**6. Simulation results**

#### **6.1 Eigenvalue analysis**

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


Table 2. Closed-loop eigenvalues

Optimal Design of Power System Controller Using Breeder Genetic Algorithm 311

speed response. P=0.5 and Q= at Xe=1.1

CPSS

BGA PSS GA PSS CPSS

BGA PSS GA PSS CPSS

0 1 2 3 4 5 6

GA PSS

time(s)

speed response. P=0.9 and Q= at Xe=0.9

Fig. 4. Speed responses of case 2 under three-phase fault

BGA PSS

0.999 0.9995

1

1.0005 1.001 1.0015 1.002 1.0025 1.003 1.0035 1.004

speed (p.u)

Fig. 5. Speed responses of case 3 under three-phase fault

BGA PSS

Breeder Genetic Algorithms is an extremely versatile and effective function optimizer. The main advantage of BGA over GA is the simplicity of the selection method and the fewer

0 1 2 3 4 5 6

GA PSS

CPSS

time(s)

**7. Conclusion** 

0.998

0.999

1.001

1.002

speed (p.u)

1.003

1.004

1.005

1.006

1.007

1

It should be mentioned that a maximum damping ratio of 50% was imposed on the BGA and GA, otherwise, their damping ratios could have been higher. If the damping of the electromechanical mode is too high this could negatively affect other modes in the system.

### **6.2 Large disturbance**

A large disturbance was considered by applying a three-phase fault to the system at 0.1 seconds. The fault was applied at the sending-end of the system (near bus 1 on line 2) for 200ms. The fault was cleared by disconnecting line 2. Fig. 3 to Fig. 5 show the speed responses of the system.

Figure 3 shows the speed responses of the generator for case 1. When the system is equipped with GA-PSS and BGA-PSS it settles around 3 seconds. On the other hand, the settling time of the system equipped with the CPSS is more than doubled (6 seconds). In addition, the subsequent oscillations are larger than those of BGA and GA PSSs.

Figure 4 shows the speed responses for case 2. The system equipped with CPSS is seen to have bigger oscillations as compared to the system equipped with BGA-PSS and GA-PSS. With both BGA and GA PSSs, the system settled in approximately 3.5 sec., whereas CPSS takes more than 6 sec. to settle down. The performances of the BGA-PSS and GA-PSS are quite similar, even though the BGA-PSS performs slightly better than the GA- PSS.

Figure 5 shows the speed responses of the system for case 3. It can be seen that the system equipped with BGA and GA PSS settled in less than 4 sec compared to more than 6 sec. for the CPSS. With CPSS, the system has large overshoots and undershoots.

Fig. 3. Speed response of case 1 under three-phase fault

speed response. P=0.5 and Q= at Xe=1.1

Fig. 4. Speed responses of case 2 under three-phase fault

Fig. 5. Speed responses of case 3 under three-phase fault

#### **7. Conclusion**

310 Bio-Inspired Computational Algorithms and Their Applications

It should be mentioned that a maximum damping ratio of 50% was imposed on the BGA and GA, otherwise, their damping ratios could have been higher. If the damping of the electromechanical mode is too high this could negatively affect other modes in the

A large disturbance was considered by applying a three-phase fault to the system at 0.1 seconds. The fault was applied at the sending-end of the system (near bus 1 on line 2) for 200ms. The fault was cleared by disconnecting line 2. Fig. 3 to Fig. 5 show the speed

Figure 3 shows the speed responses of the generator for case 1. When the system is equipped with GA-PSS and BGA-PSS it settles around 3 seconds. On the other hand, the settling time of the system equipped with the CPSS is more than doubled (6 seconds). In

Figure 4 shows the speed responses for case 2. The system equipped with CPSS is seen to have bigger oscillations as compared to the system equipped with BGA-PSS and GA-PSS. With both BGA and GA PSSs, the system settled in approximately 3.5 sec., whereas CPSS takes more than 6 sec. to settle down. The performances of the BGA-PSS and GA-PSS are

