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Battery Technologies
Characterisitics of Rechargeable Batteries

Typical PHEV Battery
The typical battery in a PHEV will have the following features:


 * Range: 16-65 km in electric-only mode
 * Battery size: 10 kWh for daily driving
 * Charging at residential locations: 120 V and 240 V

Source: Impacts Assessment of PHEV Charge Profiles on Generation Expansion Using National Energy Modeling System

Battery Model
Battery Voltage:


 * $${{V}_{bat}}={{V}_{OC}}-{{R}_{i}}{{I}_{bat}}$$

where:


 * $${{V}_{OC}} \,$$ is the open circuit voltage (measured when the engine is off and no loads are connected)
 * $${{R}_{i}} \,$$ is the internal resistance
 * $${{I}_{bat}} \,$$ is the battery current

Required Battery Power:


 * $${{P}_{bat}}={{V}_{bat}}{{I}_{bat}}$$

Battery Current:


 * $${{I}_{bat}}=\frac{{{V}_{0}}-\sqrt{V_{0}^{2}-4{{P}_{bat}}{{R}_{i}}}}{2{{R}_{i}}}$$

If regenerative braking (power is dissipated into the battery) is utilized, the current flowing into the battery becomes:


 * $${{I}_{bat}}=\frac{-{{V}_{0}}+\sqrt{V_{0}^{2}+4{{P}_{bat}}{{R}_{i}}}}{2{{R}_{i}}}$$

Maximum Power Delivered (to a DC load when $$R_{l}=R_{i}$$):


 * $${{P}_{\max }}=\frac{V_{0}^{2}}{4{{R}_{i}}}$$

Models of energy sources for EV and HEV: fuel cells, batteries, ultracapacitors, flywheels and engine-generators

Total Charge Removed from the (plates of the) Battery:


 * $$C{{R}_{n+1}}=C{{R}_{n}}+\frac{\left( \delta t \right)\left( {{I}^{k}} \right)}{3600}\text{ Ah}$$

where:


 * $$\delta t$$ is the time stamp in seconds

Total Charge Supplied (by the battery to the vehicle's electrics):


 * $$C{{S}_{n+1}}=C{{S}_{n}}+\frac{\left( \delta t \right)\left( I \right)}{3600}\text{ Ah}$$

Depth of Discharge (value from 0 to 1):


 * $$\text{DO}{{\text{D}}_{n}}=\frac{C{{R}_{n}}}$$

Source: Electric Vehicle Technology Explained

Batteries in an Electric Vehicle
Total Energy Used:


 * $${{E}_{tot}}=d\cdot {{E}_{c}}\cdot {{C}_{t}}\cdot {{C}_{tf}}$$

where


 * $$d \,$$ is the amount of distance expected to be travelled between charges
 * $${{C}_{t}} \,$$ represents flat, rolling or hilly terrain
 * $${{C}_{tf}} \,$$ represents either in-town stop and go traffic or continous highway speeds
 * $${{E}_{c}} \,$$ is vehicle efficiency
 * $${{V}_{pack}} \,$$ is battery pack voltage

Battery C Rate required (to 100% DOD):


 * $$C=\frac$$

Desired C Rate:


 * $${{C}_{desired}}=\frac{C}{0.5}$$

Estimate Distance:


 * $$d=\left( \frac{C}{0.5} \right)\left( \frac \right)$$

Battery Physics

Peukert's Law
Peukert's equation is a formula that shows how the availble capcity of a battery changes according to the rate of discharge.


 * $$C_p = I^k t \,$$

where:
 * $$C_p \,$$ is the capacity according to Peukert, at a one-ampere discharge rate, expressed in A·h.
 * $$I \,$$ is the discharge current, expressed in A.
 * $$k \,$$ is the Peukert constant, dimensionless.
 * $$t \,$$ is the time of discharge, expressed in h.

However, more commonly, manufacturers rate the capacity of a battery with reference to a discharge time. Therefore, the following equation should be used:


 * $$t = H{\left(\frac{C}{I H}\right)^k}\,$$

where H is the hour rating that the battery is specified against and C is the rated capacity at that discharge rate. Note that $$C_p \,$$ no longer appears in this equation.

Peukert's Law

Calculating the Peukert Coefficient

The Peukert Coefficient ($$k$$ value) can be calculated for any battery type as long the battery capacity at two different discharge times are known. One can solve for the Peukert Coefficient in the following manner:

The two different ratings yields two different currents:


 * $${{I}_{1}}=\frac\text{ and }{{I}_{2}}=\frac$$


 * $${{C}_{p}}=I_{1}^{k}{{T}_{1}}\text{ and }{{C}_{p}}=I_{2}^{k}{{T}_{2}}$$

Since the Peukert Capacity is constant on both sides, the two equations can be set equal to each other:


 * $$I_{1}^{k}{{T}_{1}}=I_{2}^{k}{{T}_{2}}$$


 * $${{\left( \frac{{{I}_{1}}}{{{I}_{2}}} \right)}^{k}}=\frac$$

After taking the natural log of both sides, the Peukert Coefficient is:


 * $$k=\frac{\left( \ln {{T}_{2}}-\ln {{T}_{1}} \right)}{\left( \ln {{I}_{1}}-\ln {{I}_{2}} \right)}$$

Source: Electric Vehicle Technology Explained