**Optimal Design of Power System Controller Using Breeder Genetic Algorithm**

K. A. Folly and S. P. Sheetekela *University of Cape Town Private Bag., Rondebosch 7701 South Africa* 

#### **1. Introduction**

302 Bio-Inspired Computational Algorithms and Their Applications

[8] Wang Xiaoping, Cao Liming. Genetic algorithm - theory, application and software implementation [M]. Xi'an:Xi'an Jiaotong University Press,( 2002).

[9] Syed Khairuzzaman Tanbeer, Chowdhury Farhan Ahmed, Byeong-Soo Jeong. Sliding

[10] Joong Hyuk Chang, Won Suk Lee. A sliding window method for finding recently

[11] Joong Hyuk Chang, Won Suk Lee.Finding Recent Frequent Itemsets Adaptively over

[12] F.Molnar Jr.,T.Szakaly,R.Meszaros,I.Lagzi..Air pollution modelling using a Graphics

[14] Harish P, Narayanan PJ. Accelerating large graph algorithms on the GPU using

International Journal, Vol.179,(2009) 3843 - 3865.

[13] NVIDIA Corporation.CUDA programming guide[Z].2008.

CUDA[C].Springer Heidelberg,2007:367-390.

Engineering ,Vol.20,(2004)753 - 762.

492.

2010,181(1):105-112.

window-based frequent pattern mining over data streams. Information Sciences: an

frequent itemsets over online data streams. Journal of Information Science and

Online Data Streams[C]. Washington: Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining.( 2003) 487 –

Processing Unit with CUDA[J]. Computer Physics Communications,

Genetic Algorithms (GAs) have recently found extensive applications in solving global optimization problems (Mitchell, 1996). GAs are search algorithms that use models based on natural biological evolution (Goldberg, 1989). They are intrinsically robust search and optimization mechanisms and offer several advantages over traditional optimization techniques, including the ability to effectively search large space without being caught in local optimum. GAs do not require the objective function to have properties such as continuity or smoothness and make no use of hessians or gradient estimates.

In the last few years, Genetic Algorithms (GAs) have shown their potentials in many fields, including in the field of electrical power systems. Although GAs provide robust and powerful adaptive search mechanism, they have several drawbacks (Mitchell, 1996). Some of these drawbacks include the problem of "genetic drift" which prevents GAs from maintaining diversity in its population. Once the population has converged, the crossover operator becomes ineffective in exploring new portions of the search space. Another drawback is the difficulty to optimize the GAs' operators (such as population size, crossover and mutation rates) one at a time. These operators (or parameters) interact with one another in a nonlinear manner. In particular, optimal population size, crossover rate, and mutation rate are likely to change over the course of a single run (Baluja, 1994). From the user's point of view, the selection of GAs' parameters is not a trivial task. Since the 'classical' GA was first proposed by Holland in 1975 as an efficient, easy to use tool which can be applicable to a wide range of problems (Holland, 1975), many variant forms of GAs have been suggested often tailored to specific problems (Michalewicz, 1996). However, it is not always easy for the user to select the appropriate GAs parameters for a particular problem at hand because of the huge number of choices available. At present, there is a little theoretical guidance on how to select the suitable GAs parameters for a particular problem (Michalewicz, 1996). Still another problem is that the natural selection strategy used by GAs is not immune from failure. To cope with the above limitations, an extremely versatile and effective function optimizer called Breeder Genetic Algorithm (BGA) was recently proposed (Muhlenbein, 1994). BGA is inspired by the science of breeding animals. The main idea is to use a selection strategy based on the concept of animal breeding instead of "natural selection" (Irhamah & Ismail, 2009). The assumption behind this strategy is as follows: "*mating two individuals with high fitness is more likely to produces an offspring of high fitness than mating two randomly selected individuals*".

