# US 10,769,236 B2 · Battery models, systems, and methods using robust fail-safe iteration free approach for solving differential algebraic equations

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- Number: US 10,769,236 B2
- Document ID: US10769236B2
- Status: Granted
- Kind: B2
- Application number: 15/745,984
- Filed: 2016-07-20
- Published: 2018-07-26
- Granted: 2020-09-08
- Earliest claimed: 2015-07-20
- Assignees: University of Washington; Washington University
- Applicants: University of Washington; Washington University
- Inventors: Dayaram Sonawane; Manan Pathak; Venkat Subramanian; Matthew Lawder
- Web page: https://patentgrep.com/patent/US10769236B2

## CPC

- G06F 17/13: Differential equations (using digital differential analysers G06F7/64)
- G06F 30/20: Design optimisation, verification or simulation (optimisation, verification or simulation of circuit designs G06F30/30)
- H01M 10/44: Methods for charging or discharging (circuits for charging H02J7/00)
- H01M 10/441: {for several batteries or cells simultaneously or sequentially}
- H02J 7/80: including monitoring or indicating arrangements
- G06F 2111/10: Numerical modelling
- H01M 2220/20: Batteries in motive systems, e.g. vehicle, ship, plane
- H02J 7/82: Control of state of charge [SOC]
- H02J 7/825: {Detection of fully charged condition}
- Y02E 60/10: Energy storage using batteries
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)
- H01M: PROCESSES OR MEANS, e.g. BATTERIES, FOR THE DIRECT CONVERSION OF CHEMICAL ENERGY INTO ELECTRICAL ENERGY
- H02J: ELECTRIC POWER NETWORKS; CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
- Y02E: REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION

## Abstract

Battery models using robust fail-safe iteration free approach for solving Differential Algebraic Equations, and associated systems and methods are disclosed. In one embodiment, a method includes generating a model of the rechargeable battery; determining one or more initial conditions for one or more algebraic variables of the model using a solver; holding differential variables of the model static by a switch function while determining the one or more initial conditions; applying the initial conditions to the model by the switch function; and determining one or more parameters for the rechargeable battery by solving the algebraic and differential equations.

## Claims

### Claim 1

A method for charging a rechargeable battery comprising:
generating, by a computing device, a model of the rechargeable battery;
determining one or more initial conditions for one or more algebraic variables of the model using a solver based on perturbations;
holding differential variables of the model static by a switch function while determining the one or more initial conditions for one or more algebraic variables;
applying the initial conditions to differential equations of the model by the switch function;
determining one or more parameters for the rechargeable battery by solving algebraic and differential equations of the model with the solver; and
by a programmable chip, adjusting a state of the battery based on the one or more parameters by electrically connecting a DC power supply to the rechargeable battery,
wherein the switch function is a tan h function.

### Claim 2 (depends on 1)

The method of claim 1, wherein adjusting the state of the battery includes at least one of adjusting a charging/discharging current, a charging/discharging voltage, a temperature of the battery, a peak efficiency of the battery, or an optimal number of charging/discharging cycles.

### Claim 3 (depends on 1)

The method of claim 1, wherein the model comprises a set of differential algebraic equations (DAEs).

### Claim 4 (depends on 3)

The method of claim 3, further comprising separating the DAEs into at least one algebraic equation (AE) and at least one differential equation (DE).

### Claim 5 (depends on 3)

The method of claim 3, wherein the one or more parameters for the rechargeable battery comprise a solution of the set of the DAEs.

### Claim 6 (depends on 1)

The method of claim 1, wherein the initial conditions are determined by perturbations.

### Claim 7 (depends on 1)

The method of claim 1, wherein the solver is a Rosenbrock type solver.

### Claim 8 (depends on 1)

The method of claim 1, wherein the switch function is defined as:
TH=½(1+tan h(q(t−tj))),
where q is a weighting factor determining the discreteness of the function, and tj is the time allowed for the perturbation to find consistent algebraic initial conditions.

### Claim 9

A system for charging a rechargeable battery comprising:
a computing device in electrical communication with a programmable chip on an electronics board, the computing device configured to:
generate a model of the rechargeable battery;
determine one or more initial conditions for one or more algebraic variables of the model using a solver based on perturbations;
hold differential variables of the model static by a switch function while the one or more initial conditions are being determined for one or more algebraic variables; and
apply the initial conditions to differential equations of the model by the switch function, and determine one or more parameters for the rechargeable battery by solving the algebraic and differential equations of the model with the solver; and
the programmable chip of the electronics board configured to electrically connect a DC power supply to the rechargeable battery,
wherein the electronics board is configured to adjust charging or discharging of the battery based on the one or more parameters received from the computing device,
wherein the switch function is a tan h function.

### Claim 10 (depends on 9)

The system of claim 9, further comprising one or more rechargeable batteries.

### Claim 11 (depends on 10)

The system of claim 10 wherein the one or more rechargeable batteries are lithium-ion batteries.

### Claim 12 (depends on 9)

The system of claim 9, wherein the programmable chip is an EPROM carried by the electronics board, wherein the EPROM is:
configured to receive the one or more parameters received from the computing device, and
configured to adjust a charging voltage to the rechargeable battery based on the one or more parameters received from the computing device.

### Claim 13 (depends on 9)

The system of claim 9, further comprising a housing for rechargeable batteries.

### Claim 14 (depends on 9)

The system of claim 9, wherein the computing device is a controller.

### Claim 15 (depends on 14)

The system of claim 14, wherein the controller is carried by the electronics board.

### Claim 16 (depends on 9)

The system of claim 9, wherein the model comprises a set of differential algebraic equations (DAEs).

### Claim 17 (depends on 16)

The system of claim 16, wherein the one or more parameters for the rechargeable battery comprise a solution of the set of the DAEs.

### Claim 18 (depends on 9)

The system of claim 9, wherein the initial conditions are determined by perturbations.

### Claim 19 (depends on 9)

The system of claim 9, wherein the switch function is defined as:
TH=½(1+tan h(q(t−tj))),
where q is a weighting factor determining the discreteness of the function, and tj is the time allowed for the perturbation to find consistent algebraic initial conditions.

### Claim 20

A computer-readable storage device storing computer-executable instructions, the instructions causing charging a rechargeable battery by:
generating, by a computing device, a model of the rechargeable battery;
determining one or more initial conditions for one or more algebraic variables of the model using a solver based on perturbations;
holding differential variables of the model static by a switch function while determining the one or more initial conditions for one or more algebraic variables;
applying the initial conditions to differential equations of the model by the switch function;
determining one or more parameters for the rechargeable battery by solving algebraic and differential equations of the model with the solver; and
adjusting a state of the battery based on the one or more parameters,
wherein the switch function is a tan h function.

### Claim 21 (depends on 20)

The computer-readable storage device of claim 20, further comprising instructions for separating a set of differential algebraic equations (DAEs) into at least one algebraic equation (AE) and at least one differential equation (DE).

### Claim 22 (depends on 20)

The computer-readable storage device of claim 20, wherein the one or more parameters of the rechargeable battery comprise a solution of the set of the DAEs.

### Claim 23 (depends on 20)

The computer-readable storage device of claim 20, wherein the initial conditions are determined by perturbations.

### Claim 24 (depends on 20)

The computer-readable storage device of claim 20, wherein the switch function is defined as:
TH=½(1+tan h(q(t−tj))),
where q is a weighting factor determining the discreteness of the function, and tj is the time allowed for the perturbation to find consistent algebraic initial conditions.