Figure 5 shows the speed responses of the system for case 3. It can be seen that the system equipped with BGA and GA PSS settled in less than 4 sec compared to more than 6 sec. for

speed response. P=1.1 and Q= at Xe=0.7

CPSS

BGA PSS GA PSS CPSS

0 1 2 3 4 5 6

GA PSS

time(s)

addition, the subsequent oscillations are larger than those of BGA and GA PSSs.

quite similar, even though the BGA-PSS performs slightly better than the GA- PSS.

the CPSS. With CPSS, the system has large overshoots and undershoots.

Fig. 3. Speed response of case 1 under three-phase fault

BGA PSS

0.998

0.999

1.001

1.002 1.003

speed (p.u)

1.004 1.005

1.006 1.007

1

system.

**6.2 Large disturbance** 

responses of the system.

Breeder Genetic Algorithms is an extremely versatile and effective function optimizer. The main advantage of BGA over GA is the simplicity of the selection method and the fewer

Optimal Design of Power System Controller Using Breeder Genetic Algorithm 313

<sup>1</sup> ( ) *fd fd ad fd*

*<sup>L</sup>* = − ψ ψ

*<sup>L</sup>* = − ψ ψ

*<sup>L</sup>* = − ψ ψ

*<sup>L</sup>* = − ψ ψ

*Tii e d* = − ψ

*<sup>q</sup>* are the d and q axis flux linkages, respectively.

 ψ*q q d*

( ) *<sup>A</sup> fd*

*A A*

*fd ref t*

*d K E E VV dt T <sup>T</sup>* = −−

where *KA* and *TA* are the gain and time constant of the AVR. *Vt* is the terminal voltage of the

1 1 1 <sup>1</sup> ( ) *d dad d*

1 1 1 <sup>1</sup> ( ) *q qaq q*

2 2 2 <sup>1</sup> ( ) *q qaq q*

*2q, Efd* are the same as defined in section 3. *Rfd,, Lfd,* are the field winding resistance and inductance, respectively.

*i*

*i*

*i*

*i*

<sup>a</sup>*q,* are the mutual flux linkages in the d and q axis, respectively.

*2q* are defined as before

The electrical torque is expressed by the following:

*L1d* is the d-axix amortisseur inductance. *L*1*<sup>q</sup>* is the 1st q-axix amortisseur inductance. *L*2*<sup>q</sup>* is the 2nd q-axix amortisseur inductance.

In this work *KA*=200 and *TA* = 0.05 sec.

**8.1.3 Electrical torque** 

**8.1.4 AVR equations** 

*R*1*d, is* the d-axix amortisseur resistance. *R*1*q,, is* the 1st q-axix amortisseur resistance. R2*q* is the 2nd q-axix amortisseur resistance. The rotor currents are expressed a follows:

where

where

where ψ*<sup>d</sup>*, and ψ

generator.

ψ*fd,* ψ1*d,* ψ1*q,* ψ

ψ*ad,* ψ

ψ*fd,* ψ1*d,* ψ1*q,* ψ

genetic parameters. In this work, adaptive mutation has been used to deal with the problem of premature convergence in BGA. The effectiveness of the proposed approach was demonstrated by the time and frequency domain simulation results. Eigenvalue analysis shows that the BGA based controller provides a better damping to the system for all operating conditions considered than a GA based controller. The conventional controller provides the least damping to all the operating conditions considered. The robustness of the BGA controller under large disturbance was also investigated by applying a three-phase fault to the system. Further research will be carried out in the direction of using multiobjective functions in the optimization and using a more complex power system model.

### **8. Appendix**

#### **8.1 Generator and Automatic Voltage Regulator (AVR) equations**

#### **8.1.1 Swing equations**

$$\begin{aligned} \frac{d}{dt} \Delta \phi &= \frac{1}{2H} (T\_m - T\_c - K\_D \Delta \phi) \\\\ \frac{d}{dt} \Delta \delta &= a\_b \Delta \phi \end{aligned}$$

where

δ is the rotor angle in rad *ω* is the synchronous speed in per-unit (p.u.) *ω*0 is the synchronous speed in rad/sec *H* is the inertia constant in sec. *Tm* is the mechanical torque in p.u. *Te* is the mechanical torque in p.u.

