Fsolve no solution with Exitflag = 0

I am using the following main code

clear
format short
tole=1E-9;
deg=180/pi;
rad=1/deg;

%Input Values
A=1.5; Sbs=100; Ubs=220; Zbs=Ubs^2/Sbs;
r=0.0529; x=0.529; bc=3.3081*1E-6;
L12=150; L13=170; L23=180; L34=100; L45=200; L56=100;

%Bus%
nbus=6;
%Bus 1, slack
U1=220/Ubs; the1=(-1.5)*rad; PL1=50/Sbs; QL1=5/Sbs;
%Bus 2, PQ
PG2=100/Sbs; QG2=10/Sbs; PL2=(200+0.1*A)/Sbs; QL2=20/Sbs;
%Bus 3, PQ
PG3=150/Sbs; QG3=15/Sbs; PL3=300/Sbs; QL3=30/Sbs;
%Bus 4, PQ
PG4=0/Sbs; QG4=0/Sbs; PL4=0/Sbs; QL4=0/Sbs; 
Qsh4 = (25-(0.2*A)); Qsh4_pu = j*Qsh4*Zbs/Ubs^2;
%Bus 5, PQ
PG5=200/Sbs; QG5=20/Sbs; PL5=350/Sbs; QL5=(40-0.1*A)/Sbs;
%Bus 6, PQ
PG6=200/Sbs; QG6=10/Sbs; PL6=100/Sbs; QL6=20/Sbs;

%Line Impedances%
Z12=(r+j*x)*L12/Zbs; Y12=(L12*(j*bc))*Zbs/2;
Z13=(r+j*x)*L13/Zbs; Y13=(L13*(j*bc))*Zbs/2;
Z23=(r+j*x)*L23/Zbs; Y23=(L23*(j*bc))*Zbs/2;
Z34=(r+j*x)*L34/Zbs; Y34=(L34*(j*bc))*Zbs/2;
Z45=(r+j*x)*L45/Zbs; Y45=(L45*(j*bc))*Zbs/2;
Z56=(r+j*x)*L56/Zbs; Y56=(L56*(j*bc))*Zbs/2;

Z21=Z12; Y21=Y12;
Z31=Z13; Y31=Y13;
Z32=Z23; Y32=Y23;
Z43=Z34; Y43=Y34;
Z54=Z45; Y54=Y45;
Z65=Z56; Y65=Y56;

%Admittance Matrix%
%Diagonal
y11=(1/Z12)+Y12+(1/Z13)+Y13;
y22=(1/Z12)+Y12+(1/Z23)+Y23;
y33=(1/Z13)+Y13+(1/Z23)+Y23+(1/Z34)+Y34;
y44=(1/Z34)+Y34+(1/Z45)+Y45+Qsh4_pu;
y55=(1/Z45)+(1/Z56)+Y45+Y56;
y66=(1/Z56)+Y56;
%Non-Diagnoal
y12=-1/Z12; y13=-1/Z13; y14=0; y15=0; y16=0;
y21=y12; y23=-1/Z23; y24=0; y25=0; y26=0;
y31=y13; y32=Y23; y34=-1/Z34; y35=0; y36=0;
y41=y14; y42=y24; y43=y34; y45=-1/Z45; y46=0;
y51=y15; y52=y25; y53=y35; y54=y45; y56=-1/Z56;
y61=y16; y62=y26; y63=y36; y64=y46; y65=y56;

Y=[ y11 y12 y13 y14 y15 y16
    y21 y22 y23 y24 y25 y26
    y31 y32 y33 y34 y35 y36
    y41 y42 y43 y44 y45 y46
    y51 y52 y53 y54 y55 y56
    y61 y62 y63 y64 y65 y66 ];

G=real(Y); B=imag(Y);

%Bus Generation in PQ buses
PGD2=PG2-PL2; QGD2=QG2-QL2;
PGD3=PG3-PL3; QGD3=QG3-QL3;
PGD4=PG4-PL4; QGD4=QG4-QL4;
PGD5=PG5-PL5; QGD5=QG5-QL5;
PGD6=PG6-PL6; QGD6=QG6-QL6;
PGD=[PGD2; PGD3; PGD4; PGD5; PGD6]; 
QGD=[QGD2; QGD3; QGD4; QGD5; QGD6];

