Essay: Hybrid GA and PSO Approach for Transmission Expansion Planning

Abstract: Metaheuristic optimization algorithms have become popular option for intricate problems which are difficult to solve by traditional methods. Transmission expansion planning (TEP) is one of the important challenging and decision-making activities in electric utilities. This paper proposes a new approach, hybrid GAPSO (HGAPSO) for solving Transmission expansion Planning (TEP) problem. Problem of premature convergence in particle swarm optimization (PSO) and low convergence speed in genetic algorithm (GA) mitigate to the hybridization of both techniques. In this paper, modification strategies have been used in GA and PSO algorithms to achieve global convergence and the faster searching capability in HGAPSO. This improved algorithm HGAPSO described and implemented in a MATLAB environment has been compared with GAs and PSO algorithms for transmission expansion planning problem.
Keywords: Metaheuristic, GA, HGAPSO, PSO, TEP.
1. Introduction
Hybrid algorithms are a growing area of intelligent systems research, which combines the desirable properties of different approaches to mitigate their individual weaknesses. Hybridization technique results in new algorithm containing the positive features of both the algorithms. TEP problem is a complex, large-scale, difficult, and nonlinear. Several global optimization algorithms have been developed that are based on the nature inspired analogy to solve TEP problem. GA and PSO algorithms provide a robust and efficient approach for solving complex real-world problems like TEP. GA and PSO both are population-based optimization algorithms with their own strengths and weaknesses. GA method is more suitable for solving multiple objectives problems and is quite robust, but its convergence speed is slow, while the strength of the PSO algorithm, on the other hand, is its simple coding and fast convergence. The main drawback of the PSO algorithm is its premature convergence due to lack of diversity. Therefore, a better algorithm would be the one which incorporates the strengths of these two algorithms and overcomes the weaknesses of both i.e. an algorithm that has fast convergence as well as high diversity. A hybrid approach, coupling GA and PSO algorithm, was thereby proposed because combining the two search techniques seems to be a feasible approach.
Mathematical models based on classical optimization techniques, such as Linear programming [1], [2] and Bender decomposition [3]'[5] and branch and bound methods [6], [7], have been used to solve the TNEP problem. Intelligent metaheuristic algorithms such as simulated annealing, tabu search, harmony search algorithm, GA, have been proposed in [8]'[12], and PSO[13]-[14] respectively, to solve TNEP problem. The Hybridization of various techniques is mentioned in [15]-[16]. Hybrid neuro and fuzzy approach [17] for TEP have been introduced in past decades.
The paper focus on HGAPSO technique to solve TEP problem. The article has been divided into four subsections. In the next section, formulation of TEP problem has been described. The subsequent section gives the overview of the GA, PSO and HGAPSO algorithm with the implementation to TEP problem and in the last section, results obtained by the application of the HGAPSO algorithm for TEP are discussed, and its comparison with the results obtained from GAs and PSO algorithms are presented.
2. Mathematical Formulation
In this paper, static TEP problem can be formulated by using a classical DC power flow model which is a nonlinear mixed-integer problem with high complexity, especially for large-scale realistic transmission networks. The objective function is formulated as follows:
‘ min O”= ‘_(ij”)’c_ij n_ij (1)
where O,’ c’_(ij )and ‘ n’_ij represent, respectively, investment cost of the transmission, circuit cost, which is a candidate for addition to the branch i’j and the number of circuits added to the branch i’j. Here ‘ is the set of all candidate branches for expansion. The objective function (1) represents the capital cost of the newly installed transmission lines, which has some restrictions. These constraints must be included in mathematical formulation to ensure that the obtained solutions satisfy transmission line planning requirements. These constraints can be formulated in the following (2) ‘ (7).
2.1 DC power flow node balance constraint:
The conservation of power at each node is represented by this linear inequality constraint:
P=d+B?? (2)
Where P, d and B are respectively, the real power generation vector in the existing power plants, the real load demand vector in all network nodes and the susceptance matrix of the existing and added lines in the network. Here ?? is the bus voltage phase angle vector.