Optimal Design of Power System Controller Using Breeder Genetic Algorithm 305

Differential Evolutionary (Wang, *et al* 2008), hybrid Differential Evolutionary (Chuang, & Wu, 2006), (Chuang, & Wu, 2007), Particle Swarm Optimization (Eslami, *et al* 2010),

In this chapter, Breeder Genetic Algorithm (BGA) with adaptive mutation is used for the optimization of the parameters of the Power System Stabilizer (PSSs). An eigenvalue based objective function is employed in the design such that the algorithm maximizes the lowest damping ratio over specified operating conditions. A single machine infinite bus system is used to show the effectiveness of the proposed method. For comparison purposes, Genetic Algorithms (GAs) based PSS and the Conventional PSS (CPSS) are included. Frequency and time domain simulations show that BGA-PSS performs better than GA-PSS and CPSS under both small and large disturbances for all operating conditions considered in this work.

BGA is a relatively new evolution algorithm. It is similar to GAs with the exception that it uses artificial selection and has fewer parameters. Also, BGA uses real-valued representation as opposed to GAs which mainly uses binary and sometimes floating or integer representation. In this work, a modified version of BGA called Adaptive Mutation BGA is used (Green, 2005), (Sheetekela & Folly, 2010). Truncation selection method is adopted whereby a top *T*% of the fittest individuals are chosen from the current population of *N* individuals and goes through recombination and mutation to form the next generation. The rest of the individuals are discarded. In truncation method, the fittest individual in the population called an *ellist* is guaranteed a place in the next generation. The other top (*T*-1) % goes through recombination and mutation to form up the rest of the individuals in the next generation. The process is repeated until an optimal solution is obtained or the maximum

Recombination is similar to crossover in GAs (Michalewicz, 1996). The Breeder Genetic Algorithm proposed in this work allows various possible recombination methods to be used, each of them searching the space with a particular bias. Since there is no prior knowledge as to which bias is likely to suit the task at hand, it is better to include several recombination methods and allow selection to do the elimination. Two recombination

In volume recombination, a random vector *r* of the same length as the parent is generated

In other words, the child can be said to be located at a point inside the hyper box defined by

In line recombination, a single uniformly random number *r* is generated between 0 and 1,

(1) ݕሻݎ െ ሺͳ ݔݎ ൌ ݖ

methods were used in this work: volume and line recombination (Sheetekela, 2010).

and the child *zi* is produced by the following expression.

and the child is obtained by the following expression (Green, 2005).

Population-Based Incremental Learning (Folly,2006), (Sheetekela, 2010), etc.

GA-PSS in turn gives a better performance than the Conventional PSS (CPSS).

**2. Background theory to breeder genetic algorithm** 

number of iteration is reached.

where *xi* and *yi* are the two parents.

the parents as shown in Fig. 1.

**2.1 Recombination** 

Some of the features of BGA are:


The main advantage of using BGA is its simplicity with regard to the selection method (Irhamah & Ismail, 2009) and the fewer parameters to be chosen by the user. However, there is a price to pay for this simplicity. Since only the best individuals are selected in each generation to produce the children for the next generation, there is a likelihood of premature convergence. As a result, BGA may converge to local optimum rather than the desired global one. It should be mentioned that most of the Evolutionary Algorithms including GA have problems with premature convergence to a certain degree. The general way to deal with this problem is to apply mutation to a few randomly selected individuals in the population. In this work, instead of a fixed mutation rate, we have used adaptive mutation strategy (Green, 2005), (Sheetekela & Folly, 2010). This means that the mutation rate is not fixed but varies according to the convergence and performance of the population. In general, even with fixed mutation rate, BGA may still perform better than GA as discussed in (Irhamah & Ismail, 2009).

The application of Evolutionary Algorithm to design power system stabilizer for damping low frequency oscillations in power systems has received increasing attention in recent years, see for example, (Wang, *et al* 2008), (Chuang, & Wu, 2006), (Chuang, & Wu, 2007), (Eslami, *et al* 2010), (Hongesombut, *et al* 2005), (Folly, 2006), and (Hemmati, *et al* 2010).

