Batteries, spares and readiness

Why this matters

UAT 321 covered the battery register and the status of each battery. This module works at fleet level: when will this set of batteries wear out, how many spares are needed, and how many drones will be ready tomorrow? The drone knowledge base’s unit on lithium battery safety covers charging, storage, transport, swelling and fire response, which must always be followed. The course case is still the four-drone survey fleet, with hypothetical data.

When does a battery wear out?

Lithium battery capacity falls with the number of cycles. A widely used end-of-life criterion is 80% of rated capacity; for example, the USABC battery test manual (1996) defines end of life this way, although it was written for electric vehicles. Drone teams should use the battery maker’s criterion first if there is one, and choose a threshold that still leaves enough flight time for the mission plus reserve. The team measures real capacity with a constant-current discharge every 50 cycles and fits a trend to plan purchases ahead, instead of waiting until a battery cannot deliver flight time on an operational day.

import numpy as np

cycles = np.array([0, 50, 100, 150, 200, 250, 300])
capacity = np.array([100, 98.4, 96.9, 95.1, 93.6, 92.2, 90.5])   # % of rated capacity
EOL = 80.0
m, b = np.polyfit(cycles, capacity, 1)
n_eol = (EOL - b) / m
print(f"fade {m * 100:.2f} % per 100 cycles, intercept {b:.1f} %")
print(f"predicted {EOL:.0f}% at about cycle {n_eol:.0f}; about {n_eol - cycles[-1]:.0f} cycles left")
flights_per_week = 12
print(f"at {flights_per_week} cycles per week: about {(n_eol - cycles[-1]) / flights_per_week:.0f} weeks to order replacements")

fade -3.16 % per 100 cycles, intercept 100.0 %
predicted 80% at about cycle 633; about 333 cycles left
at 12 cycles per week: about 28 weeks to order replacements

How many spares?

If components fail randomly at a constant rate, the number of failures in a period follows the Poisson distribution, with mean $\lambda$ equal to the number of components in use times the operating hours divided by the MTBF. This approach is used in planning spares for the International Space Station, where the probability that spares are enough is called the probability of sufficiency (Owens and de Weck, 2018). The team chooses a stock $s$ so that the probability of demand not exceeding $s$ is above the target. Parts replaced on schedule (Module 1) are counted separately, because their use is known in advance.

import math

N, HOURS, MTBF, TARGET = 16, 300, 2000, 0.95
lam = N * HOURS / MTBF
cum = 0.0
for k in range(12):
    cum += math.exp(-lam) * lam ** k / math.factorial(k)
    print(f"stock {k}: P(demand <= {k}) = {cum:.3f}")
    if cum >= TARGET:
        break
print(f"expected failures {lam:.1f}; keep {k} spares for >= {TARGET:.0%}")

mtbf_uav, mttr = 120, 16                       # hours, per aircraft
a = mtbf_uav / (mtbf_uav + mttr)
print(f"availability per aircraft {a:.3f}; expected aircraft ready out of 4: {4 * a:.1f}")
p_all = a ** 4
p_3plus = p_all + 4 * a ** 3 * (1 - a)
print(f"P(all 4 ready) {p_all:.2f}, P(at least 3 ready) {p_3plus:.2f}")

stock 0: P(demand <= 0) = 0.091
stock 1: P(demand <= 1) = 0.308
stock 2: P(demand <= 2) = 0.570
stock 3: P(demand <= 3) = 0.779
stock 4: P(demand <= 4) = 0.904
stock 5: P(demand <= 5) = 0.964
expected failures 2.4; keep 5 spares for >= 95%
availability per aircraft 0.882; expected aircraft ready out of 4: 3.5
P(all 4 ready) 0.61, P(at least 3 ready) 0.93

The aircraft lifecycle

Lifecycle management under continued-airworthiness practice runs from receipt (acceptance inspection, recording the serial numbers of the aircraft and key components) through operation (logging flight hours, battery cycles and repairs) and modification (updating firmware and components under configuration control) to retirement (removing reusable parts with their history, disposing of batteries safely and closing the register). The data from every stage lets the next decision use real numbers, such as the fleet’s own MTBF instead of the maker’s.

Module lab

Common mistakes

Summary

• Track capacity periodically and predict the cycle at the retirement threshold, but remeasure, since fade is not always linear
• Random failures follow the Poisson distribution, and spares must exceed the mean to reach the probability target
• Availability $A=MTBF/(MTBF+MTTR)$ (NASA RCM Guide, 2008), and fleet readiness depends on how many aircraft are ready at once
• The lifecycle starts at receipt and ends at retirement, with records at every stage

Check your understanding

1. Capacity falls 0.04% per cycle from 100%. At which cycle does it reach 80%?
2. Ten components run 200 hours with an MTBF of 1,000 hours. What is the mean number of failures $\lambda$?
3. With $\lambda =1$, what is the probability that nothing fails?
4. MTBF 100 hours and MTTR 25 hours. What is the availability?
5. Why should spares exceed the mean demand?