2.2 Power flow limit on transmission lines constraint:
In order to limit the power flow for each path, the inequality constraint is as follows:
f_ij'(n_ij^0+n_ij ) f_ij^max (3)
In the DC power flow model, each element of the branch power flow in constraint (3) can be calculated by using (4):
f_ij=((n_ij^0+nij) )/x_ij *(??_i- ??_j) (4) Where f_ij, f_ij^max,n_ij, and n_ij^0, x_ij represent, respectively, the total branch power flow in the branch i’j, the maximum power flow in the branch i’j, the number of circuits which is to be added to the i-j branch, the number of circuits in the original base system and reactance in the i-j branch. Here ??_iand ??_j are the voltage phase angle of the terminal at ith and jth bus respectively.
2.3 Power generation limit constraint:
The limit of power generation has to be included in the TEP constraints and is represented as follows:
g_imin’g_i’g_imax (5)
where’ g’_i,g_iminand g_imax are the real power generation at node i.e. the lower and upper real power generation limit at node i respectively.
2.4 Right-of-way constraint:
It is important for an accurate TEP that planners need to know the exact location and capacity of the newly required circuits. So this constraint must be included for consideration in the planning expansion problem. In Mathematical form, this constraint defines the new circuit location and the maximum number of circuits that can be installed in a specified location. It can be represented as follows:
0’n_ij’n_ijmax (6)
where n_ijand n_ijmax represent the total integer number of circuits which is to be added to the i’j branch and the maximum number of circuits that can be added to the i’j branch respectively.
2.5 Bus voltage phase angle limit constraint:
The bus voltage magnitude is not a factor in this analysis since a DC power flow model is used. The voltage phase angle is included as a TEP constraint and the calculated phase angle should be less than the predefined maximum phase angle:
??_ijcal’??_ijmax (7)
2.6 Fitness Function:
Fitness Function F for the TEP problem is as follows:
F=1/(O+pf*C) (8)
Where, C=’_(i=1)^k’V_i
k is number of constraints. V_i is violation of ith constraint in percentage. pf is the penalty factor.
3. Overview of GA, PSO & HGAPSO Algorithms
This section gives overview of GA, PSO and HGAPSO algorithms and their implementation to TEP problem.
3.1 Genetic Algorithm
GA belongs to a family of computational models inspired by evolution [19].These algorithms encode a potential solution to a specific problem on a simple chromosome-like data structure and apply recombination and mutation operators to these structures so as to preserve critical information. GA generally includes the three fundamental genetic operators of reproduction, crossover and mutation. These operators conduct the chromosomes toward better fitness. The goodness of a solution is typically defined with respect to the current population. The genetic algorithm can be viewed as two stage process. It starts with the current population. Selection is applied to the current population to create an intermediate population. Then recombination and mutation are applied to the intermediate population to create the next population. The process of going from the current population to the next population constitutes one generation in the execution of a GA. Crossover is the main genetic operator that allows information to be exchanged between individuals in the population. Mutation operator is to prevent the permanent loss of any particular bit values (genes), as without mutation there is no possibility of re-introducing a bit value that is missing from the population.
3.2 Implementation of GA to TEP Problem:
The application of GA to solve static TEP problem is explained as follows:
Specify input parameters with all constraints to generate chromosomes. Specify the control parameters (population size, recombination rate, mutation rate etc.).
Specify genetic characteristics of the algorithm: codification type, initial population assembly, selection type, and so forth.
Initialize population (Line to be added) randomly satisfying all constraints and evaluate it to become the current population.
Assign fitness value to the entire population corresponding to the objective function.
Implement a selection to choose only two generating solutions. Selection operator in this analysis used is tournament selection.
Implement the recombination and preserve an offspring.
