MATLAB: How can i calculate five unknown parameters?

fsolveMATLAB

Hello everyone,

I want to calculate the five unknown parameters for the following equations. Please could you give me a hint on how to do it?

Some of the parameters i already know them so i define them n = 1; k = 1.38064852 * 10^-23; T = 8; q = 1.60217662 * 10^-19; Vt = n*k*T/q; Np = 5; Ns = 12; Voc = 36.48; Isc = 8.12; Im = 7.62; Vm = 29.47;

These are the unknown parameters i am looking for: (Iph I_o Vt Rs Rp)

These are the equations i have and i want to solve them 1) Vt = (n*k*T)/q; 2) Np*Iph – Np*I_o*exp((Voc/Ns*Vt) – 1) – (Np/Ns) * (Voc/Rp) = 0; 3) Isc == Np*Iph – Np*I_o*exp((Isc*Rs)/(Np*Vt) -1) – ((Isc*Rs)/Rp); 4) Im == Np*Iph – Np*I_o*exp((((Vm/Ns)+((Im/Np)*Rs))/Vt) – 1) – Np* (((Vm/Ns) + ((Im/Np)*Rs))/Rp); 5) (Im/Vm) == ((Np/Ns*Vt)*I_o*exp(((Vm + Im*((Ns/Np)*Rs))/(Ns*Vt)) + (1/(Ns*Np)*Rp))) / (1 + (Rs/Vt)*I_o*exp((Vm + (Im*(Ns*Np)*Rs))/(Ns*Vt)) + (Rs/Rp)); 6) (-(1/Rp)) == ((-(Np/Ns*Vt))*I_o*exp((Isc*(Ns/Np)*Rs)/(Ns*Vt)) – (1/(Ns*Np)*Rp)) / (1 + (Rs/Vt)*I_o*exp((Isc*(Ns/Np)*Rs)/(Ns*Vt)) + (Rs/Rp)), Iph);

Thanks in advance

Regards,

Charalampos

Best Answer

Use the Optimization Toolbox fsolve (link) function.

Related Solutions

MATLAB: How to speed up matrix operations with lots of scalar constants involved

If you have a modern Matlab version >= 2016b, use the auto expanding instead of an explicit repmat:

% Ipc = repmat(I(:,jt2),[1,npmax(jz)-1]);
Ipc = I(:,jt2);
... CnE(:,1:npmax(jz)-1,jl)) .* Ipc * dt / hb

I guess, that dt and hb are scalars (please do not let the readers guess!). Then remember, that Matlab evaluate the code from left to right:

Y = X * dt / hb

is equivalent to:

Temp = X * dt;
Y    = Temp / hb;

These are 2 matrix operations. But you can combine the scalar operations:

Y = X * (dt / hb)

This is 1 matrix operation and a cheap scalar division.

As Matt J has explained already, operating on subarrays as in |dCnE(:,2:npmax(jz)) is expensive. Better crop the data, such that you can omit the ugly indexing.

I cannot run it by my own, because you did not provide the input values, but compare it optically:

Ipc=repmat(I(:,jt2),[1,npmax(jz)-1]);
dCnE(:,2:npmax(jz))=(sigma0*CnZ(:,2:npmax(jz),jl)-sigma*CnE(:,2:npmax(jz),jl)+ ...
      vs2*sigma*CnE(:,1:npmax(jz)-1,jl)).*Ipc*dt/hb+CnB(:,2:npmax(jz),jl)*(1-exp(-dt/tau2))...
      +vs2*(beta0*CnZ(:,1:npmax(jz)-1,jl).*(Ipc.^2))*dt/(2*hb)-CnE(:,2:npmax(jz),jl)/tau1*dt;

or:

c1 = 1 - exp(-dt / tau2);   % Calculate constants outside the loops!
Ipc  = I(:,jt2);
dCnE = (sigma0 * CnZ(:, :, jl) - sigma * CnE(:, :, jl) + ...
       vs2 * sigma * CnE(:, :, jl)) .* (Ipc * (dt / hb)) + ...
       CnB(:,:,jl) * c1 + ...
       (vs2 * beta0 * dt / (2*hb)) * CnZ(:, :, jl) .* (Ipc .^ 2) - ...
       CnE(:, :, jl) / (tau1 * dt);

This can be simplified a little bit for the CnE term:

dCnE = (sigma0 * CnZ(:, :, jl) + ...
       CnE(:, :, jl)) .* (Ipc * ((vs2 * sigma - sigma) * dt / hb)) + ...
       CnB(:,:,jl) * c1 + ...
       (vs2 * beta0 * dt / (2*hb)) * CnZ(:, :, jl) .* (Ipc .^ 2) - ...
       CnE(:, :, jl) / (tau1 * dt);

Methods:

Combine the scalar operations to reduce the number of matrix operations.
Move constants out of the loop.
Crop the data outside the loops instead of applying an a:b indexing in each iteration.
Use auto-expanding instead of inflating by repmat. bsxfun is sometimes faster or slower than auto-expanding - try it.
Avoid repeated calculations. In your case e.g. Ipc(:,1)*dt/hb is computed several times. Using this avoids this:

C1 = I(:, jt2) * (dt/hb);

The latter includes another idea: You inflate I(:,jt2) at first by Ipc = repmat(...), and waste time with cropping out a subvector Ipc(:,1) later on. Better create the subvector once only: Ipc = I(:,jt2).

Finally: Replace

        non0=Cn0(:,jl);
        non0(non0<0)=0;
        Cn0(:,jl)=non0;

by:

        Cn0 = max(Cn0, 0);

and move it outside the loops, because you do not use the cleaned values inside the loop.

MATLAB: 5 equations with 5 unknowns

Run this and check if the equations are those that should be...

clearvars 
syms  Vmp  Voc  Isc  mp  k  q  T Rp0 Rs0 Imp
syms I0 IL n Rs Rp 
eq1 =  Rp==Voc/(IL-I0*(exp(q/(n*k*T)*Voc)-1)) 
eq2 =  Isc==IL-I0*(exp(q/(n*k*T)*Rs*Isc)-1)-Rs*Isc/Rp
eq3 =  Imp==IL-I0*(exp(q/(n*k*T)*(Vmp+Imp*Rs))-1)-(Vmp+Imp*Rs)/Rp
eq4 =  Rs0==-(q/(n*k*T)*I0*exp(q/(n*k*T)*Voc)+1/Rs)/(1+Rs*(q/(n*k*T)*I0*exp(q/(n*k*T)*Voc)+1/Rs))
eq5 =  Rp0==-(q/(n*k*T)*I0*exp(q/(n*k*T)*Rs*Isc)+1/Rs)/(1+Rs*(q/(n*k*T)*I0*exp(q/(n*k*T)*Rs*Isc)+1/Rs))
Vmp=0.57
Voc=0.66
Isc=0.0067  
Imp=0.0062
k=1.38e-23
q=1.6e-19
T=300
Rp0=-0.0032
Rs0=-0.1535
eq1=subs(eq1); eq2=subs(eq2); eq3=subs(eq3); eq4=subs(eq4); eq5=subs(eq5);
[I0,IL,n,Rs,Rp]=vpasolve( [eq1, eq2, eq3, eq4, eq5], [I0,IL,n,Rs,Rp])

I replaced solve by vpasolve and a I filled in two multiplication signs on equations 4 and 5

You will get these results:

Best Answer

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MATLAB: How to speed up matrix operations with lots of scalar constants involved

MATLAB: 5 equations with 5 unknowns

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