System of 4 non-linear equations yields Empty sym: 0-by-1

I am trying to resolve a system of 4 non-linear (exponential) equations using vpasolve. The system reprensents the 4 equations needed to characterize the electrical model of a solar cell from the values given in datasheets (, and ). However, the solutions for the 4 desired values ( and ) are all "Empty sym: 0-by-1".

The equations forming the system are:

1)

2)

3)

4)

My code is

%parameters
I_sc = 0.473;
V_oc = 2.6;
V_m = 2.32;
I_m = 0.455;
R_s = 0.001;
T = 313.5;
k = 1.38e-23;
e = 1.6e-19;
c = 1;
g = k*T;

%unknowns : a = I_in, b = I_diode, c = gamma, d = R_sh

syms a b c d

eq1=  a - b * (exp(e*(R_s*I_sc)/c*g)-1) - (1/d)*R_s*I_sc == I_sc;

eq2= a - b * (exp(e*(V_oc)/c*g)-1) - (1/d)*V_oc == 0;

eq3=  a - b * (exp(e*(V_m + R_s*I_m)/c*g)-1) - (1/d)*(V_m + R_s*I_m) == I_m;

eq4= I_m + V_m * ((-b) * (e*exp(e*(V_m + R_s * I_m)/(c * g))/c * g) - (1/d)) == 0;

sol = vpasolve(eq1,eq2,eq3,eq4);

sol.a
sol.b
sol.c
sol.d

Is there no way to obtain values for this system of equation ? The value of can vary between 0 and 0.615.

Thanks

Answer 1

[Torsten]

%parameters
I_sc = 0.473;
V_oc = 2.6;
V_m = 2.32;
I_m = 0.455;
R_s = 0.001;
T = 313.5;
k = 1.38e-23;
e = 1.6e-19;
c = 1;
g = k*T;
%unknowns : a = I_in, b = I_diode, c = gamma, d = R_sh
fun = @(a,b,c,d)[a - b * exp(e*(R_s*I_sc)/(c*g)-1) - 1/d*R_s*I_sc - I_sc;...
                 a - b * exp(e*(V_oc)/(c*g)-1) - 1/d*V_oc;...
                 a - b * exp(e*(V_m + R_s*I_m)/(c*g)-1) - 1/d*(V_m + R_s*I_m) - I_m;...
                 I_m + V_m * (-b * e*exp(e*(V_m + R_s * I_m)/(c * g))/(c * g) - 1/d)];
x0 = [2 4 6 8];
options = optimset('MaxFunEvals',100000);
x = fsolve(@(x)fun(x(1),x(2),x(3),x(4)),x0,options)

Equation solved, inaccuracy possible.

The vector of function values is near zero, as measured by the value
of the function tolerance. However, the last step was ineffective.

x = 1×4

0.4730 0.0000 2.1996 167.6364

fun(x(1),x(2),x(3),x(4))

ans = 4×1

1.0e-14 * 0 -0.2930 -0.0056 -0.2442

[Simon Kellen]

Thanks, sadly the system does not seem to yield expected values.

Answer 2

[Abderrahim. B]

Hi!

Make sure to recheck equations parentheses.

clear

% parameters
I_sc = 0.473;
V_oc = 2.6;
V_m = 2.32;
I_m = 0.455;
R_s = 0.001;
T = 313.5;
k = 1.38e-23;
e = 1.6e-19;
c = 1;
g = k*T;

% unknowns : a = I_in, b = I_diode, c = gamma, d = R_sh
syms a b c d
eq1 =  a - b * exp((e*(R_s*I_sc)/(c*g))-1) - (1/d)*R_s*I_sc == I_sc;

eq2 = a - b * exp((e*(V_oc)/(c*g))-1) - (1/d)*V_oc == 0;

eq3 =  a - b * exp((e*(V_m + R_s*I_m)/(c*g))-1) - (1/d)*(V_m + R_s*I_m) == I_m;

eq4 = I_m + V_m * ((-b * e* exp(e*(V_m + R_s * I_m)/(c*g))/(c * g) - (1/d))) == 0;

sol = vpasolve([eq1, eq2, eq3, eq4], [a, b, c,d])  ;

a = sol.a

a = 

b = sol.b

b = 

c = sol.c

c = 

d = sol.d

d = 

[Simon Kellen]

Thanks.

Other question, when I set c (γ) as 1, and then solve for a, b and d ( and ), I get Empty sym: 0-by-1 again for all solutions. Is there any way to avoid that?

[Torsten]

4 equations for 3 unknown ? This yields no solution in general.

[Abderrahim. B]

@Simon Kellen check @Torsten answer. In this case, you are like forcing you sys of eq to have more conditions than unknowns, you should expect that generically there are no solutions.