# CCCV Battery Charger

## Description

The CCCV Battery Charger block implements a generic dynamic model battery charger. This model supports three-phase wye or delta AC, one-phase AC, or DC voltage input. The model also provides an optional ambient temperature input for charging voltage temperature compensation.

The figure shows the equivalent circuit for the CCCV Battery Charger block.

### Equations

**Output characteristics**

The output current of the battery charger is

${I}_{out}={f}_{1}({I}_{out}^{\text{'}},{k}_{p},{k}_{i},{k}_{d},{I}_{ou{t}_{r}\%},{f}_{ou{t}_{ir}},t)={I}_{out}^{\text{'}}\frac{{k}_{i}}{{k}_{d}{s}^{2}+{k}_{p}s+{k}_{i}}+\frac{{I}_{ou{t}_{r}\%}}{100}\text{sin}(2\pi {f}_{ou{t}_{ir}}t)$

Variables for the output current and related equations are:

*I*is the output current command, in A._{out}*I'*is the pre-filtered current command, in A._{out}*k*is the PID filter proportional gain._{p}*k*is the PID filter integral gain._{i}*k*is the PID filter differential gain._{d}*I*is the current output ripple, in %._{outr%}*f*is the current output ripple frequency, in Hz._{outir}*t*is the time, in s.*ξ*is the dampening factor, which is limited to values between`0`

and`0.9`

.*⍵*is the is radian frequency, in rad/s._{n}*d*is the overshoot, in %._{%}*t*is the settling time, in s._{s}*I*is the pre-filtered current regulated current command, in A._{CC}*I*is the constant current command, in A._{bulk}*I*is the pre-filtered voltage regulated current command, in A._{CV}*V'*is the voltage command, in V._{out}*V'*is the voltage command, in V._{tc}*V*is the temperature compensated voltage effect, in V._{out}$\overline{{V}_{out}}$ is the mean measured output voltage, in V.

*T*is the ambient temperature, in °C ._{a}*T*is the nominal ambient temperature, in °C._{nom}*V*is the voltage compensation ratio, in V/°C._{tc}*V*is the absorption voltage, in V._{abs}*V*is the float voltage, in V._{float}*P*is the output power, in W.$\overline{{I}_{out}}$ is the mean measured output current, in A.

The control gains are:

${k}_{d}=1$

${k}_{p}=2\xi {\omega}_{n}$

${k}_{i}={\omega}_{n}^{2}$

The dampening factor is

$\xi =-\frac{\mathrm{ln}\left(\frac{{d}_{\%}}{100}\right)}{\sqrt{{\pi}^{2}+\mathrm{ln}{\left(\frac{{d}_{\%}}{100}\right)}^{2}}}$

When the output control is current regulated the radian frequency is

${\omega}_{n}=\frac{-\text{log}(0.02)}{{t}_{s}\xi}$

When the output control is voltage regulated the radian frequency is

${\omega}_{n}=\frac{-\text{log}(0.001)}{\xi}$

The pre-filtered current command,
*I' _{out}*, is provided either from the
pre-filtered current regulated current command,

*I*, or from the pre-filtered voltage regulated current command,

_{CC}*I*. The graphs show the different charging cycle phases.

_{CV}If the **Output control mode** is set to
`Constant Current only (CC)`

or
`Constant Current - Constant Voltage (CCCV)`

and
*V _{out}* is lower than

**Float voltage**or

**Absorption voltage**, while the

**Absorption end condition**isn’t met

${I}_{out}^{\text{'}}={I}_{CC}={I}_{bulk}$

If the **Output control mode** is set to
`Constant Current only (CC)`

or
`Constant Current - Constant Voltage (CCCV)`

and
*V _{out}* is equal to the

**Float voltage**or

**Absorption voltage**, while the

**Absorption end condition**isn’t met

${I}_{out}^{\text{'}}={I}_{CV}=\frac{({V}_{out}^{\text{'}}+{V}_{tc}^{\text{'}}){I}_{out}}{\overline{{V}_{out}}}$

When the **Enable absorption phase ** option is
selected and the battery charger switches from constant current to constant
voltage control, if the **Absorption end
condition** is not met, the ambient temperature voltage
compensation, *V' _{tc}* is defined
as

