我們研究了在智慧電網裡關於需量反應問題離線排程問題。我們假設在智慧
電網裡，時間線被平分成等長的時段，並且有許多等長的電力請求；有一些電
力請求有先後關係，而它們各自需要在特定時段中的某一個時段中被執行。我
們假設一個時段的發電成本是基於那個時段的電力請求數量的某種凸函數，這
個排程問題的目標便是試圖分配所有電力請求來實現總體發電成本最小化。
為了解決這個排程問題，我們提出了一個演算法；此演算法可以在時間複雜
度O(n^2*τ) 下對於這些有著先後順序的電力請求找到最佳分配方法，其中n為電
力請求的總數量，τ為時段的總數量。

We study an offline scheduling problem that arises in demand response management
in smart grid. We assume in the smart grid, the time horizon is partitioned
into unit-size time sessions, there are many unit-size power requests, some of them
are in precedence relationship, and each of them need to be done in one time session
in a set of specific time sessions. We assume the general electricity cost for
each time session is a convex function of the amount of power requests assigned
to that time session. The objective of the problem is to assign all requests with
the minimum total electricity cost. For this problem, we introduce an algorithm
to find the optimal assignment for these unit-size power requests with precedence
constraints in O(n^2*τ) time, where n = the amount of the power requests, τ = the
amount of timeslots.

Abstract (Chinese) I
Abstract II
Acknowledgements (Chinese) III
Contents IV
List of Algorithms VI
1 Introduction 1
2 Preliminaries 3
2.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Definition Of Alternative Assignment Graph . . . . . . . . . . . . . 4
2.3 Legal Path Of The Alternative Assignment Graph . . . . . . . . . . 5
2.4 A Shift Along The Legal Path Of The Alternative Assignment Graph 5
3 Our Algorithm 6
4 Correctness 8
4.1 Proof Of Invariant (I1) . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.1.1 Agreement Graph . . . . . . . . . . . . . . . . . . . . . . . . 8
4.1.2 Proof of Invariant (I1) . . . . . . . . . . . . . . . . . . . . . 12
IV
4.1.2.1 Proof Of Lemma 1 . . . . . . . . . . . . . . . . . . 13
4.1.2.2 Proof Of Lemma 2 . . . . . . . . . . . . . . . . . . 13
4.1.2.3 Proof Of Lemma 3 . . . . . . . . . . . . . . . . . . 14
4.1.2.4 Proof Of Lemma 4 . . . . . . . . . . . . . . . . . . 15
4.2 Proof Of Lemma 9 and Lemma 10 . . . . . . . . . . . . . . . . . . . 16
4.2.1 Additional Notations. . . . . . . . . . . . . . . . . . . . . . . 16
4.2.2 Proof Of Lemma 5 . . . . . . . . . . . . . . . . . . . . . . . 16
4.2.3 Proof Of Lemma 6 . . . . . . . . . . . . . . . . . . . . . . . 28
4.2.4 Proof Of Lemma 7 . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.5 Proof Of Lemma 8 . . . . . . . . . . . . . . . . . . . . . . . 31
4.2.6 Proof Of Lemma 9 . . . . . . . . . . . . . . . . . . . . . . . 35
4.2.7 Proof Of Lemma 10 . . . . . . . . . . . . . . . . . . . . . . . 36
4.3 Proof Of The Correctness . . . . . . . . . . . . . . . . . . . . . . . 36
5 Time Complexity 37
6 Conclusion 39
Bibliography 40

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