Models

The Prognostics Model Package is distributed with a few pre-constructed models that can be used in simulation or prognostics (with the prog_algs package). These models are summarized in the table below with additional detail in the following sections.

Models Summary

Battery Model - Circuit

Battery Model - Electro Chemistry

Centrifugal Pump

Pneumatic Valve

Events

End of Discharge (EOD)

  • End of Discharge (EOD)

  • Insufficient Capacity

  • Impeller Wear Failure

  • Pump Oil Overheating

  • Radial Bering Overheat

  • Thrust Beiring Overheat

  • Leak-Bottom

  • Leak-Top

  • Leak-Internal

  • Spring Failure

  • Friction Failure

Inputs / Loading

Current (i)

Current (i)

  • Ambient Temperature-K (Tamb)

  • Voltage (V)

  • Discharge Pressure-Pa (pdisch)

  • Suction Pressure-Pa (psuc)

  • Sync Rotational Speed of

  • supply voltage-rad/sec (wsync)

  • Left Pressure-Pa (pL)

  • Right Pressure-Pa (pR)

  • Bottom Port Pressure-Pa (uBot)

  • Top Port Pressure-Pa (uTop)

Outputs / Measurements

Voltage (v), Temp °C (t)

Voltage (v), Temp °C (t)

  • Discharge Flow- m^3/s (Qout)

  • Oil Temp - K (To)

  • Radial Bearing Temp - K (Tr)

  • Thrust Bearing Temp - K (Tt)

  • Mech rotation - rad/s (w)

  • Florrate (Q)

  • Is piston at bottom (iB)

  • Is piston at top (iT)

  • Pressure at bottom - Pa (pB)

  • Pressure at top - Pa (pT)

  • Position of piston - m (x)

Battery Model - Circuit

class prog_models.models.BatteryCircuit(**kwargs)

Vectorized prognostics model for a battery, represented by an equivilant circuit model as described in the following paper: M. Daigle and S. Sankararaman, “Advanced Methods for Determining Prediction Uncertainty in Model-Based Prognostics with Application to Planetary Rovers,” Annual Conference of the Prognostics and Health Management Society 2013, pp. 262-274, New Orleans, LA, October 2013. https://papers.phmsociety.org/index.php/phmconf/article/view/2253

Events: (1)

EOD: End of Discharge

Inputs/Loading: (1)

i: Current draw on the battery

States: (4)
tb : Battery Temperature (°C)
qb : Charge stored in Capacitor Cb of the equivalent circuit model
qcp : Charge stored in Capacitor Ccp of the equivalent circuit model
qcs : Charge stored in Capacitor Ccs of the equivalent circuit model
Outputs: (2)
t: Temperature of battery (°C)
v: Voltage supplied by battery
Keyword Arguments
  • process_noise (Optional, float or Dict[Srt, float]) – Process noise (applied at dx/next_state). Can be number (e.g., .2) applied to every state, a dictionary of values for each state (e.g., {‘x1’: 0.2, ‘x2’: 0.3}), or a function (x) -> x

  • process_noise_dist (Optional, String) – distribution for process noise (e.g., normal, uniform, triangular)

  • measurement_noise (Optional, float or Dict[Srt, float]) – Measurement noise (applied in output eqn). Can be number (e.g., .2) applied to every output, a dictionary of values for each output (e.g., {‘z1’: 0.2, ‘z2’: 0.3}), or a function (z) -> z

  • measurement_noise_dist (Optional, String) – distribution for measurement noise (e.g., normal, uniform, triangular)

  • V0 (float) – Nominal Battery Voltage

  • Rp (float) – Battery Parasitic Resistance

  • qMax (float) – Maximum Charge

  • CMax (float) – Maximum Capacity

  • VEOD (float) – End of Discharge Voltage Threshold

  • Cbp3 (Cb0, Cbp0, Cbp1, Cbp2,) – Battery Capacity Parameters

  • Ccp (Rs, Cs, Rcp0, Rcp1, Rcp2,) – R-C Pair Parameter

  • Ta (float) – Ambient Temperature

  • Jt (float) – Temperature parameter

  • ha (float) – Heat transfer coefficient, ambient

  • hcp (float) – Heat transfer coefficient parameter

  • hcs (float) – Heat transfer coefficient - surface

  • x0 (Dict[Str, float]) – Initial state

Note

This is quicker but also less accurate as the electrochemistry model. We recommend using the electrochemistry model, when possible.

