# Paul Gasper, NREL import numpy as np from functions.extract_stressors import extract_stressors from functions.state_functions import update_power_B_state, update_sigmoid_state import scipy.stats as stats # EXPERIMENTAL AGING DATA SUMMARY: # Aging test matrix varied temperature and state-of-charge for calendar aging, and # varied depth-of-discharge, average state-of-charge, and C-rates for cycle aging. # There is NO LOW TEMPERATURE cycling aging data, i.e., no lithium-plating induced by # kinetic limitations on cell performance; CYCLING WAS ONLY DONE AT 25 CELSIUS AND 45 CELSIUS, # so any model predictions at low temperature cannot incorporate low temperature degradation modes. # Discharge capacity # MODEL SENSITIVITY # The model predicts degradation rate versus time as a function of temperature and average # state-of-charge and degradation rate versus equivalent full cycles (charge-throughput) as # a function of average state-of-charge during a cycle, depth-of-discharge, and average of the # charge and discharge C-rates. # MODEL LIMITATIONS # There is no influence of TEMPERATURE on CYCLING DEGRADATION RATE due to limited data. This is # NOT PHYSICALLY REALISTIC AND IS BASED ON LIMITED DATA. class Lfp_Gr_SonyMurata3Ah_Battery: # Model predicting the degradation of Sony-Murata 3 Ah LFP-Gr cylindrical cells. # Data is from Technical University of Munich, reported in studies led by Maik Naumann. # Capacity model identification was conducted at NREL. Resistance model is from Naumann et al. # Naumann et al used an interative fitting procedure, but it was found that lower model error could be # achieved by simply reoptimizing all resistance growth parameters to the entire data set. # Calendar aging data source: https://doi.org/10.1016/j.est.2018.01.019 # Cycle aging data source: https://doi.org/10.1016/j.jpowsour.2019.227666 # Model identification source: https://doi.org/10.1149/1945-7111/ac86a8 # Degradation rate is a function of the aging stressors, i.e., ambient temperature and use. # The state of the battery is updated throughout the lifetime of the cell. # Performance metrics are capacity and DC resistance. These metrics change as a function of the # cell's current degradation state, as well as the ambient temperature. The model predicts time and # cycling dependent degradation. Cycling dependent degradation includes a break-in mechanism as well # as long term cycling fade; the break-in mechanism strongly influenced results of the accelerated # aging test, but is not expected to have much influence on real-world applications. # Parameters to modify to change fade rates: # q1_b0: rate of capacity loss due to calendar degradation # q5_b0: rate of capacity loss due to cycling degradation # k_ref_r_cal: rate of resistance growth due to calendar degradation # A_r_cyc: rate of resistance growth due to cycling degradation def __init__(self): # States: Internal states of the battery model self.states = { 'qLoss_LLI_t': np.array([0]), 'qLoss_LLI_EFC': np.array([0]), 'qLoss_BreakIn_EFC': np.array([1e-10]), 'rGain_LLI_t': np.array([0]), 'rGain_LLI_EFC': np.array([0]), } # Outputs: Battery properties derived from state values self.outputs = { 'q': np.array([1]), 'q_LLI_t': np.array([1]), 'q_LLI_EFC': np.array([1]), 'q_BreakIn_EFC': np.array([1]), 'r': np.array([1]), 'r_LLI_t': np.array([1]), 'r_LLI_EFC': np.array([1]), } # Stressors: History of stressors on the battery self.stressors = { 'delta_t_days': np.array([np.nan]), 't_days': np.array([0]), 'delta_efc': np.array([np.nan]), 