1. Yasha Parvini, Ardalan Vahidi
𝐴𝑚𝑒𝑟𝑖𝑐𝑎𝑛 𝐶𝑜𝑛𝑡𝑟𝑜𝑙 𝐶𝑜𝑛𝑓𝑒𝑟𝑒𝑛𝑐𝑒
Chicago, IL
July 1, 2015
Maximizing Charging Efficiency of
Lithium-Ion and Lead-Acid Batteries
Using Optimal Control Theory
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2. Motivation
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• Unlike discharging that depends on the load charging can be
controlled
• One direction is to look at maximizing the charging efficiency
• Reduce the cost of charging
• Reduce losses which turn to heat and may cause thermal runaways

3. Outline
Lithium Ion Battery
• Scenario One (Electronic Resistance)
• Scenario Two (Electronic Resistance + Polarization Resistance)
• Lithium-Ion Efficiency Analysis
Lead Acid Battery
• Optimal Charging of the Lead-Acid Battery
• Numerical results
Conclusion and Future Work
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4. Optimal Charging of Li-Ion Battery (Scenario 1)
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Chemistry LiFePO4
Nominal Voltage 3.3 V
Capacity ( ) 2.5 Ah
Rs @ 25°C 0.01Ω
x1 (State) SOC
U (Input) Current
Pontryagin’s Minimum Principle
qmax
For example charging a Li-Ion battery from zero to full charge in
1 hour will require a constant current of 2.5A

5. Optimal Charging of Li-Ion Battery (Scenario 2)
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6. Results for Scenario 2
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Consider charging a battery
cell from zero charge
SOCi = x1(0) = 0
to full charge
SOCf = x1 (𝑡𝑓) = 1
in 1 hour

7. 7/12
Fast Charging Results for Scenario 2
Consider charging a battery
cell from zero charge
SOCi = x1(0) = 0
to full charge
SOCf = x1 (𝑡𝑓) = 1
in 5 Minutes

8. Lithium-Ion Efficiency Analysis
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9. Optimal Charging of the Lead-Acid Battery
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Nominal Voltage 12 V
Capacity (qmax ) 22 Ah
x1 (State) SOC
U (Input) Current

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Numerical Results for the Lead Acid Battery
Consider the case of
charging the lead-acid
battery module from zero
to full charge in one hour
Optimal Charging
(Energy loss)
46.18 KJ
Constant Current Charging
(Energy loss)
48.9 KJ

11. Conclusion and Future Work
• Charging of lithium ion and lead acid batteries with the objective
of maximizing charging efficiency was studied.
• The analytical results using pontryagin's minimum principle
showed that for lithium-ion batteries the optimal charging
strategy considering constant electronic resistance is constant
current and including the polarization resistance will result in a
different optimal charging current.
• Constant current charging of lead acid battery results in 5.5%
higher thermal heating compared to the optimal charging strategy.
• As a future work fast charging where the temperature variation
and its effect on model parameters plays a significant role, will be
studied.
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12. Thank You!
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