General overview and concept of PSO

The PSO is a population-based optimisation randomized search technique introduced in 1995 [54]. This optimisation method is inspired by social interaction principles. It updates the population of candidate solutions called a swarm. Each candidate solution is called a particle or agen that is used in a search space to achieve an objective function (e.g. searching for optimal locations of protective devices). The PSO algorithm begins with a randomly generated population and each point in the population has a velocity vector as shown in Figure 2-13. Consider a movement of a single particle in a two dimensional search space. Three velocity components contribute to move the particle towards the global best position at time steps t and t+1 respectively. Each particle is moving with an associated velocity to each possible point in space. The objective function at each point in the population is evaluated. Referring to the valuated population, the new population of points is created together with a new set of velocities. Each particle keeps track of the best position based on the objective function. Under this condition, the best position of an individual particle is called “personal best” (pbest) and the overall best position among all positions encountered by the entire population is named as a “global best” (gbest). The particles keep interacting with each other and updating their velocities according to their pbest as well as gbest. The velocity of each

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particle is changed at each time step towards a combination of its pbest and gbest locations as shown in Figure 2-13 (b).

Cognitive velocity X2 X1 Best position of neighbours, Gbest Personal best position, Pt best,i New position, Xit+1 Social velocity Initial position, Xit Inertia velocity, Vit New velocity, Vit+1

(a) Time step t

X2 X1 (b) Time step t+1 Xit Xit+1 Pt+1 best,i Gbest Xit+2 Vit+1

Figure 2-13: Velocity and position update for a particle in a two-dimensional search space [97]

Inertia velocity for (t+1) illustrated in Figure 2-13(b) calculated depend on inertia velocity at t and change upon each iteration. The velocity is weighted by a random term, with separate random numbers being generated for velocities towards pbest and gbest locations. Thus, the particles are drawn both to their own personal best positions as well as to the best position of the entire swarm as shown in Figure 2-14.

Optimum position _Optimum position Gbest Gbest X2 X1 X2 X1

(a) at time t = 0 (b) at time t = 1

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The optimum position is denoted as “star” and given Gbest for more than one particle. At t = 0 in Figure 2-14 (a), the velocity and direction of component for all of particles is zero, and the particles are only attracted towards the best position by the social component. After the first iteration, all the particles move to new positions with a new global best positions as shown in Figure 2-14(b). The stopping criterion is met when the algorithm reaches a certain number of iterations or by achieving a certain objective function value. The values of velocities and locations updated for each particles by using Equations (2-3) and (2-4) respectively.

𝑉𝑖(𝑖𝑡+1)= [𝑊(𝑖𝑡)+ (𝐶1× 𝑅𝑛𝑑(0,1)× (𝑝𝑏𝑒𝑠𝑡(𝑖𝑡)− 𝑋𝑖(𝑖𝑡)))

+ (𝐶2× 𝑅𝑛𝑑(0,1)× (𝑔𝑏𝑒𝑠𝑡(𝑖𝑡)− 𝑋𝑖(𝑖𝑡)))] (2-3) 𝑋_{𝑖(𝑖𝑡+1)}= 𝑋_{𝑖(𝑖𝑡)}+ 𝑉_{𝑖(𝑖𝑡+1)} (2-4)

where:

 it is the iteration number  W is the inertia weight  i is the particle’s index  Xi is the position of particle i

 Vi is the velocity of particle i

 C1 is the acceleration constant for the cognitive component

 Rnd is a random value between 0 and 1

 C2 is the acceleration constant for the social component

 pbest is the best fitness of all visited locations of particle i

 gbest is the global optimum location of the particle i through iteration The weighting function (W), is a weighting factor related to the velocity of the particle during the previous iteration. It controls the amount of the previous velocity that particles take in the next iteration. This weighting function is obtained as:

52 𝑊(𝑖𝑡) = 𝑊_𝑚𝑎𝑥 −(𝑊𝑚𝑎𝑥− 𝑊𝑚𝑖𝑛) × 𝑖𝑡

𝑖𝑡_𝑚𝑎𝑥 (2-5)

 it is the current iteration number  itmax is the maximum iteration number

 Wmax is the maximum weight (0.9)

 Wmin is the minimum weight (0.4)

Reducing the weight can give better performance [98]. According to[99], [100] and [101], faster convergence is achieved if the value of the weight is between 0.9 to 0.4 [102]. So, this range of value is used in the research of this thesis.

The second parameter that has a significant impact on speed of convergence are the acceleration coefficients ( 𝐶1 𝑎𝑛𝑑 𝐶2). These values have an effect on the step size

of a particle in a single iteration. The values of C1 and C2 regulate the maximum step

size of a particle in the direction of pbest and gbest [102] respectively as shown in (2- 2). As mentioned in [100], a time varying acceleration coefficients automation strategy for both C1 and C2 linearly decreased time during the course of a run. Simulation results

show that fixed acceleration coefficients of 2.0 generated better solutions.

To avoid the particle velocity from escaping the search space, the equation of

Vmax (2-6) is used. The search space is defined by the bounds [Xmin, Xmax]. The value of Xmin and Xmax is the range of number of substations on a network. For example, if the

total number of substations in a network is equal to 10,Xmin = 1 and Xmax = 10. The value

of Vmax is typically set to:

𝑉𝑚𝑎𝑥 = 𝑘 × (𝑋𝑚𝑎𝑥− 𝑋𝑚𝑖𝑛) (2-6)

where;

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The PSO is usually performed several times with different values of key parameters of k to achieve satisfactory results. The value of k may be valued from 0.1 until 1.0. A suitable k value is evaluated from 0.1 to 1.0 (increased by 0.1 for each step) and suitable k is selected when the simulation produces faster results with a correct answer.

In [103], a modification was made to the updating of the position of the particle by changing the direction for reaching the search space boundary to improve the speed of convergence. The direction of the particle modified to keeps the particle inside its range when the velocity takes the particle out of its boundary [Xmin, Xmax]. So the new

position of the particle will be updated based on Equation (2-7) instead of Equation (2- 4).

𝑋_{𝑖(𝑖𝑡+1)} = 𝑋_{𝑖(𝑖𝑡)}− 𝑉_{𝑖(𝑖𝑡+1)} (2-7) If the optimal solution is close to the boundary, the ‘α’ factor helps to reach the solution faster by using Equation (2-8) where α is an optimally chosen number between 0 and 1. A value of α chosen for this study was selected after several trials to be matched with different topologies of the network like the way of previous parameter selected (k).

𝑋𝑖(𝑖𝑡+1)= 𝑋𝑖(𝑖𝑡)− 𝛼𝑉𝑖(𝑖𝑡+1) (2-8)

The advantages of using the PSO in handling the distribution system reconfiguration problem is manifold according to [104]. The PSO-based method achieves fast convergence with few required tunes parameters for suitable solutions. This statement is also supported by [105] where both PSO and GA were applied in searching of optimisation in the feeder reconfiguration problem. It was demonstrated that the PSO has faster convergence than the GA, and a better global search ability compared to GA, according to [106].

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