Phase-locked loops
The input to a synchronous reference frame PLL (see figure) is$V_a = 325\,\sin\,(\omega \,t)\,$V,
$V_b = 320\,\sin\,(\omega \,t - 2\,\pi/3)\,$V,
$V_c = 325\,\sin\,(\omega \,t + 2\,\pi/3)\,$V,
with $f=50\,$Hz. The filter transfer function is given by
$G(s) = K_p\,\displaystyle\frac{1+s\,\tau}{s\,\tau}$.
The $abc$ to $dq$ transformation is given below.
$ \begin{equation} \left[ \begin{array}{c} u_d\cr u_q\cr u_0\cr \end{array} \right] = \displaystyle\frac{2}{3} \left[ \begin{array}{cccc} \cos(\omega t) & \cos(\omega t - 2\pi /3) & \cos(\omega t + 2\pi /3) \cr -\sin(\omega t) & -\sin(\omega t - 2\pi /3) & -\sin(\omega t + 2\pi /3) \cr 1/2 & 1/2 & 1/2 \end{array} \right] \left[ \begin{array}{c} u_a\cr u_b\cr u_c\cr \end{array} \right] \end{equation} $
Plot $\omega$, $V_d$, and $V_q$ versus time (in the steady state) for three values of bandwidth: $100\,$Hz, $50\,$Hz, and $10\,$Hz. What did you observe in the magnitude of the oscillations?
In [1]:
from IPython.display import Image
Image(filename =r'pll_1_fig_1.png', width=450)
Out[1]:
In [2]:
# run this cell to view the circuit file.
%pycat pll_3_orig.in
In [3]:
import numpy as np
import gseim_calc as calc
import os
import dos_unix
import matplotlib.pyplot as plt
from matplotlib.ticker import (MultipleLocator, AutoMinorLocator)
from setsize import set_size
import math
l_bw = [100.0, 50.0, 10.0]
l_vd = []
l_vq = []
l_omg = []
slv = calc.slv("pll_3_orig.in")
i_slv = 0
i_out = 0
filename = slv.l_filename_all[i_slv][i_out]
print('filename:', filename)
Vg = 325.0
zeta = 0.707
for k, bw in enumerate(l_bw):
wb = 2*math.pi*bw # Bandwidth (in rad/s)
Kp = (2*zeta*wb)/Vg
tau = (2*zeta)/wb
print("bandwidth: %11.4e"% bw)
print("Kp: %11.4e"%Kp)
print("tau: %11.4e"%tau)
s_Kp = "%11.4E"%Kp
s_tau = "%11.4E"%tau
l = [
('$Kp', s_Kp),
('$tau', s_tau),
]
calc.replace_strings_1("pll_3_orig.in", "pll_3.in", l)
# uncomment for windows:
#dos_unix.d2u("pll_3.in")
os.system('run_gseim pll_3.in')
u = np.loadtxt(filename)
t = u[:, 0]
t_end = t[-1]
# va, vb, vc do not change; we need to read them only once
if k == 0:
va = slv.get_array_double(i_slv, i_out, 'va', u)
vb = slv.get_array_double(i_slv, i_out, 'vb', u)
vc = slv.get_array_double(i_slv, i_out, 'vc', u)
vd = slv.get_array_double(i_slv, i_out, 'vd', u)
vq = slv.get_array_double(i_slv, i_out, 'vq', u)
omg = slv.get_array_double(i_slv, i_out, 'w', u)
l_vd.append(vd)
l_vq.append(vq)
l_omg.append(omg)
color1='red'
color2='goldenrod'
color3='blue'
color4='green'
color5='crimson'
color6='cornflowerblue'
fig, ax = plt.subplots(4, sharex=False)
plt.subplots_adjust(wspace=0, hspace=0.0)
set_size(5.5, 7, ax[0])
for i in range(4):
ax[i].set_xlim(left=0.0, right=t_end*1e3)
ax[i].grid(color='#CCCCCC', linestyle='solid', linewidth=0.5)
ax[0].plot(t*1e3, va, color=color1, linewidth=1.0, label="$V_a$")
ax[0].plot(t*1e3, vb, color=color2, linewidth=1.0, label="$V_b$")
ax[0].plot(t*1e3, vc, color=color3, linewidth=1.0, label="$V_c$")
ax[0].set_ylabel('voltage', fontsize=12)
ax[1].plot(t*1e3, l_omg[0], color=color4, linewidth=1.0, label="BW: "+"%3.0f"%l_bw[0])
