A single machine infinite bus power system excitation control design with extended reduced-order observer

Abstract

This paper presents an extended reduced-order observer (EROO) in order to improve the stability and voltage regulation of a synchronous machine connected to an infinite bus power system through a transmission line. The EROO-based control scheme is implemented with an automatic voltage regulator to enhance the damping of low frequency power system oscillations, so that the deviations in terminal voltage are reduced. The proposed observer reconstructs the unmeasurable states of the system and state of the disturbance model simultaneous to implement full state feedback and using the estimated disturbance state to compensate the system efficiently in presence of external and parameter uncertainties. The closed-loop poles (CLPs) of the system have been assigned by the symmetric root locus technique, with the desired level of system damping provided by the dominant CLPs. The performance of the system is analyzed through simulating at different operating conditions. The control method is not only capable of providing zero estimation error in steady-state, but also shows robustness in tracking the reference command under parametric variations and external disturbances. Illustrative examples have been provided to demonstrate the effectiveness of the developed methodology.

Appendices

Appendix A: Power system data

Parameter	Value
Line impedance, $\hbox {R}_{\mathrm{e}}+\hbox {jX}_{\mathrm{e}}$	$0.02+j0.40$
D-axis reactance, $\hbox {x}_{\mathrm{d}}$	1.70
Q-axis reactance, $\hbox {x}_{\mathrm{q}}$	1.64
Armature resistance, r	0.001096
Infinite bus voltage, V	1.00
Inertia constant, $\hbox {H}^{{\prime }}$	2.37
Rated speed, $\omega _{0}$	314.1593 rad/s
Field circuit time constant, $\tau ^{{\prime }}_{\mathrm{do}}$	8 s
Damping factor, $\hbox {K}_{\mathrm{D}}$	0
Regulator gain, $\hbox {K}_{\mathrm{A}}$	50
Exciter time constant, $\tau _{\mathrm{E}}$	0.5
Exciter gain, $\hbox {K}_{\mathrm{E}}$	$-$ 0.05
Effective field time constant, $\tau _{3}$	2.4576

