MATLAB, DSP, Julia, Power quality, Engineering, Metrology
How to Evaluate Voltage Unbalance Using Symmetrical Components Method, and Determine Magnitudes by Known K2U (K0U)
To evaluate voltage unbalance we can use Symmetrical components method proposed by Charles Fortescue.
Vector of three phase voltages can be expressed as a sum of three vectors (phasors): zero- positive- and negative-sequence.
From these phasors we can derive negative-sequence voltage unbalance ratio K2U and zero-sequence voltage unbalance ratio K0U
K2U=U2U1K0U=U0U1
Here is MATLAB script that produces the desired level of both negative- and -zero voltage unbalance ratio, then sets respective phasor magnitudes.
After that we estimate unbalance using Symmetrical components method and draw a phasor diagramm.
clearall;clc;% K2U and K0U calculator% in three phase system we accept that UB lags UA, and UC leads UA% in balanced three phase system all angles are 120 deg=2*pi/3 rad% color coding: UA=red UB=green UC=blueK2U_ref=0.05;% set K2U = 5%% magnitudes of the three phase voltages (rms)mag_UA=100;mag_UB=mag_UA*(1+2*K2U_ref)/(1-K2U_ref);% set magnitude so that K2U=K0U=K2U_refmag_UC=100;% angles of voltagesphi_A=degtorad(0);% absolute angle of UA phasorphi_AB=degtorad(-120);% relative angle between UA phasor and UB phasor% = phi_B-phiA phi_AC=degtorad(120);% relative angle between UA phasor and UC phasor% = phi_C-phiAUA=mag_UA*exp(1i*(phi_A));UB=mag_UB*exp(1i*(phi_A+phi_AB));UC=mag_UC*exp(1i*(phi_A+phi_AC));% calculate symmetrical componentsa=exp(1i*2*pi/3);% complex angle constanta2=a^2;PSP=1/3*(1*UA+a*UB+a2*UC);% Positive sequence phasorNSP=1/3*(1*UA+a2*UB+a*UC);% Negative sequence phasorZSP=1/3*(1*UA+1*UB+1*UC);% Zero sequence phasor% calculate K2U and K0U based on symmetrical componentsK2U=abs(NSP)/abs(PSP)*100% negative sequence ratioK0U=abs(ZSP)/abs(PSP)*100% zero sequence ratioROTATE_UI=angle(UA);% draw phasors relative to UAfigure;h=compass(UA*exp(-1i*ROTATE_UI),'r');holdon;set(h,'LineWidth',2);h=compass(UB*exp(-1i*ROTATE_UI),'g');set(h,'LineWidth',2);h=compass(UC*exp(-1i*ROTATE_UI),'b');set(h,'LineWidth',2);legend('UA','UB','UC');
To generate signal with known level of unbalance K2U we can change magnitude of one of the voltages in the balanced three phase system.
So we set all angles between voltages = 120 deg. Magnitudes of Ua and Uc should be equal (Ua=Uc). Ub should be set as
Ub=Ua1+2K2U1−K2U
This equation can be easily derived if we notice that