ax[1].plot(t*1e3, l_omg[1], color=color5, linewidth=1.0, label="BW: "+"%3.0f"%l_bw[1])
ax[1].plot(t*1e3, l_omg[2], color=color6, linewidth=1.0, label="BW: "+"%3.0f"%l_bw[2])
ax[1].set_ylabel("$\\omega$", fontsize=12)
ax[2].plot(t*1e3, l_vd[0], color=color4, linewidth=1.0, label="BW: "+"%3.0f"%l_bw[0])
ax[2].plot(t*1e3, l_vd[1], color=color5, linewidth=1.0, label="BW: "+"%3.0f"%l_bw[1])
ax[2].plot(t*1e3, l_vd[2], color=color6, linewidth=1.0, label="BW: "+"%3.0f"%l_bw[2])
ax[2].set_ylabel("$V_d$", fontsize=12)
ax[3].plot(t*1e3, l_vq[0], color=color4, linewidth=1.0, label="BW: "+"%3.0f"%l_bw[0])
ax[3].plot(t*1e3, l_vq[1], color=color5, linewidth=1.0, label="BW: "+"%3.0f"%l_bw[1])
ax[3].plot(t*1e3, l_vq[2], color=color6, linewidth=1.0, label="BW: "+"%3.0f"%l_bw[2])
ax[3].set_ylabel("$V_q$", fontsize=12)
ax[3].set_xlabel('time (msec)', fontsize=12)
for i in range(4):
ax[i].legend(loc = 'lower right',frameon = True, fontsize = 10, title = None,
markerfirst = True, markerscale = 1.0, labelspacing = 0.5, columnspacing = 2.0,
prop = {'size' : 12},)
plt.tight_layout()
plt.show()
filename: pll_3.dat bandwidth: 1.0000e+02 Kp: 2.7337e+00 tau: 2.2505e-03 Circuit: filename = pll_3.in main: i_solve = 0 solve_ssf: ssf_iter_newton=0, rhs_ssf_norm=1.86475, ssf_period_1_compute=0.02 solve_ssf: ssf_iter_newton=1, rhs_ssf_norm=0.044041, ssf_period_1_compute=0.02 solve_ssf: ssf_iter_newton=2, rhs_ssf_norm=4.22232e-06, ssf_period_1_compute=0.02 solve_ssf: ssf_iter_newton=3, rhs_ssf_norm=2.1441e-10, ssf_period_1_compute=0.02 solve_ssf: ssf_iter_newton=4, rhs_ssf_norm=4.34918e-14, ssf_period_1_compute=0.02 solve_ssf: convergence reached. solve_ssf: calling ssf_solve_trns for one more trns step GSEIM: Program completed. bandwidth: 5.0000e+01 Kp: 1.3668e+00 tau: 4.5009e-03 Circuit: filename = pll_3.in main: i_solve = 0 solve_ssf: ssf_iter_newton=0, rhs_ssf_norm=5.70936, ssf_period_1_compute=0.02 solve_ssf: ssf_iter_newton=1, rhs_ssf_norm=0.921612, ssf_period_1_compute=0.02 solve_ssf: ssf_iter_newton=2, rhs_ssf_norm=0.00035094, ssf_period_1_compute=0.02 solve_ssf: ssf_iter_newton=3, rhs_ssf_norm=7.91054e-08, ssf_period_1_compute=0.02 solve_ssf: ssf_iter_newton=4, rhs_ssf_norm=6.00572e-11, ssf_period_1_compute=0.02 solve_ssf: convergence reached. solve_ssf: calling ssf_solve_trns for one more trns step GSEIM: Program completed. bandwidth: 1.0000e+01 Kp: 2.7337e-01 tau: 2.2505e-02 Circuit: filename = pll_3.in main: i_solve = 0 solve_ssf: ssf_iter_newton=0, rhs_ssf_norm=31.1973, ssf_period_1_compute=0.02 solve_ssf: ssf_iter_newton=1, rhs_ssf_norm=1.29224, ssf_period_1_compute=0.02 solve_ssf: ssf_iter_newton=2, rhs_ssf_norm=0.00117394, ssf_period_1_compute=0.02 solve_ssf: ssf_iter_newton=3, rhs_ssf_norm=2.57822e-07, ssf_period_1_compute=0.02 solve_ssf: ssf_iter_newton=4, rhs_ssf_norm=5.14543e-11, ssf_period_1_compute=0.02 solve_ssf: convergence reached. solve_ssf: calling ssf_solve_trns for one more trns step GSEIM: Program completed.
This notebook was contributed by Prof. Nakul Narayanan K, Govt. Engineering College, Thrissur. He may be contacted at nakul@gectcr.ac.in.