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dfn_sensitivity_Federico.m
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dfn_sensitivity_Federico.m
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%% Sensitivity Eqn Analysis for DFN Model
% Created Feb 20, 2014 by Scott Moura
% Modified Apr 15, 2014 by Federico Bribiesca
%
% This code analyzes the sensitivity of all states in the DFN model to
% perturbations in the uncertain parameters. These parameters include
%
% UNCERTAIN PARAMETERS, theta
% 1 : D_s_n
% 2 : D_s_p
% 3 : D_e_n
% 4 : D_e_s
% 5 : D_e_p
% 6 : (1-t_plus)
% 7 : 1/sig_n
% 8 : 1/sig_p
% 9 : 1/kappa
% 10 : (1 + d ln f_ca / d ln c_e)
% 11 : k_n
% 12 : k_p
% 13 : R_f_n
% 14 : R_f_p
% 15 : epsilon_e_n
% 16 : epsilon_e_s
% 17 : epsilon_e_p
% 18 : c_s_n_max
% 19 : c_s_p_max
% 20 : h
% 21 : 1/(rho_avg * C_p)
%
% OUTPUTS OF INTEREST, y
% 1 : Volt
% 2 : SOC
% 3 : T
clc;
clear;
tic;
%% Load DFN Data
% DFN Data filename
%fn = 'data/sensitivity/zero_dfn.mat';
%fn = 'data/sensitivity/Federico_test_newBC.mat';
%fn='data/sensitivity/Federico_test_newBC_UDDS_500s.mat';
fn='data/sensitivity/Federico_test_newcs_UDDS_500s.mat';
%fn='data/sensitivity/Federico_test_newcs_constant_discharge.mat';
load(fn);
disp(['Loaded DFN data file: ' fn]);
% Parse output data
t = out.time;
p = out.p;
Cur = out.cur;
x = out.x;
z = out.z;
SOC = out.soc;
Volt = out.volt;
% Vector Lengths
NT = length(t); % Length of time
Nt = 21; % Number of params
Ny = 3; % Number of outputs
Ncsn = p.PadeOrder * (p.Nxn-1);
Ncsp = p.PadeOrder * (p.Nxp-1);
Nce = p.Nx - 3;
Nc = Ncsn+Ncsp+Nce;
Nn = p.Nxn - 1;
Np = p.Nxp - 1;
Nnp = Nn+Np;
% Indices
ind_csn = 1:Ncsn;
ind_phi_s_n = 1:Nn;
ind_phi_s_p = Nn+1:Nnp;
ind_T = Nc+1;
clear out;
% Plot DFN Response
fs = 15;
figure(1); clf;
set(gcf,'Position',[311, 16, 613, 684]);
subplot(311)
plot(t,Cur,'LineWidth',2)
ylabel('Current [A]','FontSize',fs);
xlim([0 t(end)]);
set(gca,'Fontsize',fs);
subplot(312)
plot(t,SOC,'LineWidth',2)
ylabel('SOC','FontSize',fs);
xlim([0 t(end)]);
set(gca,'Fontsize',fs);
subplot(313)
plot(t,Volt,'LineWidth',2)
ylabel('Volt','FontSize',fs);
xlabel('Time [sec]','FontSize',fs)
xlim([0 t(end)]);
set(gca,'Fontsize',fs);
%% Nominal Parameters
theta0 = [p.D_s_n;...
