This variant works reasonably well. We sum absolute values of discrepancies (instead of squares).

eqs = {7.61 == 
    0. - (26.3 + 7.6*Rs)/Rp - 
     1. (-1 + E^((38.91741123737115*(26.3 + 7.61*Rs))/A)) Subscript[i,
        o] + Subscript[i, ph], 
   8.21 == 0. - (8.21*Rs)/Rp - 
     1. (-1 + E^((319.51194625881715*Rs)/A)) Subscript[i, o] + 
     Subscript[i, ph], 
   0 == 0. - 32.9/Rp - 
     1. (-1 + E^(1280.3828297095108/A)) Subscript[i, o] + 
     Subscript[i, ph], 
   6.62 == 0. - (6.62*Rs)/Rp - 
     1. (-1 + E^((239.92927434638457*Rs)/A)) Subscript[i, o] + 
     Subscript[i, ph], 
   0 == 0. - 29.9/Rp - 
     1. (-1 + E^(1083.668474766903/A)) Subscript[i, o] + 
     Subscript[i, ph]};
exprs = Apply[Subtract, eqs, {1}];
obj = Total[Abs[exprs]];
vars = {Rp, Rs, Subscript[i, o], Subscript[i, ph], A};

One can try various methods in NMinimize. I had best results from simulated annealing, though possibly that was just luck.

SeedRandom[1111]; Quiet[
 res = Table[
   NMinimize[obj, vars, 
    Method -> {"SimulatedAnnealing", 
      "RandomSeed" -> RandomInteger[100000]}], {8}]]

(* Out[128]= {{1.87674549158, {Rp -> -22.1775022222, Rs -> 27.1223664498,
    Subscript[i, o] -> -1.27242985435, 
   Subscript[i, ph] -> -0.211055515253, 
   A -> -0.0309130227249}}, {1.64764584763, {Rp -> -36.2638152639, 
   Rs -> 42.3746667989, Subscript[i, o] -> -1.06207524175, 
   Subscript[i, ph] -> -0.053467535256, 
   A -> -1.5217022262}}, {0.019707815217, {Rp -> -1107.10143767, 
   Rs -> 1110.76425648, Subscript[i, o] -> 0.806310397178, 
   Subscript[i, ph] -> -0.83347298941, 
   A -> -0.0264490342706}}, {0.736993699664, {Rp -> -45.9270205782, 
   Rs -> 49.3700088231, Subscript[i, o] -> 0.348644661474, 
   Subscript[i, ph] -> -0.999677531674, 
   A -> -0.462683666199}}, {0.137667678614, {Rp -> -1453.74805012, 
   Rs -> 1469.63492588, Subscript[i, o] -> -38.3974516083, 
   Subscript[i, ph] -> 38.3251067397, 
   A -> -0.368567703104}}, {1.12405053251, {Rp -> -29.1065949187, 
   Rs -> 32.7292298606, Subscript[i, o] -> -2.26521557848, 
   Subscript[i, ph] -> 1.23795697332, 
   A -> -1.34684133875}}, {0.0936124366274, {Rp -> -559.623069958, 
   Rs -> 565.599281418, Subscript[i, o] -> -3.10557765226, 
   Subscript[i, ph] -> 3.03488269939, 
   A -> -2.72662194692}}, {1.43175063852, {Rp -> -23.0445680163, 
   Rs -> 27.0491806745, Subscript[i, o] -> -3.23518621627, 
   Subscript[i, ph] -> 1.80847818916, A -> -1.44644488489}}} *)

The third has a small residual so we'll check how close we come with it.

exprs /. {Rp -> -1107.1014376711053`, 
  Rs -> 1110.7642564780667`, Subscript[i, o] -> 0.8063103971775648`, 
  Subscript[i, ph] -> -0.833472989409748`, A -> -0.026449034270631236`}

(* Out[130]= {-0.0117375546656, 
 2.2635415764*10^-9, -0.0025546485558, 0.00526047949255, \
0.000155130239452} *)

An alternative is to use a sum of squares and give NMinimize some nondefault settings.

obj = exprs.exprs;

SeedRandom[1111]; Quiet[
 res = Table[
   NMinimize[obj, vars, 
    Method -> {"RandomSearch", 
      "RandomSeed" -> RandomInteger[100000]}], {8}]]