Implement the mutation of the preserved offspring.
Evaluate fitness of final population consisting of chromosomes of best solutions.
Check generation count, if it reaches its maximum then go to step 10, else go to step 5.
3.3 Particle swarm optimization
PSO incorporates swarming behaviors observed in flocks of birds, schools of fish, or swarms of bees, and even human social behavior, from which the idea is emerged [20].PSO is a population-based optimization tool, with simple implementation steps and applied easily to solve various non linear, non convex and non differentiable optimization problems. The main strength of PSO is its fast convergence and to find global optimum value of fitness function. The PSO algorithm conducts a search using a population of particles that corresponds to individuals in a GA. In PSO, the position of each agent is represented in X’Y plane with position (s_x,s_y),vx (velocity along X-axis), and vy (velocity along Y-axis). Modification of the agent position is realized by the position and velocity information. Bird blocking optimizes a certain objective function. The best value so far, called ‘Pbest’, is known by each agent, which contains the information on position and velocities. This information is the analogy of personal experience of each agent. Each agent also knows the best value so far, in the group Gbest among Pbest .This information is the analogy of knowledge, how the other neighboring agent shave performed. Each agent tries to modify its position by considering current positions (s_x, s_y), current velocities (vx, v_y), the individual intelligence (Pbest), and the group intelligence (Gbest).
The following equations are utilized, in computing the position and velocities, in the X’Y plane:
v_(ik+1)= ”?v’_ik+C_(1 )??rand_1??(P_besti-s_ik )+C_2??rand_2 ?? (G_best-s_ik) (9)
s_(ik+1)=s_ik+v_(ik+1) (10)
where v_(ik+1) is the velocity of (k+1)th iteration of ith individual, v_ik is the velocity of kth iteration of ith individual, ?? is the inertial weight, C_(1 ), C_2 are the positive constants, having values [0, 2], rand_1, rand_2 are the random numbers selected between 0 and 1, P_besti is the best position of the ith individual, G_best is the best position among the individual (group best) and s_ik is the position of ith individual at kth iteration.
The velocity of each agent is modified according to (9) and the position is modified according to (10). The weighting factor ?? is modified using (11) to enable quick convergence:
?? =??_max-(??_max-??_min)/iter_max ??iter (11)
??_max is the initial weight, ??_min is the final weight, iter is the current iteration number and iter_max is the maximum iteration number.
3.3 Implementation of PSO to TEP Problem
This section provides application of PSO algorithm to solve Static TEP problem as follows:
Define input parameters with all constraints for the swarm.
Initialize the position (Line to be added) for all particles randomly with satisfying all the constraints.
Calculate the fitness value (cost) of each particle in the swarm using equation (8).
Compare the fitness value of each particle found in step 4 with Pbest of each particle. Update Pbest of a particle if its fitness is greater than its Pbest.
Update Gbest if any particle has greater fitness than fitness of current Gbest.
Update the inertia weight ”?? by using (11).
Modify the velocity of each particle by (9).
Modify the position of each particle by using (10) with the updated velocity in step 7.
Check iteration counter, if it reaches its maximum then go to step 10, else go to step 3.
The swarm that generates the latest Gbest in step 5 is the optimal value
3.4 Hybrid PSO with GA
The drawback of PSO is that it suffers from premature convergence and the local optima trapping. In addition to this, PSO performance is problem-dependent. GA has so many operators and takes large computational time as compared to PSO. To overcome the limitations of GA and PSO. The GA-PSO algorithm was proposed by Kao and Zahara [21].The basis behind this is that such a hybrid approach is expected to have merits of PSO with those of GA. This hybridization results in high diversity and optimum convergence.
There are three different hybrid approaches which are as follows:
PSO-GA (Type 1): The gbest particle position does not change its position over some designated time steps; the crossover operation is performed on gbest particle with chromosome of GA. In this model both PSO and GA are run in parallel.