Low frequency oscillations in power systems arise due to several causes. One of these is the heavy transfer of power over long distance. In the last few years, the problems of low frequency oscillations are becoming more and more important. Some of the reasons for this are:


For several years, traditional control methods such as phase compensation technique (Hemmati *et al*, 2010), root locus (Kundur, 1994), pole placement technique (Shahgholian & Faiz, 2010), etc. have been used to design Conventional PSSs (CPSSs). These (CPSSs) are widely accepted in the industry because of their simplicity. However, conventional controllers cannot provide adequate damping to the system over a wide range of operating conditions. To cover a wide range of operating conditions when designing the PSSs several authors have proposed to use multi-power conditions, whereby the PSS parameters are optimized over a set of specified operating conditions using various optimization techniques such as sensitivity technique (Tiako & Folly, 2009), (Yoshimura& Uchida, 2000), Differential Evolutionary (Wang, *et al* 2008), hybrid Differential Evolutionary (Chuang, & Wu, 2006), (Chuang, & Wu, 2007), Particle Swarm Optimization (Eslami, *et al* 2010), Population-Based Incremental Learning (Folly,2006), (Sheetekela, 2010), etc.

In this chapter, Breeder Genetic Algorithm (BGA) with adaptive mutation is used for the optimization of the parameters of the Power System Stabilizer (PSSs). An eigenvalue based objective function is employed in the design such that the algorithm maximizes the lowest damping ratio over specified operating conditions. A single machine infinite bus system is used to show the effectiveness of the proposed method. For comparison purposes, Genetic Algorithms (GAs) based PSS and the Conventional PSS (CPSS) are included. Frequency and time domain simulations show that BGA-PSS performs better than GA-PSS and CPSS under both small and large disturbances for all operating conditions considered in this work. GA-PSS in turn gives a better performance than the Conventional PSS (CPSS).

### **2. Background theory to breeder genetic algorithm**

BGA is a relatively new evolution algorithm. It is similar to GAs with the exception that it uses artificial selection and has fewer parameters. Also, BGA uses real-valued representation as opposed to GAs which mainly uses binary and sometimes floating or integer representation. In this work, a modified version of BGA called Adaptive Mutation BGA is used (Green, 2005), (Sheetekela & Folly, 2010). Truncation selection method is adopted whereby a top *T*% of the fittest individuals are chosen from the current population of *N* individuals and goes through recombination and mutation to form the next generation. The rest of the individuals are discarded. In truncation method, the fittest individual in the population called an *ellist* is guaranteed a place in the next generation. The other top (*T*-1) % goes through recombination and mutation to form up the rest of the individuals in the next generation. The process is repeated until an optimal solution is obtained or the maximum number of iteration is reached.

#### **2.1 Recombination**

304 Bio-Inspired Computational Algorithms and Their Applications

• BGA uses real-valued representation as opposed to binary representation used in

• The selection technique used is (always) truncation, whereby a selected top *T*% of the fittest individuals are chosen from the current generation and goes through recombination and mutation to form the next generation. The rest of the individuals are

The main advantage of using BGA is its simplicity with regard to the selection method (Irhamah & Ismail, 2009) and the fewer parameters to be chosen by the user. However, there is a price to pay for this simplicity. Since only the best individuals are selected in each generation to produce the children for the next generation, there is a likelihood of premature convergence. As a result, BGA may converge to local optimum rather than the desired global one. It should be mentioned that most of the Evolutionary Algorithms including GA have problems with premature convergence to a certain degree. The general way to deal with this problem is to apply mutation to a few randomly selected individuals in the population. In this work, instead of a fixed mutation rate, we have used adaptive mutation strategy (Green, 2005), (Sheetekela & Folly, 2010). This means that the mutation rate is not fixed but varies according to the convergence and performance of the population. In general, even with fixed mutation rate, BGA may still perform better than GA as discussed

The application of Evolutionary Algorithm to design power system stabilizer for damping low frequency oscillations in power systems has received increasing attention in recent years, see for example, (Wang, *et al* 2008), (Chuang, & Wu, 2006), (Chuang, & Wu, 2007), (Eslami, *et al* 2010), (Hongesombut, *et al* 2005), (Folly, 2006), and (Hemmati, *et al* 2010).