Battery Model - Electro Chemistry

There are three different flavors of Electro Chemistry Battery Models distributed with the package, described below

End of Discharge

class prog_models.models.BatteryElectroChemEOD(**kwargs)

Vectorized prognostics model for a battery, represented by an electrochemical equations as described in the following paper: M. Daigle and C. Kulkarni, “Electrochemistry-based Battery Modeling for Prognostics,” Annual Conference of the Prognostics and Health Management Society 2013, pp. 249-261, New Orleans, LA, October 2013. https://papers.phmsociety.org/index.php/phmconf/article/view/2252. This model predicts the end of discharge event.

The default model parameters included are for Li-ion batteries, specifically 18650-type cells. Experimental discharge curves for these cells can be downloaded from the Prognostics Center of Excellence Data Repository https://ti.arc.nasa.gov/tech/dash/groups/pcoe/prognostic-data-repository/.

Events: (1)

EOD: End of Discharge

Inputs/Loading: (1)

i: Current draw on the battery

States: (8)
tb: Battery temperature (K)
Vo: Voltage Drops due to Solid-Phase Ohmic Resistances
Vsn: Negative Surface Voltage (V)
Vsp: Positive Surface Voltage (V)
qnB: Amount of Negative Ions at the Battery Bulk
qnS: Amount of Negative Ions at the Battery Surface
qpB: Amount of Positive Ions at the Battery Bulk
qpS: Amount of Positive Ions at the Battery Surface
Outputs/Measurements: (2)
t: Temperature of battery (°C)
v: Voltage supplied by battery
Keyword Arguments
  • process_noise (Optional, float or Dict[Srt, float]) – Process noise (applied at dx/next_state). Can be number (e.g., .2) applied to every state, a dictionary of values for each state (e.g., {‘x1’: 0.2, ‘x2’: 0.3}), or a function (x) -> x

  • process_noise_dist (Optional, String) – distribution for process noise (e.g., normal, uniform, triangular)

  • measurement_noise (Optional, float or Dict[Srt, float]) – Measurement noise (applied in output eqn). Can be number (e.g., .2) applied to every output, a dictionary of values for each output (e.g., {‘z1’: 0.2, ‘z2’: 0.3}), or a function (z) -> z

  • measurement_noise_dist (Optional, String) – distribution for measurement noise (e.g., normal, uniform, triangular)

  • qMobile (float) –

  • xnMax (float) – Maximum mole fraction (neg electrode)

  • xnMin (float) – Minimum mole fraction (neg electrode)

  • xpMax (float) – Maximum mole fraction (pos electrode)

  • xpMin (float) – Minimum mole fraction (pos electrode) - note xn + xp = 1

  • Ro (float) – for Ohmic drop (current collector resistances plus electrolyte resistance plus solid phase resistances at anode and cathode)

  • alpha (float) – anodic/cathodic electrochemical transfer coefficient

  • Sn (float) – Surface area (- electrode)

  • Sp (float) – Surface area (+ electrode)

  • kn (float) – lumped constant for BV (- electrode)

  • kp (float) – lumped constant for BV (+ electrode)

  • Vol (float) – total interior battery volume/2 (for computing concentrations)

  • VolSFraction (float) – fraction of total volume occupied by surface volume

  • tDiffusion (float) – diffusion time constant (increasing this causes decrease in diffusion rate)

  • to (float) – for Ohmic voltage

  • tsn (float) – for surface overpotential (neg)

  • tsp (float) – for surface overpotential (pos)

  • Ap (U0p,) – Redlich-Kister parameter (+ electrode)

  • An (U0n,) – Redlich-Kister parameter (- electrode)

  • VEOD (float) – End of Discharge Voltage Threshold

  • x0 (dict) – Initial state

End of Life (i.e., InsufficientCapacity)

class prog_models.models.BatteryElectroChemEOL(**kwargs)

Vectorized prognostics model for a battery degredation, represented by an electrochemical model as described in the following paper: M. Daigle and C. Kulkarni, “End-of-discharge and End-of-life Prediction in Lithium-ion Batteries with Electrochemistry-based Aging Models,” AIAA SciTech Forum 2016, San Diego, CA. https://arc.aiaa.org/doi/pdf/10.2514/6.2016-2132

The default model parameters included are for Li-ion batteries, specifically 18650-type cells. Experimental discharge curves for these cells can be downloaded from the Prognostics Center of Excellence Data Repository https://ti.arc.nasa.gov/tech/dash/groups/pcoe/prognostic-data-repository/.