PSO-GA (Type 2): The stagnated pbest particles are change their positions by mutation operator of GA
PSO-GA (Type 3): In this model the initial population of PSO is assigned by solution of GA. The total numbers of iterations are equally shared by GA and PSO. First half of the iterations are run by GA and the solutions are given as initial population of PSO. Remaining iterations are run by PSO.
In this paper, type 2 and type 3 hybrid approaches is used and is applied to solve the TEP problem. First, multiple solutions are generated randomly as initial population and objective function values are evaluated for each solution. After the evaluation is done, the population is divided into two subpopulations. One of these subpopulations is updated by the GA operation, while the other is updated by the PSO operation. New solutions created by each operation are combined in the next generation, and non-dominated solutions in the combined population are archived. The archive data is shared between the GA and PSO, i.e., non-dominated solutions created by the PSO can be used as parents in GA, while non-dominated solutions created by GA can be used as global guides in PSO. Flowchart of HGAPSO [22] is shown in figure1.
Fig.1 Flowchart of HGAPSO algorithm
4. Results and Discussion
Static TEP problem is solved for three test systems Garver’s 6 bus system, IEEE 14 Bus system and IEEE 24 bus system by applying proposed hybrid algorithm and is implemented in Matlab 7.9. All the necessary data of the test systems can be found in [23]. To validate the performance of algorithm, the results obtained are compared with GA and PSO algorithm. The best results obtained for three test system by proposed algorithm are after 20 trial runs and 70 iterations. Penalty factor in first two test systems is taken as 2. Fig. 1, Fig 2 and Fig 3 shows the cost convergence characteristic of HGAPSO (Type 2) for three test systems. For Garver’s 6 bus system and IEEE 14 bus system, GA pop size and swarm size is same i.e. 50 and 100, Problem dimension is 8
Fig. 2 Cost convergence characteristic of HGAPSO Fig. 3 Cost convergence characteristic of HGAPSO
for Garver’s 6 bus system for IEEE 14 bus system
and Mutation rate and crossover rate are 0.1 and 0.8 respectively. Parameters of PSO such as c1 and c2 are 2, ??_min and ??_max are taken as 0.4 and 0.9 for both above mentioned systems. Similarly for IEEE 24 bus system, GA pop size and swarm size is 100, problem dimension is 20 and all other parameters are same as that of other two systems. Pentalty factor for this test system is taken as 0.7.Maximum number of iterations in each case are 70.
Fig. 4 Cost convergence characteristic of Fig. 5 Convergence characteristic of GA, PSO
HGAPSO for IEEE 24 bus system and HGAPSO for Garver’s 6 bus system
Table 1. Comparison of GA, PSO and HGAPSO
Algorithm Running time
Best cost
GA
PSO
HGAPSO
5. Conclusion
PSO is a powerful optimization technique that has been applied to wide range of optimization problems. Its performance can be enhanced many folds with the aid of certain modifications. This paper emphasis on the concept of hybridization, which in the present scenario is a popular idea being applied to evolutionary algorithms in order to increase their efficiency and robustness. In this paper we present a brief review on the algorithms which is to be hybrised in which PSO is one of the main algorithms. PSO, which is stochastic in nature and makes use of the memory of each particles as well as the knowledge gained by the swarm as a whole, has been proved to be powerful in solving many optimization problems. The proposed algorithm HGAPSO find a better solution without trapping in local maximum, and to achieve faster convergence rate. This is because when the PSO particles stagnate, GA diversifies the particle position even though the solution is worse. In HGAPSO particle movement uses randomness in its search. Hence, it is a kind of stochastic optimization algorithm that can search a complicated and uncertain area. This makes HGAPSO more flexible and robust. Unlike standard PSO, HGAPSO is more reliable in giving better quality solutions with reasonable computational time, since the hybrid strategy avoids premature convergence of the search process to local optima and provides better exploration of the search process.
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