Enhanced GSA-Based Optimization for Minimization of Power Losses in Power System

Gravitational Search Algorithm (GSA) is a heuristic method based on Newton’s law of gravitational attraction and law of motion. In this paper, to further improve the optimization performance of GSA, the memory characteristic of Particle SwarmOptimization (PSO) is employed in GSAPSO for searching a better solution. Besides, to testify the prominent strength of GSAPSO, GSA, PSO, and GSAPSO are applied for the solution of optimal reactive power dispatch (ORPD) of power system. Conventionally, ORPD is defined as a problem ofminimizing the total active power transmission losses by setting control variables while satisfying numerous constraints. Therefore ORPD is a complicated mixed integer nonlinear optimization problem including many constraints. IEEE14bus, IEEE30-bus, and IEEE57-bus test power systems are used to implement this study, respectively. The obtained results of simulation experiments using GSAPSO method, especially the power loss reduction rates, are compared to those yielded by the other modern artificial intelligence-based techniques including the conventional GSA and PSO methods. The results presented in this paper reveal the potential and effectiveness of the proposed method for solving ORPD problem of power system.


Introduction
Optimal reactive power dispatch (ORPD), as one of the significant optimization problems in power system operation, is to minimize the given objective function such as total active power transmission losses ( Loss ) by optimizing settings of control variables while satisfying a set of constraints during the entire dispatch period.Control variables contain discrete variables such as tap positions of transformers and amount of reactive compensation and continuous variables like generator voltages.Constraints consist of a series of equality constraints and inequality constraints [1].Besides, it is worth noting that the ORPD problem in this paper is a single-objective optimization problem different from the study in [1] which researches a multiobjective optimization problem.In [1], the multiobjective ORPD problem seeks for a compromise solution for minimization of power losses and  index simultaneously, but this paper is required to find out a global optimal solution for minimization of  Loss .Thus, ORPD problem in this paper is a complex mixed integer nonlinear optimization problem with a number of constraints and has the challenge of searching for the global optimal solution.
Numerous classical approaches including gradient-based optimization algorithms and many mathematical programming methods [2][3][4][5] have been developed and applied for solving the ORPD problem in the past.However, these traditional techniques almost only optimize the differentiable objective functions and they have difficulties in dealing with the nonconvex, nonlinear, discontinuous functions with constraints [6,7].But now a number of modern artificial intelligence-based techniques with stochastic optimization such as Genetic Algorithm (GA) [8], Differential Evolution (DE) [9], Particle Swarm Optimization (PSO) [10], and Gravitational Search Algorithm (GSA) have been applied to solve different ORPD problems efficiently without the abovementioned restraints, which overcomes the defects of conventional techniques.Furthermore, different method has its peculiar strength; for instance, the process of variation and hybridization in GA increases the diversity of population, which contributes to obtain the better solutions; DE uses the differences between individuals to change the individual itself, and this operation utilizes the distribution features of population to improve the search capacity effectively; PSO gets highlighted by virtue of the memory characteristic from imitating animals' predation process containing social and individual behaviors; the pattern of movement of GSA contributes significantly to the high efficiency of the search process.Conversely, every method yet has its own weakness: GA is apt to converge prematurely and the long and complicated evolution procedures of it increase the running time; in DE, the individuals' differences decrease with the increase of number of iterations, which impacts the increase of diversity of population directly.PSO tends to be trapped into prematurity in the latter period of searching, which directly lessens the possibility of acquiring the better solution; different from the algorithms based on biology, GSA is a memoryless algorithm, which is adverse to recording of the optimal value during the process of searching.According to these features, the exertion of merging individual superiorities into a new algorithm has become wide now in various engineering fields, which avoids their own disadvantages by benefiting from each other's advantages; for example, a method composed of chaotic embedded Backtracking Search Optimization Algorithm (BSA) and Binary Charged System Search (BCSS) algorithm is proposed for solving Short-Term Hydrothermal Generation Scheduling (SHTGS) in [11]; in [12], the authors present a new hybrid evolutionary algorithm based on new fuzzy adaptive PSO algorithm and Nelder-Mead simplex search method to solve distribution feeder reconfiguration problem; the combination of the vertical search algorithm and presented lateral search algorithm is used to solve the midterm schedule for thermal power plants problem in [13].But this paper highly favors the superiorities of PSO and GSA.On the basis of GSA, PSO is merged with it as fine tuning to improve the quality of solutions, which forms GSAPSO method for solving ORPD problem.
Gravitational Search Algorithm (GSA), a heuristic evolutionary optimization algorithm, was proposed by Rashedi et al. in [14].GSA deriving from the thought of Newton's law stands out depending on the flexible and efficient movement characteristic, and naturally, its application is wide.GSAbased Photovoltaic (PV) excitation control strategy is used for single-phase operation of three-phase wind-turbine coupled induction generator [15].GSA is applied to coordinate Power System Stabilizers (PSSs) and Thyristor Controlled Series Capacitor (TCSC) controllers simultaneously, which is demonstrated to achieve good robust performance for damping the low frequency interarea oscillations [16].GSA is proposed to find the optimal solution for optimal power flow problem in a power system [17].
Particle Swarm Optimization (PSO) proposed by Kennedy and Eberhart [18] owns numerous absorbing aspects: simple thought, convenient implementation, high efficiency, powerful search ability, and so on.But what the most prominent feature is belongs to the memory peculiarity during the search process, the memory peculiarity contributing particles to record the global optimum and individual optimum in every generation.The applications of PSO are far more than the authors care to mention; for instance, enhanced PSO approach is applied for optimal scheduling of hydrosystem [19]; robust PID controller tuning based on the constrained PSO is researched in [20]; PSO is used for optimization of acoustic filters [21]; a hybrid Particle Swarm Optimization algorithm is proposed for solving the problem of optimal reactive power dispatch within a wind farm [10].
In this paper, GSAPSO is the combination of GSA and PSO, which not only retains movement characteristic in the search process of GSA but also increases capability of sharing information and memory ability.In this work, GSA, PSO, and GSAPSO have been examined and tested in IEEE14-bus, IEEE30-bus, and IEEE57-bus test systems for the solution of ORPD problem of power system with the objective minimizing total active power transmission losses ( Loss ).The obtained performances of GSA, PSO, and GSAPSO are compared.And the power loss reduction rates of the proposed GSAPSO algorithm are also compared to those of other optimization methods.The simulation results reveal that the proposed GSAPSO approach can obtain a better optimum effect than these compared algorithms and the results' distribution of it is more concentrated than conventional GSA and PSO methods; besides, the proposed GSAPSO can avoid falling into the local optimum.
The rest of this paper is organized as follows: Section 2 introduces the mathematical modeling of ORPD problem.GSAPSO algorithm is described in detail in Section 3. Section 4 presents the calculation process of GSAPSO algorithm for ORPD problem.Some simulation experiments are shown in Section 5, and Section 6 gives the conclusions.