%unknown variables [ theta2 theta3 theta4 theta5 theta6 U2 U3 U4 U5 U6 ]
X0   = [0 0 0 0 0 1 1 1 1 1]';
s_z  = size(X0);
nx   = s_z(1,1);

options_solve = optimset('Display', 'off', 'TolX', tole, 'TolFun', tole);
PAR  = [nx ; nbus ; U1 ; the1];
[X_X, FVAL, EXITFLAG, OUTPUT] = fsolve('solve_LF_A18', X0, options_solve, G, B, PGD, QGD, PAR);

if EXITFLAG~=1,
   disp('No Solution'),
   EXITFLAG=EXITFLAG,
   return
end,

the=[the1 X_X(1) X_X(2) X_X(3) X_X(4) X_X(5)]; %phase angle
U=[U1 X_X(6) X_X(7) X_X(8) X_X(9) X_X(10)]; %voltage magnitude

the_deg=the*deg;
U_kv=U*Ubs;

%Generated Active & Reactive power on non-PQ bus
g=-G; b=-B;

%at bus 1 (slack)
P12=g(1,2)*U(1)^2-U(1)*U(2)*(g(1,2)*cos(the(1)-the(2))+b(1,2)*sin(the(1)-the(2)));
P13=g(1,3)*U(1)^2-U(1)*U(3)*(g(1,3)*cos(the(1)-the(3))+b(1,3)*sin(the(1)-the(3)));
Q12=(-abs(Y12)-b(1,2))*U(1)^2-U(1)*U(2)*(g(1,2)*sin(the(1)-the(2))-b(1,2)*cos(the(1)-the(2)));
Q13=(-abs(Y13)-b(1,3))*U(1)^2-U(1)*U(3)*(g(1,3)*sin(the(1)-the(3))-b(1,3)*cos(the(1)-the(3)));

P1=P12+P13;
Q1=Q12+Q13;

PG1=P12+P13+PL1;
QG1=Q12+Q13+QL1;

PG1_MW=PG1*Sbs;
QG1_MVAr=QG1*Sbs;

%Active - Reactive power in each lines
%at bus 2
P21=g(2,1)*volt(2)^2-volt(2)*volt(1)*(g(2,1)*cos(ang(2)-ang(1))+b(2,1)*sin(ang(2)-ang(1)));
P23=g(2,3)*volt(2)^2-volt(2)*volt(3)*(g(2,3)*cos(ang(2)-ang(3))+b(2,3)*sin(ang(2)-ang(3)));
Q21=(-abs(Y21)-b(2,1))*volt(2)^2-volt(2)*volt(1)*(g(2,1)*sin(ang(2)-ang(1))-b(2,1)*cos(ang(2)-ang(1)));
Q23=(-abs(Y23)-b(2,3))*volt(2)^2-volt(2)*volt(3)*(g(2,3)*sin(ang(2)-ang(3))-b(2,3)*cos(ang(2)-ang(3)));
PG2=P21+P23+PL2;
QG2=Q21+Q23+QL2;

%at bus 3
P31=g(3,1)*volt(3)^2-volt(3)*volt(1)*(g(3,1)*cos(ang(3)-ang(1))+b(3,1)*sin(ang(3)-ang(1)));
P32=g(3,2)*volt(3)^2-volt(3)*volt(2)*(g(3,2)*cos(ang(3)-ang(2))+b(3,2)*sin(ang(3)-ang(2)));
P34=g(3,4)*U(3)^2-U(3)*U(4)*(g(3,4)*cos(the(3)-the(4))+b(3,4)*sin(the(3)-the(4)));
Q31=(-abs(Y31)-b(3,1))*volt(3)^2-volt(3)*volt(1)*(g(3,1)*sin(ang(3)-ang(1))-b(3,1)*cos(ang(3)-ang(1)));
Q32=(-abs(Y32)-b(3,2))*volt(3)^2-volt(3)*volt(2)*(g(3,2)*sin(ang(3)-ang(2))-b(3,2)*cos(ang(3)-ang(2)));
Q34=(-abs(Y34)-b(3,4))*volt(3)^2-volt(3)*volt(4)*(g(3,4)*sin(ang(3)-ang(4))-b(3,4)*cos(ang(3)-ang(4)));
PG3=P31+P32+P34+PL3;
QG3=Q31+Q32+Q34+QL3;