Low frequency oscillations in power systems arise due to several causes. One of these is the heavy transfer of power over long distance. In the last few years, the problems of low frequency oscillations are becoming more and more important. Some of the reasons for this

a. Modern power systems are required to operate close to their stability margins. A small disturbance can easily reduce the damping of the system and drive the system to

b. The deregulation and open access of the power industry has led to more power transfer across different regions. This has the effect of reducing the stability margins.

For several years, traditional control methods such as phase compensation technique (Hemmati *et al*, 2010), root locus (Kundur, 1994), pole placement technique (Shahgholian & Faiz, 2010), etc. have been used to design Conventional PSSs (CPSSs). These (CPSSs) are widely accepted in the industry because of their simplicity. However, conventional controllers cannot provide adequate damping to the system over a wide range of operating conditions. To cover a wide range of operating conditions when designing the PSSs several authors have proposed to use multi-power conditions, whereby the PSS parameters are optimized over a set of specified operating conditions using various optimization techniques such as sensitivity technique (Tiako & Folly, 2009), (Yoshimura& Uchida, 2000),

• BGA only requires a few parameters to be chosen by the user.

Some of the features of BGA are:

classical GAs.

discarded.

in (Irhamah & Ismail, 2009).

are:

instability.

Recombination is similar to crossover in GAs (Michalewicz, 1996). The Breeder Genetic Algorithm proposed in this work allows various possible recombination methods to be used, each of them searching the space with a particular bias. Since there is no prior knowledge as to which bias is likely to suit the task at hand, it is better to include several recombination methods and allow selection to do the elimination. Two recombination methods were used in this work: volume and line recombination (Sheetekela, 2010).

In volume recombination, a random vector *r* of the same length as the parent is generated and the child *zi* is produced by the following expression.

$$\mathbf{z}\_{l} = r\_{l}\mathbf{x}\_{l} + (\mathbf{1} - r\_{l})\mathbf{y}\_{l} \tag{1}$$

where *xi* and *yi* are the two parents.

In other words, the child can be said to be located at a point inside the hyper box defined by the parents as shown in Fig. 1.

In line recombination, a single uniformly random number *r* is generated between 0 and 1, and the child is obtained by the following expression (Green, 2005).

Optimal Design of Power System Controller Using Breeder Genetic Algorithm 307

The power system considered is a single machine infinite bus (SMIB) system as shown in Fig. A. 1 of Appendix 8.2.1. The generator is connected to the infinite bus through a doublecircuit transmission line. The generator is modeled using a 6th order machine model, and is equipped with an automatic voltage regulator (AVR) which is represented by a simple exciter of first order differential equation as given in the Appendix 8.1.4. The block diagram of the AVR is shown in Fig. A. 2 of Appendix 8.2.2. A supplementary controller also known as power system stabilizer (PSS) is to be designed to damp the system's oscillations. The

The non-linear differential equations of the system are linearized around the nominal

*<sup>d</sup> x Ax Bu*

(3)

*fd* is the field flux linkage deviation,

*1q* is the 1st q-axis amortisseur flux

Eigenvalues (Damping ratio)

(0.0547)

(0.0800)

(0.0572)

δis

*2q* Δ*E*fd ]; *u* = [Δ*Tm* Δ*Vref* ]; *y* = Δ*ω*; where, Δ

 = + = +

*y Cx Du*

*A* is the system state matrix, *B* is the system input matrix, *C* is the system output matrix and

*x* is the vector of the system states, *u* is the vector of the system inputs and *y* is the vector of

output voltage deviation. Δ*Tm* is the mechanical torque deviation and Δ*Vref* is the voltage

Several operating conditions were considered for the design of the controllers. These operating conditions were obtained by varying the active power output, *Pe* and the reactive power *Qe* of the generator as well as the line reactance, *Xe*. However, for simplicity, only three operating conditions will be presented in this paper. These operating conditions are listed in the Table 1 together with the open loop eigenvalues and their respective damping

1 1.1000 0.4070 0.7000 -0.2894 ± j5.2785

2 0.5000 0.1839 1.1000 -0.3472 ± j4.3271

3 0.9000 0.3372 0.9000 -0.2704 + j4.7212

Reacctive Power Qe [p.u]

