How come fsolve is exceeding evaluation limit?

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Jack Cimorelli
Jack Cimorelli on 2 Jan 2025
Commented: Torsten on 4 Jan 2025
Hello, I am trying to code and solve the 'five-paramter' solar model in MATLAB using fsolve. However the solver is not coming to a soltuion I expect. The initial values (x0) are essentially the answers I expect, and they are in the correct order. So I do not suspect the initial values the the issue.
Current error:
fsolve stopped because it exceeded the function evaluation limit,
options.MaxFunctionEvaluations = 5.000000e+02.
I have tried increasing the max function evaluations but that has not helped.
Any assistance in making this code work would be greatly appreciated!
clear; clc;
% Initial guess for the solution
x0 = [8.487 6.330e-9 1.149 5.837 5.125e-3];
% Use fsolve to find the solution
options = optimoptions('fsolve', 'Display', 'iter');%, 'MaxFunctionEvaluations', 2000);
[x, fval, exitflag] = fsolve(@mySystem, x0, options);
% Display the results
fprintf('Solution:\n');
disp(x);
fprintf('Function values at solution:\n');
disp(fval);
fprintf('Exit flag:\n');
disp(exitflag);
function F = mySystem(x)
F = zeros(5,1); % Initialize F to store 5 equations
% x(1) = I_irr_ref
% x(2) = I_o_ref
% x(3) = n_ref
% x(4) = Rp_ref
% x(5) = Rs_ref
V_oc_ref = 36.5;
I_sc_ref = 8.10;
V_mp_ref = 29.1;
I_mp_ref = 7.4;
Np = 1;
Ns = 60;
G_ref = 1000;
G = G_ref;
T_ref = 25+273.15;
T_cell = 25+273.15;
q = 1.602e-19;
k = 1.3806e-23;
alpha = 0.0005;
beta = 0;
E_g_ref = 1.12;
E_g = 1.16 - 7.02e-4*(T_cell^2/(T_cell-1108));
I_irr = x(1)*(G/G_ref)*(1 + alpha*(T_cell));
I_o = x(2)*(T_cell/T_ref)^3*exp((E_g_ref/(k*T_ref)) - (E_g/(k*T_cell)));
V_oc_T = V_oc_ref + beta*((T_cell+15)-T_ref);
Rp = x(4);
n = x(3);
% Intermediate terms:
NsNp = Ns/Np;
T1 = Ns*x(3)*k*T_ref;
T2 = Np*x(3)*k*T_ref;
T3 = V_mp_ref + (I_mp_ref*NsNp*x(5));
X1 = ((q*Np*x(2))/(Ns*x(3)*k*T_ref))*exp((q*T3)/T1) + 1/(NsNp*x(4));
X2 = 1 + ((q*x(2)*x(5))/(x(3)*k*T_ref))*exp((q*T3)/T1) + x(5)/x(4);
% 5 equations
F(1) = Np*x(1) - Np*x(2)*(exp((q*V_oc_ref)/T1)-1) - V_oc_ref/(NsNp*x(4));
F(2) = Np*x(1) - Np*x(2)*(exp((q*I_sc_ref*x(5))/T2)-1) - (I_sc_ref*NsNp*x(5))/(NsNp*x(4)) - I_sc_ref;
F(3) = Np*x(1) - Np*x(2)*(exp((q*T3)/T1)-1) - T3/(NsNp*x(4)) - I_mp_ref;
F(4) = X1/X2 - I_mp_ref/V_mp_ref;
F(5) = Np*I_irr - Np*I_o*exp(((q*V_oc_T)/(Ns*n*k*T_cell))-1) - V_oc_T/(NsNp*Rp);
end

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Answers (3)

Walter Roberson
Walter Roberson on 2 Jan 2025
Ran in:
% Initial guess for the solution
x0 = [sym(8487)/sym(10)^3, sym(6330)/sym(10)^12, sym(1149)/sym(10)^3, sym(5837)/sym(10)^3, sym(5125)/sym(10)^6];
syms x [5 1]
FF = mySystem(x)
FF = 
FFexp = expand(FF)
FFexp = 
sol = vpasolve(FFexp)
sol = struct with fields:
x1: [0x1 sym] x2: [0x1 sym] x3: [0x1 sym] x4: [0x1 sym] x5: [0x1 sym]
This result is not surprising considering the exp(6319....) term
function F = mySystem(x)
F = zeros(5,1,'sym'); % Initialize F to store 5 equations
% x(1) = I_irr_ref
% x(2) = I_o_ref
% x(3) = n_ref
% x(4) = Rp_ref
% x(5) = Rs_ref
V_oc_ref = sym(36.5);
I_sc_ref = sym(8.10);
V_mp_ref = sym(29.1);
I_mp_ref = sym(7.4);
Np = sym(1);
Ns = sym(60);
G_ref = sym(1000);
G = G_ref;
T_ref = sym(25)+sym(273.15);
T_cell = sym(25)+sym(273.15);
q = sym(1602)/sym(10)^22;
k = sym(13806)/sym(10)^27;
alpha = sym(0.0005);
beta = sym(0);
E_g_ref = sym(1.12);
E_g = sym(1.16) - sym(702)/sym(10)^6*(T_cell^2/(T_cell-sym(1108)));
I_irr = x(1)*(G/G_ref)*(1 + alpha*(T_cell));
I_o = x(2)*(T_cell/T_ref)^3*exp((E_g_ref/(k*T_ref)) - (E_g/(k*T_cell)));
V_oc_T = V_oc_ref + beta*((T_cell+15)-T_ref);
Rp = x(4);
n = x(3);
% Intermediate terms:
NsNp = Ns/Np;
T1 = Ns*x(3)*k*T_ref;
T2 = Np*x(3)*k*T_ref;
T3 = V_mp_ref + (I_mp_ref*NsNp*x(5));
X1 = ((q*Np*x(2))/(Ns*x(3)*k*T_ref))*exp((q*T3)/T1) + 1/(NsNp*x(4));
X2 = 1 + ((q*x(2)*x(5))/(x(3)*k*T_ref))*exp((q*T3)/T1) + x(5)/x(4);
% 5 equations
F(1) = Np*x(1) - Np*x(2)*(exp((q*V_oc_ref)/T1)-1) - V_oc_ref/(NsNp*x(4));
F(2) = Np*x(1) - Np*x(2)*(exp((q*I_sc_ref*x(5))/T2)-1) - (I_sc_ref*NsNp*x(5))/(NsNp*x(4)) - I_sc_ref;
F(3) = Np*x(1) - Np*x(2)*(exp((q*T3)/T1)-1) - T3/(NsNp*x(4)) - I_mp_ref;
F(4) = X1/X2 - I_mp_ref/V_mp_ref;
F(5) = Np*I_irr - Np*I_o*exp(((q*V_oc_T)/(Ns*n*k*T_cell))-1) - V_oc_T/(NsNp*Rp);
end
  2 Comments
John D'Errico
John D'Errico on 2 Jan 2025
Gosh. A simple thing like a term with exp(6319) in it, and it has numerical troubles. ;-)
Walter Roberson
Walter Roberson on 2 Jan 2025
Ran in:
I just noticed that the term is negative. If I transcribe it correctly, it is
format long g
-6319812173000000000222236818011/222236818011
ans =
-2.84372869876457e+19
Then there is
syms x5 x3
log_sigma1 = vpa(14800000/51389) * (x5/x3)
log_sigma1 = 

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KALYAN ACHARJYA
KALYAN ACHARJYA on 2 Jan 2025
Ran in:
clear; clc;
% Initial guess for the solution
x0 = [8; 1e-10; 1.2; 5; 0.01]; % Adjusted initial guess
% Options for solver
options = optimoptions('fsolve', ...
