# Can I get a better fit to data

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Matthew Hunt on 2 Feb 2023
Commented: Matthew Hunt on 14 Mar 2023 at 14:57
I've written a code to find some parameters to a mathematical model but I can't seem to get a good fit to the experimental data. I've seen that this model can model the data very well but I can't seem to get the right parameters that actually give me a reasonable fit. Are there any tricks I can use to get the right parameters to get me close to the data?
The functions and code I use are:
function y=eta_SPM(c_s,k,T,I_app)
if isrow(I_app)==true
I_app=I_app';
end
if isrow(c_s)==true
c_s=c_s';
end
R=8.314;
F=9.648e4;
g_0=2*k*sqrt(c_s.*(1-c_s));
y=(2*R*T/F)*asinh(I_app./g_0);
.
function V=volt(X,L_a,L_c,T,I_app,t)
%Compute the lithium concentration in each of the
r=linspace(0,1,100);
D_a=X(1);
D_c=X(2);
u_0_a=X(3);
u_0_c=X(4);
k_a=X(5);
k_c=X(6);
alpha_a=X(7);
alpha_c=X(8);
c_c=c(u_0_c,alpha_c*I_app,D_c,r,t);
c_a=c(u_0_a,-alpha_a*I_app,D_a,r,t);
%Compute Voltage
V=OCV_plus(c_c)+eta_SPM(c_c,k_c,T,-alpha_c*I_app/L_c)-(OCV_minus(c_a)+eta_SPM(c_a,k_a,T,alpha_a*I_app/L_a));
.
function y=OCV_plus(x)
if isrow(x)==true
x=x';
end
y=-2.613*x.^6+9.858*x.^5-16.63*x.^4+14.09*x.^3-4.952*x.^2-0.4427*x+4.27;
.
function y=OCV_minus(x)
if isrow(x)==true
x=x';
end
y=(289.7*x.^2+339.3*x+99.95)./(x.^3+4408*x.^2+4080*x+142.9);
Here I use a weight function to highlight the parts of the data I want the optimiser to take special notice of.
function y=objective_fn(V_exp,V_calc,t)
%This is the objective function
N=length(t); %gets length of vectors
if isrow(V_exp)==true
V_exp=V_exp';
end
if isrow(V_calc)==true
V_calc=V_calc';
end
if isrow(t)==true
t=t';
end
%define the weight of the objective function to concentrate on a particular
%feature
w=ones(N,1);
w(6500:6800)=2;
w(1:100)=2;
y=trapz(t,(V_exp.*w-V_calc).^2);
.
%This program finds the optimised parameters
%Define upper and lower bounds for the parameters;
D_a=0.03;
D_c=0.01;
u_0_a=0.95;
u_0_c=0.27;
k_a=1e-3;
k_c=1e2;
alpha_a=0.01;
alpha_c=0.015;
lb = [10^-4, 10^-4 ,0.7 ,0.15 ,1e-3 ,1e-3 ,1e-4 ,1e-4 ];
ub = [1e1, 1e1, 0.99, 0.5, 2 ,2 ,1 ,1 ];
x0=[D_a,D_c,u_0_a,u_0_c,k_a,k_c,alpha_a,alpha_c];
n=length(t);
T=293;
I_app=ones(1,n);
L_a=1;
L_c=1;
V_0=volt(x0,L_a,L_c,T,I_app,t);
A = [];
b = [];
Aeq = [];
beq = [];
fun=@(X) objective_fn(V_exp, volt(X,L_a,L_c,T,I_app,t) ,t);
X = fmincon(fun,x0,A,b,Aeq,beq,lb,ub);
%Compare with experimental data.
D_a=X(1);
D_c=X(2);
u_0_a=X(3);
u_0_c=X(4);
k_a=X(5);
k_c=X(6);
alpha_a=X(7);
alpha_c=X(8);
r=linspace(0,1,150);
c_c=c(u_0_c,alpha_c*I_app,D_c,r,t);
c_a=c(u_0_a,-alpha_a*I_app,D_a,r,t);
%Compute Voltage
V_op=OCV_plus(c_c)+eta_SPM(c_c,k_c,T,-alpha_c*I_app/L_c)-(OCV_minus(c_a)+eta_SPM(c_a,k_a,T,alpha_a*I_app/L_a));
plot(t,V_exp);
hold on
plot(t,V_op);
xlabel('Time(hrs)');
ylabel('Terminal Voltage');
axis([0 1.2 2.7 4.5]);
I did a curve fit to the experimental data which is valid for 0<=t<=2.05
function y=V_data(t)
if isrow(t)==true
t=t';
end
y=(-3821*t.^4+1209*t.^3-2564*t.^2+3545*t+2284)./(t.^5-657.6*t.^4*t.^3+48.25*t.^2+914.6*t+544.6);
.
function u=c(v_0,q,D,r,t)
n=length(r);m=length(t); %The dimensions of u are defined by the length os r and t.
v=zeros(m,n); %Initialise u
dr=r(2); %as both t and r start at zero, dr and dt can be defined in this way
dt=t(2);
alpha=D*dt/dr^2; %compute alpha and beta
beta=D*dt/(2*dr);
A_main=-(1+alpha)*ones(n,1); %Begin constructing the vectors to put in the diagonals
A_u=0.5*alpha+beta./r(1:n-1);
A_l=0.5*alpha-beta./r(2:n);
A=diag(A_u,1)+diag(A_l,-1)+diag(A_main,0); %Construct A
B_main=(alpha-1)*ones(n,1); %Only the main diagonal for differes by anyting more than a sign
B=diag(-A_u,1)+diag(B_main,0)+diag(-A_l,-1); %Construct B
A(1,1)=-(1+3*alpha); %Insert the inner boundary values for A and B
A(1,2)=3*alpha;
B(1,1)=3*alpha-1;
B(1,2)=-3*alpha;
A(n,n)=-(1+alpha); %Insert the outer boundary values for A and B
A(n,n-1)=alpha;
B(n,n)=-alpha;
B(n,n-1)=alpha-1;
p=zeros(n,1); %Used in the consutruction of c
p(n)=1;
B_1=A\B;
w=A\p;
gamma=-2*(dr/D)*(0.5*alpha+beta/r(n)); %For computational efficency.
v(1,:)=v_0*ones(1,n);
for i=2:m
c=gamma*w*(q(i-1)+q(i));
sol=B_1*v(i-1,:)'+c;
v(i,:)=sol';
end
u=v(:,end);
Matthew Hunt on 3 Feb 2023
Hi, I see where the issue is and the proper code for the curve fit of the experimental data is:
function y=V_data(t)
if isrow(t)==true
t=t';
end
y=(-2.376e+04*t.^4+5.47e+04*t.^3-3.137e+04*t.^2+3.294e+04*t+2.165e+04)./(t.^5-6368*t.^4+1.418e+04*t.^3-6837*t.^2+8408*t+5154);
The input is in hours and it goes from 0 to 2.03.

Yoga on 10 Mar 2023 at 10:43
I guess you found the fix for the question.
The proper code for the curve fit of the data is working fine as you have mentioned in the comments.
function y=V_data(t)
if isrow(t)==true
t=t';
end
y=(-2.376e+04*t.^4+5.47e+04*t.^3-3.137e+04*t.^2+3.294e+04*t+2.165e+04)./(t.^5-6368*t.^4+1.418e+04*t.^3-6837*t.^2+8408*t+5154);
Matthew Hunt on 14 Mar 2023 at 14:57
Actually I didn't.

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