Mathematical Modeling
The mathematical modeling of ORPD problem is composed of two parts: objectives and constraints.The former provided in the paper are to minimize  Loss , and the latter contain equality constraints and inequality constraints [1].

Objective Functions.
The objective to minimize the total active power transmission losses in reactive power optimization is expressed as follows: where  Loss represents the total active power losses in transmission lines;   is the number of network branches;   is the conductance of the th branch which connects bus  and bus ;   and   , respectively, denote the voltage magnitude of the th and th bus;   is the voltage phase between buses  and .

System Constraints.
The aforementioned objective function is subject to the system constraints which include the equality and inequality constraints.

Equality Constraints.
There are two equality constraints describing the active and reactive power balance, which are expressed as follows: where  is the number of the buses connecting with the th bus;  is the number of total buses except for swing bus;  PQ is the number of PQ buses;   and   are the active power of the th generator bus and the th load bus, respectively;   and   are, respectively, the reactive power at the th generator bus and the th load bus;   and   , respectively, represent the real part and imaginary part of   which is the element of the -bus matrix (bus admittance matrix) at the th row and the th column.Equations ( 2) are considered as the termination conditions of calculating the Jacobian matrix in Newton-Raphson load flow calculation.

Inequality Constraints.
The descriptions of inequality constraints are given based on the state variables and control variables, respectively.