'efc': np.array([0]), 'TdegK': np.array([np.nan]), 'soc': np.array([np.nan]), 'Ua': np.array([np.nan]), 'dod': np.array([np.nan]), 'Crate': np.array([np.nan]), } # Rates: History of stressor-dependent degradation rates self.rates = { 'q1': np.array([np.nan]), 'q3': np.array([np.nan]), 'q5': np.array([np.nan]), 'q7': np.array([np.nan]), 'r_kcal': np.array([np.nan]), 'r_kcyc': np.array([np.nan]), } # Nominal capacity @property def _cap(self): return 3 # Define life model parameters @property def _params_life(self): return { # Capacity fade parameters 'q2': 0.000130510034211874, 'q1_b0': 0.989687151293590, # CHANGE to modify calendar degradation rate 'q1_b1': -2881067.56019324, 'q1_b2': 8742.06309157261, 'q3_b0': 0.000332850281062177, 'q3_b1': 734553185711.369, 'q3_b2': -2.82161575620780e-06, 'q3_b3': -3284991315.45121, 'q3_b4': 0.00127227593657290, 'q8': 0.00303553871631028, 'q9': 1.43752162947637, 'q7_b0': 0.582258029148225, 'q7_soc_skew': 0.0583128906965484, 'q7_soc_width': 0.208738181522897, 'q7_dod_skew': -3.80744333129564, 'q7_dod_width': 1.16126260428210, 'q7_dod_growth': 25.4130804598602, 'q6': 1.12847759334355, 'q5_b0': -6.81260579372875e-06, # CHANGE to modify cycling degradation rate 'q5_b1': 2.59615973160844e-05, 'q5_b2': 2.11559710307295e-06, # Resistance growth parameters 'k_ref_r_cal': 3.4194e-10, # CHANGE to modify calendar degradation rate 'Ea_r_cal': 71827, 'C_r_cal': -3.3903, 'D_r_cal': 1.5604, 'A_r_cyc': -0.002, # CHANGE to modify cycling degradation rate 'B_r_cyc': 0.0021, 'C_r_cyc': 6.8477, 'D_r_cyc': 0.91882 } # Battery model def update_battery_state(self, t_secs, soc, T_celsius): # Update the battery states, based both on the degradation state as well as the battery performance # at the ambient temperature, T_celsius # Inputs: # t_secs (ndarry): vector of the time in seconds since beginning of life for the soc_timeseries data points # soc (ndarry): vector of the state-of-charge of the battery at each t_sec # T_celsius (ndarray): the temperature of the battery during this time period, in Celsius units. # Check some input types: if not isinstance(t_secs, np.ndarray): raise TypeError('Input "t_secs" must be a numpy.ndarray') if not isinstance(soc, np.ndarray): raise TypeError('Input "soc" must be a numpy.ndarray') if not isinstance(T_celsius, np.ndarray): raise TypeError('Input "T_celsius" must be a numpy.ndarray') if not (len(t_secs) == len(soc) and len(t_secs) == len(T_celsius)): raise ValueError('All input timeseries must be the same length') self.__update_states(t_secs, soc, T_celsius) self.__update_outputs() def __update_states(self, t_secs, soc, T_celsius): # Update the battery states, based both on the degradation state as well as the battery performance # at the ambient temperature, T_celsius # Inputs: # t_secs (ndarry): vector of the time in seconds since beginning of life for the soc_timeseries data points # soc (ndarry): vector of the state-of-charge of the battery at each t_sec # T_celsius (ndarray): the temperature of the battery during this time period, in Celsius units. # Extract stressors delta_t_secs = t_secs[-1] - t_secs[0] delta_t_days, delta_efc, TdegK, soc, Ua, dod, Crate, cycles = extract_stressors(t_secs, soc, T_celsius) # Grab parameters p = self._params_life # Calculate the degradation coefficients q1 = np.abs( p['q1_b0'] * np.exp(p['q1_b1']*(1/(TdegK**2))*(Ua**0.5)) * np.exp(p['q1_b2']*(1/TdegK)*(Ua**0.5)) ) q3 = np.abs( p['q3_b0'] * np.exp(p['q3_b1']*(1/(TdegK**4))*(Ua**(1/3))) * np.exp(p['q3_b2']*(TdegK**3)*(Ua**(1/4))) * np.exp(p['q3_b3']*(1/(TdegK**3))*(Ua**(1/3))) * np.exp(p['q3_b4']*(TdegK**2)*(Ua**(1/4))) ) q5 = np.abs( p['q5_b0'] + p['q5_b1']*dod + p['q5_b2']*np.exp((dod**2)*(Crate**3)) ) q7 = np.abs( p['q7_b0'] * skewnormpdf(soc, p['q7_soc_skew'], p['q7_soc_width']) * skewnormpdf(dod, p['q7_dod_skew'], p['q7_dod_width']) * sigmoid(dod, 1, p['q7_dod_growth'], 1) ) k_temp_r_cal = ( p['k_ref_r_cal'] * np.exp((-p['Ea_r_cal'] / 8.3144) * (1/TdegK - 1/298.15)) ) k_soc_r_cal = p['C_r_cal'] * (soc - 0.5)**3 + p['D_r_cal'] k_Crate_r_cyc = p['A_r_cyc'] * Crate + p['B_r_cyc'] k_dod_r_cyc = p['C_r_cyc']* (dod - 0.5)**3 + p['D_r_cyc'] # Calculate time based average of each rate q1 = np.trapz(q1, x=t_secs) / delta_t_secs q3 = np.trapz(q3, x=t_secs) / delta_t_secs #q5 = np.trapz(q5, x=t_secs) / delta_t_secs # no time varying inputs q7 = np.trapz(q7, x=t_secs) / delta_t_secs # no time varying inputs k_temp_r_cal = np.trapz(k_temp_r_cal, x=t_secs) / delta_t_secs k_soc_r_cal = np.trapz(k_soc_r_cal, x=t_secs) / delta_t_secs # no time varying inputs #k_Crate_r_cyc = np.trapz(k_Crate_r_cyc, x=t_secs) / delta_t_secs # no time varying inputs #k_dod_r_cyc = np.trapz(k_dod_r_cyc, x=t_secs) / delta_t_secs # no time varying inputs # Calculate incremental state changes states = self.states # Capacity dq_LLI_t = update_sigmoid_state(states['qLoss_LLI_t'][-1], delta_t_days, q1, p['q2'], q3) dq_LLI_EFC = update_power_B_state(states['qLoss_LLI_EFC'][-1], delta_efc, q5, p['q6']) if delta_efc / delta_t_days > 2: # only evalaute if more than 2 full cycles per day dq_BreakIn_EFC = update_sigmoid_state(states['qLoss_BreakIn_EFC'][-1], delta_efc, q7, p['q8'], p['q9']) else: dq_BreakIn_EFC = 0 # Resistance dr_LLI_t = k_temp_r_cal * k_soc_r_cal * delta_t_secs dr_LLI_EFC = k_Crate_r_cyc * k_dod_r_cyc * delta_efc / 100 # Accumulate and store states dx = np.array([dq_LLI_t, dq_LLI_EFC, dq_BreakIn_EFC, dr_LLI_t, dr_LLI_EFC]) for k, v in zip(states.keys(), dx): x = self.states[k][-1] + v self.states[k] = np.append(self.states[k], x) # Store stressors t_days = self.stressors['t_days'][-1] + delta_t_days efc = self.stressors['efc'][-1] + delta_efc stressors = np.array([delta_t_days, t_days, delta_efc, efc, np.mean(TdegK), np.mean(soc), np.mean(Ua), dod, Crate]) for k, v in zip(self.stressors.keys(), stressors): self.stressors[k] = np.append(self.stressors[k], v) # Store rates rates = np.array([q1, q3, q5, q7, k_temp_r_cal * k_soc_r_cal, k_Crate_r_cyc * k_dod_r_cyc]) for k, v in zip(self.rates.keys(), rates): self.rates[k] = np.append(self.rates[k], v) def __update_outputs(self): # Calculate outputs, based on current battery state states = self.states p = self._params_life # Capacity q_LLI_t = 1 - states['qLoss_LLI_t'][-1] q_LLI_EFC = 1 - states['qLoss_LLI_EFC'][-1] q_BreakIn_EFC = 1 - states['qLoss_BreakIn_EFC'][-1] q = 1 - states['qLoss_LLI_t'][-1] - states['qLoss_LLI_EFC'][-1] - states['qLoss_BreakIn_EFC'][-1] # Resistance r_LLI_t = 1 + states['rGain_LLI_t'][-1] r_LLI_EFC = 1 + states['rGain_LLI_EFC'][-1] r = 1 + states['rGain_LLI_t'][-1] + states['rGain_LLI_EFC'][-1] # Assemble output out = np.array([q, q_LLI_t, q_LLI_EFC, q_BreakIn_EFC, r, r_LLI_t, r_LLI_EFC]) # Store results for k, v in zip(list(self.outputs.keys()), out): self.outputs[k] = np.append(self.outputs[k], v) def sigmoid(x, alpha, beta, gamma): return 2*alpha*(1/2 - 1/(1 + np.exp((beta*x)**gamma))) def skewnormpdf(x, skew, width): x_prime = (x-0.5)/width return 2 * stats.norm.pdf(x_prime) * stats.norm.cdf(skew * (x_prime))