如何有效的管理大型電池系統在近期越來越顯得重要。在之前的一些研究當中，大致上都包含以下三個設計目標:可靠度、效率和系統延續性。然而，系統彈性和複雜度(擴充性)卻很少被考慮到。
為了考量到以上所有五點，我們設計及分析了一個多階電池交換網路(multistage battery switching network)，透過許多矩形的電池組(battery pack)來串接而成。每一個電池組裡面含有一個電壓值和電量值。我們證明了此多階電池交換網路(multistage battery switching network)可以提供Lmax個負載使用，只要所有負載的總電壓值不超過Vmax即可。此外，每個電池組的最佳電壓值可以透過解一個同時整數表達問題(Simultaneous Integer Representation problem)來找出。為了找出每個電池組的電量值，我們提出了一個大小公平電池配置演算法(max-min fairness battery allocation algorithm)並且透過電腦模擬顯示出此演算法比平均配置方法(uniform allocation scheme)還有效率。
我們也提出了一個容錯電池交換網路(fault tolerant battery switching network)，就算壞掉了Fmax個電池組，系統依然可以正常運作。這樣的一個容錯電池交換網路可以搭配實作加入一個最大剩餘電量優先(Largest Remaining Capacity First)規則。如此一來，我們就不需要預先知道負載的詳細資訊(load profile)。

How to effectively manage large-scale battery systems has received a lot of attention
recently. There are several design issues, such as reliability, efficiency and sustainability,
that have been previously addressed in early works. However, the
exibility issue and
the complexity (scalability) issue are rarely addressed. To address these ve design
issues, we design and analyze a multistage battery switching network constructed by a
concatenation of various rectangular \shapes" of battery packs. The shape of each battery
pack is specied by its voltage and its capacity. We show that our multistage battery
switching network can support a maximum number of Lmax loads under the constraint
that the total voltages of these loads do not exceed a design constant Vmax. Moreover,
the voltage of each battery pack can be determined optimally by solving a Simultaneous
Integer Representation (SIR) problem. To determine the capacity of each battery pack,
we propose a max-min fairness battery allocation algorithm, and show by computer
simulations that such an algorithm outperforms the uniform allocation scheme. We also
propose a fault tolerant battery switching network that can still be operated properly
even after Fmax battery packs fail. Such a fault tolerant battery switching network enables
a battery system to implement the Largest Remaining Capacity First (LRCF) policy that
does not require the knowledge of the load prole.

[1] G. Castelli, A. Macii, E. Macii, and M. Poncino, \Current-controlled policies for
battery-driven dynamic power management," in Electronics, Circuits and Systems,
2001. ICECS 2001. The 8th IEEE International Conference on, vol. 2. IEEE, 2001,
pp. 959{962.
[2] T. Stuart, F. Fang, X. Wang, C. Ashtiani, and A. Pesaran, \A modular battery
management system for hevs," SAE Technical Paper, Tech. Rep., 2002.
[3] S. Ci, J. Zhang, H. Sharif, and M. Alahmad, \A novel design of adaptive recon-
gurable multicell battery for power-aware embedded networked sensing systems,"
in Global Telecommunications Conference, 2007. GLOBECOM'07. IEEE. IEEE,
2007, pp. 1043{1047.
[4] H. Visairo and P. Kumar, \A recongurable battery pack for improving power con-
version efficiency in portable devices," in Devices, Circuits and Systems, 2008. IC-
CDCS 2008. 7th International Caribbean Conference on. IEEE, 2008, pp. 1{6.
[5] M. Alahmad, H. Hess, M. Mojarradi, W. West, and J. Whitacre, \Battery switch
array system with application for jpl's rechargeable micro-scale batteries," Journal
of Power Sources, vol. 177, no. 2, pp. 566{578, 2008.
[6] H. Kim and K. G. Shin, \On dynamic reconguration of a large-scale battery sys-
tem," in Real-Time and Embedded Technology and Applications Symposium, 2009.
RTAS 2009. 15th IEEE. IEEE, 2009, pp. 87{96.
[7] ||, \Dependable, efficient, scalable architecture for management of large-scale
batteries," in Proceedings of the 1st ACM/IEEE International Conference on Cyber-
Physical Systems. ACM, 2010, pp. 178{187.
[8] ||, \Efficient sensing matters a lot for large-scale batteries," in Proceedings of
the 2011 IEEE/ACM Second International Conference on Cyber-Physical Systems.
IEEE Computer Society, 2011, pp. 197{205.
[9] T. Kim, W. Qiao, and L. Qu, \A series-connected self-recongurable multicell bat-
tery capable of safe and effective charging/discharging and balancing operations,"
in Applied Power Electronics Conference and Exposition (APEC), 2012 Twenty-
Seventh Annual IEEE. IEEE, 2012, pp. 2259{2264.
[10] S. Ci, J. Zhang, H. Sharif, and M. Alahmad, \Dynamic recongurable multi-cell
battery: A novel approach to improve battery performance," in Applied Power Elec-
tronics Conference and Exposition (APEC), 2012 Twenty-Seventh Annual IEEE.
IEEE, 2012, pp. 439{442.
[11] F. Jin and K. G. Shin, \Pack sizing and reconguration for management of large-
scale batteries," in Cyber-Physical Systems (ICCPS), 2012 IEEE/ACM Third Inter-
national Conference on. IEEE, 2012, pp. 138{147.
[12] L. He, L. Gu, L. Kong, Y. Gu, C. Liu, and T. He, \Exploring adaptive reconguration
to optimize energy efficiency in large-scale battery systems," in Real-Time Systems
Symposium (RTSS), 2013 IEEE 34th. IEEE, 2013, pp. 118{127.
[13] [Online]. Source resistance: the efficiency killer in DC-DC converter circuits.
http://www.maxim-ic.com.
[14] [Online]. Understanding the Terms and Denitions of LDO Voltage Regulators.
http://www.ti.com/lit/an/slva079/slva079.pdf.
[15] [Online]. Dimension Engineering.
https://www.dimensionengineering.com/info/switching-regulators.
[16] S.-Y. R. Li, Algebraic switching theory and broadband applications. Academic Press,
Inc., 2000.
[17] F. K. Hwang and F. Hwang, The mathematical theory of nonblocking switching net-
works. World Scientic, 2004, vol. 15.
[18] R. Alter and J. A. Barnett, \A postage stamp problem," American Mathematical
Monthly, pp. 206{210, 1980.
[19] E. S. Selmer, \On the postage stamp problem with the three stamp denominations."
Mathematica Scandinavica, vol. 47, pp. 29{71, 1980.
[20] C.-C. Chou, C.-S. Chang, D.-S. Lee, and J. Cheng, \A necessary and sufficient
condition for the construction of 2-to-1 optical fo multiplexers by a single crossbar
switch and ber delay lines," Information Theory, IEEE Transactions on, vol. 52,
no. 10, pp. 4519{4531, 2006.
[21] R. Nelson, Probability, stochastic processes, and queueing theory: the mathematics
of computer performance modeling. Springer Science & Business Media, 1995.