(i) Inequality Constraints of Control Variables
(a) The limit for generator bus voltages: where  PV denotes the number of PV buses and   is the voltages at the th generator bus.
(b) The limit for tap positions of transformers: where   is the number of transformers;   is the tap positions of the th transformer, which is a discrete variable.
(c) The limit for amount of reactive compensation: where   is the number of the banks of capacitor or inductor;   denotes the reactive compensation capacity of the th bank of capacitor or inductor.

(ii) Inequality Constraints of State Variables
(a) The limit for voltages at PQ bus: where   is the voltage at the th load bus.(b) The limit for reactive compensation capacity at PV bus: where   is the reactive compensation capacity at the generator .(c) The limit for apparent power of transmission line: where   is the apparent power of the transmission line between buses  and .

Handling of Constraints.
What is worth mentioning is that, during the process of optimization, equality constraints and inequality constraints of control and state variables are satisfied as the following explanations, respectively.
(i) The two equality constraints are satisfied by Newton-Raphson power flow algorithm in load flow calculation.(ii) The generator bus voltages (  ), tap positions of transformers (  ), and amount of reactive compensation (  ) are the control variables which can be selfrestricted according to their limits by the algorithm.(iii) The limits on active power generation at the swing bus ( swing ), voltages at PQ bus (  ), reactive compensation capacity at PV bus (  ), and apparent power of transmission line (  ) are state variables which are restricted by the objective function combining the penalty function.

Formulation.
In short, the ORPD problem can be formulated as a complex nonlinear constrained optimization mathematical model, and a compact expression is given in where  and  denote the vector of control variables and the vector of state variables, respectively; (, ) and (, ), respectively, represent the equality constraints and inequality constraints of system.In this paper,  and  are expressed as follows: = [ 1 , . . .,   PQ ,  1 , . . .,   PV ,  1 , . . .,    ] , (11) where "" denotes transposition.[14].

Description of GSAPSO Algorithm
In GSA, a series of agents are considered as objects and their performances are measured by their masses, and all these objects attract each other by the gravity force, while this force causes a global movement of all objects towards the objects with heavier masses [22].The description of GSA about how to solve the problem is as follows.
Assume there are  agents distributed in space and the position of the th agent is defined as in where    denotes the position of the th agent in the th dimension and  is the dimension of the search space.
The mass of every agent is computed based on the current agents' fitness as follows: where   () and fit  () represent the mass and fitness value of the th agent at iteration ;  best () and  worst (), respectively, denote the best and worst fitness value among the  agents at iteration , which is defined as follows: best () = min In accordance with the law of gravity, the force acting on agent  by agent  is computed as follows: where () is the gravitational constant at iteration ;  represents a small constant which can avoid the denominator equal to zero;   (), defined as   () = ‖  (),   ()‖ 2 , denotes the Euclidian distance between agent  and agent .
We always use   () rather than   () 2 in (15) because of the better performance of   () in most cases based on many simulation experiments.The better performance refers to the lower power losses in this paper.And () is reduced from an initial value with iteration  as follows: where  0 is the initial gravitational constant;  is a constant greater than zero;  is the current number of iterations; and  max represents the maximum number of iterations.
On the basis of (15), the total force acting on agent  can be given as where rand represents random number drawn uniformly on (0, 1);  best denotes the set of the first  agents with the best fitness value and biggest mass, which is a function and reduced with time from the initial value  0 .
Based on the law of motion, the acceleration of the th agent is computed as follows: The updates of velocity and position of agent  at the next iteration are computed as follows: where V   () and    () are the velocity and position of agent  at iteration  in the th dimension.

Memory Characteristic of PSO.
Schools of fish and flocks of birds always find foods, which is attributed to the communication among individuals and the memory ability for individual best direction and global best direction.PSO algorithm simulates the behaviors of animals, whose update of velocity is defined as follows: where  is the inertia weight;  1 and  2 represent the acceleration factors;   best and    , respectively, denote the best position of particle  and the best position in swarm in the th dimension at iteration .

