Loss of excitation of SMIB power system equipped with high photovoltaic penetration with constant injected voltage

Abstract

The effect of high photovoltaic (PV) penetration with constant injected voltage on the loss of excitation phenomenon of single machine infinite bus power system with two transmission lines has been investigated. The PV modules are integrated to the power system at the same bus of the synchronous generator via DC–DC buck–boost switch mode converter, three-phase DC–AC switch mode inverter, LC filter and transformer. The constant injected voltage from the PV generator at the common coupling point at all system running conditions is achieved by controlling the duty cycle the DC–DC converter. To ensure the robustness of this integration, the system response based on a complete nonlinear dynamical mathematical model in dq stationary reference frame after step changes on the synchronous generator input mechanical power at different practical solar irradiance levels is first observed. Then, the response of the system after successive step changes on the solar irradiance levels is highlighted and compared. The loss of excitation of the synchronous generator is detected by extracting the trajectory of the impedance of the synchronous generator seen across its terminals at two practical solar irradiance levels as compared with the case of conventionally powered system without PV generator. It is concluded that the setting of the distance mho relay which works properly in case the power system is conventionally powered by the synchronous generator alone cannot detect the loss of excitation when the PV generator is integrated with the power system at all realistic solar irradiance levels. Therefore, a new setting for the distance relay has to be carried out. This new setting must be accompanied with a shift to the right such that the trajectory of the synchronous generator terminal impedance can enter the circle of the mho distance relay which in turn results in shorter tripping time in case of conventionally powered power system. All numerical simulations have been executed using MATLAB environment by constructing the code required.

Appendices

Appendix 1

The following are the physical explanations of all system variables:

D : :

Duty cycle of the DC–DC converter.

$\frac{\text {d}}{\text {d}t}:$ :

Differential operator with respect to time.

E : :

Synchronous generator excitation voltage.

$i_{dg} , i_{qg} :$ :

d-Axis and q-axis components of the synchronous generator armature current, respectively.

$i_{dinj} , i_{qinj} :$ :

d-Axis and q-axis components of the PV generator injected current, respectively.

$i_{dPV} , i_{qPV} :$ :

d-Axis and q-axis components of the PV generator current through the inductance of the LC filter, respectively.

$i_{fd} :$ :

d-Axis field winding current of the synchronous generator.

$i_{kd} , i_{kq} :$ :

d-Axis and q-axis components of the damper winding currents of the synchronous generator, respectively.

$L_C :$ :

Inductance of the DC–DC converter.

$C_C :$ :

Capacitance of the DC–DC converter.

$m_a :$ :

Amplitude modulation index of the PWM of the inverter.

$m_f :$ :

Frequency modulation index of the PWM of the inverter.

$M_g :$ :

Synchronous generator and prime mover moment of inertia.

$R_1 , X_1 :$ :

Resistance and reactance of the first transmission line, respectively.

$R_2 , X_2 :$ :

Resistance and reactance of the second transmission line, respectively.

$R_{eq} , X_{eq} :$ :

Equivalent resistance and reactance of the synchronous generator transformer and transmission lines, respectively.

$R_{Tg} , X_{Tg} :$ :

Resistance and reactance of the synchronous generator transformer, respectively.

$R_{TPV} , X_{TPV} :$ :

Resistance and reactance of the PV generator transformer, respectively.

$r_a :$ :

Synchronous generator armature resistance.

$r_{fd} :$ :

d-Axis resistance of the field winding of the synchronous generator.

$r_{kd} , r_{kq} :$ :

d-Axis and q-axis resistances of the synchronous generator damper windings, respectively.

$T_D :$ :

Time constant of the duty cycle of the DC–DC converter.

$T_m :$ :

Synchronous generator input mechanical torque.

$T_s :$ :

Switching interval of the DC–DC converter switch.

$t_{on} :$ :

On-state time interval of the DC–DC converter switch.

$V_C :$ :

Magnitude of the voltage across the capacitor of the LC filter.

$V_d , I_d :$ :

Terminal DC voltage and output DC current of the PV generator, respectively.

$V_{dC} , V_{qC} :$ :

d-Axis and q-axis components of the voltage across the capacitor of the LC filter, respectively.

$V_{dinj} , V_{qinj} :$ :

d-Axis and q-axis components of the PV generator injected voltage at the common coupling point with the power system, respectively.

