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 Introduction 3
2 Construction of Battery Switching Networks 7
2.1 Constructing a battery switching network by using a single crossbar switch 8
2.2 Optimal selection of the basis set . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Alternating C-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Multi-stage feedforward battery switching network . . . . . . . . . . . . . 18
3 Capacity Assignment and Battery Allocation 21
3.1 Average energy consumption rate . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Max-min fairness allocation . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Fault Tolerant Battery Switching Networks 25
5 Simulation 29
5.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Comparison between the max-min fairness allocation scheme and the uni-
form allocation scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.3 The LRCF scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6 Conclusions 34

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