'Display', 'iter', ...
'MaxFunctionEvaluations', 5000, ...
'MaxIterations', 1000, ...
'FunctionTolerance', 1e-8);
% Use fsolve to find the solution
[x, fval, exitflag] = fsolve(@mySystem, x0, options);
Norm of First-order Trust-region Iteration Func-count ||f(x)||^2 step optimality radius 0 6 144.074 2.95e+09 1 1 7 144.074 1 2.95e+09 1 2 8 144.074 0.25 2.95e+09 0.25 3 14 101.245 0.0625 2.5e+09 0.0625 4 20 83.1048 0.0625 2.68e+06 0.0625 5 26 83.0406 0.15625 1.77e+05 0.156 6 32 82.8592 0.390625 4.45e+05 0.391 7 38 82.2151 0.976562 3.48e+05 0.977 8 44 74.6459 2.44141 1.69e+08 2.44 9 45 74.6459 6.10352 1.69e+08 6.1 10 46 74.6459 1.52588 1.69e+08 1.53 11 52 68.5614 0.38147 1.06e+09 0.381 12 53 68.5614 0.953674 1.06e+09 0.954 13 54 68.5614 0.238419 1.06e+09 0.238 14 60 65.7214 0.0596046 2.11e+07 0.0596 15 61 65.7214 0.149012 2.11e+07 0.149 16 67 64.2682 0.0372529 2.85e+08 0.0373 17 73 62.7138 0.0931323 6.13e+09 0.0931 18 74 62.7138 0.0931323 6.13e+09 0.0931 19 80 58.3362 0.0232831 1.83e+09 0.0233 20 81 58.3362 0.0582077 1.83e+09 0.0582 21 87 57.3372 0.0145519 1.42e+09 0.0146 22 93 56.6423 0.0363798 1.31e+10 0.0364 23 99 53.6716 0.0363798 7.22e+09 0.0364 24 100 53.6716 0.0909495 7.22e+09 0.0909 25 106 52.8965 0.0227374 1.18e+10 0.0227 26 112 52.2236 0.0227374 1.73e+10 0.0227 27 118 51.6902 0.0227374 2.83e+10 0.0227 28 124 51.238 0.0227374 4.64e+10 0.0227 29 130 50.8912 0.0227374 8.39e+10 0.0227 30 136 50.6493 0.0227374 1.65e+11 0.0227 31 137 50.6493 0.0227374 1.65e+11 0.0227 32 143 50.2461 0.00568434 1.12e+10 0.00568 33 144 50.2461 0.0142109 1.12e+10 0.0142 34 150 50.2059 0.00355271 1.02e+10 0.00355 35 156 50.1406 0.00888178 7.77e+10 0.00888 36 162 50.0573 0.00888178 8.78e+10 0.00888 37 168 50.0097 0.00888178 1.38e+11 0.00888 38 174 49.9736 0.00888178 1.92e+11 0.00888 39 180 49.9494 0.00888178 2.7e+11 0.00888 40 186 49.9276 0.00888178 3.76e+11 0.00888 41 192 49.9093 0.00888178 5.3e+11 0.00888 42 198 49.893 0.00888178 7.51e+11 0.00888 43 204 49.8794 0.00888178 1.07e+12 0.00888 44 210 49.8662 0.00888178 1.54e+12 0.00888 45 216 49.8606 0.00888178 2.23e+12 0.00888 46 222 49.8128 0.00888178 1.58e+13 0.00888 47 228 49.6639 0.00888178 2.52e+12 0.00888 48 234 49.4976 0.00888178 2.44e+11 0.00888 49 240 49.3636 0.0222045 8.02e+12 0.0222 50 246 49.0296 0.0222045 9.78e+12 0.0222 51 252 48.9172 0.0222045 2.36e+13 0.0222 52 258 48.7467 0.0222045 4.49e+13 0.0222 53 259 48.7467 0.0222045 4.49e+13 0.0222 54 265 48.3553 0.00555112 3.07e+12 0.00555 55 266 48.3553 0.0138778 3.07e+12 0.0139 56 272 48.3192 0.00346945 2.73e+12 0.00347 57 278 48.2589 0.00867362 2.11e+13 0.00867 58 284 48.1684 0.00867362 2.41e+13 0.00867 59 290 48.0972 0.00867362 3.78e+13 0.00867 60 296 48.0264 0.00867362 5.52e+13 0.00867 61 302 47.9632 0.00867362 8.42e+13 0.00867 62 308 47.9056 0.00867362 1.29e+14 0.00867 63 314 47.8549 0.00867362 2.02e+14 0.00867 64 320 47.811 0.00867362 3.2e+14 0.00867 65 326 47.774 0.00867362 5.14e+14 0.00867 66 332 47.7439 0.00867362 8.36e+14 0.00867 67 338 47.7205 0.00867362 1.38e+15 0.00867 68 344 47.7034 0.00867362 2.3e+15 0.00867 69 350 47.6925 0.00867362 3.9e+15 0.00867 70 351 47.6925 0.00867362 3.9e+15 0.00867 71 357 47.4809 0.0021684 1.92e+14 0.00217 72 358 47.4809 0.00542101 1.92e+14 0.00542 73 364 47.4753 0.00135525 1.74e+14 0.00136 74 370 47.4717 0.00338813 1.26e+15 0.00339 75 376 47.4563 0.00338813 1.28e+15 0.00339 76 382 47.4453 0.00338813 1.67e+15 0.00339 77 388 47.4341 0.00338813 2.07e+15 0.00339 78 394 47.4236 0.00338813 2.59e+15 0.00339 79 400 47.4136 0.00338813 3.24e+15 0.00339 80 406 47.404 0.00338813 4.07e+15 0.00339 81 412 47.395 0.00338813 5.13e+15 0.00339 82 418 47.3865 0.00338813 6.47e+15 0.00339 83 424 47.3784 0.00338813 8.19e+15 0.00339 84 430 47.3707 0.00338813 1.04e+16 0.00339 85 436 47.3635 0.00338813 1.32e+16 0.00339 86 442 47.3567 0.00338813 1.68e+16 0.00339 87 448 47.3502 0.00338813 2.15e+16 0.00339 88 454 47.3442 0.00338813 2.76e+16 0.00339 89 460 47.3386 0.00338813 3.55e+16 0.00339 90 466 47.3333 0.00338813 4.57e+16 0.00339 91 472 47.3283 0.00338813 5.91e+16 0.00339 92 478 47.3237 0.00338813 7.66e+16 0.00339 93 484 47.3195 0.00338813 9.96e+16 0.00339 94 490 47.3155 0.00338813 1.3e+17 0.00339 95 496 47.3119 0.00338813 1.7e+17 0.00339 96 502 47.3086 0.00338813 2.23e+17 0.00339 97 508 47.3055 0.00338813 2.93e+17 0.00339 98 514 47.3027 0.00338813 3.87e+17 0.00339 99 520 47.3003 0.00338813 5.13e+17 0.00339 100 526 47.298 0.00338813 6.82e+17 0.00339 101 532 47.2961 0.00338813 9.09e+17 0.00339 