Optimal power flow solution with current injection model of generalized interline power flow controller using ameliorated ant lion optimization

Optimal power flow (OPF) solutions with generalized interline power flow controller (GIPFC) devices play an imperative role in enhancing the power system’s performance. This paper used a novel ant lion optimization (ALO) algorithm which is amalgamated with Lévy flight operator, and an effectual algorithm is proposed named as, ameliorated ant lion optimization (AALO) algorithm. It is being implemented to solve single objective OPF problem with the latest flexible alternating current transmission system (FACTS) controller named as GIPFC. GIPFC can control a couple of transmission lines concurrently and it also helps to control the sending end voltage. In this paper, current injection modeling of GIPFC is being incorporated in conventional Newton-Raphson (NR) load flow to improve voltage of the buses and focuses on minimizing the considered objectives such as generation fuel cost, emissions, and total power losses by fulfilling equality, in-equality. For optimal allocation of GIPFC, a novel Lehmann-Symanzik-Zimmermann (LSZ) approach is considered. The proposed algorithm is validated on single benchmark test functions such as Sphere, Rastrigin function then the proposed algorithm with GIPFC has been testified on bus


INTRODUCTION
Optimal power flow problem aims to identify the best operating condition of a power with the fulfillment to the demand at the load side by fulfilling the considered security and practical constraints. The authors are performed particle swarm optimization (PSO) algorithm and gravitational search algorithm when tackling single-objective numerical optimization [1], [2]. Flexible alternating current transmission system (FACTS) devices plays an imperative role in optimal power flow (OPF), devices like static synchronous compensator (STATCOM), Scottish social services council (SSSC), interline power flow controller (IPFC), unified power flow controller (UPFC) and generalized interline power flow controller (GIPFC) have the better operating performance as compared to static Var compensator (SVC), thyristor controlled series capacitor (TCSC) and thyristor controlled phase shifter (TCPS) as stated in [3]- [8]. GIPFC is the latest controller which can control real and reactive power of multiple lines simultaneously, which helps to share

Current injection modeling of GIPFC
The current based model of GIPFC is shown in Figure 2, and the two flowing currents can be written as (1) where,

Power mismatches equations of GIPFC
The proposed current injection model of GIPFC is easily incorporated into the system by modifying the Jacobian elements and power mismatch equations related to device connected buses. The existing Jacobian elements obtained from NR is modified by adding the variational derivative of real and reactive power occurring because of the incorporation of GIPFC. The final equation of NR load flow with GIPFC can be expressed as (12): where, ΔP, ΔQ: the vectors representing real and reactive power mismatches, Δδ, ΔV: the vectors of incremental change in the angles and voltages, H, N, J, L: the partial derivative of P and Q with respect to δ and V.  Where, vi, vj are the sending and receiving end voltages respectively, Zij is the impedance of transmission line, Si, Sj is the sending and receiving end apparent powers respectively. Thus, LSZ formula obtained from [11] is: In this, weakest transmission line of power system is being identified which could be in critical state and more likely to collapse and has the largest value for the LSZ formula. This identified transmission required external support thus it would be considered as the optimal line to place the GIPFC device. In this paper, some heuristic rules are considered for reducing the number of possible locations for GIPFC: i) device should not be placed at PV buses; ii) there should not be any shunt compensating device present; and iii) lines in which tap changing transformers are already present should be avoided

PROBLEM FORMULATION OF OPF
Basically, OPF problem is implemented for optimizing the objective function under the consideration of some below mentioned constraints. OPF can expressed as: where, u and v represent equality and inequality constraints respectively, a represents state vector of dependent variables and b represents control vector of system and Of represents total number of objective functions. The state vector can be represented by: where, 1 , 1 , 1 , 1 and 1 are the active power, load bus voltage, reactive power, apparent power, and generator voltage of slack bus respectively. NPQ, NPV, NTL, NC and NT are the total number of PQ buses, PV buses, transmission lines, shunt compensators and off-nominal taps transformers respectively.