p.D_s_p;
1; 1; 1;
(1-p.t_plus);
1/p.sig_n;
1/p.sig_p;
1/1;
1 + 0;
p.k_n;
p.k_p;
p.R_f_n;
p.R_f_p;
p.epsilon_e_n;
p.epsilon_e_s;
p.epsilon_e_p;
p.c_s_n_max;
p.c_s_p_max;
p.h;
(1/p.rho_avg*p.C_p)];
p.theta0 = theta0;
%% Precompute Data
% Solid concentration matrices
[A_csn,B_csn,A_csp,B_csp,C_csn,C_csp, A_csn_normalized, A_csp_normalized] = c_s_mats(p);
p.A_csn = A_csn;
p.A_csn_normalized=A_csn_normalized;
p.B_csn = B_csn;
p.A_csp = A_csp;
p.A_csp_normalized=A_csp_normalized;
p.B_csp = B_csp;
p.C_csn = C_csn;
p.C_csp = C_csp;
% Electrolyte concentration matrices
c_ex = x(Ncsn+Ncsp+1,1) * ones(p.Nx+4,1);
[trash_var,trash_var,C_ce] = c_e_mats_federico(p,c_ex);
p.C_ce = C_ce;
% Solid Potential
[F1_psn,F1_psp,F2_psn,F2_psp,G_psn,G_psp,...
C_psn,C_psp,D_psn,D_psp] = phi_s_mats(p);
p.F1_psn = F1_psn;
p.F1_psp = F1_psp;
p.F2_psn = F2_psn;
p.F2_psp = F2_psp;
p.G_psn = G_psn;
p.G_psp = G_psp;
p.C_psn = C_psn;
p.C_psp = C_psp;
p.D_psn = D_psn;
p.D_psp = D_psp;
% Electrolyte Current
[F1_ien,F1_iep,F2_ien,F2_iep,F3_ien,F3_iep] = i_e_mats(p);
p.F1_ien = F1_ien;
p.F1_iep = F1_iep;
p.F2_ien = F2_ien;
p.F2_iep = F2_iep;
p.F3_ien = F3_ien;
p.F3_iep = F3_iep;
% Jacobian w.r.t. states
[A11, A12, A21, A22] = jac_dfn_pre(p);
p.A11 = A11;
p.A12 = A12;
p.A21 = A21;
p.A22 = A22;
clear A11 A12 A21 A22
%% Matrices for output equations
CCx = zeros(Ny, size(x,1));
CCz = zeros(Ny, size(z,1));
DD = zeros(Ny, Nt);
% Voltage
CCz(1,ind_phi_s_p) = p.C_psp(2,:);
CCz(1,ind_phi_s_n) = -p.C_psn(1,:);
% SOC
for i = 1:Nn
CCx(2,(1:3)+3*(i-1)) = 1/(Nn*p.c_s_n_max) * p.C_csn(2,:);
end
% Temperautre
CCx(3,ind_T) = 1;
%% Preallocate Sensitivity Vars
S1 = zeros(size(x,1), Nt, NT); % Sensitivity of x vars
S2 = zeros(size(z,1), Nt, NT); % Sensitivity of z vars
S3 = zeros(Ny, Nt, NT); % Sensitivity of outputs
%% Solve Sensitivity Eqns
disp('Solving Sensitivity Eqns...');
for k = 1:(NT-1)
%%% Jacobian w.r.t. states, @ current and nxt time step
[A11, A12, A21, A22] = jac_dfn_federico(x(:,k),z(:,k),Cur(k),p.A11,p.A12,p.A21,p.A22,p);
[A_corr_p, A_corr_n, A_corr_s] = jac_dfn_sensitivity_federico(x(:,k),z(:,k),Cur(k),p.A11,p.A12,p.A21,p.A22,p);
[A11_nxt, A12_nxt, A21_nxt, A22_nxt] = ...