102 538 47.2944 0.00338813 1.22e+18 0.00339 103 539 47.2944 0.00338813 1.22e+18 0.00339 104 545 47.2646 0.000847033 3.36e+15 0.000847 105 546 47.2646 0.00211758 3.36e+15 0.00212 106 552 47.2642 0.000529396 3.89e+16 0.000529 107 558 47.2641 0.00132349 2.8e+17 0.00132 108 564 47.2628 0.00132349 2.83e+17 0.00132 109 570 47.2618 0.00132349 3.23e+17 0.00132 110 576 47.2607 0.00132349 3.62e+17 0.00132 111 582 47.2597 0.00132349 4.08e+17 0.00132 112 588 47.2588 0.00132349 4.59e+17 0.00132 113 594 47.2578 0.00132349 5.17e+17 0.00132 114 600 47.2569 0.00132349 5.83e+17 0.00132 115 606 47.256 0.00132349 6.57e+17 0.00132 116 612 47.2551 0.00132349 7.41e+17 0.00132 117 618 47.2542 0.00132349 8.37e+17 0.00132 118 624 47.2534 0.00132349 9.45e+17 0.00132 119 630 47.2526 0.00132349 1.07e+18 0.00132 120 636 47.2518 0.00132349 1.21e+18 0.00132 121 642 47.251 0.00132349 1.37e+18 0.00132 122 648 47.2503 0.00132349 1.55e+18 0.00132 123 654 47.2496 0.00132349 1.75e+18 0.00132 124 660 47.2488 0.00132349 1.99e+18 0.00132 125 666 47.2482 0.00132349 2.26e+18 0.00132 126 672 47.2475 0.00132349 2.56e+18 0.00132 127 678 47.2468 0.00132349 2.91e+18 0.00132 128 684 47.2462 0.00132349 3.31e+18 0.00132 129 690 47.2456 0.00132349 3.76e+18 0.00132 130 696 47.245 0.00132349 4.28e+18 0.00132 131 702 47.2444 0.00132349 4.87e+18 0.00132 132 708 47.2438 0.00132349 5.55e+18 0.00132 133 714 47.2433 0.00132349 6.33e+18 0.00132 134 720 47.2428 0.00132349 7.22e+18 0.00132 135 726 47.2422 0.00132349 8.24e+18 0.00132 136 732 47.2417 0.00132349 9.42e+18 0.00132 137 738 47.2413 0.00132349 1.08e+19 0.00132 138 744 47.2408 0.00132349 1.23e+19 0.00132 139 750 47.2403 0.00132349 1.41e+19 0.00132 140 756 47.2399 0.00132349 1.62e+19 0.00132 141 762 47.2395 0.00132349 1.85e+19 0.00132 142 768 47.2391 0.00132349 2.12e+19 0.00132 143 774 47.2387 0.00132349 2.44e+19 0.00132 144 780 47.2383 0.00132349 2.8e+19 0.00132 145 786 47.2379 0.00132349 3.22e+19 0.00132 146 792 47.2375 0.00132349 3.7e+19 0.00132 147 798 47.2372 0.00132349 4.26e+19 0.00132 148 804 47.2369 0.00132349 4.91e+19 0.00132 149 810 47.2365 0.00132349 5.66e+19 0.00132 150 816 47.2362 0.00132349 6.54e+19 0.00132 151 822 47.2359 0.00132349 7.55e+19 0.00132 152 828 47.2356 0.00132349 8.72e+19 0.00132 153 834 47.2354 0.00132349 1.01e+20 0.00132 154 840 47.2351 0.00132349 1.17e+20 0.00132 155 846 47.2348 0.00132349 1.35e+20 0.00132 156 852 47.2346 0.00132349 1.57e+20 0.00132 157 858 47.2344 0.00132349 1.82e+20 0.00132 158 864 47.2341 0.00132349 2.11e+20 0.00132 159 870 47.2339 0.00132349 2.45e+20 0.00132 160 876 47.2337 0.00132349 2.85e+20 0.00132 161 882 47.2335 0.00132349 3.32e+20 0.00132 162 888 47.2333 0.00132349 3.86e+20 0.00132 163 894 47.2332 0.00132349 4.5e+20 0.00132 164 900 47.233 0.00132349 5.26e+20 0.00132 165 906 47.2328 0.00132349 6.14e+20 0.00132 166 912 47.2327 0.00132349 7.17e+20 0.00132 167 913 47.2327 0.00132349 7.17e+20 0.00132 168 919 47.2298 0.000330872 2.28e+19 0.000331 169 920 47.2298 0.000827181 2.28e+19 0.000827 170 926 47.2298 0.000206795 1.96e+19 0.000207 171 927 47.2298 0.000516988 1.96e+19 0.000517 172 933 47.2297 0.000129247 7.67e+18 0.000129 173 939 47.2297 0.000323117 5.2e+19 0.000323 174 945 47.2297 0.000323117 5.24e+19 0.000323 175 951 47.2296 0.000323117 5.45e+19 0.000323 176 957 47.2296 0.000323117 5.67e+19 0.000323 177 963 47.2295 0.000323117 5.89e+19 0.000323 178 969 47.2295 0.000323117 6.12e+19 0.000323 179 975 47.2294 0.000323117 6.36e+19 0.000323 180 981 47.2294 0.000323117 6.61e+19 0.000323 181 987 47.2293 0.000323117 6.87e+19 0.000323 182 993 47.2293 0.000323117 7.15e+19 0.000323 183 999 47.2292 0.000323117 7.43e+19 0.000323 184 1005 47.2292 0.000323117 7.72e+19 0.000323 185 1011 47.2291 0.000323117 8.03e+19 0.000323 186 1017 47.2291 0.000323117 8.35e+19 0.000323 187 1023 47.229 0.000323117 8.68e+19 0.000323 188 1029 47.229 0.000323117 9.03e+19 0.000323 189 1035 47.229 0.000323117 9.39e+19 0.000323 190 1041 47.2289 0.000323117 9.77e+19 0.000323 191 1047 47.2289 0.000323117 1.02e+20 0.000323 192 1053 47.2288 0.000323117 1.06e+20 0.000323 193 1059 47.2288 0.000323117 1.1e+20 0.000323 194 1065 47.2287 0.000323117 1.14e+20 0.000323 195 1071 47.2287 0.000323117 1.19e+20 0.000323 196 1077 47.2287 0.000323117 1.24e+20 0.000323 197 1083 47.2286 0.000323117 1.29e+20 0.000323 198 1089 47.2286 0.000323117 1.34e+20 0.000323 199 1095 47.2285 0.000323117 1.39e+20 0.000323 200 1101 47.2285 0.000323117 1.45e+20 0.000323 201 1107 47.2285 0.000323117 1.51e+20 