電力潮流方程組是電力系統分析中重要的一環。它描述一個電網各節點的狀態。一個電網在相同的供給與負載下也存在著不同的穩定態。找出其他的解將可降低營運風險與成本。本文使用牛頓灑點法與同倫連續法求解方程組。牛頓撒點法隨機給定初始值執行牛頓疊代法，若收斂即為一解。同倫連續法對於欲求根的方程組 P(x) 建立一個類似的方程組 Q(x) 經過同倫連到 P(x)。使用同倫連續法求解電力潮流方程組。根據同倫連續法性質，將不只得到其他解還會得到所有解。

近幾十年來，電腦逐漸朝多核心跟顯示卡加速發展。如何有效地調用電腦資源將大幅影響計算速度。CPU 核心數量 10 核、20 核已為高階處理器，核心性能強大。GPU 有數以百計的核心，核心性能較弱。文末執行牛頓撒點法與同倫連續法程式並比較 CPU 與 GPU 實際運算時間。

The power flow equations are an important part of the power system analysis. It describes the status of nodes in an electrical grid. A grid has different solutions under the same supply and load. Identifying other solutions will reduce operational risk and cost.

This paper uses the Newton iteration method and the homotopy continuation method to solve the equations. To solve the roots for a system of equations P(x) by the homotopy continuous method is making an equation system Q(x). The roots of Q(x) and P(x) are connected by homotopy paths. After tracing all the roots of Q(x) to P(x), we will obtain all the roots of P(x).

In recent years, multi-core CPU and the GPGPU (General-purpose computing on graphics processing units) are developed. Effective call to computer resources will significantly affect the speed of calculation. A CPU has less but stronger cores, while a GPU has many but weak cores. At the end of this paper, the numerical solutions of the power flow equations are computed by parallel computing on CPU and GPU. This paper shows the efficiency of parallel computing.

1 簡介 1
2 電力潮流問題 1
2.1 電網簡介 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2.2 導纳矩陣 Admittance Matrix . . . . . . . . . . . . . . . . . . . . . . . . 1
2.3 電力潮流方程式 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.4 節點種類 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.5 電力潮流方程組 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 計算方式 7
3.1 牛頓撒點法 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1.1 牛頓法 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1.2 高維度牛頓法 . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1.3 演算法 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 同倫連續法 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2.1 計算方法 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2.2 全次數同倫 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.3 係數同倫 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 演算法 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 平行計算 13
4.1 CPU 平行計算：pthread/openMP . . . . . . . . . . . . . . . . . . . . . 13
4.1.1 牛頓灑點 CPU 平行演算法 . . . . . . . . . . . . . . . . . . . . 13
4.1.2 同倫連續法 CPU 平行演算法 . . . . . . . . . . . . . . . . . . . 15
4.2 GPGPU 顯示卡加速：CUDA . . . . . . . . . . . . . . . . . . . . . . . 17
4.2.1 牛頓灑點 GPU 平行演算法 . . . . . . . . . . . . . . . . . . . . 17
4.2.2 同倫連續法 GPU 平行演算法 . . . . . . . . . . . . . . . . . . 20
5 結果 21
5.1 Cyclic-5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2 Power flow: bus-6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6 結論 25
Reference 27

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