Application: consider the following power supply network, consisting of powersup

Question

Application: consider the following power supply network, consisting of powersupply stations (denoted S1 and S2) and power recipient nodes (N1 to N5). The arrows representlines of power flow between supply stations and demand nodes, where flow (fi) in the direction of the arrow has a positive value (and flow in the opposite direction has a negative value).

Assuming that power supply stations must run at their capacity, that power demand at each nodemust be met, and that there are no losses in the network, a flow balance on each node gives thefollowing equation fout = fin + p, where p is the power gained or lost at that node, such the valueof p is positive for supply capacity and negative for demand. For example, for supply station S1,the flow balance gives the equation f1 = pS1, while for recipient node N1, it gives the equationf3+f6 = f4 +pN1. Because there are seven nodes overall, this results in a system of seven linearequations for the seven unknown flows.a) Write out this system of equations and set it up as a matrix problem;b) Write a MATLAB program to solve for the unknown flows, given that pS1 = 10 MW, pS2= 10 MW, pN1 = -4 MW, pN2 = -4 MW, pN3 = -4 MW, pN4 = -4 MW, and pN5 = -4 MW.Note: Be sure to check for existence and uniqueness of the solution, and use theappropriate numerical approach. If the solution is unique, your program should report theunique solution. If the solution exists but is not unique, your program should plot theflows as a function of any free parameters (for a reasonable range of the free parameterflows). If the solution does not exist, your program should provide that information and make a best-fit estimate.

Explanation / Answer

f1 = pS1
f2 - f3 - f1 = pN4
f2 = pN3
f3 + f6 - f4 = pN1
f4 + f5 = pS2
f7 - f5 = pN2
f7 + f6 = pN5

Can be further written as:

f1 + 0*f2 + 0*f3 + 0*f4 + 0*f5 + 0*f6 + 0*f7 = pS1
-f1 + f2 - f3 + 0*f4 + 0*f5 + 0*f6 + 0*f7 = pN4
0*f1 + f2 + 0*f3 + 0*f4 + 0*f5 + 0*f6 + 0*f7 = pN3
0*f1 + 0*f2 + f3 - f4 + 0*f5 + f6 + 0*f7 = pN1
0*f1 + 0*f2 + 0*f3 + f4 + f5 + 0*f6 + 0*f7 = pS2
0*f1 + 0*f2 + 0*f3 + 0*f4 - f5 + 0*f6 + f7 = pN2
0*f1 + 0*f2 + 0*f3 + 0*f4 + 0f5 + f6 + f7 = pN5

#matlab code is(using Symbolic Math Toolbox):

syms f1 f2 f3 f4 f5 f6 f7

eqn1 = f1 + 0*f2 + 0*f3 + 0*f4 + 0*f5 + 0*f6 + 0*f7 == 10;
eqn1 = -f1 + f2 - f3 + 0*f4 + 0*f5 + 0*f6 + 0*f7 == -4;
eqn1 = 0*f1 + f2 + 0*f3 + 0*f4 + 0*f5 + 0*f6 + 0*f7 == -4;
eqn1 = 0*f1 + 0*f2 + f3 - f4 + 0*f5 + f6 + 0*f7 == -4;
eqn1 = 0*f1 + 0*f2 + 0*f3 + f4 + f5 + 0*f6 + 0*f7 == 10;
eqn1 = 0*f1 + 0*f2 + 0*f3 + 0*f4 - f5 + 0*f6 + f7 == -4;
eqn1 = 0*f1 + 0*f2 + 0*f3 + 0*f4 + 0f5 + f6 + f7 == 4;

sol = solve([eqn1, eqn2, eqn3, eqn4, eqn5, eqn6, eqn7], [f1, f2, f3, f4, f5, f6, f7]);

sol.f1
sol.f2
sol.f3
sol.f4
sol.f5
sol.f6
sol.f7

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