0.000323 202 1113 47.2284 0.000323117 1.57e+20 0.000323 203 1119 47.2284 0.000323117 1.63e+20 0.000323 204 1125 47.2283 0.000323117 1.7e+20 0.000323 205 1131 47.2283 0.000323117 1.77e+20 0.000323 206 1137 47.2283 0.000323117 1.84e+20 0.000323 207 1143 47.2282 0.000323117 1.92e+20 0.000323 208 1149 47.2282 0.000323117 2e+20 0.000323 209 1155 47.2282 0.000323117 2.08e+20 0.000323 210 1161 47.2281 0.000323117 2.17e+20 0.000323 211 1167 47.2281 0.000323117 2.25e+20 0.000323 212 1173 47.228 0.000323117 2.35e+20 0.000323 213 1179 47.228 0.000323117 2.45e+20 0.000323 214 1185 47.228 0.000323117 2.55e+20 0.000323 215 1191 47.2279 0.000323117 2.65e+20 0.000323 216 1197 47.2279 0.000323117 2.76e+20 0.000323 217 1203 47.2279 0.000323117 2.88e+20 0.000323 218 1209 47.2278 0.000323117 3e+20 0.000323 219 1215 47.2278 0.000323117 3.12e+20 0.000323 220 1221 47.2278 0.000323117 3.25e+20 0.000323 221 1227 47.2277 0.000323117 3.39e+20 0.000323 222 1233 47.2277 0.000323117 3.53e+20 0.000323 223 1239 47.2277 0.000323117 3.68e+20 0.000323 224 1245 47.2276 0.000323117 3.83e+20 0.000323 225 1251 47.2276 0.000323117 3.99e+20 0.000323 226 1257 47.2276 0.000323117 4.16e+20 0.000323 227 1263 47.2275 0.000323117 4.34e+20 0.000323 228 1269 47.2275 0.000323117 4.52e+20 0.000323 229 1275 47.2275 0.000323117 4.71e+20 0.000323 230 1281 47.2274 0.000323117 4.91e+20 0.000323 231 1287 47.2274 0.000323117 5.12e+20 0.000323 232 1293 47.2274 0.000323117 5.34e+20 0.000323 233 1299 47.2274 0.000323117 5.56e+20 0.000323 234 1305 47.2273 0.000323117 5.8e+20 0.000323 235 1311 47.2273 0.000323117 6.05e+20 0.000323 236 1317 47.2273 0.000323117 6.3e+20 0.000323 237 1323 47.2272 0.000323117 6.57e+20 0.000323 238 1329 47.2272 0.000323117 6.86e+20 0.000323 239 1335 47.2272 0.000323117 7.15e+20 0.000323 240 1341 47.2272 0.000323117 7.46e+20 0.000323 241 1347 47.2271 0.000323117 7.78e+20 0.000323 242 1353 47.2271 0.000323117 8.11e+20 0.000323 243 1359 47.2271 0.000323117 8.46e+20 0.000323 244 1365 47.227 0.000323117 8.83e+20 0.000323 245 1371 47.227 0.000323117 9.21e+20 0.000323 246 1377 47.227 0.000323117 9.61e+20 0.000323 247 1383 47.227 0.000323117 1e+21 0.000323 248 1389 47.2269 0.000323117 1.05e+21 0.000323 249 1395 47.2269 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0.000323 292 1653 47.226 0.000323117 7.23e+21 0.000323 293 1659 47.226 0.000323117 7.57e+21 0.000323 294 1665 47.226 0.000323117 7.92e+21 0.000323 295 1671 47.226 0.000323117 8.29e+21 0.000323 296 1677 47.2259 0.000323117 8.67e+21 0.000323 297 1683 47.2259 0.000323117 9.08e+21 0.000323 298 1689 47.2259 0.000323117 9.5e+21 0.000323 299 1695 47.2259 0.000323117 9.95e+21 0.000323 300 1701 47.2259 0.000323117 1.04e+22 0.000323 301 1707 47.2259 0.000323117 1.09e+22 0.000323 302 1713 47.2258 0.000323117 1.14e+22 0.000323 303 1719 47.2258 0.000323117 1.2e+22 0.000323 304 1725 47.2258 0.000323117 1.25e+22 0.000323 305 1731 47.2258 0.000323117 1.31e+22 0.000323 306 1737 47.2258 0.000323117 1.37e+22 0.000323 307 1743 47.2258 0.000323117 1.44e+22 0.000323 308 1749 47.2258 0.000323117 1.51e+22 0.000323 309 1755 47.2257 0.000323117 1.58e+22 0.000323 310 1761 47.2257 0.000323117 1.65e+22 0.000323 311 1767 47.2257 0.000323117 1.73e+22 0.000323 312 1773 47.2257 0.000323117 1.82e+22 0.000323 313 1779 47.2257 0.000323117 1.9e+22 0.000323 314 1785 47.2257 0.000323117 1.99e+22 0.000323 315 1791 47.2257 0.000323117 2.09e+22 0.000323 316 1797 47.2256 0.000323117 2.19e+22 0.000323 317 1803 47.2256 0.000323117 2.3e+22 0.000323 318 1809 47.2256 0.000323117 2.41e+22 0.000323 319 1815 47.2256 0.000323117 2.52e+22 0.000323 320 1821 47.2256 0.000323117 2.64e+22 0.000323 321 1827 47.2256 0.000323117 2.77e+22 0.000323 322 1833 47.2256 0.000323117 2.91e+22 0.000323 323 1839 47.2256 0.000323117 3.05e+22 0.000323 324 1845 47.2255 0.000323117 3.2e+22 0.000323 325 1851 47.2255 0.000323117 3.35e+22 0.000323 326 1857 47.2255 0.000323117 3.52e+22 0.000323 327 1863 47.2255 0.000323117 3.69e+22 0.000323 328 1869 47.2255 0.000323117 3.87e+22 0.000323 329 1875 47.2255 0.000323117 4.06e+22 0.000323 330 1881 47.2255 0.000323117 4.26e+22 0.000323 331 1887 47.2255 0.000323117 4.47e+22 0.000323 332 1893 47.2254 0.000323117 4.69e+22 0.000323 333 1899 47.2254 0.000323117 4.92e+22 0.000323 334 1905 47.2254 0.000323117 5.16e+22 0.000323 335 1911 47.2254 0.000323117 5.42e+22 0.000323 336 1917 47.2254 0.000323117 5.69e+22 0.000323 337 1923 47.2254 0.000323117 5.97e+22 0.000323 338 1929 47.2254 0.000323117 6.27e+22 0.000323 