Fast-slow bursting behaviors of hydroelectric governing system with double periodic excitations

ABSTRACT

In this article, we integrate the nonelastic water column model of the hydro-turbine with the third-order nonlinear generator model. Further, we introduce the periodic functions of the hydraulic derivative coefficient and the electric field voltage. Based on the novel integrated nonlinear mathematical model of the hydroelectric governing system and the double periodic excitations claimed from the fast-slow analysis method, the fast-slow bursting behaviors of the system are found. The nonlinear dynamic behaviors of the system regarding the derivative gain, excitation frequency, and excitation amplitude are illustrated via bifurcation diagrams, time waveforms, phase trajectories, and power spectrums. The results show that the governing system sustains distinct kinds of nonlinear dynamic behaviors depending on the sensitive parameter values. The system can escape from the fast-slow bursting phenomena when ${\mathrm{k}}_{\mathrm{d}}$ grows larger. The increase of $\mathrm{\Omega }$ leads the system to the stable state. However, the increase of $\mathrm{B}$ leads the system to the robust fast-slow bursting state. Finally, the analytical method and the results of this article provide principal references for the sensitive parameter setting to guard the hydroelectric governing system from the fast-slow bursting behaviors and ensure the safe and stable operation of hydroelectric power stations.

KEYWORDS:

• Hydroelectric governing system
• periodic excitation
• fast-slow bursting behaviors
• sensitive parameters

Disclosure statement

No potential conflict of interest was reported by the authors.

Nomenclature

$A$ and $B$amplitude of the periodic excitation

$D$damping coefficient

$e$derivative coefficient of the hydraulic system, p.u.

$e_{mx}$, $e_{my}$, $e_{mh}$partial derivatives of the mechanical torque on the hydro-turbine speed, guide vane, and head, p.u.

$e_{qx}$, $e_{qy}$, $e_{qh}$partial derivatives of the turbine discharge on the hydro-turbine speed, guide vane, and head, p.u.

$E_f$electric field voltage, p.u.

$E_q^{\prime }$quadrature axis transient electric potential, p.u.

$h$hydro-turbine head deviation, p.u.

$i_d$direct axis armature current, p.u.

$k_d$, $k_i$, and $k_p$derivative, integral, and proportional gain

$m_e$generator electric torque, p.u.

$m_t$hydro-turbine mechanical torque deviation, p.u.

$P_e$electric power, p.u.

$q$hydro-turbine discharge deviation, p.u.

$T_{ab}$mechanical starting time, s

$T_{d0}^{\prime }$direct axis transient open-circuit time constant, s

$T_w$water time constant, s

$T_y$engager relay time constant, s

$u$control signal of the governor system

$V_s$infinite-bus voltage, p.u.

$x$hydro-turbine speed deviation, p.u.

$x_d$direct axis synchronous reactance, p.u.

$x_d^{\prime }$direct axis transient reactance, p.u.

$x_L$transmission-line reactance, p.u.

$x_q$quadrature axis synchronous reactance, p.u.

$x_T$transformer reactance, p.u.

$y$guide vane opening deviation, p.u.

$\delta$rotor angle relative deviation, p.u.

$\mathrm{\Omega }$frequency of the periodic excitation

$\omega$rotor speed relative deviation, p.u.

${\omega }_0$base angular speed, rad/s