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 TODO:  Complete missing sections

Overview

The purpose of developing Particle Swam Optimization (PSO) in GridLAB-D is to determine the tuning parameters of Proportional Integral Derivative (PID) controller based price control mechanisms with energy resources on a power system. This model has a lot of flexibility and can easily be applied for single and multi-object optimization problems.

General Description

PSO is a population based stochastic optimization technique [1], [2]. It is inspired by social behavior of bird flocking, fish schooling and swarming theory. The implementation of PSO paradigms is not computationally intensive in terms of memory requirements and speed. PSO can be used to solve many of the same kinds of problems as genetic algorithms. In PSO, the potential solutions are called particles and they move through the search-space by following current optimum particles. Each particle keeps track of its coordinates in the search-space. The particle with the best solution (fitness) is stored and this value is called pbest. PSO also keeps track of the overall best value obtained so far by any particle in the population. This is called gbest. In each step, PSO calculates particle velocities and updates their positions. This process continues until the number of iterations reaches the maximum value or a minimum error criterion is satisfied. The final output indicates the fitness of the given candidate solution.

Modeling Assumptions

PSO

Interfacing overview

Equations

Table 1: Equations
Equation Number
[math]\begin{align} Vel_{p,d}^{(j+1)} &= Vel_{p,d}^{(j)}+C_1rand()\left(pBest_{p,d}^{(j)}-present_{p,d}^{(j)}\right)+C_2rand()\left(gBest_{p,d}^{(j)}-present_{p,d}^{(j)}\right)\end{align}[/math] 1.1
[math]\begin{align} present_{p,d}^{(j+1)} &= present_{p,d}^{(j)}+Vel_{p,d}^{(j)}\end{align}[/math] 1.2



Table 2: Equation Notation
Variable Definition
[math]p[/math] Population size
[math]d[/math] Dimension
[math]j[/math] Iteration count
[math]Vel[/math] Particle velocity
[math]pBest[/math] Personal best
[math]gBest[/math] Global best
[math]present[/math] Current value
[math]rand()[/math] Standard uniform distribution (uniformly distributed random numbers) on the open interval (0,1)
[math]C_1, C_2[/math] Positive constants


PSO published inputs

Table 3 : PSO inputs
Variable Type Units Value (default) Allowable values Definition
number_agents double NA 10 Value > 1 The number of agents (particles) in a swarm.
number_unknowns double NA 3 Value >= 1 This is the dimension of solution space. It is equal to the number of unknowns in a variable to be optimized.
maximum_iterations double NA 250 Value > 1 number of iteration to execute the optimization.
maximum_weight double NA 0.9 0< Value < 1 The upper limit of the time varying inertia weight.
minimum_weight double NA 0.4 0< Value < 1 The lower limit of the time varying inertia weight.
maximum_velocity double NA 20 0< Value < 100 It is the maximum change that particle can take during one iteration.
learning_factor_C1 double NA 2 0< Value < 2 Hookes's coefficient.
learning_factor_C2 double NA 2 0< Value < 2 Hookes's coefficient.
minimum_error double NA 0.001 0< Value < 0.1 This is used as one of the stop conditions.
update_period double sec 900 Value > 1 How often should PSO run to obtain fitness of the given candidate solution.
cost_function TBD

PSO published outputs

Table 4: PSO outputs
Variable Type Units Definition
gbest double NA Best value obtained so far

PSO timing

PSO finds the fitness of the given candidate solution for every user defined time period. The values of the decision variables are set in presync (). The objective function value is calculated in sync () and it is compared with the best solution so far in postsync(). If the current solution is better than the best solution so far, then keep the current solution as the best else do not update the best. Update particle velocity and position with the current best for each particle and update the fitness of the function for each iteration. This process repeats until it reaches the maximum iteration limit.

PSO passes

For obtaining the fitness of the given candidate solution, PSO proceeds as:

 1.	Solve the fitness of the objective function for each particle
 2.	Find the best fitness for each particle (pBest)
 3.	Find the best fitness among all the particles (gBest)
 4.	Solve for new velocity and position for each particle using 1.1 and 1.2
 5.	Update the fitness of each particle with the new particle position. This procedure continues until it reaches the maximum iteration limit.

Pseudo code for basic PSO

* Step1: Initialize the particles and their velocity components (init ())
 For each particle:
    For all dimensions:
        particle position (particle, dimension) = rand
        particle velocity (particle, dimension) = rand
        pbest (particle, dimension) = particle position (particle, dimension)
    End
 End
* Step 2: Initialize the pbest fitness array (init ())
 For each particle
     pbest fitness (particle) = -1000
 End
* Step 3: Main PSO routine (presync ())
  For each iteration(as long as number of iterations are less than maximum iterations)
      * Step 3A: Find the fitness of each particle
                 For each particle
                     value = particle position (particle)
                     solution = f (value)(declaration of an objective function) 
                    (the calculation of the objective function takes place in sync (inrinsic sync()) using runtime classes)
                     If Solution is NOT zero
                        current fitness = 1/absolute (solution)
                     Else
                        current fitness = 1000
                     End
                 End
       * Step 3B: Decide pbest among all the particles
                   For each particle
                       If current fitness (particle) is greater than pbest fitness (particle)
              		   pbest fitness (particle) = current fitness (particle)
                          For all dimensions
                              pbest (particle, dimension) = particle position (particle, dimension)
                          End
                       End
                   End        
       * Step 3C: Decide gbest among all the particles
                   [gbest, gbest index] = maximum of current fitness
                       For all dimensions
                           gbest (dimension) = particle position (gbest index, dimension)
                       End
       * Step 3D: Update the position and velocity components
                      For each particle
                          For all dimensions
                              present (dimension) = particle position (particle, dimension)
                         End
                         For all dimensions
                             calculate particle velocity using 1.1
                             calculate particle position using 1.2
                         End
                      End
 End 
  * Step 4: Publish outputs
            gbest

Testing and Validation

……Testing and validation is still being finalized and will be updated shortly. Please try back later …….

References

1.	Eberhart, R. C. and Kennedy, J. A new optimizer using particle swarm theory. “Proceedings of the Sixth International Symposium on Micromachine and Human
        Science”, Nagoya, Japan. pp. 39-43, 1995 
2.	Kennedy, J. and Eberhart, R. C. Particle swarm optimization. “Proceedings of IEEE International Conference on Neural Networks”, Piscataway, NJ. pp.1942-
        1948,1995

See also