Objective functions
In this paper, three single objective functions are considered for minimization, which is mathematically expressed: − Generation fuel cost minimization where, , and are the fuel cost coefficients of ℎ unit.
− Emission minimization where, , , , and are the emission coefficients of ℎ unit. − Total power loss minimization where, is the real power loss in ℎ line.

Constraints
Basically, OPF with FACTS device is subjected to equality, in-equality, and device constraints. The in-equality constraints include generator, transformer, shunt compensator operating constraints depending upon various system parameters. In this paper, practical constraints are also considered such as ramp-rate limits and prohibited operating zone (POZ).

Equality constraints
The optimization of objective functions must satisfy the equality constraint as given bellow.
where, , and , are the real and reactive demands and losses respectively.

Inequality constraints
The said objective function optimization the following in-equality constraints must satisfy. The inequality constraints as given bellow. a) Generator constraints. All the buses with generators including slack bus are bounded by the voltages, real and reactive powers limits as expressed: ∀ ∈ ≤ ∀ ∈ e) GIPFC constraints. GIPFC is restricted in terms of injected series and shunt voltages magnitudes, phase angles and reactance.
The ramp-rate limits of ℎ generator in MW/hour are as shown: where, 0 is the generated power in previous hour, 1 and 2 are the up-rate and down-rate limit of ℎ generator. g) POZ limits. In power systems, POZ limits generator operation in certain ranges from the entire range of possible generation. For ℎ generator POZ at each time interval can be formulated as: (1) where, n is the number of prohibited zones, p is the index for prohibited zones, ( ) and ( ) are the lower and upper limit of ℎ prohibited zone for ℎ generator. With these constraints a penalty function is also incorporated to the objective function; it will allocate the initial values of penalty weights if constraints violate their limits [32].

AMELIORATED ANT LION OPTIMIZATION ALGORITHM 5.1. Existing ant lion optimization algorithm
ALO algorithm is an optimization algorithm inspired by natural phenomenon and proposed in [31]. It mimics the hunting mechanism of ant lions. It is a novel meta-heuristic algorithm that is dwelt mathematically through the interaction between ants (prey) and ant lions (predator). Ant lions are the doodlebugs which basically belong to the Myrmeleontidae family, they possess two phases of life in 3 years of their total life span i.e., larvae for just 3-5 weeks and remaining is adult phase. In the phase of larvae their hunting mechanism is very fascinating. Initially, they start building cone shaped traps by throwing sand out with its jaw, and then they wait for ants at the edge of the conical structure to be get trapped into it, as shown in Figure 4. The stimulating features of these ant lions for digging traps are the hunger level and shapes of the moon. They used to dig a bigger trap in hunger and in full-shaped moon, which enhances the probability of their survival. ALO yields for global optimum solutions for constraint bounded optimization problems. It also maintains equilibrium among exploration and exploitation, with faster convergence [36].

Proposed ameliorated ant lion optimization algorithm
Basically ALO is implemented in five stages, which includes the complete mechanism of antlions' hunting and further in the existing algorithm the Lévy flight operator of Cuckoo Search algorithm [27] has been added to ameliorate the overall performance. Main steps of proposed algorithm are being described with mathematical equations below.

Stochastic walk of ants
Initially, ants move erratically in search of food, which can be modeled: where, cs stands for cumulative sum, r is the step of random walks, IT maximum number of iterations and w(t) can be considered as (17): where, matrix which have position of each ant, represents the value of ℎ ants for ℎ dimension, N and D are total number of ants (search variables) and dimension respectively. Therefore, the random walks can be normalized so that they are inside the search space.
where, and are the minimum and maximum of walk of ℎ variable, and are the minimum and maximum of walk of ℎ variable at ℎ iteration.

