Calculating features of a 3 phase power timeseries

regarding Total, active, reactive Power and $cos phi$

1. Estimate the energy in a timeperiod T for every timestep ($t in [t_0,t_0+T]$) by using the formula

$$P = frac{1}{T}sum_{t=i} I_{A,i} V_{A,i}dt + frac{1}{T}sum_{t=i} I_{B,i} V_{B,i}dt + frac{1}{T}sum_{t=i} I_{C,i} V_{C,i}dt$$

where:

• $dt$: is the timestep (e.g. 1.25ms)
• $T$ is the integration period
• $frac{1}{T}sum_{t=i} I_{k,i} V_{k,i}dt$: the active power of each line k $in {A,B,C}$

2. Find the RMS values for Voltage and Current ($V_{k, RMS}, I_{k, RMS}$) for each line (A,B,C) in the time period.

The total power for each line is $$P_{k,T} = V_{RMS} * I_{RMS}$$

make sure you don't confuse line with phase (see star and delta configurations)

3. The reactive power will be the difference between the two (Total-Active).

4. if you want the $cos phi$ for each line then just do

$$phi_k = arccos left(frac{P_{k,Active}}{P_{k,Total}}right)$$

Caveat: This is a numerical estimation. Depending on the duration of integration, you might get some strange results (e.g. Total Power less than Active). That is why you should prefer complete periods if possible only one at a time (longer time periods will tend to average too much the data).

Regarding the grid frequency:

what you should do is perform an fft and find the peak frequencies. For a signal like that, you would need to add a window (usually Hanning, or Hamming) and also get a longer time period (e.g. 10 or more standard grid frequency periods).

Python Code

I am also adding some python code, just for verification of the above:

#%%
import numpy as np
import matplotlib.pyplot as plt
# %%

Tmax = 1/50 #integratin period
f_g = 50  # grid frequency 
dt = 0.00125

ts = np.arange(0, Tmax, step=dt)


# %%  Plot 3 line voltages
Vmax = 10
V_As = Vmax*np.cos(2*np.pi*f_g*ts)
V_Bs = Vmax*np.cos(2*np.pi*(f_g*ts + 1/3))
V_Cs = Vmax*np.cos(2*np.pi*(f_g*ts + 2/3))

plt.plot(ts, V_As, label='A')
plt.plot(ts, V_Bs, label='B')
plt.plot(ts, V_Cs, label='C')
plt.legend()

# %%  Estimation of Power in line A, for a given impedance R_A (complex)
R_A = 2+1/2*np.pi*1j
angle = np.angle(R_A)  # actual cos phi
print(angle)
I_As  = np.real(Vmax/np.abs(R_A) * np.exp(1j* (2*np.pi*f_g*ts + np.angle(R_A))))

# plots V and I for verification
plt.plot(ts, V_As, label='V_A')
plt.plot(ts, I_As, label='I_A')
plt.legend()
plt.grid()
# %%
def calc(Vs,Is):
    P_activ = np.sum(Vs*Is)*dt/Tmax
    Vrms = np.std(Vs)
    Irms = np.std(Is)
    P_total = Irms*Vrms
    phi = np.arccos(P_activ/P_total)
    print ('Active :{:.3f}'.format(P_activ))
    print ('Vrms  :{:.3f}'.format(Vrms))
    print ('Irms  :{:.3f}'.format(Irms))
    print ('P_total  :{:.3f}'.format(P_total))
    print ('phi  :{:.3f}[rad] = {:.3f}'.format(phi, phi*180/np.pi))
    
calc(V_As, I_As)