(a, b) = different equality constraints set, h(a, b) = different inequality constraints set.

Figure 1 .
Figure 1.Single line illustration of 30-bus IEEE test system.

Figure 2 .
Figure 2. Fuel cost variations with different algorithms.

Figure
Figure 3. Voltage deviation minimization with different algorithms.

Figure 4 .
Figure 4. L max variations with different algorithms.

Figure
Figure 5. Minimization of active power transmission losses with different algorithms.

Figure
Figure 6.Minimization of reactive power transmission losses with different algorithms.

Figure 5
Figure5display the tendency for reducing the total real power losses objective function using the different techniques.The active power losses obtained with different techniques are shown in Table8which made sense that the results obtained by ALO give better values than the other methods.The optimal values of control variables obtained by different algorithms for case 4 are displayed in Table9.By means of the same settings the results achieved in case 4 with the ALO technique are compared to some other methods and it displays that the real power transmission losses are greatly reduced compared to the initial case(Bouchekara, 2013) from 5.821 to 2.891.
Figure5display the tendency for reducing the total real power losses objective function using the different techniques.The active power losses obtained with different techniques are shown in Table8which made sense that the results obtained by ALO give better values than the other methods.The optimal values of control variables obtained by different algorithms for case 4 are displayed in Table9.By means of the same settings the results achieved in case 4 with the ALO technique are compared to some other methods and it displays that the real power transmission losses are greatly reduced compared to the initial case(Bouchekara, 2013) from 5.821 to 2.891.
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Table 6 . Comparison of L max index obtained with different algorithms
is produced by the partially inverting of Y bus , H LL , H LG , H GL and H GG are co-a matrix of H, V G , I G , V L and I L are voltage and current vector of PV buses and PQ buses, respectively.