Ants trapped in ant lions' traps
Ant lion's pits have their impact on the erratic walks of ants. To mathematically model this assumption the following equations are used: where, and minimum and maximum of all variables at ℎ iteration respectively. is the position of selected ℎ ant lion at ℎ iteration.

Building the traps
Roulette wheel selection is used for building the traps. Basically, in optimization the roulette wheel operator helps in selecting the ant lions based on their fitness. The chance of ants getting caught by the fitter ant lions is more.

Slipping ants in the direction of ant lion
Ant lions always build the trap accordance to their fitness and the ants usually reach the search space randomly. When the ants enter into the pit the ant lions throw the sand outwards so that ants can slip towards ant lions. The radius of ant's erratic walk is represented as (21), (22): where, = 10 , in which t and IT are current and maximum number of iterations respectively. K is a constant weight defined based on the basis of current iteration (K=2 when t>0.1IT, K=3 when t>0.5IT, K=4 when t>0.75IT, K=5 when t>0.9IT, K=6 when t>0.95IT).

Hunting ant then revamp the pit
The terminating stage will occur when ant being caught by the ant lions, and it is made possible only when the ant becomes fitter than the corresponding ant lion. Then antlion should amend their position towards current position where ant gets hunted so that it enhances the probability of catching another ant. This phenomenon can be expressed as (23): where, is the position of selected ℎ ant at ℎ iteration.

Elitism
Elitism is a salient operation of the evolutionary algorithms. It basically facilitates to obtain the optimal solution for the considered problems. For each iteration, best ant lion will be identified and contemplated as Elite, which will affect the movements of ants. It can be mathematically modeled as (24): where, and random walk around the ant lion and elite selected at ℎ iteration.

Lévy flight operator
Random walk of ants is being updated using Lévy flight operator from CSA, which enhances performance of existing algorithm. As the Lévy flight operator makes exploration more effective, it can be calculated as (25): where, α is a constant whose value considered to be equal to 1.5 [37] and is a gamma distribution function. Thus, it updates the random walk of ants by (26):

RESULTS AND ANALYSIS 6.1. Benchmark test functions
In this section, two benchmark test functions are preferred to validate the proposed algorithm. First function F (1) is a Sphere function, and second function F (2) is a Rastrigin function, both the functions are continuous, differential, separable and can be defined for N-dimension space, only difference is that sphere function is a unimodal function while Rastrigin function is a multimodal function. Parameters setting for both the test functions are listed in Table 1. From Table 2, it can be seen that the results achieved for both the test functions when proposed AALO algorithm is implemented are less than the existing algorithms. For the sphere function, output is 4.0681e-11 which is approximately 2.1832e-10 less than the output obtained through ALO algorithm. Similarly, for the Rastrigin function output is 7.437e-07 which is less than the output obtained through ALO algorithm. In Table 2   be observed that convergence curve of proposed algorithm shows a smooth variation while for ALO algorithm curve has stepped variations. The optimum value obtained for both the functions are achieved in less iteration as compared to existing algorithms. Thus, it can be concluded that with the proposed AALO algorithm converges very fast for multi-modal as well as unimodal test function and it efficiently solve the multi-modal test function problems with the avoidance of local optima, so for further analysis AALO algorithm is preferred.

Electrical test system
In order to justify the robustness and effectiveness of the proposed algorithm, IEEE-30 bus system is considered for solving the single objective OPF problem, in which 6 generators are placed at 1, 2, 5, 8, 11 and 13 and remaining are the PQ buses. It consists of 41 transmission lines, 18 control variables, 4 transformers are placed between buses 6-9, 6-10, 4-12, 28-27 and two shunt devices are located at buses 10 and 24. Required bus data, line data, load data, cost data and generation data has been considered from [38].
Optimal power flow results for generation fuel cost with implementation of proposed AALO algorithm is tabulated in Table 3, with its comparison from the existing algorithms. It can be seen that generation fuel cost is minimized to 789.462$/h which is by far the better solution obtained in comparison to other existing algorithms. The proposed algorithm compared with existing method such as fruit fly algorithm (FFA), hybrid cuckoo search algorithm (HCSA), modified sine-cosine algorithm (MSCA).