${V}_{tc}^{\text{'}}=\left({T}_{a}-{T}_{nom}\right){V}_{tc}$

Otherwise, the ambient temperature voltage compensation,
*V' _{tc}* is defined as

${V}_{tc}^{\text{'}}=0$

When the **Enable absorption phase ** option is
selected and the battery charger switches from constant current to constant
voltage control, if the **Absorption end
condition** is not met, the ambient temperature voltage
compensation, *V' _{out}* is defined
as

${V}_{out}^{\text{'}}={V}_{abs}$

Otherwise, the ambient temperature voltage compensation,
*V' _{out}* is defined as

${V}_{out}^{\text{'}}={V}_{float}$

The output power is defined as

$P=\overline{{V}_{out}}\overline{{I}_{out}}$

**Input characteristics**

The input current of the battery charger is

${I}_{in}={f}_{2}({P}^{\text{'}},{f}_{eff}\left({P}^{\text{'}}\right),{f}_{THD}\left({P}^{\text{'}}\right),{f}_{PF}\left({P}^{\text{'}}\right),{f}_{HARMS})$

Variables for the input current and related equations are:

*I*is the input current command, in A._{in}*P'*is the normalized output power.*P*is the output power, in W.*P*is the nominal output power, in W._{nom}*I'*is the normalized harmonic amplitude._{n}*f*is the input voltage frequency, in Hz._{in}*f*is the harmonic frequency, in Hz._{n}*t*is the time, in s.*V*is the input voltage delayed by the fifth of its period, in V._{inA}*V*is the input voltage of phase A delayed by the fifth of its period, in V._{in}*I*is the input current ripple, in %._{inr%}*f*is the input current ripple frequency, in Hz._{inir}*θ*is the input voltage angle, in rad._{Vin}*θ*is the input voltage angle of phase A, in rad._{VinA}

${P}^{\text{'}}=\frac{P}{{P}_{nom}}$

Where, *f _{eff}*, is a polynomial
function following the parameters

**Charger efficiency**and

**Efficiency usage factor**. This polynomial order is half of the number of data pairs entered. For input values of,

*P'*, between 0 and 1, the polynomial must return values between 0 and 1. Otherwise, the polynomial order is reduced until this condition is met. If the order reaches 0, the output will remain constant for the mean value of the dataset.

Where, *f _{THD}*, is a polynomial
function following the parameters

**Total harmonic distortion**and

**THD usage factor**. This polynomial order is half of the number of data pairs entered. For input values of,

*P'*, between 0 and 1, the polynomial must return values between 0 and 1. Otherwise, the polynomial order is reduced until this condition is met. If the order reaches 0, the output will remain constant for the mean value of the dataset.

Where, *f _{PF}*, is a polynomial
function following the parameters

**Power factor**and

**PF usage facto**r. This polynomial order is half of the number of data pairs entered. For input values of,

*P'*, between 0 and 1, the polynomial must return values between 0 and 1. Otherwise, the polynomial order is reduced until this condition is met. If the order reaches 0, the output will remain constant for the mean value of the dataset.

Where, *f _{HARMS}*, is the sum of sine
waves specified with the parameters

**Harmonics amplitude**and

**Harmonics frequency**following the expression

${f}_{HARMS}={{\displaystyle \sum}}^{\text{}}{I}_{n}^{\text{'}}\text{sin}(2\pi {f}_{in}{f}_{n}t)$

When the **Type** parameter is set to
`DC`

,

${I}_{in}=\frac{{P}^{\text{'}}}{{f}_{eff}{V}_{in}}+{I}_{in\_r\%}\text{sin}(2\pi {f}_{i{n}_{ir}}t)$

When the **Type** parameter is set to ```
1-phase AC
```

,

${I}_{in}=\frac{\sqrt{2}{P}^{\text{'}}}{{f}_{eff}{f}_{PF}{V}_{in}}\left({f}_{THD}{f}_{HARMS}+\mathrm{sin}\left({\theta}_{{V}_{in}}-\text{acos}\left({f}_{PF}\right)\right)\right)$

When the **Type** parameter is set to ```
3-phase AC
(wye)
```