Events: (1)

InsufficientCapacity: Insufficient battery capacity

Inputs/Loading: (1)

i: Current draw on the battery

States: (3)
qMax: Maximum battery capacity
Ro : for Ohmic drop (current collector resistances plus electrolyte resistance plus solid phase resistances at anode and cathode)
D : diffusion time constant (increasing this causes decrease in diffusion rate)

Outputs/Measurements: (0)

Keyword Arguments
  • process_noise (Optional, float or Dict[Srt, float]) – Process noise (applied at dx/next_state). Can be number (e.g., .2) applied to every state, a dictionary of values for each state (e.g., {‘x1’: 0.2, ‘x2’: 0.3}), or a function (x) -> x

  • process_noise_dist (Optional, String) – distribution for process noise (e.g., normal, uniform, triangular)

  • measurement_noise (Optional, float or Dict[Srt, float]) – Measurement noise (applied in output eqn). Can be number (e.g., .2) applied to every output, a dictionary of values for each output (e.g., {‘z1’: 0.2, ‘z2’: 0.3}), or a function (z) -> z

  • measurement_noise_dist (Optional, String) – distribution for measurement noise (e.g., normal, uniform, triangular)

  • qMaxThreshold (float) – Threshold for qMax (for threshold_met and event_state), after which the InsufficientCapacity event has occured. Note: Battery manufacturers specify a threshold of 70-80% of qMax

  • wd (wq, wr,) – Wear rate for qMax, Ro, and D respectively

  • x0 (dict) – Initial state

End of Discharge, End of Life (i.e., InsufficientCapacity & EOD)

prog_models.models.BatteryElectroChem

alias of prog_models.models.battery_electrochem.BatteryElectroChemEODEOL

class prog_models.models.BatteryElectroChemEODEOL(**kwargs)

Prognostics model for a battery degredation and discharge, represented by an electrochemical model as described in the following papers:

  1. M. Daigle and C. Kulkarni, “End-of-discharge and End-of-life Prediction in Lithium-ion Batteries with Electrochemistry-based Aging Models,” AIAA SciTech Forum 2016, San Diego, CA. https://arc.aiaa.org/doi/pdf/10.2514/6.2016-2132

  2. M. Daigle and C. Kulkarni, “Electrochemistry-based Battery Modeling for Prognostics,” Annual Conference of the Prognostics and Health Management Society 2013, pp. 249-261, New Orleans, LA, October 2013. https://papers.phmsociety.org/index.php/phmconf/article/view/2252

The default model parameters included are for Li-ion batteries, specifically 18650-type cells. Experimental discharge curves for these cells can be downloaded from the Prognostics Center of Excellence Data Repository https://ti.arc.nasa.gov/tech/dash/groups/pcoe/prognostic-data-repository/.

Events: (2)
EOD: End of Discharge
InsufficientCapacity: Insufficient battery capacity
Inputs/Loading: (1)

i: Current draw on the battery

States: (11)

See BatteryElectroChemEOD, BatteryElectroChemEOL

Outputs/Measurements: (2)
t: Temperature of battery (°C)
v: Voltage supplied by battery

Note

For keyword arguments, see BatteryElectroChemEOD, BatteryElectroChemEOL

Pump Model

There are two variants of the pump model based on if the wear parameters are estimated as part of the state. The models are described below

Pump Model (Base)

class prog_models.models.CentrifugalPumpBase(**kwargs)

Prognostics model for a Centrifugal Pump as described in the following paper: M. Daigle and K. Goebel, “Model-based Prognostics with Concurrent Damage Progression Processes,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 43, no. 4, pp. 535-546, May 2013. https://www.researchgate.net/publication/260652495_Model-Based_Prognostics_With_Concurrent_Damage_Progression_Processes

