Politecnico di Torino C.so Duca degli Abruzzi, 24 ... 9500 Gilman Drive. La Jolla, CA 92093-0407 ... conditions. A pulsed discharge allows charge recovery dur-.
A Model for Battery Pulsed Discharge with Recovery Effect Carla F. Chiasserini Dipartimento di Elettronica Politecnico di Torino C.so Duca degli Abruzzi, 24 10129 Torino - Italy
Ramesh R. Rao Dept. of Electrical and Computer Engineering, 0407 University of California at San Diego 9500 Gilman Drive La Jolla, CA 92093-0407 Abstract - This paper introduces a stochastic model of battery behavior, that emulates electrochemical mechanisms that are key to battery performance under pulsed discharge conditions. A pulsed discharge allows charge recovery during the idle periods. Recovery depends on the state of charge of the battery and on the duration of the rest time period. Using the postulated model, we derive the improvement to battery lifetime that results from pulsed current discharge driven by bursty stochastic transmissions. The results emphasize the role of traffic shaping in the quest to enhance battery behavior.
I. I NTRODUCTION One of the challenging issues in wireless communications is energy consumption management. In order to support users mobility, it is necessary to make available light and reliable batterypowered apparatus. Since the advances in battery technology are much slower than the market evolution, effective solutions to extend battery lifetime at reasonable cost are still needed. Two findings suggest that there can be room to dramatically improve the performance of communication devices. On one hand, several papers [1], [2], [3], [4], [5] have shown that battery efficiency can be improved by using a pulsed current discharge instead of a constant current discharge due to the charge recovery process that takes place in the battery during the idle time periods, called rest time. On the other hand, the major amount of current drained from the battery is consumed to supply the power amplifier during packet transmissions. Our idea is to exploit the benefits of the pulsed discharge in conjunction with the burstiness that often characterizes the traffic sources. In [6] a summary of the major results regarding the battery capacity and the electrochemical mechanisms that effect the discharge process is presented. Batteries store chemically active materials and deliver energy through electrochemical reactions. When a current is drawn off the battery, two concurrent phenomena occur: i) the concentration of the active materials around
On leave at the Center for Wireless Communications, La Jolla, CA.
the electrode drops (polarization effect); ii) the active materials move toward the depletion region due to the diffusion mechanism and reduce the concentrations gradient. As the current is drained, the polarization effect overcomes the diffusion process leading to the battery discharge before the active materials are exhausted. This battery discharge is faster when the drained current is high. However, if the current is interrupted, the polarization effect can be overcome and the battery may recover some of its charge. Fig. 1 shows the voltage behavior of a lead acid battery when impulses of current are drawn off [1]. As it can be clearly seen, the battery is able to recover the initial value of voltage during the rest time periods. However, the recovery process depends on the idle time duration and on the state of charge of the battery; in particular, as the battery is discharged, the recovery effect decreases until all the active materials are consumed, i.e., the theoretical capacity of the battery is exhausted. In the experiment shown in Fig. 1, a slight reduction of the battery voltage can be already observed after four current impulses were drained. In [6] is proposed a simple model of the battery behavior during the discharge process, that considers the recovery mechanism depending on the rest time duration only. Here, a more accurate battery model is developed taking into account the degradation of the recovery capability of the battery as the battery state of charge decreases. The advantages of the pulsed discharge with respect to the constant discharge are derived through the analysis of the presented model. Then, a simple traffic shaping technique is applied showing that discharge shaping can be a determinant of the battery efficiency. II. T HE BATTERY M ODEL We consider the theoretical capacity of the battery as the maximum number of packets that the battery is able to deliver for a given value of current. Assuming that the amount of capacity necessary to transmit a packet is one charge unit, we set the theoretical capacity equal to charge units and we set the initial value of charge of the battery to charge units. is also assumed to be the number of charge units that can be drained from the battery under constant discharge.
r1 p1
0
q
1
rN
r N-1
q
pN-2
. . .
q
pN-1
N-1
N
q
Fig. 2. Markov chain representing the battery behavior.
otherwise the battery may recover one charge unit or remain in the same state. The recovery effect is represented as a decreasing exponential function of the state of charge of the battery. Such a model was used in [7], where the behavior of the state of charge of lead-acid batteries was studied. The recovery probability at state is as follows
!!"#
$
(1)
where % is a parameter that depends on the battery technology characteristics: as smaller % , as greater the recovery capability is. The value of % has to be chosen accordingly to the internal battery resistance and the value of current drawn off the battery. Note that in [6], is considered a constant value for all , that is the degradation of the battery recovery capability is neglected. The probability to remain in the same state of charge is
& ' &
() "!"#
* (2)
As a measure of the battery efficiency, we consider the ratio of the mean number of packets, +-, , transmitted during the discharge process, to the theoretical capacity of the battery. In order to calculate + , , we notice that in the chain shown in Fig. 2, a packet transmission corresponds to a left transition, while a charge recovery corresponds to a right transition. Denoting by . the number of time units to reach state starting from state , and by / the number of right transitions, we find that
.0 2143/-5
(3)
moreover, the number of left and stationary transitions are equal to /768 and .' 3)/ , respectively. Fig. 1. Performance of a bipolar lead-acid battery subjected to six current impulses. Pulse length=3 ms, rest period=22 ms.
We represent the battery behavior as a discrete time Markov process with initial state equal to and one absorbing state, , corresponding to complete discharge (see Fig. 2). In case of constant current discharge the battery transmits successive packets and goes from to in time units. Using a pulsed discharge, the battery can partially recover its charge during the idle time, and thus transmit a number of packets greater than before reaching state . Let us assume a Bernoulli packet arrival process with arrival probability . Starting from , at each time unit, called slot, one charge unit is lost if a packet arrives and has to be transmitted,
Since the probability of making a right transition and a stationary transition are state dependent, the joint probability to reach in . time units making / right transitions results as
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OQP S / EJ)L N O P 6\ P WX = P S @AB#CYD!D"D @B#C D!D"D EJ N OF] P OU] ] 6 ] 6\ S D!D"D
O R 6 .' )3 / P P P P S D"D!D .' " D ! D D 3 / M D!D"D R @ @ _ ^ & P@ & ] & V WX =F GIGIG EJ D"D!D D"D!D N
O P /
/01
O"a
A. 14`683/
1.2
1
0.8
(4)
where the indices (for b c) !"!# 3 ) correspond to the number of right made from state b to state bd6 , while a transitions e
! !"# ) denote the number of the indices (for b transitions from a state b to itself.
Gd
6\+ +
Sih
N f
j
B#C
6n+ +
Sih j
0.2
0.3 0.4 0.5 0.6 0.7 average packet arrival rate
0.8
Fig. 3. vs. the average packet arrival rate for =100,
=0.1, and
sequence in (5) results as
m
0.6
0.4
Now, we recognize the second part of (4) as the convolution among sequences that have the following form
Ngf
PD - N=10 CD - N=10 PD - N=50 CD - N=50
0.9
varying.
1.2
o h j ;
P
(6)
PD - N=10 CD - N=10 PD - N=50 CD - N=50
1
we obtain
> = < ; = p& P p& ? @ B#C D!D!D ?A@[GIGIG ? D!D"D N4OUP OFR =F > J)K O6 P S D!D"D ? B#C ? @ N MJ) L O P O P / O D!D!D R
