# Paul Gasper, NREL # This model is fit to Panasonic 18650B NCA-Gr cells. # Calendar data is reported by Keil et al (https://dx.doi.org/10.1149/2.0411609jes) # Cycling data is reported by Preger et al (https://doi.org/10.1149/1945-7111/abae37) and # is available at batteryarchive.com. # I'm not aware of any study conducting both calendar aging and cycle aging of these cells. import numpy as np from functions.extract_stressors import extract_stressors from functions.state_functions import update_power_state # EXPERIMENTAL AGING DATA SUMMARY: # Calendar aging widely varied SOC at 25, 40, and 50 Celsius. 300 days max. # Cycle aging varied temperature and C-rates, and DOD. Some accelerating fade is observed # at room temperature and high DODs but isn't modeled well here. That's not a huge problem, # because the modeled lifetime is quite short anyways. # 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 C-rate, temperature, and depth-of-discharge (DOD dependence is assumed to be linear, no aging data) # MODEL LIMITATIONS # Cycle degradation predictions WILL NOT PREDICT KNEE-POINT due to limited data. # Cycle aging is only modeled at 25, 35, and 45 Celsius, PREDICTIONS OUTSIDE THIS # TEMPERATURE RANGE MAY BE OPTIMISTIC. class Nca_Gr_Panasonic3Ah_Battery: def __init__(self): # States: Internal states of the battery model self.states = { 'qLoss_t': np.array([0]), 'qLoss_EFC': np.array([0]), } # Outputs: Battery properties derived from state values self.outputs = { 'q': np.array([1]), 'q_t': np.array([1]), 'q_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]), 'dod': np.array([np.nan]), 'Crate': np.array([np.nan]), } # Rates: History of stressor-dependent degradation rates self.rates = { 'k_cal': np.array([np.nan]), 'k_cyc': np.array([np.nan]), } # Nominal capacity @property def _cap(self): return 3.2 # Define life model parameters @property def _params_life(self): return { # Capacity fade parameters 'qcal_A': 75.4, 'qcal_B': -3.34e+03, 'qcal_C': 353, 'qcal_p': 0.512, 'qcyc_A': 1.86e-06, 'qcyc_B': 4.74e-11, 'qcyc_C': 0.000177, 'qcyc_D': 3.34e-11, 'qcyc_E': 2.81e-09, 'qcyc_p': 0.699, } # 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 k_cal = p['qcal_A'] * np.exp(p['qcal_B']/TdegK) * np.exp(p['qcal_C']*soc/TdegK) k_cyc = ( (p['qcyc_A'] + p['qcyc_B']*Crate + p['qcyc_C']*dod) * (np.exp(p['qcyc_D']/TdegK) + np.exp(-p['qcyc_E']/TdegK)) ) # Calculate time based average of each rate k_cal = np.trapz(k_cal, x=t_secs) / delta_t_secs k_cyc = np.trapz(k_cyc, x=t_secs) / delta_t_secs # Calculate incremental state changes states = self.states # Capacity dq_t = update_power_state(states['qLoss_t'][-1], delta_t_days, k_cal, p['qcal_p']) dq_EFC = update_power_state(states['qLoss_EFC'][-1], delta_efc, k_cyc, p['qcyc_p']) # Accumulate and store states dx = np.array([dq_t, dq_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), 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([k_cal, k_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 # Capacity q_t = 1 - states['qLoss_t'][-1] q_EFC = 1 - states['qLoss_EFC'][-1] q = 1 - states['qLoss_t'][-1] - states['qLoss_EFC'][-1] # Assemble output out = np.array([q, q_t, q_EFC]) # Store results for k, v in zip(list(self.outputs.keys()), out): self.outputs[k] = np.append(self.outputs[k], v)