GSAPSO.
Different from PSO, every agent in GSA determines the direction by the total force from other agents but lacks the communication with others so as to miss the memory ability.In this paper, the proposed GSAPSO is an enhanced GSA-based optimization algorithm, combining the memory characteristic of PSO based on the GSA, which is helpful to the agents to move to the global best position.The difference between GSA and GSAPSO is the update modes of velocity and position which are crucial for the artificial intelligence-based algorithms.And the updates of velocity and position in GSAPSO combining the law of gravity and memory characteristic of PSO are defined as follows: It is worth pointing out that the inertia weight impacting velocity in ( 20) is not introduced in ( 21), because GSA can determine the direction by the total force from other agents and only need to reinforce the impact of the memory for the best position of every agent and the best position in all the agents.And the steps of GSAPSO are depicted as follows.
Step 1. Generate the initial population.
Step 2. Compute the fitness of every agent, and record   best and    .
Step 5. Update the velocity and position by using ( 21) and (22).
Step 6 (check stop criterion).Go to the next step if the number of iterations reaches the maximum number of iterations; otherwise go back and continue Step 2.
Step 7. Select the solution with the best fitness as the global best solution.
And the computational flow of the GSAPSO algorithm is shown in Figure 1.

The Calculation Process of GSAPSO Algorithm for ORPD Problem
In the application of GSAPSO for solving the ORPD problem, the fitness value of every agent is the objective function value ( Loss ), and the position of the agent is the solution which is a set of control variables containing the generator bus voltages, tap positions of transformers, and amount of reactive compensation.These control variables are applied in Newton-Raphson load flow calculation to obtain the total active power transmission losses, constraint violations of variables, reactive compensation capacity at PV bus, and so on.The optimization procedures of GSAPSO for ORPD problem are as follows.
Step 1. Set the parameters of power system and those of GSAPSO.
Step 2. Generate the initial population.
The vector of control variables expressed in (10) is as an agent to represent a potential solution for ORPD problem in GSAPSO.The initial population is made up of  agents, and the th particle in the th dimension is generated based on where  is the size of population,  is the dimension of control variables, and  ×  can represent the dimension of the population.Besides, the initial velocity of population is set as a zero matrix with  ×  dimension.
Step 6. Update the velocity and position by using ( 21) and (22), and generate a new population.
Step 7 (check stop criterion).Go to the next step if the number of iterations reaches the maximum number of iterations; otherwise go back and continue Step 2.
Step 8.The solution with the best fitness (the lowest  Loss ) is the global best solution.

Simulation Experiments
In order to verify the effectiveness of the proposed GSAPSO algorithm compared to the traditional GSA and PSO, they have been examined and tested, respectively, in three IEEE test systems to solve the ORPD problem of power system with the objective minimizing total active power transmission losses ( Loss ).The system data of test systems is depicted in Table 1.All the code of the abovementioned algorithms is written by MATLAB R2013b programming language and run on PC with Intel(R) Core 2 Duo CPU E7500 @ 2.93 GHz with 2 GB RAM.

IEEE14-Bus
System.The IEEE14-bus test system is taken from [23], whose single-line diagram is shown in Figure 2. The reactive power limits of generators are given in Table 2,  and Table 3 lists the settings of control variables.This network consists of 20 branches, 3 transformers, and 1 capacitor bank.The capacitor bank is set at bus 9.The 3 transformers are connected to branches 4-7, 4-9, and 5-6.In the 14 buses of this test system, bus 1 is the swing bus, 2, 3, 6, and 8 are regarded as the PV buses, and the remaining 9 are the PQ buses.

IEEE30-Bus
System.The IEEE30-bus test system is taken as the second test system, whose detailed data is given in [24,25].As shown in Figure 3

IEEE57-Bus System.
In order to verify the applicability of the proposed GSAPSO algorithm for the larger scale system, it has been applied in the IEEE57-bus system whose detailed data is given in [23].As shown in Figure 4, the network has 80 branches, 17 transformers, and 3 capacitor banks which are set at buses 8, 25, and 53.The  the swing bus, 2, 3, 6, 8, 9, and 12 are taken as PV buses, and the remaining 50 are PQ buses.The modified reactive power limits of generators are listed in Table 6, and the settings of control variables are seen in Table 7.

