Nodal Analysis

To construct a reliable power system we have to design it in a way that by damaging one element we wont destroy the whole system. To get a better understanding of this and the nodal analysis, we modeled a circuit by replacing cable resistances and loads by  equivalent resistors and batteries by voltage sources.
The modeled circuit is as follow:

 
to use the node voltage analysis we put the ground on the bottom and labeled the top two corners and the two junctions.

 

since we have 4 unknowns we have to write 4 equations.

 Given

R_C1=100Ω, R_C2=R_C1=220Ω, R_L1=R_L2=1KΩ

V_bat1=12V, V_bat2=9V

and the equations we should be able to solve the problem for  V₁, V₂,V₃, V₄. 
 In this case V₁ and V₄ are easy to find since they are just equal to the battery voltages.
 V₁ = 12V and V₄ = 9V.
Solving the equations gives,  V₂ = 10.2554V and V₃ = 8.6736V.

Next to be able to construct the circuit we have to know the exact values of all the elements.

The following table is the measured values for the resistors. 

After constructing the circuit these are the values obtained:

Here is a closer look at our circuit setup.

V_bat1 = V₂ + ( I_bat1×R_C1) = 10.39 + 0.01729×100 =12.119 ± 0.5 V  
V_bat2 = V₃ + ( I_bat2×R_C3) = 8.75 + 0.0015×220 =9.08 ± 0.5 V
P_bat1 = 209.5 ± 2 mW
P_bat2 = 13.62 ± 2 mW

Now if we want V₂ = V₂ = 9V then by carrying the equation wrote earlier we obtain these values:

In order to get the precise values of the battery voltages we used the variable voltage sources.

The small amount of error in the results  comes from the fact as always to make calculations easier we ignored the effects of wiring, which is minimal but still big enough to interfere.