Events (4)
ImpellerWearFailure: Failure of the impeller due to wear
PumpOilOverheat: Overheat of the pump oil
RadialBearingOverheat: Overheat of the radial bearing
ThrustBearingOverheat: Overhead of the thrust bearing
Inputs/Loading: (5)
Tamb: Ambient Temperature (K)
V: Voltage
pdisch: Discharge Pressure (Pa)
psuc: Suction Pressure (Pa)
wsync: Syncronous Rotational Speed of Supply Voltage (rad/sec)
States: (9)
A: Impeller Area (m^2)
Q: Flow Rate into Pump (m^3/s)
To: Oil Temperature (K)
Tr: Radial Bearing Temperature (K)
Tt: Thrust Bearing Temperature (K)
rRadial: Radial (thrust) Friction Coefficient
rThrust: Rolling Friction Coefficient
w: Rotational Velocity of Pump (rad/sec)
QLeak: Leak Flow Rate (m^3/s)
Outputs/Measurements: (5)
Qout: Discharge Flow (m^3/s)
To: Oil Temperature (K)
Tr: Radial Bearing Temperature (K)
Tt: Thrust Bearing Temperature (K)
w: Rotational Velocity of Pump (rad/sec)
Keyword Arguments
  • process_noise (Optional, float or Dict[Srt, float]) – Process noise (applied at dx/next_state). Can be number (e.g., .2) applied to every state, a dictionary of values for each state (e.g., {‘x1’: 0.2, ‘x2’: 0.3}), or a function (x) -> x

  • process_noise_dist (Optional, String) – distribution for process noise (e.g., normal, uniform, triangular)

  • measurement_noise (Optional, float or Dict[Srt, float]) – Measurement noise (applied in output eqn). Can be number (e.g., .2) applied to every output, a dictionary of values for each output (e.g., {‘z1’: 0.2, ‘z2’: 0.3}), or a function (z) -> z

  • measurement_noise_dist (Optional, String) – distribution for measurement noise (e.g., normal, uniform, triangular)

  • pAtm (float) – Atmospheric pressure

  • a2 (a0, a1,) – empirical coefficients for flow torque eqn

  • A (float) – impeller blade area

  • b (float) –

  • n (float) – Pole Phases

  • p (float) – Pole Pairs

  • I (float) – impeller/shaft/motor lumped inertia

  • r (float) – lumped friction parameter (minus bearing friction)

  • R2 (R1,) –

  • L1 (float) –

  • FluidI (float) – Pump fluid inertia

  • c (float) – Pump flow coefficient

  • cLeak (float) – Internal leak flow coefficient

  • ALeak (float) – Internal leak area

  • mcThrust (float) –

  • HThrust2 (HThrust1,) –

  • mcRadial (float) –

  • HRadial2 (HRadial1,) –

  • mcOil (float) –

  • HOil3 (HOil1, HOil2,) –

  • wThrust (wA, wRadial,) – Wear rates. See also CentrifugalPumpWithWear

  • lim (dict) – Parameter limits

  • x0 (dict) – Initial state

Pump Model (With Wear)

prog_models.models.CentrifugalPump

alias of prog_models.models.centrifugal_pump.CentrifugalPumpWithWear

class prog_models.models.CentrifugalPumpWithWear(**kwargs)

Prognostics model for a centrifugal pump with wear parameters as part of the model state. This is identical to CentrifugalPumpBase, only CentrifugalPumpBase has the wear params as parameters instead of states

This class implements a Centrifugal Pump model as described in the following paper: M. Daigle and K. Goebel, “Model-based Prognostics with Concurrent Damage Progression Processes,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 43, no. 4, pp. 535-546, May 2013. https://www.researchgate.net/publication/260652495_Model-Based_Prognostics_With_Concurrent_Damage_Progression_Processes

Events (4)

See CentrifugalPumpBase

Inputs/Loading: (5)

See CentrifugalPumpBase

States: (12)
States from CentrifugalPumpBase +
wA: Wear Rate for Impeller Area
wRadial: Wear Rate for Radial (thrust) Friction Coefficient
wRadial: Wear Rate for Rolling Friction Coefficient
Outputs/Measurements: (5)

See CentrifugalPumpBase

Model Configuration Parameters:

See CentrifugalPumpBase

Pneumatic Valve

There are two variants of the valve model based on if the wear parameters are estimated as part of the state. The models are described below

Pneumatic Valve (Base)

class prog_models.models.PneumaticValveBase(**kwargs)

Prognostics model for a Pneumatic Valve model as described in the following paper: M. Daigle and K. Goebel, “A Model-based Prognostics Approach Applied to Pneumatic Valves,” International Journal of Prognostics and Health Management, vol. 2, no. 2, August 2011. https://papers.phmsociety.org/index.php/ijphm/article/view/1359

