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Alex Nazarovsky

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.

Uabc=[Ua Ub Uc]=[Ua,0 Ub,0 Uc,0]+[Ua,1 Ub,1 Uc,1]+[Ua,2 Ub,2 Uc,2] αe23πi Uabc=[U0 U0 U0]+[U1 α2U1 αU1]+[U2 αU2 α2U2]=  =AU012

From these phasors we can derive negative-sequence voltage unbalance ratio K2U and zero-sequence voltage unbalance ratio K0U

K2U=U2U1 K0U=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.

K2U and K0U calculation
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clear all; 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=blue
K2U_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_ref
mag_UC=100;
% angles of voltages
phi_A  = degtorad(0);    % absolute angle of UA phasor
phi_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-phiA

UA=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 components
a=exp(1i*2*pi/3); % complex angle constant
a2=a^2;
PSP = 1/3* (1*UA  + a *UB  + a2*UC); % Positive sequence phasor
NSP = 1/3* (1*UA  + a2*UB  + a *UC); % Negative sequence phasor
ZSP = 1/3* (1*UA  + 1 *UB  + 1 *UC); % Zero sequence phasor
% calculate K2U and K0U based on symmetrical components
K2U=abs(NSP)/abs(PSP)*100 % negative sequence ratio
K0U=abs(ZSP)/abs(PSP)*100 % zero sequence ratio

ROTATE_UI=angle(UA); % draw phasors relative to UA
figure;
h=compass(UA*exp(-1i*ROTATE_UI),'r'); hold on;
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+2K2U1K2U

This equation can be easily derived if we notice that

U1K2U=13(Ub+2Ua)K2U=13(UbUa)=U2