339 1935 47.2254 0.000323117 6.58e+22 0.000323 340 1941 47.2254 0.000323117 6.91e+22 0.000323 341 1947 47.2254 0.000323117 7.25e+22 0.000323 342 1953 47.2253 0.000323117 7.61e+22 0.000323 343 1959 47.2253 0.000323117 8e+22 0.000323 344 1965 47.2253 0.000323117 8.4e+22 0.000323 345 1971 47.2253 0.000323117 8.82e+22 0.000323 346 1977 47.2253 0.000323117 9.26e+22 0.000323 347 1983 47.2253 0.000323117 9.73e+22 0.000323 348 1989 47.2253 0.000323117 1.02e+23 0.000323 349 1995 47.2253 0.000323117 1.07e+23 0.000323 350 2001 47.2253 0.000323117 1.13e+23 0.000323 351 2007 47.2253 0.000323117 1.19e+23 0.000323 352 2013 47.2252 0.000323117 1.25e+23 0.000323 353 2019 47.2252 0.000323117 1.31e+23 0.000323 354 2025 47.2252 0.000323117 1.38e+23 0.000323 355 2031 47.2252 0.000323117 1.45e+23 0.000323 356 2037 47.2252 0.000323117 1.52e+23 0.000323 357 2043 47.2252 0.000323117 1.6e+23 0.000323 358 2049 47.2252 0.000323117 1.68e+23 0.000323 359 2055 47.2252 0.000323117 1.77e+23 0.000323 360 2061 47.2252 0.000323117 1.86e+23 0.000323 361 2067 47.2252 0.000323117 1.95e+23 0.000323 362 2073 47.2252 0.000323117 2.05e+23 0.000323 363 2079 47.2252 0.000323117 2.16e+23 0.000323 364 2085 47.2251 0.000323117 2.27e+23 0.000323 365 2091 47.2251 0.000323117 2.39e+23 0.000323 366 2097 47.2251 0.000323117 2.52e+23 0.000323 367 2103 47.2251 0.000323117 2.65e+23 0.000323 368 2109 47.2251 0.000323117 2.78e+23 0.000323 369 2115 47.2251 0.000323117 2.93e+23 0.000323 370 2121 47.2251 0.000323117 3.08e+23 0.000323 371 2127 47.2251 0.000323117 3.24e+23 0.000323 372 2133 47.2251 0.000323117 3.41e+23 0.000323 373 2139 47.2251 0.000323117 3.59e+23 0.000323 374 2145 47.2251 0.000323117 3.78e+23 0.000323 375 2151 47.2251 0.000323117 3.98e+23 0.000323 376 2157 47.2251 0.000323117 4.19e+23 0.000323 377 2163 47.2251 0.000323117 4.42e+23 0.000323 378 2169 47.225 0.000323117 4.65e+23 0.000323 379 2175 47.225 0.000323117 4.9e+23 0.000323 380 2181 47.225 0.000323117 5.16e+23 0.000323 381 2187 47.225 0.000323117 5.43e+23 0.000323 382 2193 47.225 0.000323117 5.72e+23 0.000323 383 2199 47.225 0.000323117 6.03e+23 0.000323 384 2205 47.225 0.000323117 6.35e+23 0.000323 385 2211 47.225 0.000323117 6.69e+23 0.000323 386 2217 47.225 0.000323117 7.05e+23 0.000323 387 2223 47.225 0.000323117 7.43e+23 0.000323 388 2229 47.225 0.000323117 7.83e+23 0.000323 389 2235 47.225 0.000323117 8.25e+23 0.000323 390 2241 47.225 0.000323117 8.7e+23 0.000323 391 2247 47.225 0.000323117 9.17e+23 0.000323 392 2253 47.225 0.000323117 9.66e+23 0.000323 393 2259 47.225 0.000323117 1.02e+24 0.000323 394 2265 47.2249 0.000323117 1.07e+24 0.000323 395 2271 47.2249 0.000323117 1.13e+24 0.000323 396 2277 47.2249 0.000323117 1.19e+24 0.000323 397 2283 47.2249 0.000323117 1.26e+24 0.000323 398 2289 47.2249 0.000323117 1.33e+24 0.000323 399 2295 47.2249 0.000323117 1.4e+24 0.000323 400 2301 47.2249 0.000323117 1.48e+24 0.000323 401 2307 47.2249 0.000323117 1.56e+24 0.000323 402 2313 47.2249 0.000323117 1.65e+24 0.000323 403 2319 47.2249 0.000323117 1.74e+24 0.000323 404 2325 47.2249 0.000323117 1.83e+24 0.000323 405 2331 47.2249 0.000323117 1.93e+24 0.000323 406 2337 47.2249 0.000323117 2.04e+24 0.000323 407 2343 47.2249 0.000323117 2.16e+24 0.000323 408 2349 47.2249 0.000323117 2.28e+24 0.000323 409 2355 47.2249 0.000323117 2.4e+24 0.000323 410 2361 47.2249 0.000323117 2.54e+24 0.000323 411 2367 47.2249 0.000323117 2.68e+24 0.000323 412 2373 47.2249 0.000323117 2.83e+24 0.000323 413 2379 47.2249 0.000323117 2.99e+24 0.000323 414 2385 47.2249 0.000323117 3.16e+24 0.000323 415 2391 47.2248 0.000323117 3.33e+24 0.000323 416 2397 47.2248 0.000323117 3.52e+24 0.000323 417 2403 47.2248 0.000323117 3.72e+24 0.000323 418 2409 47.2248 0.000323117 3.93e+24 0.000323 419 2415 47.2248 0.000323117 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5.05e-05 441 2532 47.2247 0.000126218 2.71e+23 0.000126 442 2538 47.2247 3.15544e-05 1.07e+23 3.16e-05 443 2544 47.2247 7.88861e-05 6.87e+23 7.89e-05 444 2550 47.2247 7.88861e-05 6.89e+23 7.89e-05 445 2556 47.2247 7.88861e-05 6.99e+23 7.89e-05 446 2562 47.2247 7.88861e-05 7.09e+23 7.89e-05 447 2568 47.2247 7.88861e-05 7.19e+23 7.89e-05 448 2574 47.2247 7.88861e-05 7.29e+23 7.89e-05 449 2580 47.2247 7.88861e-05 7.39e+23 7.89e-05 450 2586 47.2247 7.88861e-05 7.49e+23 7.89e-05 451 2592 47.2247 7.88861e-05 7.6e+23 7.89e-05 452 2598 47.2247 7.88861e-05 7.7e+23 7.89e-05 453 2604 47.2247 7.88861e-05 7.81e+23 7.89e-05 454 2610 47.2247 7.88861e-05 7.92e+23 