*KD* is the damping coefficient in torque/ p.u.

#### **8.1.2 Rotor circuit equations**

$$\begin{aligned} \frac{d}{dt}\mathcal{Y}\_{\beta i} &= a\_0(E\_{\beta i} - \frac{R\_{\beta i}}{L\_{\beta i}}\dot{\mathbf{i}}\_{\beta i}) \\\\ \frac{d}{dt}\mathcal{Y}\_{1d} &= -a\_0 R\_{1d}\dot{\mathbf{i}}\_{1d} \\\\ \frac{d}{dt}\mathcal{Y}\_{1q} &= -a\_0 R\_{1q}\dot{\mathbf{i}}\_{1q} \\\\ \frac{d}{dt}\mathcal{Y}\_{2q} &= -a\_0 R\_{2q}\dot{\mathbf{i}}\_{2q} \end{aligned}$$

#### where

312 Bio-Inspired Computational Algorithms and Their Applications

genetic parameters. In this work, adaptive mutation has been used to deal with the problem of premature convergence in BGA. The effectiveness of the proposed approach was demonstrated by the time and frequency domain simulation results. Eigenvalue analysis shows that the BGA based controller provides a better damping to the system for all operating conditions considered than a GA based controller. The conventional controller provides the least damping to all the operating conditions considered. The robustness of the BGA controller under large disturbance was also investigated by applying a three-phase fault to the system. Further research will be carried out in the direction of using multiobjective functions in the optimization and using a more complex power system model.

<sup>1</sup> ( ) <sup>2</sup> *me D*

0

<sup>0</sup>( ) *fd fd fd fd*

*E i*

1 011 *d d d <sup>d</sup> R i*

1 0 11 *q q q <sup>d</sup> R i*

2 022 *q q q <sup>d</sup> R i*

 ω= −

 ω= −

 ω= −

*d R*

*dt L*

ψ ω= −

*dt*ψ

*dt*ψ

*dt*ψ *fd*

 ω

*<sup>d</sup> T TK*

Δ= − − Δ

Δ= Δ δ ωω

*dt H*

*d dt*

ω

**8.1 Generator and Automatic Voltage Regulator (AVR) equations** 

**8. Appendix** 

where

δ

**8.1.1 Swing equations** 

is the rotor angle in rad

*H* is the inertia constant in sec. *Tm* is the mechanical torque in p.u. *Te* is the mechanical torque in p.u.

**8.1.2 Rotor circuit equations** 

*ω* is the synchronous speed in per-unit (p.u.) *ω*0 is the synchronous speed in rad/sec

*KD* is the damping coefficient in torque/ p.u.

ψ*fd,* ψ1*d,* ψ1*q,* ψ*2q, Efd* are the same as defined in section 3. *Rfd,, Lfd,* are the field winding resistance and inductance, respectively. *R*1*d, is* the d-axix amortisseur resistance. *R*1*q,, is* the 1st q-axix amortisseur resistance.

R2*q* is the 2nd q-axix amortisseur resistance.

The rotor currents are expressed a follows:

$$\begin{aligned} i\_{\beta l} &= \frac{1}{L\_{\beta l}} (\boldsymbol{\mu}\_{\beta l} - \boldsymbol{\mu}\_{\omega l}) \\\\ i\_{1d} &= \frac{1}{L\_{1d}} (\boldsymbol{\mu}\_{1d} - \boldsymbol{\mu}\_{\omega l}) \\\\ i\_{1q} &= \frac{1}{L\_{1q}} (\boldsymbol{\mu}\_{1q} - \boldsymbol{\mu}\_{\omega q}) \\\\ i\_{2q} &= \frac{1}{L\_{2q}} (\boldsymbol{\mu}\_{2q} - \boldsymbol{\mu}\_{\omega q}) \end{aligned}$$