%at bus 4
P43=g(4,3)*volt(4)^2-volt(4)*volt(3)*(g(4,3)*cos(ang(4)-ang(3))+b(4,3)*sin(ang(4)-ang(3)));
P45=g(4,5)*volt(4)^2-volt(4)*volt(5)*(g(4,5)*cos(ang(4)-ang(5))+b(4,5)*sin(ang(4)-ang(5)));
Q43=(-abs(Y43)-b(4,3))*volt(4)^2-volt(4)*volt(3)*(g(4,3)*sin(ang(4)-ang(3))-b(4,3)*cos(ang(4)-ang(3)));
Q45=(-abs(Y45)-b(4,5))*volt(4)^2-volt(4)*volt(5)*(g(4,5)*sin(ang(4)-ang(5))-b(4,5)*cos(ang(4)-ang(5)));
PG4=P43+P45+PL4;
QG4=Q43+Q45+QL4;

%at bus 5
P54=g(5,4)*volt(5)^2-volt(5)*volt(4)*(g(5,4)*cos(ang(5)-ang(4))+b(5,4)*sin(ang(5)-ang(4)));
P56=g(5,6)*volt(5)^2-volt(5)*volt(6)*(g(5,6)*cos(ang(5)-ang(6))+b(5,6)*sin(ang(5)-ang(6)));
Q54=(-abs(Y54)-b(5,4))*volt(5)^2-volt(5)*volt(4)*(g(5,4)*sin(ang(5)-ang(4))-b(5,4)*cos(ang(5)-ang(4)));
Q56=(-abs(Y56)-b(5,6))*volt(5)^2-volt(5)*volt(6)*(g(5,6)*sin(ang(5)-ang(6))-b(5,6)*cos(ang(5)-ang(6)));
PG5=P54+P56+PL5;
QG5=Q54+Q56+QL5;

%at bus 6
P65=g(6,5)*volt(6)^2-volt(6)*volt(5)*(g(6,5)*cos(ang(6)-ang(5))+b(6,5)*sin(ang(6)-ang(5)));
Q65=(-abs(Y65)-b(6,5))*volt(6)^2-volt(6)*volt(5)*(g(6,5)*sin(ang(6)-ang(5))-b(6,5)*cos(ang(6)-ang(5)));
PG6=P65+PL6;
QG6=Q65+QL6;

%Losses%
Ploss_tot=(PG1+PG2+PG3+PG4+PG5+PG6)-(PL1+PL2+PL3+PL4+PL5+PL6);
Ploss_tot_MW=Ploss_tot*Sbs

Ploss_sys1=(PG1+PG2+PG3)-(PL1+PL2+PL3+P34);
Ploss_sys1_MW=Ploss_sys1*Sbs

Ploss_sys2=(PG4+PG5+PG6+P34)-(PL4+PL5+PL6);
Ploss_sys2_MW=Ploss_sys2*Sbs

and fsolve function as below

function [g_x] = solve_LF_A18(X,G,B,PGD,QGD,PAR);

nx=PAR(1); nbus=PAR(2); U1=PAR(3); the1=PAR(4);
PGD2=PGD(1); PGD3=PGD(2); PGD4=PGD(3); PGD5=PGD(4); PGD6=PGD(5);
QGD2=QGD(1); QGD3=QGD(2); QGD4=QGD(3); QGD5=QGD(4); QGD6=QGD(5);
the2=X(1); the3=X(2); the4=X(3); the5=X(4); the6=X(5);
U2=X(6); U3=X(7); U4=X(8); U5=X(9); U6=X(10);

the  = [the1 the2 the3 the4 the5 the6]';
U    = [U1 U2 U3 U4 U5 U6]';

g_x=zeros(nx,1);