Table 1. Selected operating conditions with open-loop eigenvalues

ψ

ψ

*2q* is the 2nd q-axis amortisseur flux linkage deviation, Δ*Efd* is the exciter

Line reactance Xe [p.u]

*dt*

block diagram of the PSS is shown in Fig. A.3 of Appendix 8.2.3.

operating condition to form a set of linear equations as follows:

**3. Test model** 

where:

Δψ

*D* is the feed-forward matrix

δ Δ*ω* Δψ*fd* Δψ<sup>1</sup>*<sup>d</sup>* Δψ*1q* Δψ

ψ

the rotor angle deviation, Δ*ω* is the speed deviation, Δ

<sup>1</sup>*<sup>d</sup>* is d-axis amortisseur flux linkage deviation, Δ

the system outputs.

In this work, *x*= [Δ

linkage deviation, Δ

reference deviation.

ratios in % in brackets.

case Active Power

Pe [p.u]

$$\mathbf{z}\_l = r\mathbf{x}\_l + (1 - r)y\_l \tag{2}$$

where *xi* and *yi* are the two parents.

In light of this, a child can be said to be located at a randomly chosen point on a line connecting the two parents as shown in Fig.2.

Fig. 1. Volume recombination

Fig. 2. Line recombination

#### **2.2 Mutation**

One problem that has been of concern in GAs is premature convergence, whereby a good but not optimal solution will come to dominate the population. In other words, the search may well converge to local optimum than the desired global one. This problem can be eliminated by adding a small vector of normally-distributed zero-mean random numbers (say with a standard deviation *R)* to each child before inserting it into the population. The magnitude of the standard deviation *R* of the vector is very critical, as small *R* might lead to premature converge and large *R* might impair the search and reduce its ability to converge optimally. Therefore, it's better to use an adaptive approach whereby the rate of mutation is modified during the course of the search. We set *R* to the nominal rate *Rnom. T*he population is divided into two halves *X* and *Y*. A mutation rate of *2Rnom* is applied to *X* whereas a mutation of *Rnom/2* is applied to Y. The mutation rate *Rnom* is adjusted depending on the population (*X* or *Y*) that is producing better and fitter solutions on average. If *X* individuals are fitter, then the mutation rate *Rnom* is increased slightly by say l0%. If *Y* is fitter then the mutation rate, *Rnom* is reduced by a similar amount.

#### **3. Test model**

306 Bio-Inspired Computational Algorithms and Their Applications

In light of this, a child can be said to be located at a randomly chosen point on a line

One problem that has been of concern in GAs is premature convergence, whereby a good but not optimal solution will come to dominate the population. In other words, the search may well converge to local optimum than the desired global one. This problem can be eliminated by adding a small vector of normally-distributed zero-mean random numbers (say with a standard deviation *R)* to each child before inserting it into the population. The magnitude of the standard deviation *R* of the vector is very critical, as small *R* might lead to premature converge and large *R* might impair the search and reduce its ability to converge optimally. Therefore, it's better to use an adaptive approach whereby the rate of mutation is modified during the course of the search. We set *R* to the nominal rate *Rnom. T*he population is divided into two halves *X* and *Y*. A mutation rate of *2Rnom* is applied to *X* whereas a mutation of *Rnom/2* is applied to Y. The mutation rate *Rnom* is adjusted depending on the population (*X* or *Y*) that is producing better and fitter solutions on average. If *X* individuals are fitter, then the mutation rate *Rnom* is increased slightly by say l0%. If *Y* is fitter then the

where *xi* and *yi* are the two parents.

 Fig. 1. Volume recombination

Fig. 2. Line recombination

mutation rate, *Rnom* is reduced by a similar amount.

**2.2 Mutation** 

connecting the two parents as shown in Fig.2.

(2) ݕሻݎ െ ሺͳ ݔݎ ൌ ݖ

The power system considered is a single machine infinite bus (SMIB) system as shown in Fig. A. 1 of Appendix 8.2.1. The generator is connected to the infinite bus through a doublecircuit transmission line. The generator is modeled using a 6th order machine model, and is equipped with an automatic voltage regulator (AVR) which is represented by a simple exciter of first order differential equation as given in the Appendix 8.1.4. The block diagram of the AVR is shown in Fig. A. 2 of Appendix 8.2.2. A supplementary controller also known as power system stabilizer (PSS) is to be designed to damp the system's oscillations. The block diagram of the PSS is shown in Fig. A.3 of Appendix 8.2.3.