7.89e-05 455 2616 47.2247 7.88861e-05 8.03e+23 7.89e-05 456 2622 47.2247 7.88861e-05 8.15e+23 7.89e-05 457 2628 47.2247 7.88861e-05 8.26e+23 7.89e-05 458 2634 47.2247 7.88861e-05 8.38e+23 7.89e-05 459 2640 47.2247 7.88861e-05 8.49e+23 7.89e-05 460 2646 47.2247 7.88861e-05 8.61e+23 7.89e-05 461 2652 47.2247 7.88861e-05 8.73e+23 7.89e-05 462 2658 47.2247 7.88861e-05 8.86e+23 7.89e-05 463 2664 47.2247 7.88861e-05 8.98e+23 7.89e-05 464 2670 47.2247 7.88861e-05 9.11e+23 7.89e-05 465 2676 47.2247 7.88861e-05 9.24e+23 7.89e-05 466 2682 47.2247 7.88861e-05 9.37e+23 7.89e-05 467 2688 47.2247 7.88861e-05 9.5e+23 7.89e-05 468 2694 47.2247 7.88861e-05 9.63e+23 7.89e-05 469 2700 47.2247 7.88861e-05 9.77e+23 7.89e-05 470 2706 47.2247 7.88861e-05 9.91e+23 7.89e-05 471 2712 47.2247 7.88861e-05 1e+24 7.89e-05 472 2718 47.2247 7.88861e-05 1.02e+24 7.89e-05 473 2724 47.2247 7.88861e-05 1.03e+24 7.89e-05 474 2730 47.2247 7.88861e-05 1.05e+24 7.89e-05 475 2736 47.2247 7.88861e-05 1.06e+24 7.89e-05 476 2742 47.2247 7.88861e-05 1.08e+24 7.89e-05 477 2748 47.2247 7.88861e-05 1.09e+24 7.89e-05 478 2754 47.2247 7.88861e-05 1.11e+24 7.89e-05 479 2760 47.2247 7.88861e-05 1.12e+24 7.89e-05 480 2766 47.2247 7.88861e-05 1.14e+24 7.89e-05 481 2772 47.2247 7.88861e-05 1.16e+24 7.89e-05 482 2778 47.2247 7.88861e-05 1.17e+24 7.89e-05 483 2784 47.2247 7.88861e-05 1.19e+24 7.89e-05 484 2790 47.2247 7.88861e-05 1.21e+24 7.89e-05 485 2796 47.2247 7.88861e-05 1.22e+24 7.89e-05 486 2802 47.2247 7.88861e-05 1.24e+24 7.89e-05 487 2808 47.2247 7.88861e-05 1.26e+24 7.89e-05 488 2814 47.2247 7.88861e-05 1.28e+24 7.89e-05 489 2820 47.2247 7.88861e-05 1.29e+24 7.89e-05 490 2826 47.2247 7.88861e-05 1.31e+24 7.89e-05 491 2832 47.2247 7.88861e-05 1.33e+24 7.89e-05 492 2838 47.2247 7.88861e-05 1.35e+24 7.89e-05 493 2844 47.2247 7.88861e-05 1.37e+24 7.89e-05 494 2850 47.2247 7.88861e-05 1.39e+24 7.89e-05 495 2856 47.2247 7.88861e-05 1.41e+24 7.89e-05 496 2862 47.2247 7.88861e-05 1.43e+24 7.89e-05 497 2868 47.2247 7.88861e-05 1.45e+24 7.89e-05 498 2874 47.2247 7.88861e-05 1.47e+24 7.89e-05 499 2880 47.2247 7.88861e-05 1.49e+24 7.89e-05 500 2886 47.2247 7.88861e-05 1.51e+24 7.89e-05 501 2892 47.2247 7.88861e-05 1.54e+24 7.89e-05 502 2898 47.2247 7.88861e-05 1.56e+24 7.89e-05 503 2904 47.2247 7.88861e-05 1.58e+24 7.89e-05 504 2910 47.2247 7.88861e-05 1.6e+24 7.89e-05 505 2916 47.2247 7.88861e-05 1.62e+24 7.89e-05 506 2922 47.2247 7.88861e-05 1.65e+24 7.89e-05 507 2928 47.2247 7.88861e-05 1.67e+24 7.89e-05 508 2934 47.2247 7.88861e-05 1.7e+24 7.89e-05 509 2940 47.2247 7.88861e-05 1.72e+24 7.89e-05 510 2946 47.2247 7.88861e-05 1.74e+24 7.89e-05 511 2952 47.2247 7.88861e-05 1.77e+24 7.89e-05 512 2958 47.2247 7.88861e-05 1.8e+24 7.89e-05 513 2964 47.2247 7.88861e-05 1.82e+24 7.89e-05 514 2970 47.2247 7.88861e-05 1.85e+24 7.89e-05 515 2976 47.2247 7.88861e-05 1.87e+24 7.89e-05 516 2982 47.2247 7.88861e-05 1.9e+24 7.89e-05 517 2988 47.2247 7.88861e-05 1.93e+24 7.89e-05 518 2994 47.2247 7.88861e-05 1.96e+24 7.89e-05 519 3000 47.2247 7.88861e-05 1.98e+24 7.89e-05 520 3006 47.2247 7.88861e-05 2.01e+24 7.89e-05 521 3012 47.2247 7.88861e-05 2.04e+24 7.89e-05 522 3018 47.2247 7.88861e-05 2.07e+24 7.89e-05 523 3024 47.2247 7.88861e-05 2.1e+24 7.89e-05 524 3030 47.2247 7.88861e-05 2.13e+24 7.89e-05 525 3036 47.2247 7.88861e-05 2.16e+24 7.89e-05 526 3042 47.2247 7.88861e-05 2.19e+24 7.89e-05 527 3048 47.2247 7.88861e-05 2.23e+24 7.89e-05 528 3054 47.2247 7.88861e-05 2.26e+24 7.89e-05 529 3060 47.2247 7.88861e-05 2.29e+24 7.89e-05 530 3066 47.2247 7.88861e-05 2.32e+24 7.89e-05 531 3072 47.2247 7.88861e-05 2.36e+24 7.89e-05 532 3078 47.2247 7.88861e-05 2.39e+24 7.89e-05 533 3084 47.2247 7.88861e-05 2.43e+24 7.89e-05 534 3090 47.2246 7.88861e-05 2.46e+24 7.89e-05 535 3096 47.2246 7.88861e-05 2.5e+24 7.89e-05 536 3102 47.2246 7.88861e-05 2.53e+24 7.89e-05 537 3108 47.2246 7.88861e-05 2.57e+24 7.89e-05 538 3114 47.2246 7.88861e-05 2.61e+24 7.89e-05 539 3120 47.2246 7.88861e-05 2.65e+24 7.89e-05 540 3126 47.2246 7.88861e-05 2.68e+24 7.89e-05 541 3132 47.2246 7.88861e-05 2.72e+24 7.89e-05 542 3138 47.2246 7.88861e-05 2.76e+24 7.89e-05 543 3144 47.2246 7.88861e-05 2.8e+24 7.89e-05 544 3150 47.2246 7.88861e-05 2.84e+24 7.89e-05 545 3156 47.2246 7.88861e-05 2.88e+24 7.89e-05 546 3162 47.2246 7.88861e-05 2.93e+24 7.89e-05 547 3168 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7.37e+25 7.89e-05 760 4446 47.2246 7.88861e-05 7.48e+25 7.89e-05 761 4452 47.2246 7.88861e-05 