Proposed control design method 21 System uncertainties

System nonlinear characteristics, variations of system configuration due to unpredictable disturbances, loading conditions etc., cause various uncertainties in the power system. A controller which is designed without considering system uncertainties in the system modeling, the robustness of the controller against system uncertainties can not be guaranteed. As a result, the controller may fail to operate and lose stabilizing effect under various operating conditions. To enhance the robustness of power system damping controller against system uncertainties, the inverse additive perturbation (Gu et.al. 2005) is applied to represent all possible unstructured system uncertainties. The concept of enhancement of robust stability margin is used to formulate the optimization problem of controller parameters.

The feedback control system with inverse additive perturbation is shown in Fig.1. G is the nominal plant. K is the designed controller. For unstructured system uncertainties such as various generating and loading conditions, variation of system parameters and

Fig. 1. Feedback system with inverse additive perturbation.

nonlinearities etc., they are represented by AA which is the additive uncertainty model. Based on the small gain theorem, for a stable additive uncertainty AA , the system is stable if then,

The right hand side of equation (2) implies the size of system uncertainties or the robust stability margin against system uncertainties. By minimizing 11Gj(1 - GK )|| , the robust stability margin of the closed-loop system is a maximum or near maximum.

2.2 Implementation 2.2.1 Objective function

To optimize the stabilizer parameters, an inverse additive perturbation based-objective function is considered. The objective function is formulated to minimize the infinite norm of ||G/(1 - GK)|| . Therefore, the robust stability margin of the closed-loop system will increase to achieve near optimum and the robust stability of the power system will be improved. As a result, the objective function can be defined as

It is clear that the objective function will identify the minimum value of ||G/(1 - GK)|| for nominal operating conditions considered in the design process.

2.2.2 Optimization problem

In this study, the problem constraints are the controller parameters bounds. In addition to enhance the robust stability, another objective is to increase the damping ratio and place the closed-loop eigenvalues of hybrid wind-diesel power system in a D-shape region (Abdel-Magid et.al. 1999). the conditions will place the system closed-loop eigenvalues in the D-shape region characterized by Z - Zspec and c < crspec as shown in Fig. 2. Therefore, the design problem can be formulated as the following optimization problem.

Minimize

Fig. 2. D-shape region in the s-plane where a < cr and Ç >Çs

Subject to

mm _ max where Ç and Çsvec are the actual and desired damping ratio of the dominant mode, respectively; a and <rsvec are the actual and desired real part, respectively; Kmax and Kmin are the maximum and minimum controller gains, respectively; Tmax and Tmin are the maximum and minimum time constants, respectively. This optimization problem is solved by GA (GAOT, 2005) to search the controller parameters.

2.3 Genetic algorithm 2.3.1 Overview

GA is a type of meta-heuristic search and optimization algorithms inspired by Darwin's principle of natural selection. GA is used to try and solving search problems or optimize existing solutions to a certain problem by using methods based on biological evolution. It has many applications in certain types of problems that yield better results than the common used methods.

According to Goldberg (Goldberg,1989), GA is different from other optimization and search procedures in four ways:

1. GA searches a population of points in parallel, not a single point.

2. GA does not require derivative information or other auxiliary knowledge; only the objective function and corresponding fitness levels influence the directions of search.

3. GA uses probabilistic transition rules, not deterministic ones.

4. GA works on an encoding of the parameter set rather than the parameter set itself (except in where real-valued individuals are used).

It is important to note that the GA provides a number of potential solutions to a given problem and the choice of final solution is left to the user.

2.3.2 GA algorithm

A. Representation of Individual.

Individual representation scheme determines how the problem is structured in the GA and also determines the genetic operators that are used. Each individual is made up of a sequence of genes. Various types of representations of an individual are binary digits, floating point numbers, integers, real values, matrices, etc. Generally, natural representations are more efficient and produce better solutions. Encoding is used to transform the real problem to binary coding problem which the GA can be applied.

B. GA Operators.

The basic search mechanism of the GA is provided by the genetic operators. There are two basic types of operators: crossover and mutation. These operators are used to produce new solutions based on existing solutions in the population. Crossover takes two individuals to be parents and produces two new individuals while mutation alters one individual to produce a single new solution (S. Panda,2009).

In crossover operator, individuals are paired for mating and by mixing their strings new individuals are created. This process is depicted in Fig. 3.

Parent 1

11010

01100

Parent 2

10110

11011

Child 1

1 1010

11011

Child 2

10110

01100

Fig. 3. Crossover operator

In natural evolution, mutation is a random process where one point of individual is replaced by another to produce a new individual structure. The effect of mutation on a binary string is illustrated in Fig. 4 for a 10-bit chromosome and a mutation point of 5 in the binary string. Here, binary mutation flips the value of the bit at the loci selected to be the mutation point (Andrew C et.al).

Parent 1 11010 01100 Child 1 11010 11100

Fig. 4. Mutation operator

C. Selection for Reproduction

To produce successive generations, selection of individuals plays a very significant role in a GA. The selection function determines which of the individuals will survive and move on to the next generation. A probabilistic selection is performed based upon the individual's fitness such that the superior individuals have more chance of being selected (S. Panda et.al ,2009). There are several schemes for the selection process: roulette wheel selection and its extensions, scaling techniques, tournament, normal geometric, elitist models and ranking methods. Roulette wheel selection method has simple method. The basic concept of this method is " High fitness, high chance to be selected".

2.3.3 Parameters optimization by GA

In this section, GA is applied to search the controller parameters with off line tuning. Each step of the proposed method is explained as follows.

Step 1. Generate the objective function for GA optimization.

In this study, the performance and robust stability conditions in inverse additive perturbation design approach is adopted to design a robust controller as mention in equation (4) and (5).

Step 2. Initialize the search parameters for GA. Define genetic parameters such as population size, crossover, mutation rate, and maximum generation. Step 3. Randomly generate the initial solution.

Step 4. Evaluate objective function of each individual in equation (4) and (5).

Step 5. Select the best individual in the current generation. Check the maximum generation.

Step 6. Increase the generation.

Step 7. While the current generation is less than the maximum generation, create new population using genetic operators and go to step 4. If the current generation is the maximum generation, then stop.

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