Appendix B: Coefficients of observer system

$$\begin{aligned}&\left. {\begin{array}{l} {a}_{{21}} {t}_{{12}} -{ t}_{{21}} +{ a}_{{31}} {t}_{{13}} +{ a}_{{41}} {t}_{{14}} =K_5 \\ {a}_{{22}} {t}_{{12}} -{ t}_{{22}} +{ a}_{{32}} {t}_{{13}} +{ a}_{{42}} {t}_{{14}} =K_6 \\ {a}_{{23}} {t}_{{12}} -{ t}_{{23}} +{ a}_{{33}} {t}_{{13}} ={ }0 \\ {a}_{{44}} {t}_{{14}} -{ t}_{{24}} +{ t}_{{11}} \omega _{{0 }} ={ }0{ } \\ {b}_{{13}} {t}_{{13}} -{ t}_{{25}} =0 \\ {a}_{{21}} {t}_{{22}} -{ t}_{{31}} +{ a}_{{31}} {t}_{{23}} +{ a}_{{41}} {t}_{{24}} =0 \\ {a}_{{22}} {t}_{{22}} -{ t}_{{32}} +{ a}_{{32}} {t}_{{23}} +{ a}_{{42}} {t}_{{24}} =0 \\ {a}_{{23}} {t}_{{22}} -{ t}_{{33}} +{ a}_{{33}} {t}_{{23}} ={ }0{ } \\ {a}_{{44}} {t}_{{24}} -{ t}_{{34}} +{ t}_{{21}} \omega _{0} ={ }0 { } \\ {b}_{{13}} {t}_{{23}} -{ t}_{{35}} =0 \\ {a}_{{21}} {t}_{{32}} +{ a}_{{31}} {t}_{{33}} +{ a}_{{41}} {t}_{{34}} +{ d}_{0} {t}_{{11}} +{ d}_{1} {t}_{{21}} +{ d}_{2} {t}_{{31}} =0 \\ {a}_{{22}} {t}_{{32}} +{ a}_{{32}} {t}_{{33}} +{ a}_{{42}} {t}_{{34}} +{ d}_{0} {t}_{{12}} +{ d}_{1} {t}_{{22}} +{ d}_{2} {t}_{{32}} =0 \\ {a}_{{23}} {t}_{{32}} +{ a}_{{33}} {t}_{{33}} +{ d}_{0} {t}_{{13}} +{ d}_{1} {t}_{{23}} +{ d}_{2} {t}_{{33}} =0 \\ {a}_{{44}} {t}_{{34}} +{ d}_{0} {t}_{{14}} +{ d}_{1} {t}_{{24}} +{ d}_{2} {t}_{{34}} +{ t}_{{31}} \omega _{0} ={ }0 \\ {b}_{{13}} {t}_{{33}} +{ d}_{0} {t}_{{15}} +{ d}_{1} {t}_{{25}} +{ d}_{2} {t}_{{35}} =0 \\ \end{array}} \right\} \quad \hbox {T-matrices for Eq}.~(28) \\&\left. {\begin{array}{l} {a}_{{o3}} = -{ h}_{{11}} \\ {a}_{{o2}} ={K}_{5} {m}_{{11}} -{ d}_{2} {h}_{{11}} \\ {a}_{{o1}} ={K}_{5} {d}_{2} {m}_{{11}} -{ K}_{5} {d}_{0} {m}_{{13}} -{ d}_{1} {h}_{{11}} \\ {a}_{{o0}} = -{ d}_{0} {h}_{{11}} -{ K}_{5} {d}_{0} {m}_{{12}} +{ K}_{5} {d}_{1} {m}_{{11}} \\ {b}_{{o3}} = -{ h}_{{12}} \\ {b}_{{o2}} ={K}_{6} {m}_{{11}} -{ d}_{2} {h}_{{12}} \\ {b}_{{o1}} ={K}_{6} {d}_{2} {m}_{{11}} -{ K}_{6} {d}_{0} {m}_{{13}} -{ d}_{1} {h}_{{12}} \\ {b}_{{o0}} = -{ d}_{0} {h}_{{12}} -{ K}_{6} {d}_{0} {m}_{{12}} +{ K}_{6} {d}_{1} {m}_{{11}} \\ {c}_{{o2}} ={(K}_{A} {m}_{{11}} {t}_{{13}} +{ K}_{A} {m}_{{12}} {t}_{{23}} +{ K}_{A} {m}_{{13}} {t}_{{33}} {)/}\tau _{E} \\ {c}_{{o1}} ={(K}_{A} {m}_{{11}} {t}_{{23}} +{ K}_{A} {m}_{{12}} {t}_{{33}} -{ K}_{A} {d}_{0} {m}_{{13}} {t}_{{13 }} +{ K}_{A} {d}_{2} {m}_{{11}} {t}_{{13}} \\ \quad \quad \quad -{ K}_{A} {d}_{1} {m}_{{13}} {t}_{{23 }} +{ K}_{A} {d}_{2} {m}_{{12}} {t}_{{23}} {)/}\tau _{E} \\ {c}_{{o0}} ={(K}_{A} {m}_{{11}} {t}_{{33}} -{ K}_{A} {d}{ }_{0}{m}_{{12}} {t}_{{13 }} +{ K}_{A} {d}_{1} {m}_{{11}} {t}_{{13}} -{ K}_{A} {d}_{0} {m}_{{13}} {t}{ }_{{23}} \\ \quad \quad \quad +{ K}_{A} {d}_{2} {m}_{{11}} {t}_{{23}} {)/}\tau _{E} \\ \end{array}} \right\} \quad \hbox {Coefficients of Eq}.~(38) \\&\left. {\begin{array}{l} {d}_{{o3}} = -{ h}_{{21}} \\ {d}_{{o2}} ={K}_{5} {m}_{{21}} -{ d}_{2} {h}_{{21}} \\ {d}_{{o1}} ={K}_{5} {d}_{2} {m}_{{21 }} -{ K}_{5} {d}_{0} {m}_{{23}} -{ d}_{1} {h}_{{21}} \\ {d}_{{o0}} = -{ d}_{0} {h}_{{21}} -{ K}_{5} {d}_{0} {m}_{{22}} +{ K}_{5} {d}_{1} {m}_{{21}} \\ {e}_{{o3}} = -{ h}{ }_{{22}} \\ {e}_{{o2}} ={K}_{6} {m}_{{21}} -{ d}_{2} {h}_{{22}} \\ {e}_{{o1}} ={K}_{6} {d}_{2} {m}_{{21}} -{ K}_{6} {d}_{0} {m}_{{23}} -{ d}_{1} {h}_{{22}} \\ {e}_{{o0}} = -{ d}_{0} {h}_{{22}} -{ K}_{6} {d}_{0} {m}_{{22}} +{ K}_{6} {d}_{1} {m}_{{21}} \\ {f}_{{o2}} ={(K}_{_{A} } {m}{ }_{{21}}{t}_{{13}} +{ K}_{A} {m}_{{22}} {t}_{{23}} +{ K}_{A} {m}_{{23}} {t}_{{33}} {)/}\tau _{E} \\ {f}_{{o1}} ={(K}_{A} {m}_{{21}} {t}_{{23 }} +{ K}_{A} {m}_{{22}} {t}_{{33 }} -{ K}_{A} {d}_{0} {m}_{{23}} {t}_{{13}} \\ \quad \quad \quad +{ K}_{A} {d}_{2} {m}_{{21}} {t}_{{13}} -{ K}_{A} {d}_{1} {m}_{{23}} {t}_{{23}} +{ K}_{A} {d}_{2} {m}_{{22}} {t}_{{23}} {)/}\tau _{E} \\ {f}_{{o0}} ={(K}_{A} {m}_{{21}} {t}_{{33}} -{ K}_{A} {d}_{0} {m}_{{22}} {t}_{{13}} +{ K}_{A} {d}_{1} {m}_{{21}} {t}_{{13}} -{ K}_{A} {d}_{0} {m}_{{23}} {t}_{{23}} \\ \quad \quad \quad +{ K}_{A} {d}_{2} {m}_{{21}} {t}_{{23}} {)/}\tau _{E} \\ \end{array}} \right\} \quad \hbox {Coefficients of Eq}.~(39) \end{aligned}$$