Incorporation of GIPFC device in IEEE-30 bus system
Now we will incorporate the GIPFC devices in our electric test system but before that we need to identify the weakest buses in IEEE-30 bus system so that GIPFC can be optimally places at those buses to enhance the system parameters. For determination of weakest line, LSZ method is applied, and whichever lines get maximum LSZ value will be considered to be the weakest bus and rank high in order. For all the lines LSZ is calculated which are mentioned in Table 5, from which we can see that line 7 between the buses 5-7 is the weakest bus in test system but as per considered heuristic rule, it cannot be considered to place GIPFC as bus 5 is a PV bus, so next weakest line is 39 which is between buses 29-30. Next lines 5 and 3 which are between buses 2-5 and 2-4 respectively also cannot be considered as bus 2 is also a PV bus. Then second line which can be selected is 32 which are between buses 24-25. So, lines between the buses 24-25 and 29-30 are taken as the weakest line and optimal location to place GIPFC.
After determining the weakest buses, GIPFC is incorporated in NR load flow to overcome the shortcomings caused due to the weak lines. The OPF solution for considered objectives are mentioned in Table 6, in this table three cases are compared, one without placing any FACTS device, second are with incorporation of IPFC and third is with incorporation of GIPFC. It can be seen from the Table 6, that objectives get reduced to lesser value when IPFC [9] and GIPFC are incorporated in comparison to the system without any FACTS controller. We can also observe and conclude that among both the devices i.e., IPFC and GIPFC, OPF solution for considered objectives are least when GIPFC is incorporated. Generation fuel cost got minimized to 800.3032$/h, emission got minimized to 0.20496 ton/h and total power loss got minimized to 3.271884 MW when GIPFC is incorporated (all results obtained is under both the practical constraints). From this result GIPFC controller is superior and more efficient as compared to IPFC controller. So GIPFC is being carried forward for further analysis.

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As in section 2, current injection modeling of GIPFC is mathematically derived and it is being observed that in comparison to power injection modeling of GIPFC, current injection modeling is better technique in terms of mathematical observation. In this section, PIM and CIM model of GIPFC is incorporated in IEEE-30 bus system and from Table 7, they can be easily compared as OPF solution for considered objectives are solved using both the modeling techniques. We can see that generation fuel cost is 798.881$/h, emission is 0.202061 ton/h and total power loss is 3.18369 MW when CIM of GIPFC is incorporated, which is less than the objectives minimized by PIM of GIPFC and by far the best result obtained for the objectives under the effect of practical constraints. Convergence curve for both the technique is plotted for all three objectives in Figures 8-10. So, it is an evident that with the CIM of GIPFC using proposed AALO algorithm under practical constraints is best approach to solve the optimal power flow problems. It fastens the convergence and initial iteration starts with less objectives value.

CONCLUSION
In this paper, ameliorated ant lion optimization algorithm had been proposed for minimizing the single as well as multi-objectives problem of OPF including objectives such as generation fuel cost, emission, and total power losses with the incorporation of GIPFC in power system. The proposed algorithm had been tested on Sphere and Rastrigin test functions, which in results proves that the addition of Lévy flight operator in existing ant lion optimization Algorithm revamp the performance and yields better solutions with fast convergence rate. In addition to it, current injection modeling of GIPFC had been derived and implemented in NR load flow method and observed that it is simple and better approach for modeling a series controller as it leads to wider and faster the convergence. For optimal allocation of GIPFC, LSZ formula had been implemented which helps to maintain the voltage stability in power system as it identify the weakest lines which are at a verge to get collapse and those weakest lines had been considered for placing the device. Thus, the proposed algorithm with CIM of GIPFC had been validated on IEEE-30 bus system and hence it can be contemplated as a better alternative approach for solving OPF problems more effectively and efficiently.