Tailored flower pollination (TFP) algorithm for diminution of real power loss

In this paper, Tailored Flower Pollination (TFP) algorithm is proposed to solve the optimal reactive power problem. Comprising of the elements of chaos theory, Shuffled frog leaping search and Levy Flight, the performance of the flower pollination algorithm has been improved. Proposed TFP algorithm has been tested in standard IEEE 118 & practical 191 bus test systems and simulation results show clearly the better performance of the proposed algorithm in reducing the real power loss


INTRODUCTION
Optimal reactive power dispatch (ORPD) problem is to minimize the real power loss and bus voltage deviation. Various numerical methods like the gradient method [1][2], Newton method [3] and linear programming [4][5][6][7] have been adopted to solve the optimal reactive power dispatch problem. Both the gradient and Newton methods have the complexity in managing inequality constraints. If linear programming is applied then the input-output function has to be uttered as a set of linear functions which mostly lead to loss of accuracy. The problem of voltage stability and collapse play a major role in power system planning and operation [8]. Evolutionary algorithms such as genetic algorithm have been already proposed to solve the reactive power flow problem [9][10][11]. Evolutionary algorithm is a heuristic approach used for minimization problems by utilizing nonlinear and non-differentiable continuous space functions. In [12], Hybrid differential evolution algorithm is proposed to improve the voltage stability index. In [13] Biogeography Based algorithm is projected to solve the reactive power dispatch problem. In [14], a fuzzy based method is used to solve the optimal reactive power scheduling method. In [15], an improved evolutionary programming is used to solve the optimal reactive power dispatch problem. In [16], the optimal reactive power flow problem is solved by integrating a genetic algorithm with a nonlinear interior point method. In [17], a pattern algorithm is used to solve ac-dc optimal reactive power flow model with the generator capability limits. In [18], F. Capitanescu proposes a two-step approach to evaluate Reactive power reserves with respect to operating constraints and voltage stability. In [19], a programming based approach is used to solve the optimal reactive power dispatch problem. In [20], A. Kargarian et al present a probabilistic algorithm for optimal reactive power provision in hybrid electricity markets with uncertain loads. This paper proposes Tailored Flower Pollination (TFP) algorithm is proposed to solve the reactive power problem. The basic idea of flower pollination process which leads to the formulation of flower pollination algorithm (FPA) [21] is first introduced and subsequently, chaos

PROBLEM FORMULATION
The optimal power flow problem is treated as a general minimization problem with constraints, and can be mathematically written in the following form: and where f(x,u) is the objective function. g(x.u) and h(x,u) are respectively the set of equality and inequality constraints. x is the vector of state variables, and u is the vector of control variables. The state variables are the load buses (PQ buses) voltages, angles, the generator reactive powers and the slack active generator power: x = (P g1 , θ 2 , . . , θ N , V L1 , . , V LNL , Q g1 , . . , Q gng ) The control variables are the generator bus voltages, the shunt capacitors/reactors and the transformers tapsettings: or u = (V g1 , … , V gng , T 1 , . . , T Nt , Q c1 , . . , Q cNc ) T (6) where ng, nt and nc are the number of generators, number of tap transformers and the number of shunt compensators respectively.

OBJECTIVE FUNCTION 3.1. Active power loss
The objective of the reactive power dispatch is to minimize the active power loss in the transmission network, which can be described as follows: or where gk : is the conductance of branch between nodes i and j, Nbr: is the total number of transmission lines in power systems. Pd: is the total active power demand, Pgi: is the generator active power of unit i, and Pgsalck: is the generator active power of slack bus.

Voltage profile improvement
For minimizing the voltage deviation in PQ buses, the objective function becomes: where ωv: is a weighting factor of voltage deviation. VD is the voltage deviation given by:

Equality Constraint
The equality constraint g(x,u) of the ORPD problem is represented by the power balance equation, where the total power generation must cover the total power demand and the power losses: This equation is solved by running Newton Raphson load flow method, by calculating the active power of slack bus to determine active power loss.