P2=  U(2)*(U(1)*(G(2,1)*cos(the(2)-the(1))+B(2,1)*sin(the(2)-the(1)))...
    +U(3)*(G(2,3)*cos(the(2)-the(3))+B(2,3)*sin(the(2)-the(3)))...
    +U(2)*(G(2,2)*cos(the(2)-the(2))+B(2,2)*sin(the(2)-the(2))));
P3=  U(3)*(U(1)*(G(3,1)*cos(the(3)-the(1))+B(3,1)*sin(the(3)-the(1)))...
    +U(2)*(G(3,2)*cos(the(3)-the(2))+B(3,2)*sin(the(3)-the(2)))...
    +U(4)*(G(3,4)*cos(the(3)-the(4))+B(3,4)*sin(the(3)-the(4)))...
    +U(3)*(G(3,3)*cos(the(3)-the(3))+B(3,3)*sin(the(3)-the(3))));
P4=  U(4)*(U(3)*(G(4,3)*cos(the(4)-the(3))+B(4,3)*sin(the(4)-the(3)))...
    +U(5)*(G(4,5)*cos(the(4)-the(5))+B(4,5)*sin(the(4)-the(5)))...
    +U(4)*(G(4,4)*cos(the(4)-the(4))+B(4,4)*sin(the(4)-the(4))));
P5=  U(5)*(U(4)*(G(5,4)*cos(the(5)-the(4))+B(5,4)*sin(the(5)-the(4)))...
    +U(6)*(G(5,6)*cos(the(5)-the(6))+B(5,6)*sin(the(5)-the(6)))...
    +U(5)*(G(5,5)*cos(the(5)-the(5))+B(5,5)*sin(the(5)-the(5))));
P6=  U(6)*(U(5)*(G(6,5)*cos(the(6)-the(5))+B(6,5)*sin(the(6)-the(5)))...
    +U(6)*(G(6,6)*cos(the(6)-the(6))+B(6,6)*sin(the(6)-the(6)))); 

Q2=  U(2)*(U(1)*(G(2,1)*sin(the(2)-the(1))-B(2,1)*cos(the(2)-the(1)))...
    +U(3)*(G(2,3)*sin(the(2)-the(3))-B(2,3)*cos(the(2)-the(3)))...
    +U(2)*(G(2,2)*sin(the(2)-the(2))-B(2,2)*cos(the(2)-the(2))));
Q3=  U(3)*(U(1)*(G(3,1)*sin(the(3)-the(1))-B(3,1)*cos(the(3)-the(1)))...
    +U(2)*(G(3,2)*sin(the(3)-the(2))-B(3,2)*cos(the(3)-the(2)))...
    +U(4)*(G(3,4)*sin(the(3)-the(4))-B(3,4)*cos(the(3)-the(4)))...
    +U(3)*(G(3,3)*sin(the(3)-the(3))-B(3,3)*cos(the(3)-the(3))));
Q4=  U(4)*(U(3)*(G(4,3)*sin(the(4)-the(3))-B(4,3)*cos(the(4)-the(3)))...
    +U(5)*(G(4,5)*sin(the(4)-the(5))-B(4,5)*cos(the(4)-the(5)))...
    +U(4)*(G(4,4)*sin(the(4)-the(4))-B(4,4)*cos(the(4)-the(4))));
Q5=  U(5)*(U(4)*(G(5,4)*sin(the(5)-the(4))-B(5,4)*cos(the(5)-the(4)))...
    +U(6)*(G(5,6)*sin(the(5)-the(6))-B(5,6)*cos(the(5)-the(6)))...
    +U(5)*(G(5,5)*sin(the(5)-the(5))-B(5,5)*cos(the(5)-the(5))));
Q6=  U(6)*(U(5)*(G(6,5)*sin(the(6)-the(5))-B(6,5)*cos(the(6)-the(5)))...
    +U(6)*(G(6,6)*sin(the(6)-the(6))-B(6,6)*cos(the(6)-the(6)))); 

g_x(1)  =   P2-PGD2;
g_x(2)  =   P3-PGD3;
g_x(3)  =   P4-PGD4;
g_x(4)  =   P5-PGD5;
g_x(5)  =   P6-PGD6;

g_x(6)  =   Q2-QGD2;
g_x(7)  =   Q3-QGD3;
g_x(8)  =   Q4-QGD4;
g_x(9)  =   Q5-QGD5;
g_x(10) =   Q6-QGD6;

end

When I run, it comes off as

>> LF_A18
No Solution

EXITFLAG =

     0

I take the function solution is divergent, therefore no solution, but I believe my parameters are properly defined

thanks,

[John D'Errico]

Literally impossible for us to track down. While you BELIEVE things are proerly defined, we do not know they are. And apparently you believe the code is written properly. Of course, there could be a bug in code, written with virtually no comments, and the few comments I see are little more than hints to anyone but you.

Does a solution exist? It might. But not all nonlinear systems of equations have simply found solutions. In fact, most systems of nonlinear equations tend to have multiple solutions, and depending on the starting values, might lead to a divergent non-solution. So are your starting values any good? How do we know that?

Lets see...

X0   = [0 0 0 0 0 1 1 1 1 1]';

So a set of starting values that are probably completely useless, not much better than a completely random guess.

For a 10 variable problem? I'm not at all surprised that fsolve gave up the ghost.