Battery-Powered Systems:
Efficiency, Control, Economics
ECEN 2060
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 might
expect that the battery would run longer (10 hours) before becoming
discharged. In practice, the relationship between battery capacity and
discharge current is not linear, and less energy is recovered at faster discharge
rates.
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
ECEN2060
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Example
Our lab batteries
k = 1.15
C = 36 A
Cp = 63 A-hr
Prediction of Peukert
equation is plotted at
left
What the manufacturer’s
data sheet specified:
ECEN2060
Nominal capacity: A-hrs @ 25˚C to 1.75 V/cell
1 hr
2 hr
4 hr
8 hr
24 hr
36 A-hr
45 A-hr
46 A-hr
49 A-hr
56 A-hr
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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
I DT D
= voltage efficiency coulomb efficiency
I CT C
Coulomb efficiency = (discharge A-hrs)/(charge A-hrs)
Voltage efficiency = (discharge voltage)/(charge voltage)
Rdischarge(SOC)
Ibatt
+
V(SOC) +–
Rcharge(SOC)
Ideal diodes
Vbatt
–
ECEN2060
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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. Can
improve 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%
ECEN2060
5
Charge profile
A typical good charge profile:
1. Bulk charging at maximum power
Terminate when battery is 80%
charged (when a voltage set point
is reached)
2. Charging at constant voltage
The current will decrease
This reduces gassing and improves
charge efficiency
3. Trickle charging / float mode
Equalizes the charge on seriesconnected cells without significant
gassing
Prevents discharging of battery by
leakage currents
Occasional pulsing helps reverse
sulfation of electrodes
ECEN2060
The three-step charge profile used
by the chargers in our power lab
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Battery charge controller
PV
array
Charge
controller
• 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
ECEN2060
Inverter
AC
loads
Direct energy transfer
Charge battery by direct connection
to PV array
MPPT
Connect dc-dc converter between
PV array and battery; control this
converter with a maximum power
point tracker
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Direct energy transfer
Inverter
PV
array
Ipv
AC
loads
Charge controller may simply be
series switches
Battery
characteristic
Bulk charge: connect battery directly
to PV array
PV
characteristic
Other charge modes: pulse current on
and off to reduce average current
Nighttime disconnect of PV from
battery
Vpv
ECEN2060
Disconnect inverter when state of
charge is low
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Maximum power point tracking
PV
array
Inverter
Buck
converter
AC
loads
Insert buck converter into charge controller, and perform maximum
power point tracking in bulk charging mode
Battery reaches full charge earlier in the day
Battery
state of
charge
100%
MPPT
DET
0%
Sunrise
Sunset
time of day
In a closed system, the input energy must always equal the load
consumption. Excess generated energy must be dumped.
ECEN2060
• Can you adjust the load consumption?
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Economics of battery storage
Example: the deep discharge
batteries 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 discharge
cycle:
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
ECEN2060
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
ECEN2060
$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.
2.
3.
ECEN2060
Must generate the power with PV; est. cost $0.21/kWh
Must store in batteries, est $0.10 to $0.4 / kWh
May additionally need other backup power sources, with additional
costs but substantially improves reliability
Utility bill has a charge for kWh consumed only
Reliability is worth at least as much per kWh as the
energy itself, but is not included in current pricing
schemes approved by PUC
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