Parameter Settings.
Based on the reduplicative experiments, we find that the smaller population size of particles cannot guarantee the diversity of particle, and the larger population size of particles increases the computational complexity and total computing time.Therefore, by synthesizing each kind of factor, the parameter settings of different algorithms for test systems are listed in Table 8.  9-15.Besides, the power loss reduction rates ( save ) of the compared algorithms are also seen in Tables 9-11.
As seen in Figures 5-7, the proposed GSAPSO can obtain better searching effect than PSO and GSA.For example, in IEEE14-bus test system, PSO, GSA, and GSAPSO get converged, respectively, at about the 20th, 120th, and 300th iteration, which indicates PSO and GSA easily get into the local optimum.And in test systems, compared with PSO and GSA, the proposed GSAPSO can find the solution with lower power losses.In Tables 9-11, the power loss reduction rates of GSAPSO in IEEE14-bus, IEEE30-bus, and IEEE57-bus test systems are 7.05%, 17.69%, and 19.51%, respectively, which denotes that the proposed GSAPSO method has a better optimum effect for ORPD problem than other modern artificial intelligence-based techniques including conventional  PSO and GSA algorithms.Furthermore, the CPU times shown in Table 12 make it clear that the run times of three different algorithms are not much different for these test systems, which shows that the effectiveness of the proposed method does not mean the low efficiency.Owning to the higher complexity of IEEE57-bus test system, its CPU time is  10, it is obvious that the distribution of the results of GSAPSO was more concentrated in a smaller range than that of PSO and GSA, which manifests that the result uniformity of the proposed GSAPSO is better than GSA and PSO.

Conclusions
GSAPSO is a novel heuristic stochastic optimization algorithm, which combined the memory characteristic of PSO based on the traditional GSA algorithm.The proposed GSAPSO, PSO, and GSA algorithms have been successfully introduced to solve the ORPD problem based on IEEE14bus, IEEE30-bus, and IEEE57-bus test power systems in this paper.The simulation results demonstrate the potential and effectiveness of the proposed GSAPSO approach for solving  ORPD problem of power system, especially in the larger IEEE57-bus test system.The proposed GSAPSO method can obtain bigger power loss reduction rates than the compared algorithms, and its results' distribution is more concentrated compared to that of GSA and PSO under the condition that these algorithms take almost the same computing time.Besides, the proposed GSAPSO can avoid falling into the local optimum.

Figure 4 :
Figure 4: The single-line diagram of IEEE57-bus test system.

Figure 5 :
Figure 5: The average convergence curves of different algorithms for IEEE14-bus test system.

Figure 6 :
Figure 6: The average convergence curves of different algorithms for IEEE30-bus test system.

Figure 7 :Figure 8 :
Figure 7: The average convergence curves of different algorithms for IEEE57-bus test system.

Figure 9 :Figure 10 :
Figure 9: The distribution of the results of different algorithms for IEEE30-bus test system.
3.1.Brief Introduction of GSA.GSA proposed by Rashedi et al. in 2009 is a newly developed stochastic search algorithm which is inspired by the law of gravity and law of motion 3. Compute the objective function value ( Loss ) of every agent by Newton-Raphson load flow calculation, and compare those of agents, and then record   best and   Update ,  best ,  worst , and  of the population based on (

Table 1 :
System data of test systems.

Table 2 :
Limits of reactive power of generators in IEEE14-bus test system.Bus 1 is the swing bus, 2, 5, 8, 11, and 13 are taken as PV buses, and the remaining 24 are the PQ buses.The reactive power limits of generators are seen in Table4, and Table5lists the settings of control variables.

Table 3 :
Settings of control variables in IEEE14-bus test system.

Table 4 :
Limits of reactive power of generators in IEEE30-bus test system.

Table 5 :
Settings of control variables in IEEE30-bus test system.

Table 6 :
Limits of reactive power of generators in IEEE57-bus test system.

Table 7 :
Settings of control variables in IEEE57-bus test system.

Table 9 :
Comparison of power losses of different algorithms for IEEE14-bus test systems.

Table 10 :
Comparison of power losses of different algorithms for IEEE30-bus test systems.

Table 11 :
Comparison of power losses of different algorithms for IEEE57-bus test systems.

Table 12 :
CPU time of different algorithms for test system (s).

Table 13 :
Best solutions of different algorithms for IEEE14-bus test system.

Table 14 :
Best solutions of different algorithms for IEEE30-bus test system.

Table 15 :
Best solutions of different algorithms for IEEE57-bus test system.