jac_dfn_federico(x(:,k+1),z(:,k+1),Cur(k+1),p.A11,p.A12,p.A21,p.A22,p);
[A_corr_p_nxt, A_corr_n_nxt, A_corr_s_nxt] = jac_dfn_sensitivity_federico(x(:,k+1),z(:,k+1),Cur(k+1),p.A11,p.A12,p.A21,p.A22,p);
%%% Jacobian w.r.t. params, @ current and nxt time step
[B1, trash_var, B2, trash_var] = jac_p_dfn_federico(x(:,k),z(:,k),Cur(k),p);
[B1_nxt, trash_var, B2_nxt, trash_var] = jac_p_dfn_federico(x(:,k+1),z(:,k+1),Cur(k+1),p);
%%% Assemble Matrices %% Sensitivity BCs enter as a correction to the
%%% A*S term
A = A11 - A12*(A22\A21);
B = B1 - A12*(A22\B2);
A_nxt = A11_nxt - A12_nxt*(A22_nxt\A21_nxt);
B_nxt = B1_nxt - A12_nxt*(A22_nxt\B2_nxt);
C_nxt = -(A22_nxt\A21_nxt);
D_nxt = -(A22_nxt\B2_nxt);
MM = speye(size(x,1)) - p.delta_t/2 * A_nxt;
AA = speye(size(x,1)) + p.delta_t/2 * A;
BB = p.delta_t/2 * (B + B_nxt);
AA_corr_p=p.delta_t/2 * (A_corr_p + A_corr_p_nxt);
AA_corr_n=p.delta_t/2 * (A_corr_n + A_corr_n_nxt);
AA_corr_s=p.delta_t/2 * (A_corr_s + A_corr_s_nxt);
%%% Output sensitivity matrices
% Jacobian of SOC (y=2) w.r.t. c_s_n_max (theta=18)
c_s_n =x(1:Ncsn,k+1);
c_s_n_mat = reshape(c_s_n,p.PadeOrder,p.Nxn-1);
DD(2,18) = -1/(p.c_s_n_max^2) * mean(p.C_csn(2,:) * c_s_n_mat);
% Jacobian of SOC (y=2) w.r.t. D_s_n (theta=1)
dGdD =[0, 0 ,0]; %[3/p.R_s_n*6930*p.D_s_n/p.R_s_n^4, 3/p.R_s_n*189/p.R_s_n^2, 0];
DD(2,1) = 1/(p.c_s_n_max) * mean(dGdD * c_s_n_mat);
%%% Time-stepping
%%%%%%%%%%%%%%%%%%%%%%%% Commented by Federico %%%%%%%%%%%%%%%%%%%%%%%%
% % Not sure about stability of this method
% % Forward Euler (Faster, Less Accurate)
% S1(:,:,k+1) = AA*S1(:,:,k) + p.delta_t/2 * B;
% S2(:,:,k+1) = C_nxt*S1(:,:,k+1) + D_nxt;
% S3(:,:,k+1) = CCx*S1(:,:,k+1) + CCz*S2(:,:,k+1) + DD*diag(p.theta0);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Central Difference in time (More Accurate, Slower)
S1(:,:,k+1) = (MM\AA)*S1(:,:,k) + (MM\BB);
%%% Correct for the effect of BCs in sensitivity
S1(:,3,k+1) = S1(:,3,k+1) + (MM\AA_corr_n);
S1(:,15,k+1) = S1(:,15,k+1) + (MM\AA_corr_n);
S1(:,4,k+1) = S1(:,4,k+1) + (MM\AA_corr_s);
S1(:,16,k+1) = S1(:,16,k+1) + (MM\AA_corr_s);
S1(:,5,k+1) = S1(:,5,k+1) + (MM\AA_corr_p);
S1(:,17,k+1) = S1(:,17,k+1) + (MM\AA_corr_p);
S2(:,:,k+1) = C_nxt*S1(:,:,k+1) + D_nxt;
%%%%%%%%%%%%%%%%%% Addition by Federico %%%%%%%%%%%%%%%%%%%%%%%
S3(:,:,k+1) = CCx*S1(:,:,k+1) + CCz*S2(:,:,k+1) + DD*diag(p.theta0);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Error Checking
if(isnan(any(S1(:,:,k+1))))
disp('Nan Error');
disp(S1(:,:,k+1));
end
%%% Output Progress to Command Prompt
fprintf(1,'Time : %3.2f sec | Current : %2.4f A/m^2 | SOC : %1.3f | Voltage : %2.4fV\n',...
t(k),Cur(k+1),SOC(k+1),Volt(k+1));
end
%%
simTime = toc;
fprintf(1,'Simulation Time : %3.2f min\n',simTime/60);