7.6e+25 7.89e-05 762 4458 47.2246 7.88861e-05 7.73e+25 7.89e-05 763 4464 47.2246 7.88861e-05 7.85e+25 7.89e-05 764 4470 47.2246 7.88861e-05 7.97e+25 7.89e-05 765 4476 47.2246 7.88861e-05 8.1e+25 7.89e-05 766 4482 47.2246 7.88861e-05 8.23e+25 7.89e-05 767 4488 47.2245 7.88861e-05 8.36e+25 7.89e-05 768 4494 47.2245 7.88861e-05 8.5e+25 7.89e-05 769 4500 47.2245 7.88861e-05 8.63e+25 7.89e-05 770 4506 47.2245 7.88861e-05 8.77e+25 7.89e-05 771 4512 47.2245 7.88861e-05 8.91e+25 7.89e-05 772 4518 47.2245 7.88861e-05 9.06e+25 7.89e-05 773 4524 47.2245 7.88861e-05 9.2e+25 7.89e-05 774 4530 47.2245 7.88861e-05 9.35e+25 7.89e-05 775 4536 47.2245 7.88861e-05 9.5e+25 7.89e-05 776 4542 47.2245 7.88861e-05 9.65e+25 7.89e-05 777 4548 47.2245 7.88861e-05 9.81e+25 7.89e-05 778 4554 47.2245 7.88861e-05 9.97e+25 7.89e-05 779 4560 47.2245 7.88861e-05 1.01e+26 7.89e-05 780 4566 47.2245 7.88861e-05 1.03e+26 7.89e-05 781 4572 47.2245 7.88861e-05 1.05e+26 7.89e-05 782 4578 47.2245 7.88861e-05 1.06e+26 7.89e-05 783 4584 47.2245 7.88861e-05 1.08e+26 7.89e-05 784 4590 47.2245 7.88861e-05 1.1e+26 7.89e-05 785 4596 47.2245 7.88861e-05 1.11e+26 7.89e-05 786 4602 47.2245 7.88861e-05 1.13e+26 7.89e-05 787 4608 47.2245 7.88861e-05 1.15e+26 7.89e-05 788 4614 47.2245 7.88861e-05 1.17e+26 7.89e-05 789 4620 47.2245 7.88861e-05 1.19e+26 7.89e-05 790 4626 47.2245 7.88861e-05 1.21e+26 7.89e-05 791 4632 47.2245 7.88861e-05 1.23e+26 7.89e-05 792 4638 47.2245 7.88861e-05 1.25e+26 7.89e-05 793 4644 47.2245 7.88861e-05 1.27e+26 7.89e-05 794 4650 47.2245 7.88861e-05 1.29e+26 7.89e-05 795 4656 47.2245 7.88861e-05 1.31e+26 7.89e-05 796 4662 47.2245 7.88861e-05 1.33e+26 7.89e-05 797 4668 47.2245 7.88861e-05 1.35e+26 7.89e-05 798 4674 47.2245 7.88861e-05 1.37e+26 7.89e-05 799 4680 47.2245 7.88861e-05 1.4e+26 7.89e-05 800 4686 47.2245 7.88861e-05 1.42e+26 7.89e-05 801 4692 47.2245 7.88861e-05 1.44e+26 7.89e-05 802 4698 47.2245 7.88861e-05 1.47e+26 7.89e-05 803 4704 47.2245 7.88861e-05 1.49e+26 7.89e-05 804 4710 47.2245 7.88861e-05 1.51e+26 7.89e-05 805 4716 47.2245 7.88861e-05 1.54e+26 7.89e-05 806 4722 47.2245 7.88861e-05 1.56e+26 7.89e-05 807 4728 47.2245 7.88861e-05 1.59e+26 7.89e-05 808 4734 47.2245 7.88861e-05 1.61e+26 7.89e-05 809 4740 47.2245 7.88861e-05 1.64e+26 7.89e-05 810 4746 47.2245 7.88861e-05 1.67e+26 7.89e-05 811 4752 47.2245 7.88861e-05 1.69e+26 7.89e-05 812 4758 47.2245 7.88861e-05 1.72e+26 7.89e-05 813 4764 47.2245 7.88861e-05 1.75e+26 7.89e-05 814 4770 47.2245 7.88861e-05 1.78e+26 7.89e-05 815 4776 47.2245 7.88861e-05 1.81e+26 7.89e-05 816 4782 47.2245 7.88861e-05 1.84e+26 7.89e-05 817 4788 47.2245 7.88861e-05 1.87e+26 7.89e-05 818 4794 47.2245 7.88861e-05 1.9e+26 7.89e-05 819 4800 47.2245 7.88861e-05 1.93e+26 7.89e-05 820 4806 47.2245 7.88861e-05 1.96e+26 7.89e-05 821 4812 47.2245 7.88861e-05 1.99e+26 7.89e-05 822 4818 47.2245 7.88861e-05 2.03e+26 7.89e-05 823 4824 47.2245 7.88861e-05 2.06e+26 7.89e-05 824 4830 47.2245 7.88861e-05 2.09e+26 7.89e-05 825 4836 47.2245 7.88861e-05 2.13e+26 7.89e-05 826 4842 47.2245 7.88861e-05 2.16e+26 7.89e-05 827 4848 47.2245 7.88861e-05 2.2e+26 7.89e-05 828 4854 47.2245 7.88861e-05 2.23e+26 7.89e-05 829 4860 47.2245 7.88861e-05 2.27e+26 7.89e-05 830 4866 47.2245 7.88861e-05 2.31e+26 7.89e-05 831 4872 47.2245 7.88861e-05 2.35e+26 7.89e-05 832 4878 47.2245 7.88861e-05 2.39e+26 7.89e-05 833 4884 47.2245 7.88861e-05 2.43e+26 7.89e-05 834 4890 47.2245 7.88861e-05 2.47e+26 7.89e-05 835 4896 47.2245 7.88861e-05 2.51e+26 7.89e-05 836 4902 47.2245 7.88861e-05 2.55e+26 7.89e-05 837 4908 47.2245 7.88861e-05 2.59e+26 7.89e-05 838 4914 47.2245 7.88861e-05 2.63e+26 7.89e-05 839 4920 47.2245 7.88861e-05 2.68e+26 7.89e-05 840 4926 47.2245 7.88861e-05 2.72e+26 7.89e-05 841 4932 47.2245 7.88861e-05 2.76e+26 7.89e-05 842 4938 47.2245 7.88861e-05 2.81e+26 7.89e-05 843 4944 47.2245 7.88861e-05 2.86e+26 7.89e-05 844 4950 47.2245 7.88861e-05 2.9e+26 7.89e-05 845 4956 47.2245 7.88861e-05 2.95e+26 7.89e-05 846 4962 47.2245 7.88861e-05 3e+26 7.89e-05 847 4968 47.2245 7.88861e-05 3.05e+26 7.89e-05 848 4974 47.2245 7.88861e-05 3.1e+26 7.89e-05 849 4980 47.2245 7.88861e-05 3.15e+26 7.89e-05 850 4986 47.2245 7.88861e-05 3.21e+26 7.89e-05 851 4992 47.2245 7.88861e-05 3.26e+26 7.89e-05 852 4998 47.2245 7.88861e-05 3.31e+26 7.89e-05 853 5004 47.2245 7.88861e-05 3.37e+26 7.89e-05 Solver stopped prematurely. fsolve stopped because it exceeded the function evaluation limit, options.MaxFunctionEvaluations = 5.000000e+03.