Inequality Constraints
The inequality constraints h(x,u) reflect the limits on components in the power system as well as the limits created to ensure system security. Upper and lower bounds on the active power of slack bus, and reactive power of generators: Upper and lower bounds on the bus voltage magnitudes: Upper and lower bounds on the transformers tap ratios: Upper and lower bounds on the compensators reactive powers: where N is the total number of buses, NT is the total number of Transformers; Nc is the total number of shunt reactive compensators.

FLOWER POLLINATION ALGORITHM
Generally we use the following systems in Flower Pollination Algorithm (FPA),  System 1. Biotic and cross-pollination has been treated as global pollination process, and pollen-carrying pollinators travel in a way which obeys Levy flights.  System 2. For local pollination, A-biotic and self-pollination has been utilized.  System 3. Pollinators such as insects can develop flower reliability, which is equivalent to a reproduction probability and it is proportional to the similarity of two flowers implicated.  System 4. The communication of local pollination and global pollination can be controlled by a control probability p ∈ [0, 1], with a slight bias towards local pollination. System 1 and flower reliability can be represented mathematically as where is the pollen i or solution vector xi at iteration t, and * is the current best solution found among all solutions at the current generation/iteration. Here γ is a scaling factor to control the step size. L(λ) is the parameter that corresponds to the strength of the pollination, which essentially is also the step size. Since insects may move over a long distance with various distance steps, we can use a Levy flight to mimic this characteristic efficiently. We draw L > 0 from a Levy distribution here, Γ(λ) is the standard gamma function, and this distribution is valid for large steps s > 0. Then, to model the local pollination, for both system 2 and system 3 can be represented as where and are pollen from different flowers of the same plant species. This essentially mimics the flower reliability in a limited neighbourhood. Mathematically, if and comes from the same species or selected from the same population, this equivalently becomes a local random walk if we draw ∈ from a uniform distribution in [0,1]. Though Flower pollination performance can occur at all balance, local and global, neighbouring flower patch or flowers in the not-so-far-away neighbourhood are more likely to be pollinated by local flower pollen than those far away. In order to mimic this, we can effectively use a control probability (system 4) or proximity probability p to switch between common global pollination to intensive local pollination. To start with, we can use a raw value of p = 0.8 as an initially value.
The simplest method is to use a weighted sum to combine all multiple objectives into a composite single objective where m is the number of objectives and wi(i = 1, ...,m) are non-negative weights. FP Algorithm for solving optimal reactive power optimization Step 1. Objective min of (x), x = (x1, x2, ..., xd) Step 2. Initialize a population of n flowers Step 3. Find the best solution * in the initial population Step

CHAOTIC MAPS
Chaos is a random state found in the non-linear dynamical deterministic system, possesses non-period, non-converging and bounded properties. The use of chaotic sequences is more beneficial than the random sequences due to its non-repetition and ergodicity properties. Borrowing the advantages of ergodicity, nonrepetition and randomness of the chaotic sequences, the chaotic map is replacing the random sequences in generating the initial population in the FPA in this study. This is to ensure that the diversity of the initial population can be improved, where the distribution of the initial population is more uniform. Ten different chaotic maps are and circle map is selected for the integration with FPA.