:

${I}_{inA}=\frac{\sqrt{2}{P}^{\text{'}}}{3{f}_{eff}{f}_{PF}{V}_{inA}}\left({f}_{THD}{f}_{HARMS}+\mathrm{sin}\left({\theta}_{{V}_{inA}}-\text{acos}\left({f}_{PF}\right)\right)\right)$

${I}_{inB}=\frac{\sqrt{2}{P}^{\text{'}}}{3{f}_{eff}{f}_{PF}{V}_{inA}}\left({f}_{THD}{f}_{HARMS}+\mathrm{sin}\left({\theta}_{{V}_{inA}}-\text{acos}\left({f}_{PF}\right)+\frac{1}{3{f}_{in}}\right)\right)$

${I}_{inC}=\frac{\sqrt{2}{P}^{\text{'}}}{3{f}_{eff}{f}_{PF}{V}_{inA}}\left({f}_{THD}{f}_{HARMS}+\mathrm{sin}\left({\theta}_{{V}_{inA}}-\text{acos}\left({f}_{PF}\right)+\frac{2}{3{f}_{in}}\right)\right)$

When the **Type** parameter is set to ```
3-phase AC
(delta)
```

:

${I}_{inA}=\frac{\sqrt{2}{P}^{\text{'}}}{3{f}_{eff}{f}_{PF}{V}_{inA}}\left(\frac{1}{3}{f}_{THD}{f}_{HARMS}\text{'}+\frac{1}{\sqrt{3}}\mathrm{sin}\left({\theta}_{{V}_{inA}}+\frac{\pi}{6}-\text{acos}\left({f}_{PF}\right)\right)\right)$

${I}_{inB}=\frac{\sqrt{2}{P}^{\text{'}}}{3{f}_{eff}{f}_{PF}{V}_{inA}}\left(\frac{1}{3}{f}_{THD}{f}_{HARMS}\text{'}\text{'}+\frac{1}{\sqrt{3}}\mathrm{sin}\left({\theta}_{{V}_{inA}}+\frac{\pi}{6}-\text{acos}\left({f}_{PF}\right)+\frac{1}{3{f}_{in}}\right)\right)$

Where:

${f}_{THD}{f}_{HARMS}\text{'}={f}_{THD}{f}_{HARMS}\left(\omega t\right)-{f}_{THD}{f}_{HARMS}\left(\omega t+\frac{1}{3{f}_{in}}\right)$

${f}_{THD}{f}_{HARMS}\text{'}\text{'}={f}_{THD}{f}_{HARMS}\left(\omega t+\frac{1}{3{f}_{in}}\right)-{f}_{THD}{f}_{HARMS}\left(\omega t+\frac{2}{3{f}_{in}}\right)$

## Assumptions and Limitations

**Model Assumptions**

The output load consists of an adequately sized battery.

Three-phase alternative current inputs are balanced, synchronized, and without jitter.

The ambient temperature does not affect the charger’s parameters.

**Limitations**

The output power is independent from the input power.

## Ports

### Input

### Output

### Conserving

## Parameters

## References

[1] Cope, R.C., and Y.
Podrazhansky. *The Art of Battery Charging*. Proc. 14th Annual
Battery Conference on Applications and Advances, pp. 233-235. Long Beach, CA:
1999.

[2] Dubey, A., Santoso, S., and
M.P. Cloud. *Average-Value Model of Electric Vehicle Chargers*.
IEEE Transactions on Smart Grid Vol. 4 No. 3. Pp. 1549-1557. Argonne, IL: IEEE Power
& Energy Society, 2013.

[3] Elias, M., Nor, K., and A.
Arof. *Design of Smart Charger for Series Lithium-Ion Batteries*.
Proceedings from IEEE International Conference on Power Electronics and Drive Systems,
pp. 1485-1490. Kuala Lumpur: PEDS, 2005.

[4] Hussein A.A.-H., and I.
Batarseh. *A Review of Charging Algorithms for Nickel and Lithium Battery
Chargers*. IEEE Transactions on Vehicular Technology. Sendai, Japan: IEEE
Vehicular Technology Society, 2011.

## Version History

**Introduced in R2019a**