Events: (5)
Bottom Leak: Failure due to a leak at the bottom pneumatic port
Top Leak: Failure due to a leak at the top pneumatic port
Internal Leak: Failure due to an internal leak at the seal surrounding the piston
Spring Failure: Failure due to spring weakening with use
Friction Failure: Failure due to increase in friction along the piston with wear
Inputs/Loading: (4)
pL: Fluid pressure at the left side of the plug (Pa)
pR: Fluid pressure at the right side of the plug (Pa)
uBot: input pressure at the bottom pneumatic port (Pa)
uTop: input pressure at the botton pneumatic port (Pa)
States: (10)
Aeb: Area of the leak at the bottom pneumatic port
Aet: Area of the leak at the top pneumatic port
Ai: Area of the internal leak
k: Spring Coefficient
mBot: Mass on bottom of piston (kg)
mTop: Mass on top of the piston (kg)
r: Friction Coefficient
v: Velocity of the piston (m/s)
x: Poisition of the piston (m)
pDiff: Difference in pressure between the left and the right
Outputs/Measurements: 6
Q: Flowrate
iB: Is the piston at the bottom (bool)
iT: Is the piston at the top (bool)
pB: Pressure at the bottom (Pa)
pT: Pressure at the top (Pa)
x: Position of piston (m)
Keyword Arguments
  • process_noise (Optional, float or Dict[Srt, float]) – Process noise (applied at dx/next_state). Can be number (e.g., .2) applied to every state, a dictionary of values for each state (e.g., {‘x1’: 0.2, ‘x2’: 0.3}), or a function (x) -> x

  • process_noise_dist (Optional, String) – distribution for process noise (e.g., normal, uniform, triangular)

  • measurement_noise (Optional, float or Dict[Srt, float]) – Measurement noise (applied in output eqn). Can be number (e.g., .2) applied to every output, a dictionary of values for each output (e.g., {‘z1’: 0.2, ‘z2’: 0.3}), or a function (z) -> z

  • measurement_noise_dist (Optional, String) – distribution for measurement noise (e.g., normal, uniform, triangular)

  • g (float) – Acceleration due to gravity (m/s^2)

  • pAtm (float) – Atmospheric pressure (Pa)

  • m (float) – Plug mass (kg)

  • offsetX (float) – Spring offset distance (m)

  • Ls (float) – Stroke Length (m)

  • Ap (float) – Surface area of piston for gas contact (m^2)

  • Vbot0 (float) – Below piston “default” volume (m^3)

  • Vtop0 (float) – Above piston “default” volume (m^3)

  • indicatorTol (float) – tolerance bound for open/close indicators

  • pSupply (float) – Supply Pressure (Pa)

  • Av (float) – Surface area of plug end (m^2)

  • Cv (float) – flow coefficient assuming Cv of 1300 GPM

  • rhoL (float) – density of LH2 in kg/m^3

  • gas_mass (float) – Molar mass of supply gas (kg/mol)

  • gas_temp (float) – Temperature of supply gas (K)

  • gas_R (gas_gamma, gas_z,) – Supply gas parameters

  • Cb (At, Ct, Ab,) –

  • AbMax (float) – Max limit for state Aeb

  • AtMax (float) – Max limit for state Aet

  • AiMax (float) – Max limit for state Ai

  • kMin (float) – Min limit for state k

  • rMax (float) – Max limit for state r

  • x0 (Dict[str, float]) – Initial state

  • wb (float) – Wear parameter for bottom leak

  • wi (float) – Wear parameter for internal leak

  • wt (float) – Wear parameter for top leak

  • wk (float) – Wear parameter for spring

  • wr (float) – Wear parameter for friction

Note

Supply gas parameters (gas_mass, gas_temp, gas_gamme, gas_z, gas_R) are for Nitrogen by default

Pneumatic Valve (With Wear)

prog_models.models.PneumaticValve

alias of prog_models.models.pneumatic_valve.PneumaticValveWithWear

class prog_models.models.PneumaticValveWithWear(**kwargs)

Prognostics model for a pneumatic valve with wear parameters as part of the model state. This is identical to PneumaticValveBase, only PneumaticValveBase has the wear params as parameters instead of states

This class implements a Pneumatic Valve model as described in the following paper: M. Daigle and K. Goebel, “A Model-based Prognostics Approach Applied to Pneumatic Valves,” International Journal of Prognostics and Health Management, vol. 2, no. 2, August 2011. https://www.phmsociety.org/node/602

Events (4)

See PneumaticValveBase

Inputs/Loading: (5)

See PneumaticValveBase

States: (12)

States from PneumaticValveBase + wb, wi, wk, wr, wt

Outputs/Measurements: (5)

See PneumaticValveBase

Model Configuration Parameters:

See PneumaticValveBase