# Paul Gasper, NREL import numpy as np from functions.extract_stressors import extract_stressors from functions.state_functions import update_power_state, update_sigmoid_state # EXPERIMENTAL AGING DATA SUMMARY: # Aging test matrix varied primarly temperature, with small DOD variation. # Calendar and cycle aging were performed between 0 and 55 Celsius. C-rates always at 1C, # except for charging at 0 Celsius, which was conducted at C/3. Depth-of-discharge was 80% # for nearly all tests (3.4 V - 4.1 V), with one 100% DOD test (3 V - 4.2 V). # Reported relative capacity was measured at C/5 rate at the aging temperatures. Reported # relative DC resistance was measured by HPPC using a 10s, 1C DC pulse, averaged between # charge and discharge, calculated using a simple ohmic fit of the voltage response. # 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 temperature and depth-of-discharge. Sensitivity to cycling degradation rate # at low temperature is inferred from physical insight due to limited data. # MODEL LIMITATIONS # There is NO C-RATE DEPENDENCE for degradation in this model. THIS IS NOT PHYSICALLY REALISTIC # AND IS BASED ON LIMITED DATA. class Nmc111_Gr_Kokam75Ah_Battery: # Model predicting the degradation of a Kokam 75 Ah NMC-Gr pouch cell. # https://ieeexplore.ieee.org/iel7/7951530/7962914/07963578.pdf # It is uncertain if the exact NMC composition is 1-1-1, but it this is definitely not a high nickel (>80%) cell. # 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, using Loss of Lithium Inventory (LLI) and Loss of Active # Material (LAM) degradation modes that interact competitively (cell performance is limited by # one or the other.) # Parameters to modify to change fade rates: # Calendar capacity loss rate: q1_0 # Cycling capacity loss rate (LLI): q3_0 # Cycling capacity loss rate (LAM): q5_0, will also effect resistance growth onset due to LAM. # Calendar resistance growth rate (LLI), relative to capacity loss rate: r1 # Cycling resistance growth rate (LLI), relative to capacity loss rate: r3 def __init__(self): # States: Internal states of the battery model self.states = { 'qLoss_LLI_t': np.array([0]), # relative Li inventory change, time dependent (SEI) 'qLoss_LLI_EFC': np.array([0]), # relative Li inventory change, charge-throughput dependent (SEI) 'qLoss_LAM': np.array([1e-8]), # relative active material change, charge-throughput dependent (electrode damage) 'rGain_LLI_t': np.array([0]), # relative SEI growth, time dependent (SEI) 'rGain_LLI_EFC': np.array([0]), # relative SEI growth, charge-throughput dependent (SEI) } # Outputs: Battery properties derived from state values self.outputs = { 'q': np.array([1]), # relative capacity 'q_LLI': np.array([1]), # relative lithium inventory 'q_LLI_t': np.array([1]), # relative lithium inventory, time dependent loss 'q_LLI_EFC': np.array([1]), # relative lithium inventory, charge-throughput dependent loss 'q_LAM': np.array([1.01]), # relative active material, charge-throughput dependent loss 'r': np.array([1]), # relative resistance 'r_LLI': np.array([1]), # relative SEI resistance 'r_LLI_t': np.array([1]), # relative SEI resistance, time dependent growth 'r_LLI_EFC': np.array([1]), # relative SEI resistance, charge-throughput dependent growth 'r_LAM': np.array([1]), # relative electrode resistance, q_LAM dependent growth } # 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]), } # Rates: History of stressor-dependent degradation rates self.rates = { 'q1': np.array([np.nan]), 'q3': np.array([np.nan]), 'q5': np.array([np.nan]), } # Nominal capacity @property def _cap(self): return 75 # Define life model parameters @property def _params_life(self): return { 'q1_0' : 2.66e7, # CHANGE to modify calendar degradation rate (larger = faster degradation) 'q1_1' : -17.8, 'q1_2' : -5.21, 'q2' : 0.357, 'q3_0' : 3.80e3, # CHANGE to modify cycling degradation rate (LLI) (larger = faster degradation) 'q3_1' : -18.4, 'q3_2' : 1.04, 'q4' : 0.778, 'q5_0' : 1e4, # CHANGE to modify cycling degradation rate (LAM) (accelerating fade onset) (larger = faster degradation) 'q5_1' : 153, 'p_LAM' : 10, 'r1' : 0.0570, # CHANGE to modify change of resistance relative to change of capacity (calendar degradation) 'r2' : 1.25, 'r3' : 4.87, # CHANGE to modify change of resistance relative to change of capacity (cycling degradation) 'r4' : 0.712, 'r5' : -0.08, 'r6' : 1.09, } # 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) TdegC = TdegK - 273.15 TdegKN = TdegK / (273.15 + 35) # normalized temperature UaN = Ua / 0.123 # normalized anode-to-reference potential # Grab parameters p = self._params_life # Calculate degradation rates q1 = p['q1_0'] * np.exp(p['q1_1'] * (1 / TdegKN)) * np.exp(p['q1_2'] * (UaN / TdegKN)) q3 = p['q3_0'] * np.exp(p['q3_1'] * (1/TdegKN)) * np.exp(p['q3_2'] * np.exp(dod**2)) q5 = p['q5_0'] + p['q5_1'] * (TdegC - 55) * dod # 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 # Calculate incremental state changes states = self.states # Capacity dq_LLI_t = update_power_state(states['qLoss_LLI_t'][-1], delta_t_days, 2*q1, p['q2']) dq_LLI_EFC = update_power_state(states['qLoss_LLI_EFC'][-1], delta_efc, q3, p['q4']) dq_LAM = update_sigmoid_state(states['qLoss_LAM'][-1], delta_efc, 1, 1/q5, p['p_LAM']) # Resistance dr_LLI_t = update_power_state(states['rGain_LLI_t'][-1], delta_t_days, p['r1']*q1, p['r2']) dr_LLI_EFC = update_power_state(states['rGain_LLI_EFC'][-1], delta_efc, p['r3']*q3, p['r4']) # Accumulate and store states dx = np.array([dq_LLI_t, dq_LLI_EFC, dq_LAM, 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]) 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]) 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 = 1 - states['qLoss_LLI_t'][-1] - states['qLoss_LLI_EFC'][-1] q_LLI_t = 1 - states['qLoss_LLI_t'][-1] q_LLI_EFC = 1 - states['qLoss_LLI_EFC'][-1] q_LAM = 1.01 - states['qLoss_LAM'][-1] q = np.min(np.array([q_LLI, q_LAM])) # Resistance r_LLI = 1 + states['rGain_LLI_t'][-1] + states['rGain_LLI_EFC'][-1] r_LLI_t = 1 + states['rGain_LLI_t'][-1] r_LLI_EFC = 1 + states['rGain_LLI_EFC'][-1] r_LAM = p['r5'] + p['r6'] * (1 / q_LAM) r = np.max(np.array([r_LLI, r_LAM])) # Assemble output out = np.array([q, q_LLI, q_LLI_t, q_LLI_EFC, q_LAM, r, r_LLI, r_LLI_t, r_LLI_EFC, r_LAM]) # Store results for k, v in zip(list(self.outputs.keys()), out): self.outputs[k] = np.append(self.outputs[k], v)