$$\begin{aligned}&\left. {\begin{array}{l} {g}_{{o3}} = -{ h}_{{31}} \\ {g}_{{o2}} ={K}_{5} {m}_{{31}} -{ d}_{2} {h}_{{31}} \\ {g}_{{o1}} ={K}_{5} {d}_{2} {m}_{{31}} -{ K}_{5} {d}_{0} {m}_{{33}} -{ d}_{1} {h}_{{31}} \\ {g}_{{o0}} = -{ d}_{0} {h}_{{31}} -{ K}_{5} {d}_{0} {m}_{{32}} +{ K}_{5} {d}_{1} {m}_{{31}} \\ {h}_{{o3}} = -{ h}_{{32}} \\ {h}_{{o2}} ={K}_{6} {m}_{{31}} -{ d}_{2} {h}_{{32}} \\ {h}_{{o1}} ={K}_{6} {d}_{2} {m}_{{31}} -{ K}_{6} {d}_{0} {m}_{{33}} -{ d}_{1} {h}_{{32}} \\ {h}_{{o0}} = -{ d}_{0} {h}_{{32}} -{ K}_{6} {d}_{0} {m}_{{32}} +{ K}_{6} {d}_{1} {m}_{{31}} \\ {i}_{{o2}} ={(K}_{A} {m}_{{31}} {t}_{{13 }} +{ K}_{A} {m}_{{32}} {t}_{{23}} +{ K}_{A} {m}_{{33}} {t}_{{33}} {)/}\tau _{E} \\ {i}_{{o1}} ={(K}_{A} {m}_{{31}} {t}_{{23}} +{ K}_{A} {m}_{{32}} {t}_{{33}} -{ K}_{A} {d}_{0} {m}_{{33}} {t}_{{13}} +{ K}_{A} {d}_{2} {m}_{{31}} {t}_{{13}} \\ \quad \quad \quad -{ K}_{A} {d}_{1} {m}_{{33}} {t}_{{23 }} +{ K}_{A} {d}_{2} {m}_{{32}} {t}_{{23}} {)/}\tau _{E} \\ {i}_{{o0}} ={(K}_{A} {m}_{{31}} {t}_{{33}} -{ K}_{A} {d}_{0} {m}_{{32}} {t}_{{13}} +{ K}_{A} {d}_{1} {m}_{{31}} {t}_{{13}} -{ K}_{A} {d}_{0} {m}_{{33}} {t}_{{23 }} \\ \quad \quad \quad +{ K}_{A} {d}_{2} {m}_{{31}} {t}_{{23}} {)/}\tau _{E} \\ \end{array}} \right\} \quad \hbox {Coefficients of Eq}.~(40) \end{aligned}$$