where

ψ*fd,* ψ1*d,* ψ1*q,* ψ*2q* are defined as before ψ*ad,* ψ<sup>a</sup>*q,* are the mutual flux linkages in the d and q axis, respectively. *L1d* is the d-axix amortisseur inductance. *L*1*<sup>q</sup>* is the 1st q-axix amortisseur inductance. *L*2*<sup>q</sup>* is the 2nd q-axix amortisseur inductance.

#### **8.1.3 Electrical torque**

The electrical torque is expressed by the following:

$$T\_c = \Psi\_d i\_q - \Psi\_q i\_d$$

where ψ*<sup>d</sup>*, and ψ*<sup>q</sup>* are the d and q axis flux linkages, respectively.

#### **8.1.4 AVR equations**

$$\frac{d}{dt}E\_{fl} = \frac{K\_A}{T\_A}(V\_{ref} - V\_t) - \frac{E\_{fl}}{T\_A}$$

where *KA* and *TA* are the gain and time constant of the AVR. *Vt* is the terminal voltage of the generator.

In this work *KA*=200 and *TA* = 0.05 sec.

Optimal Design of Power System Controller Using Breeder Genetic Algorithm 315

*Xl* =0.0742 p.u, , *Xd*=1.72 p.u,, *X'd*=0.45 p.u,, *X"d*=0.33 p.u,*T'd0*=6.3sec., *T"d0* = 0.033 p.u,,

**8.3 Generator's parameters** 

*T"q0* = 0.033sec., *H* = 4.0sec

**Initialize** mutation rate *Rnom* **While** termination criterion not met

**for** I =1 **to** N-1 **do**

**end** 

**9. Acknowledgment** 

**10. References** 

**end** 

**end**

**Begin** 

*Xq* =1.68 p.u,, *X'q* =0.59 p.u,, *X"q* =0.33 p.u, *T'q0* =0.43 sec

**8.4 Pseudo code for BGA generator's parameters** 

**Randomly initialize** a population of N individuals;

**save** the best individual in the new population

**select** the best T% individuals and discarding the rest;

**divide** the new population into two halves (X and Y) **apply** mutation rate rnom/2 to X and 2 *Rnom* to Y

If X performs better than Y; assign r= *Rnom* -0.1 rnom; If Y performs better than X; assign r= *Rnom* + 0.1 rnom;

*CMU*-CS-94-163, Carnegie Mellon University.

*Technology.* Vol..14, No. 2, pp. 84-92.

Addison-Wesley; 1989.

**recombine** the two parents to obtain one offspring

**randomly** select two individuals among the T% best individual

**evaluate** the average fitness value for the two half population (X and Y)

The authors would like to acknowledge the financial support of THRIP and TESP.

Baluja, S. (1994). Population-Based Incremental Learning: A method for Integrating Genetic

Chuang, Y.S . & Wu, C. J. (2006). A Damping Constant Limitation Design of Power System

Chuang, Y.S . & Wu, C. J. (2007). Novel Decentralized Pole Placement Design of Power

Eslami, M., Shareef, H., Mohamed, A., & Ghohal, S. P. (2010). Tuning of Power System

*Mathematics and Computer in Simulation.* Vol..1, No. 4, pp. 410-418.

*International Journal of Physical Sciences.* Vol..5, No. 17, pp. 2574-2589. Goldberg, D. E. (1989). *Genetic Algorithms in Search, Optimization & Machine Learning*.

Search Based Function Optimization and Competitive Learning. *Technical Report* 

Stabilizer using Hybrid Differential Evolution. *Journal of Marine Science and* 

System Stabilizers using Hybrid Differential Evolution. *International Journal of* 

Stabilizer using Particle Swarm Optimization with Passive Congregation.

**evaluate** goodness of each individuals