SHUFFLED FROG LEAPING ALGORITHM
Shuffled frog leaping algorithm is a biological evolution algorithm based on swarm intelligence. The algorithm simulates a group of frogs in the wetland passing thought and foraging by classification of ethnic groups. In the execution of the algorithm, F frogs are generated at first to form a group, for N-dimensional optimization problem, frog i of the group is represented as = ( 1 , 2 , . . , ) then individual frogs in the group are sorted in descending order according to fitness values, to find the global best solution Px. The group is divided into m ethnic groups, each ethnic group including n frogs, satisfying the relation F = m × n. The rule of ethnic group division is: the first frog into the first sub-group, the second frog into the second sub-group, frog m into sub-group m, frog m + 1 into the first sub-group again, frog m + 2 into the second sub-group, and so on, until all the frogs are divided, then find the best frog in each sub-group, denoted by Pb; get a worst frog correspondingly, denoted by Pw. Its iterative formula can be expressed as: where rand() represents a random number between 0 and 1, Pb represents the position of the best frog, Pw represents the position of the worst frog, D represents the distance moved by the worst frog, _− is the improved position of the frog, Dmax represents the step length of frog leaping.
In the execution of the algorithm, if the updated _− is in the feasible solution space, calculate the corresponding fitness value of _− , if the corresponding fitness value of _− is worse than the corresponding fitness value of Pw, then use Pw to replace Pb in equation (22) and re-update _− ; if there is still no improvement, then randomly generate a new frog to replace Pw; repeat the update process until satisfying stop conditions.

LEVY FLIGHT
Levy flight is a rank of non-Gaussian random processes whose arbitrary walks are drawn from Levy stable distribution. This allocation is a simple power-law formula L(s) ~ |s| -1-β where 0 < ß < 2 is an index. Mathematically exclamation, a easy version of Levy distribution can be defined as , where > 0 parameter is scale (controls the scale of distribution) parameter, μ parameter is location or shift parameter. In general, Levy distribution should be defined in terms of Fourier transform where α is a parameter within [-1,1] interval and known as scale factor. An index of o stability β ∈ [0, 2] is also referred to as Levy index. In particular, for β = 1, the integral can be carried out analytically and is known as the Cauchy probability distribution. One more special case when β= 2, the distribution correspond to Gaussian distribution. β and α parameters take a key part in determination of the distribution. The parameter β controls the silhouette of the probability distribution in such a way that one can acquire different shapes of probability distribution, especially in the tail region depending on the parameter β. Thus, the smaller β parameter causes the distribution to make longer jumps since there will be longer tail. It makes longer jumps for smaller values whereas it makes shorter jumps for bigger values. By Levy flight, new-fangled state of the particle is designed as, α is the step size which must be related to the scales of the problem of interest. In the proposed method α is random number for all dimensions of particles.
the product ⊕ means entry-wise multiplications. A non-trivial scheme of generating step size s samples are summarized as follows, here Г is standard Gamma function. One of the important points to be considered while performing distribution by Levy flights is the value taken by the β parameter and it substantially affects distribution.

TAILORED FLOWER POLLINATION (TFP) ALGORITHM
In the TFP algorithm, the initial population is generated using the circle map, frog leaping local search is performed by each solution and when rand>p, modified Levy flight with integration of inertia weight in global pollination is performed on that particular solution. The steps involved in the TFP are as follows: Step

SIMULATION RESULTS
At first Tailored Flower Pollination (TFP) algorithm has been tested in standard IEEE 118-bus test system [22].The system has 54 generator buses, 64 load buses, 186 branches and 9 of them are with the tap setting transformers. The limits of voltage on generator buses are 0.95 -1.1 per-unit., and on load buses are 0.95 -1.05 per-unit. The limit of transformer rate is 0.9 -1.1, with the changes step of 0.025. The limitations of reactive power source are listed in Table 1, with the change in step of 0.01. The statistical comparison results of 50 trial runs have been list in Table 2 and the results clearly show the better performance of proposed TFP algorithm.

CONCLUSION
Tailored Flower Pollination (TFP) algorithm has been effectively applied for solving reactive power problem. And it has been tested in standard IEEE 118 & practical 191 bus test systems. Performance comparisons with well-known population-based algorithms give improved results. Tailored Flower Pollination (TFP) algorithm emerges to find good solutions when compared to that of other reported algorithms. The simulation results presented in previous section prove the capability of TFP approach to arrive at near to global optimal solution.