% Display the results
fprintf('Solution:\n');
Solution:
disp(x);
7.7572 0.0000 0.3461 0.2833 -0.0068
fprintf('Function values at solution:\n');
Function values at solution:
disp(fval);
0.0007 -0.1484 -1.1773 -0.1940 6.7660
fprintf('Exit flag:\n');
Exit flag:
disp(exitflag);
0
function F = mySystem(x)
F = zeros(5,1); % Initialize F to store 5 equations
% Problem parameters
V_oc_ref = 36.5;
I_sc_ref = 8.10;
V_mp_ref = 29.1;
I_mp_ref = 7.4;
Np = 1;
Ns = 60;
G_ref = 1000;
T_ref = 25+273.15;
T_cell = 25+273.15;
q = 1.602e-19;
k = 1.3806e-23;
alpha = 0.0005;
beta = 0;
E_g_ref = 1.12;
E_g = 1.16 - 7.02e-4*(T_cell^2/(T_cell-1108));
I_irr = x(1)*(G_ref/G_ref)*(1 + alpha*(T_cell));
I_o = x(2)*(T_cell/T_ref)^3*exp((E_g_ref/(k*T_ref)) - (E_g/(k*T_cell)));
V_oc_T = V_oc_ref + beta*((T_cell+15)-T_ref);
Rp = x(4);
n = x(3);
NsNp = Ns/Np;
T1 = Ns*x(3)*k*T_ref;
T2 = Np*x(3)*k*T_ref;
T3 = V_mp_ref + (I_mp_ref*NsNp*x(5));
X1 = ((q*Np*x(2))/(Ns*x(3)*k*T_ref))*exp((q*T3)/T1) + 1/(NsNp*x(4));
X2 = 1 + ((q*x(2)*x(5))/(x(3)*k*T_ref))*exp((q*T3)/T1) + x(5)/x(4);
% Define equations
F(1) = Np*x(1) - Np*x(2)*(exp((q*V_oc_ref)/T1)-1) - V_oc_ref/(NsNp*x(4));
F(2) = Np*x(1) - Np*x(2)*(exp((q*I_sc_ref*x(5))/T2)-1) - (I_sc_ref*NsNp*x(5))/(NsNp*x(4)) - I_sc_ref;
F(3) = Np*x(1) - Np*x(2)*(exp((q*T3)/T1)-1) - T3/(NsNp*x(4)) - I_mp_ref;
F(4) = X1/X2 - I_mp_ref/V_mp_ref;
F(5) = Np*I_irr - Np*I_o*exp(((q*V_oc_T)/(Ns*n*k*T_cell))-1) - V_oc_T/(NsNp*Rp);
end
  1 Comment
John D'Errico
John D'Errico on 2 Jan 2025
I might point out several things, at least.
First, fsolve does not appear to have found a solution. We see this result, which is not very close to a vector of zeros.
disp(fval);
0.0007
-0.1484
-1.1773
-0.1940
6.7660
Are they effectively zero, when compared to where they were the start point? If not, then fsolve has not actually converged here, despite it thinking it has done its best. Remember, fsolve is a ROOTFINDER!
Part of the problem may be the wide disparity in parameter scaling. That will often mess around with the linear algebra. We see 10 powers of 10 between the different variables. That is coming a little too close for comfort to the dynamic range of a double, so possibly making the iterations work poorly. This is worth checking.
There are many other points in the code to worry about in this respect. This code fragment, for example.
exp((E_g_ref/(k*T_ref)) - (E_g/(k*T_cell)))
What is k? And the other variables in that sub-expression?
T_ref = 25+273.15;
k = 1.3806e-23;
E_g_ref = 1.12;
Yup. That is going to blow the exponential out of the water. The result will likely be numerical garbage.
I would suggest that trying to use fsolve here is likely to be a failure, even if the code itself is "correcty" written otherwise. There will be too many problems with the dynamic range of these variables.

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Matt J
Matt J on 2 Jan 2025
Edited: Matt J on 3 Jan 2025
Ran in:
I don't quite understand the physics of your problem, but I have the impression that x(3:5) are meant to be non-negative. Hence, the more appropriate solver is lsqnonlin which allow you to place bounds on the x(i). Below, I have reimplemented using lsqnonlin with scaling of the initial Jacobian,
x=solveIt();
Optimization stopped because the norm of the current step, 4.873213e-13, is less than options.StepTolerance = 1.000000e-12. Solution: 8.4458 0.0000 0.4835 0.0944 0.0000 Inital and Final Resnorms: 101.3033 27.5020 Exit flag: 2
function x=solveIt()
V_oc_ref = 36.5;
I_sc_ref = 8.10;
V_mp_ref = 29.1;
I_mp_ref = 7.4;
Np = 1;
Ns = 60;
G_ref = 1000;
G = G_ref;
T_ref = 25+273.15;
T_cell = 25+273.15;
q = 1.602e-19;
k = 1.3806e-23;
alpha = 0.0005;
beta = 0;
E_g_ref = 1.12;
E_g = 1.16 - 7.02e-4*(T_cell^2/(T_cell-1108));
V_oc_T = V_oc_ref + beta*((T_cell+15)-T_ref);
NsNp = Ns/Np;
% Initial guess for the solution
x0 = [8.487 6.330e-9 1.149 5.837 5.125e-3];
% Use fsolve to find the solution
options = optimoptions('lsqnonlin', 'Display','off','MaxIter',1,'DiffMaxChange',1e-10);%, 'MaxFunctionEvaluations', 2000);
s=1;
fun=@(z) mySystem(z./s);
[~,~,~,~,~,~,jacobian] = lsqnonlin(fun, x0, [],[], options);
s=full(max(abs(jacobian)));
fun=@(z) mySystem(z./s);
options.MaxIterations=2000;
options.MaxFunEvals=Inf;
options.FunctionTolerance=1e-12;
options.StepTolerance=1e-12;
options.OptimalityTolerance=1e-12;
options.Display='final-detailed';
resnorm0=norm(fun(x0.*s)).^2;
[z,resnorm,~,exitflag] = lsqnonlin(fun, x0.*s,[-inf,-inf,0,0,0],[], options);
x=z./s;
% Display the results
fprintf('Solution:\n');
disp(x);
fprintf('Inital and Final Resnorms:\n');
disp([resnorm0, resnorm]);
fprintf('Exit flag:\n');
disp(exitflag);
function F = mySystem(x)
F = zeros(5,1); % Initialize F to store 5 equations
% x(1) = I_irr_ref
% x(2) = I_o_ref
% x(3) = n_ref
% x(4) = Rp_ref
% x(5) = Rs_ref
I_irr = x(1)*(G/G_ref)*(1 + alpha*(T_cell));
I_o = x(2)*(T_cell/T_ref)^3*exp((E_g_ref/(k*T_ref)) - (E_g/(k*T_cell)));
Rp = x(4);
n = x(3);
% Intermediate terms:
T1 = Ns*x(3)*k*T_ref;
T2 = Np*x(3)*k*T_ref;
T3 = V_mp_ref + (I_mp_ref*NsNp*x(5));
X1 = ((q*Np*x(2))/(Ns*x(3)*k*T_ref))*exp((q*T3)/T1) + 1/(NsNp*x(4));
X2 = 1 + ((q*x(2)*x(5))/(x(3)*k*T_ref))*exp((q*T3)/T1) + x(5)/x(4);
% 5 equations
F(1) = Np*x(1) - Np*x(2)*(exp((q*V_oc_ref)/T1)-1) - V_oc_ref/(NsNp*x(4));
F(2) = Np*x(1) - Np*x(2)*(exp((q*I_sc_ref*x(5))/T2)-1) - (I_sc_ref*NsNp*x(5))/(NsNp*x(4)) - I_sc_ref;
F(3) = Np*x(1) - Np*x(2)*(exp((q*T3)/T1)-1) - T3/(NsNp*x(4)) - I_mp_ref;
F(4) = X1/X2 - I_mp_ref/V_mp_ref;
F(5) = Np*I_irr - Np*I_o*exp(((q*V_oc_T)/(Ns*n*k*T_cell))-1) - V_oc_T/(NsNp*Rp);
end
end
  5 Comments
Show 3 older comments
Matt J
Matt J on 4 Jan 2025
I suspect that this is a constraint that should hold for the final set of parameters, too.
Possibly, but lsqnonlin reached a positive solution in all parameters
Solution: 8.4458 0.0000 0.4835 0.0944 0.0000
so there was no need to impose that constraint explicitly.
Torsten
Torsten on 4 Jan 2025
@Matt J
It was just a comment to Alex Shaw's solution.

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