A Review of High-Performance Computing Methods for Power Flow Analysis
Abstract
:1. Introduction
2. Background
2.1. PF Model and Analysis
2.2. PF Methods
2.3. Linear System Solver
2.3.1. LU Factorization
- Elimination: Factorize matrix A into lower and upper triangular matrices L and U and obtain permutation matrix P, which can be expressed as follows:
- 2.
- Forward: Generate the intermediate vector y depending on P and L in Equation (17), and they can be written as:
- 3.
- Backward: solving a triangular system, based on the vector y of Equation (18), as follows:
2.3.2. QR Factorization
2.3.3. Iterative Method
3. Parallel Method and Performance
3.1. The Multi-Core CPU Architecture
3.2. The Multi-Core Parallel in PF Studies
Algorithm 1 Solving dx = J\F [65] |
1 #pragma omp parallel for schedule (guided, chunk_size) … |
2 for k = 0, 1, …, n{ |
3 for j = k + 1, k + 2, …, n { |
4 x = Jik/Jkk; |
5 for I = k, k + 1, …, n |
6 Jji = Jji − x × Jki; |
7 Jik = x; |
8 } |
9 } |
10 #paragma omp parallel for schedule (guided, chunk_size) |
11 for m = 0, 1, …, n { |
12 d[m] = 1.0; |
13 if(m! = 0) d[m − 1] = 0; |
14 for i = 0, 1, …, n { |
15 sum = 0.0; |
16 for j = 0, 1, …, i − 1 sum = sum + Jij × yj; |
17 yi = di − sum; |
18 } |
19 for i = n, n − 1, …, 0 { |
20 sum = 0.0; |
21 for j = i + 1, i + 2, …, n sum = sum + Jij × dxjm; |
22 dxjm = (yi − sum)/Jii; |
23 } |
24 } |
3.3. The GPU Architecture
3.4. The Hybrid CPU-GPU Parallel in PF Studies
3.5. The FPGA Architecture
3.6. The FPGA Parallel in PF Studies
4. Results and Comparison
- The direct method is usually more robust than the iterative method.
- The direct method of sparse format is suited for small-memory platforms, such as PCs, laptops, and workstations.
- The iterative method performs better on almost all kinds of platforms, but it needs a preconditioner for enhanced stability in PF calculation.
- The direct method of dense format can provide an overall performance improvement for large-scale power systems on large-memory parallel platforms.
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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References | Method | Speedup | Matrix Format | Hardware Platform |
---|---|---|---|---|
[65] | Direct method | 2.0× (MATPOWER counterpart) | Dense | CPU: Intel Core i5-2400 RAM: 12GB |
[67] | Direct method | 3.0× (CPU counterpart) | Dense | HPC cluster self-built RAM: not available |
[26] | Iterative method | 2.86× (MATPOWER counterpart) | Sparse | CPU: Intel Xeon E5607 GPU: NVIDIA Tesla M2070 RAM: 24GB |
[27] | Iterative method | 4× (MATPOWER counterpart) | Sparse | CPU: Intel Xeon E5607 GPU: NVIDIA Tesla M2070 RAM: 24GB |
[28] | Iterative method | 8× (MATPOWER counterpart) | Sparse | CPU: Intel Xeon E5607 GPU: NVIDIA Tesla M2070 RAM: 24GB |
[16] | Direct method | 10×–40× (CPU counterpart) | Dense | CPU: Intel i7-4500U GPU: NVIDIA GeForce GT745 RAM: not available |
[31] | Direct method | 7× (CPU counterpart) | Dense | CPU: Intel Xeon Quad GPU: NVIDIA TESLA C1006 RAM: 8GB |
[21] | Direct method | 4.16× (MATPOWER counterpart) | Sparse | CPU: Intel Xeon E5-2620 GPU: NVIDIA Ge-Force Titan RAM: 32GB |
[35] | Direct method | 2.68× (CPU counterpart) | Sparse | CPU: Intel i9 GPU: NVIDIA RTX400 RAM: 16GB |
[24] | Direct method | 100× (PandaPower counter) | Sparse | CPU: Intel i7-8700k GPU: NVIDIA GTX 1080 GPU: NVIDIA GTX 1080 RAM: 32GB |
[47] | Direct method | 9×–33× (Single GPU counterpart) | Dense | CPU: Intel Xeon GPU: NVIDIA Tesla V100 (4 nodes) RAM: 32GB |
[75] | Direct method | 3× (CPU counterpart) | Sparse | FPGA RAM: not available |
[81] | Direct method | 7× (EP20KE1500 counterpart) | Sparse | ES2S180 RAM: 9,383,040 bits |
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Alawneh, S.G.; Zeng, L.; Arefifar, S.A. A Review of High-Performance Computing Methods for Power Flow Analysis. Mathematics 2023, 11, 2461. https://doi.org/10.3390/math11112461
Alawneh SG, Zeng L, Arefifar SA. A Review of High-Performance Computing Methods for Power Flow Analysis. Mathematics. 2023; 11(11):2461. https://doi.org/10.3390/math11112461
Chicago/Turabian StyleAlawneh, Shadi G., Lei Zeng, and Seyed Ali Arefifar. 2023. "A Review of High-Performance Computing Methods for Power Flow Analysis" Mathematics 11, no. 11: 2461. https://doi.org/10.3390/math11112461