(* Out[150]= {{3.53513331148*10^-6, {Rp -> -2822.92221197, 
   Rs -> 2827.13076594, Subscript[i, o] -> -0.0332358721578, 
   Subscript[i, ph] -> 0.0222353109126, 
   A -> -31.9630925697}}, {6.24332180567*10^-7, {Rp -> -2700.49545384,
    Rs -> 2705.67999858, Subscript[i, o] -> 0.000018778654236, 
   Subscript[i, ph] -> -0.0116263048577, 
   A -> 160412.601173}}, {3.53513331148*10^-6, {Rp -> -2822.92221197, 
   Rs -> 2827.13076594, Subscript[i, o] -> -0.512019293981, 
   Subscript[i, ph] -> 0.501018732736, 
   A -> -0.00380045704701}}, {3.53513331148*10^-6, {Rp -> \
-2822.92221197, Rs -> 2827.13076594, 
   Subscript[i, o] -> 0.353615439225, 
   Subscript[i, ph] -> -0.36461600047, 
   A -> -0.689503356585}}, {3.58492039403*10^-6, {Rp -> \
-2905.05796609, Rs -> 2909.30416484, Subscript[i, o] -> -38.346593732,
    Subscript[i, ph] -> 38.3359024824, 
   A -> -8.35575763632}}, {3.53513331148*10^-6, {Rp -> -2822.92221197,
    Rs -> 2827.13076594, Subscript[i, o] -> -0.165396243166, 
   Subscript[i, ph] -> 0.154395681921, 
   A -> -0.648193065037}}, {3.53513331148*10^-6, {Rp -> \
-2822.92221197, Rs -> 2827.13076594, 
   Subscript[i, o] -> -0.01209512964, 
   Subscript[i, ph] -> 0.00109456839478, 
   A -> -1.01228544162}}, {3.53513331148*10^-6, {Rp -> -2822.92221197,
    Rs -> 2827.13076594, Subscript[i, o] -> -2.27722923527, 
   Subscript[i, ph] -> 2.26622867403, A -> -1.20741804005}}} *)

We check the quality of the best of these.

exprs /. {Rp -> -2700.4954538433844`, 
  Rs -> 2705.6799985799325`, 
  Subscript[i, o] -> 0.000018778654235991685`, 
  Subscript[i, ph] -> -0.01162630485766281`, A -> 160412.60117315938`}

(* Out[149]= {0.0000695635876515, -0.0000413467732426, \
-0.000556494251506, -0.0000273792092713, 0.000554389807918} *)

So now our residuals are all under 1/1000.

I need to look into the memory allocation failure when using a sum of squares wit h default method.

Tried that and got a SystemException[MemoryAllocationFailure...]. Perhaps the problem is singular?

In[21]:= NMinimize[{0, 
  7.61 == 0. - (26.3 + 7.6*Rs)/Rp - 
    1. (-1 + E^((38.91741123737115* (26.3 + 7.61* Rs))/A)) Subscript[
     i, o] + Subscript[i, ph],
     8.21 == 
   0. - (8.21* Rs)/Rp - 
    1. (-1 + E^((319.51194625881715* Rs)/A)) Subscript[i, o] + 
    Subscript[i, ph], 
     0 == 0. - 32.9/Rp - 
    1. (-1 + E^(1280.3828297095108/A)) Subscript[i, o] + Subscript[i, 
    ph],
     6.62 == 
   0. - (6.62* Rs)/Rp - 
    1. (-1 + E^((239.92927434638457 *Rs)/A)) Subscript[i, o] + 
    Subscript[i, ph],
     0 == 0. - 29.9/Rp - 
    1. (-1 + E^(1083.668474766903/A)) Subscript[i, o] + Subscript[i, 
    ph]},
       {Subscript[i, ph], Subscript[i, o], A, Rs, Rp}]

During evaluation of In[21]:= NMinimize::nosat: Obtained solution does not satisfy the following constraints within Tolerance -> 0.001`: {0. +29.9/Rp+1. (-1+E^(1083.67/A)) Subscript[i, o]-Subscript[i, ph]==0,0. +32.9/Rp+1. (-1+E^(1280.38/A)) Subscript[i, o]-Subscript[i, ph]==0,6.62 +<<4>>==0,8.21 +(8.21 Rs)/Rp+1. (-1+E^(<<1>>/A)) Subscript[i, o]-Subscript[i, ph]==0,7.61 +(26.3 +7.6 Rs)/Rp+1. (-1+E^((38.9174 (<<5>> +<<1>>))/A)) Subscript[i, o]-Subscript[i, ph]==0}. >>

Out[21]= {0., {Subscript[i, ph] -> 1.42325, 
  Subscript[i, o] -> 36.4052, A -> -3.62571, Rs -> 0.00234389, 
  Rp -> 0.841729}}

I reran NMinimize, giving it 0 as an objective function, along with the constraints. It returned an answer, but with a warning message that the obtained solution did not satisfy the constraints within tolerance. The answer obtained was

{0., {Subscript[i, ph] -> 1.42325, Subscript[i, o] -> 36.4052, 
  A -> -3.62571, Rs -> 0.00234389, Rp -> 0.841729}}

Those values might serve as starting values for FindRoot.

This is the file, can you help me?

Attachments:

I worked with FindRoot too, but the answer was not validate. I have problem when work with FindRoot because it need initials values