$V_{inj} :$ :

Magnitude of the injected voltage from the PV generator at the common coupling point with the power system.

$V_o :$ :

Magnitude of the infinite bus voltage.

$V_{PV} :$ :

PV generator voltage (across the output terminals of the inverter).

$V_{set} :$ :

Pre-defined set voltage value.

$X_{afd} :$ :

d-Axis mutual inductance between armature and field winding of the synchronous generator.

$X_{akd} , X_{akq} :$ :

d-Axis and q-axis mutual inductance between synchronous generator armature and damper windings, respectively.

$X_d , X_q :$ :

d-Axis and q-axis synchronous generator armature reactances, respectively.

$X_d^\backslash :$ :

Transient reactance of synchronous generator.

$X_{ffd} :$ :

d-Axis self inductance of the field winding of the synchronous generator.

$X_{fkd} :$ :

d-Axis mutual inductance between field and damper windings of the synchronous generator.

$X_{kkd} , X_{kkq} :$ :

d-Axis and q-axis self inductance of the damper windings of the synchronous generator, respectively.

$X_{LF} , X_{CF} :$ :

Inductive and capacitive reactances of the LC filter, respectively.

$\alpha _n :$ :

polynomial curve fitting constants of the output characteristics of the PV generator.

$\delta _g :$ :

Delta (load) angle of the synchronous generator.

$\delta _{inj} :$ :

Delta angle of the injected voltage from the PV generator at the common coupling point with the power system.

$\delta _{PV} :$ :

Delta angle of the PV generator voltage (across the output terminals of the inverter).

$\omega _g :$ :

Synchronous generator rotational speed.

$\omega _o :$ :

Nominal system frequency (377 rad/s).

The following are the physical explanations of all system acronyms:

DC::

Direct current.

AC::

Alternative current.

pu::

Per-unit.

LC::

Inductor–capacitor low-pass filter.

PV::

Photovoltaic.

PWM::

Pulse width modulation.

SMIB::

Single machine infinite bus.

Appendix 2

The following are the numerical values of the system parameters in pu on the base of the synchronous generator which are 600 MVA and 22 kV:

$M_g =5.3703, \quad R_1 =0.0444, \quad X_1 =0.4800, \quad R_2 =0.0402, \quad X_2 =0.4434, \quad R_{eq} =0.0211, \quad X_{eq} =0.2305, R_{Tg} =0.0012, \quad X_{Tg} =0.1200, \quad R_{TPV} =0.0010, X_{TPV} =0.1000, \quad r_a =0.0045, \quad r_{fd} =0.00096, \quad r_{kd} =0.0160, \quad r_{kq} =0.0116, \quad T_D =0.0010\hbox {s}$, $V_{set} =1.0800$, $X_{afd} =1.5100$, $X_{akd} =1.5100$, $X_{akq} =1.4500$, $X_d =1.6500$, $X_q =1.5900$, $X_{ffd} =1.6286$, $X_{fkd} =1.5100$, $X_{kkd} =1.6420$, $X_{kkq} =1.5238$, $X_{LF} =0.0012$, $X_{CF} =0.1250$.

Appendix 3

The following table shows the values of the constants of the polynomial curves fitting of the output characteristics of the PV generator at the three solar irradiance levels considered in this study.

Constants	At full solar irradiance	At 75 % of full solar irradiance	At 50 % of full solar irradiance
$\alpha _1$	$-$80936.0525872129	$-$1365378.281869848	$-$72933095.7073894
$\alpha _2$	308715.9958017707	3905992.161045680	139095079.0684451
$\alpha _3$	$-$502063.9075526456	$-$4764227.592489641	$-$113104957.0934593
$\alpha _4$	453813.5228187755	3229774.449151762	51117555.2103065
$\alpha _5$	$-$249149.6497895185	$-$1329891.793297683	$-$14032108.2761456
$\alpha _6$	85260.4644182314	341322.550582337	2400934.6780174
$\alpha _7$	$-$17951.1087488245	$-$53897.649976095	$-$252751.6111834
$\alpha _8$	2208.2248065138	4972.595120346	15545.9026379
$\alpha _9$	$-$142.3967094320	$-$240.492220404	$-$501.2364172
$\alpha _{10}$	3.4631760374	4.386689648	6.0951898
$\alpha _{11}$	1.4611834147	1.388124244	1.2858414