Towards a Robust Fractional Order PID Stabilizer for Electric Power Systems

Abstract

This chapter deals with the design and application of a robust Fractional Order PID (FOPID) power system stabilizer tuned by Genetic Algorithm (GA). The system’s robustness is assured through the application of Kharitonov’s theorem to overcome the effect of system parameter’s changes within upper and lower pounds. The FOPID stabilizer has been simplified during the optimization using the Oustaloup’s approximation for fractional calculus and the “nipid” toolbox of Matlab during simulation. The objective is to keep robust stabilization with maximum attained degree of stability against system’s uncertainty. This optimization will be achieved with the proper choice of the FOPID stabilizer’s coefficients (k_p, k_i, k_d, λ, and δ) as discussed later in this chapter. The optimization has been done using the GA which limits the boundaries of the tuned parameters within the allowable domain. The calculations have been applied to a single machine infinite bus (SMIB) power system using Matlab and Simulink. The results show superior behavior of the proposed stabilizer over the traditional PID.

Appendices

Appendix A: Derivation of k-Constants

All the variables with subscript 0 are values of variables evaluated at their pre-disturbance steady-state operating point from the known values of P₀, Q₀ and V_t0.

$$ i_{q0} = \frac{{P_{0} V_{to} }}{{\sqrt {(P_{0} x_{q} )^{2} + (V_{t0}^{2} + Q_{0} x_{q} )^{2} } }} $$

(A1)

$$ v_{d0} = i_{q0} x_{q} $$

(A2)

$$ v_{qo} = \sqrt {V_{t0}^{2}} - v_{t0}^{2} $$

(A3)

$$ i_{d0} = \frac{{Q_{0} + x_{q} i_{q0}^{2} }}{{v_{q0}^{{}} }} $$

(A4)

$$ E_{q0} = v_{q0} + i_{d0} x_{q} $$

(A5)

$$ E_{0} = \sqrt {(v_{d0} + x_{e} i_{q0} )^{2} + (v_{q0} - x_{e} i_{d0} )^{2} } $$

(A6)

$$ \delta_{0} = \tan^{ - 1} \frac{{(v_{d0} + x_{e} i_{q0} )}}{{(v_{q0} - x_{e} i_{d0} )}} $$

(A7)

$$ {\text{K}}_{ 1} = \frac{{x_{q} - x{{\prime }}_{d} }}{{x_{e} + x{{\prime }}_{d} }}i_{q0} E_{0} \,\sin \delta_{0} + \frac{{E_{q0} E_{0} \,\cos \delta_{0} }}{{x_{e} + x_{q} }} $$

(A8)

$$ {\text{K}}_{ 2} = \frac{{E_{0} \sin \delta_{0} }}{{x_{e} + x{{\prime }}_{d} }} $$

(A9)

$$ {\text{K}}_{ 3} = \frac{{x{{\prime }}_{d} + x_{e} }}{{x_{d} + x_{e} }} $$

(A10)

$$ {\text{K}}_{ 4} = \frac{{x_{q} - x{{\prime }}_{d} }}{{x_{e} + x{{\prime }}_{d} }}E_{0} \sin \delta_{0} $$

(A11)

$$ {\text{K}}_{ 5} = \frac{{x_{q} }}{{x_{e} + x_{q} }}\frac{{v_{d0} }}{{V_{t0} }}E_{0} \cos \delta_{0} - \frac{{x{{\prime }}d }}{{x_{e} + x{{\prime }}_{d} }}\frac{{v_{q0} }}{{V_{t0} }}E_{0} \sin \delta_{0} $$

(A12)

$$ {\text{K}}_{ 6} = \frac{{x_{e} }}{{x_{e} + x{{\prime }}_{d} }}\frac{{v_{q0} }}{{V_{t0} }} $$

(A13)

Appendix B

Nomenclature

All quantities are per unit on machine base.

D:

Damping Torque Coefficient

M:

Inertia constant

ω:

Angular speed

$\updelta$ :

Rotor angle

I_d, I_q :

Direct and quadrature components of armature current

$x_{d} \,{\text{and}}\,x_{q}$ :

Synchronous reactance in d and q axis

$x{{\prime }}_{d} \,{\text{and}}\,x{{\prime }}_{q}$ :

Direct axis and Quadrature axis transient reactance

E _fd :

Equivalent excitation voltage

K_E :

Exciter gain

T_E :

Exciter time constant

T_m and T_e :

Mechanical and Electrical torque

$T{{\prime}}_{do}$ :

Field open circuit time constant

V_d and V_q :

Direct and quadrature components of terminal voltage

K₁ :

Change in T_e for a change in $\updelta$ with constant flux linkages in the d axis

K₂ :

Change in T_e for a change in d axis flux linkages with constant $\updelta$

K₃ :

Impedance factor

K₄ :

Demagnetizing effect of a change in rotor angle

K₅ :

Change in V_t with change in rotor angle for constant $E{{\prime}}_{q}$

K₆ :

Change in V_t with change in $E{{\prime}}_{q}$ constant rotor angle

Appendix C

The system data are as follows:

Machine (p.u):

$$ \begin{aligned} x_{d} & = 1.6\quad x{{\prime }}_{d} = 0.32 \\ x_{q} & = 1.55\quad T{{\prime }}_{d0} = 6\,{\text{s}} \\ {\text{D}} & { = 0} . 0\quad {\text{M = 10}}\,{\text{s}} \\ \end{aligned} $$

(C1)

Transmission line (p.u):

$$ {\text{r}}_{\text{e}} = 0.0\quad {\text{x}}_{\text{e}} = 0. 4 $$

(C2)

Exciter:

$$ {\text{K}}_{\text{E}} = 2 5.00\quad {\text{T}}_{\text{E}} = 0.0 5\,{\text{s}} $$

(C3)

Nominal Operating point:

$$ \begin{aligned} V_{t0} = 1.0\quad P_{0} = 0. 8\hfill \\ Q_{0} = 0. 3 \quad \delta_{0} = 4 5^{\circ} \hfill \\\upomega_{0} = 314 \hfill \\ \end{aligned} $$

(C4)

Others

$$ \begin{aligned} {\text{k}}_{3} & = 1/2.78 \\ {\text{v}} & = 1.0 \\ {\text{T}}_{\text{w}} & = 5 \\ \end{aligned} $$

(C5)