Battery-Powered Systems: Efficiency, Control, Economicsecee.colorado.edu/~ecen2060/materials/lecture_notes/Battery3.pdf · If we discharge the battery more slowly, say at a current

Battery-Powered Systems:Efficiency, Control, Economics

ECEN 2060

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Battery capacity

The quantity C is defined as the current that discharges the battery in 1 hour,so that the battery capacity can be said to be C Ampere-hours (units confusion)

If we discharge the battery more slowly, say at a current of C/10, then we mightexpect that the battery would run longer (10 hours) before becomingdischarged. In practice, the relationship between battery capacity anddischarge current is not linear, and less energy is recovered at faster dischargerates.

Peukert’s Law relates battery capacity to discharge rate:

Cp = Ik t

where Cp is the amp-hour capacity at a 1 A discharge rate

I is the discharge current in Amperes

t is the discharge time, in hours

k is the Peukert coefficient, typically 1.1 to 1.3

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Example

Our lab batteries

k = 1.15

C = 36 A

Cp = 63 A-hr

Prediction of Peukertequation is plotted atleft

Nominal capacity: A-hrs @ 25˚C to 1.75 V/cell

36 A-hr

1 hr

56 A-hr49 A-hr46 A-hr45 A-hr

24 hr8 hr4 hr2 hr

What the manufacturer’sdata sheet specified:

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Energy efficiency

Efficiency = ED/EC

EC = Total energy during charging = vbatt (-ibatt) dt VCICTC

ED = Total energy during discharging = vbatt ibatt dt VDIDTD

Energy efficiency =VD

VC

IDTDICTC

= voltage efficiency coulomb efficiency

+–V(SOC)

Ideal diodes

Rcharge(SOC)

Rdischarge(SOC)

+

Vbatt

–

Ibatt

Coulomb efficiency = (discharge A-hrs)/(charge A-hrs)

Voltage efficiency = (discharge voltage)/(charge voltage)

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Energy efficiency

Energy is lost during charging when reactions other than reversal of sulfation occur

• At beginning of charge cycle, coulomb efficiency is near 100%

• Near end of charge cycle, electrolysis of water reduces coulomb efficiency. Canimprove this efficiency by reducing charge rate (taper charging)

• Typical net coulomb efficiency: 90%

• Approximate voltage efficiency: (2V)/(2.3V) = 87%

Energy efficiency = (87%)(90%) = 78%

Commonly quoted estimate: 75%

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Charge profile

A typical good charge profile:

1. Bulk charging at maximum power

Terminate when battery is 80%charged (when a voltage set pointis reached)

2. Charging at constant voltage

The current will decrease

This reduces gassing and improvescharge efficiency

3. Trickle charging / float mode

Equalizes the charge on series-connected cells without significantgassing

Prevents discharging of battery byleakage currents

Occasional pulsing helps reversesulfation of electrodes

The three-step charge profile usedby the chargers in our power lab

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Battery charge controller

PVarray

Chargecontroller

Inverter ACloads

• Prevent sulfation of battery

• Low SOC disconnect

• Float or trickle charge mode

• Control charge profile

• Multi-mode charging, set points

• Nightime disconnect of PV panel

Direct energy transfer

Charge battery by direct connectionto PV array

MPPT

Connect dc-dc converter betweenPV array and battery; control thisconverter with a maximum powerpoint tracker

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Direct energy transfer

Vpv

Ipv

PVcharacteristic

Batterycharacteristic

PVarray

Inverter ACloads

Charge controller may simply beseries switches

Bulk charge: connect battery directlyto PV array

Other charge modes: pulse current onand off to reduce average current

Nighttime disconnect of PV frombattery

Disconnect inverter when state ofcharge is low

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Maximum power point tracking

PVarray

Inverter ACloadsBuck

converter

Insert buck converter into charge controller, and perform maximumpower point tracking in bulk charging mode

Battery reaches full charge earlier in the day

Batterystate ofcharge100%

0%time of daySunrise Sunset

DET

MPPT

In a closed system, the input energy must always equal the loadconsumption. Excess generated energy must be dumped.

• Can you adjust the load consumption?

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Economics of battery storage

Example: the deep dischargebatteries in our lab

Retail cost: $150

Assumptions:50% depth of discharge

20 hour uniform discharge

Average voltage 12.4 V

1000 cycles

Energy of each dischargecycle:

I = (Cp/t)1/k = (63/20)1/1.15 = 2.7A

ED = (2.7A)(12.4V)(20hrs) = 0.67 kWh

Battery capital cost per kWh:

($150)/[(0.67 kWh)(1000 cycles)] = $0.22/kWh

Typically $0.10/kWh for large optimized installations

Battery costs more than the energy it stores!

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Backup gas generation

Cost of gasoline

Estimated 5 kWh/gallon, $3/gal

$0.60/kWh

Capital cost

$0.50 to $1.00 per Watt

Adds another $0.05 to $0.10 per Watt if amortized over 19 years

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The value of grid energy

Power supplied by the utility is available

Estimated cost $0.10/kWh in Colorado

Available on demand, very high reliability

To reproduce this in a standalone PV system:

1. Must generate the power with PV; est. cost $0.21/kWh

2. Must store in batteries, est $0.10 to $0.4 / kWh

3. May additionally need other backup power sources, with additionalcosts but substantially improves reliability

Utility bill has a charge for kWh consumed only

Reliability is worth at least as much per kWh as theenergy itself, but is not included in current pricingschemes approved by PUC