The non-linear differential equations of the system are linearized around the nominal operating condition to form a set of linear equations as follows:

$$\begin{cases} \frac{d}{dt}\mathbf{x} = A\mathbf{x} + Bu\\ \mathbf{y} = \mathbf{C}\mathbf{x} + Du \end{cases} \tag{3}$$

where:

*A* is the system state matrix, *B* is the system input matrix, *C* is the system output matrix and *D* is the feed-forward matrix

*x* is the vector of the system states, *u* is the vector of the system inputs and *y* is the vector of the system outputs.

In this work, *x*= [Δδ Δ*ω* Δψ*fd* Δψ<sup>1</sup>*<sup>d</sup>* Δψ*1q* Δψ*2q* Δ*E*fd ]; *u* = [Δ*Tm* Δ*Vref* ]; *y* = Δ*ω*; where, Δδ is the rotor angle deviation, Δ*ω* is the speed deviation, Δψ*fd* is the field flux linkage deviation, Δψ<sup>1</sup>*<sup>d</sup>* is d-axis amortisseur flux linkage deviation, Δψ*1q* is the 1st q-axis amortisseur flux linkage deviation, Δψ*2q* is the 2nd q-axis amortisseur flux linkage deviation, Δ*Efd* is the exciter output voltage deviation. Δ*Tm* is the mechanical torque deviation and Δ*Vref* is the voltage reference deviation.

Several operating conditions were considered for the design of the controllers. These operating conditions were obtained by varying the active power output, *Pe* and the reactive power *Qe* of the generator as well as the line reactance, *Xe*. However, for simplicity, only three operating conditions will be presented in this paper. These operating conditions are listed in the Table 1 together with the open loop eigenvalues and their respective damping ratios in % in brackets.


Table 1. Selected operating conditions with open-loop eigenvalues

Optimal Design of Power System Controller Using Breeder Genetic Algorithm 309

More information on the selection, crossover and mutation can be found in (Michalewicz,

The Conventional PSS (CPSS) was designed at the nominal operating condition using the phase compensation method. The phase lag of the system was first obtained, which was found to be 20o, thus only a single lead-lag block was used for the PSS. After obtaining the phase lag, a PSS with a phase lead was designed using the phase compensation technique. The final phase lead obtained was approximately 18o, thus giving the system a slight phase lag of 2o. Once the phase lag is improved, then the damping needed to be improved as well by varying the gain *KS*. The parameters of the CPSS are given in *Table A.*1 of Appendix 8.2.3.

Under the assumption of small-signal disturbance (i.e, small change in *Vref* or *Tm*), the eigenvalues of the system are obtained and the stability of the system investigated. Table 2 shows the eigenvalues of the system for the different PSSs. The damping ratios are shown in brackets. For all of the cases, it can be seen that on average, BGA-PSS provides more damping to the system than GA-PSS. On the other hand, GA-PSS performs better than CPSS. For example for case 1, BGA-PSS provides a damping ratio of 50% as compared to 48.85% for GA-PSS and 44.93% for CPSS. This means that, BGA gives the best performance.

case BGA-PSS GA-PSS CPSS







The following GA parameters have been used during the design

The parameters of the GA-PSS are given in *Table A.*1 of Appendix 8.2.3.

Likewise, BGA provides better damping ratios for cases 2 and 3.

1 -3.0664 ±j 5.3117 (0.5000)

2 -1.2793 ± j4.3024 (0.2850)

3 -2.1245 + j4.6503 (0.4155)

Table 2. Closed-loop eigenvalues

**5.2 GA-P15 Folly\_secondSS** 


1996), (Sheetekela & Folly, 2010).



**5.3 Conventional-PSS** 

**6. Simulation results 6.1 Eigenvalue analysis** 
