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SAS and PowerSAS.m: The Story
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1. What are Semi-Analysical Solutions (SAS)?
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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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Semi-analytical solutions (SAS) is a family of computational methods
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that uses certain analytical formulations (e.g., power series, fraction
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of power series, continued fractions) to approximate the solutions of
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mathematical problems. In terms of formulation, they are quite different
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from the commonly used numerical approaches e.g., Newton-Raphson method
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for solving algebraic equations, Runge-Kutta and Trapezoidal methods for
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solving differential equations. The parameters of SAS still need to be
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determined through some (easier and more robustness-guaranteed)
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numerical computation, and thus these methods are called
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semi-analytical.
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2. What are the advantages of SAS?
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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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In power system modeling and analysis, SAS has been proven to have the
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following features:
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- **High numerical robustness.** Steady-state analysis usually requires
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solving nonlinear algebraic equations. Traditional tools usually use
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Newton-Raphson method or its variants, whose results can be highly
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dependent on the selection of starting point and they suffer from
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non-convergence problem. In contrast, SAS provides much better
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convergence thanks to the high-level analytical nature.
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- **Enhanced computational performance.** In dynamic analysis, the
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traditional numerical integration approaches are essentially
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lower-order methods, which are confined to small time steps to avoid
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too-rapid error accumulation. These tiny time steps severely restrict
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the computation speed. In contrast, SAS provides high-order
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approximation, enabling much larger effective time steps and faster
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computation speed.
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- **More accurate event-driven simulation.** For complex system
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simulation, it is common to simulate discrete events. Traditional
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numerical integration methods only provide solution values on
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discrete time steps and thus may incur substantial errors predicting
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events. In contrast, SAS provides an analytical form of solution as a
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continuous function, and thus can significantly reduce event
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prediction errors.
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3. How is the performance of PowerSAS.m?
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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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3.1 Benchmarking with traditional methods on Matlab
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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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PowerSAS.m shows advantages in both computational robustness and
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efficiency over the traditional approaches.
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On **steady-state analysis**, we have done several benchmarking with
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traditional methods. For example, we test the steady-state contingency
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analysis on PowerSAS.m and Newton-Raphson (NR) method and its variants
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on Matlab. The test is performed on a reduced Eastern-Interconnection
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(EI) system and we tested on 30,000 contingency scenarios. The results
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suggest that the traditional methods have about 1% chance of failing to
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deliver correct results, while SAS has delivered all the correct
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results.
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For more details, please refer to our recent paper:
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- Rui Yao, Feng Qiu, Kai Sun, “Contingency Analysis Based on
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Partitioned and Parallel Holomorphic Embedding”, IEEE Transactions on
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Power Systems, in press.
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On **dynamic analysis**, we have compared with serveral most commonly
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used traditional numerical approaches for solving ODE/DAEs, including
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modified Euler, Runge-Kutta, and trapezoidal methods. Tests of
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transient-stability analysis on IEEE 39-bus system model and large-scale
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mdodified Polish 2383-bus system model have verified that SAS has
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significant advantages over the traditional methods in both accuracy and
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efficiency.
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**Accuracy comparison on IEEE 39-bus system (1) – Comparison with
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fixed-time-step traditional methods** |accuracy_039_1|
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**Accuracy comparison on IEEE 39-bus system (2) – Comparison with
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variable-time-step traditional method** |accuracy_039_2|
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**Computation time comparison on IEEE 39-bus system**
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.. figure:: https://user-images.githubusercontent.com/96191387/184000437-6aa9150e-d4b1-4297-b982-61e3e68bc2b8.png
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:alt: comp_time_039
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comp_time_039
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For more details, please refer to our recent paper:
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- Rui Yao, Yang Liu, Kai Sun, Feng Qiu, Jianhui Wang,“Efficient and
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Robust Dynamic Simulation of Power Systems with Holomorphic
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Embedding”, IEEE Transactions on Power Systems, 35 (2), 938 - 949,
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2020.
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3.2 Benchmarking with PSS/E
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^^^^^^^^^^^^^^^^^^^^^^^^^^^
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3.2.1 Static Security Region (SSR)
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''''''''''''''''''''''''''''''''''
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Static Security Region (SSR) is an important decision-support tool
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showing region of stable operating points. However, there are often
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challenges on convergence when computing SSRs, especially near the
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boundaries. So SSR can be used for benchmarking the numerical robustness
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of computational methods.
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We test SSR on IEEE 39-bus system by varying active power of buses 3&4.
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The active power of buses 3&4 are sampled uniformly over the interval of
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[-4000, 4000] MW. The figure below shows the SSR derived by PSS/E and
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PowerSAS.m. It shows that PSS/E result have some irregular outliers
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(about 0.1% of the samples) outside of the SSR and actually are not
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correct solutions of power flow equations. In contrast, PowerSAS.m
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correctly identifies the SSR.
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.. figure:: https://user-images.githubusercontent.com/96191387/184000532-d838e7c4-7dc3-4fd6-98ad-486a596ef33d.png
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:alt: ssa_benchmarking
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ssa_benchmarking
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3.2.2 N-k Contingency analysis
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''''''''''''''''''''''''''''''
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Contingency ananlysis also has convergence challenges due to large
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disturbances. Here we perform benchmarking between PSS/E (with and
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without non-divergence options) and PowerSAS.m on the N-25 contingency
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analysis on a reduced eastern-interconnection (EI) system with 458
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buses. We increase the load & generation level by 15%, 20%, and 20.7%,
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respectively, as 3 different loading scenarios (loading margin is
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20.791%). In each scenario, we randomly choose 5000 N-25 contingency
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samples.
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.. figure:: https://user-images.githubusercontent.com/96191387/184000600-6ac3101f-d8bc-49bb-b85d-4cea43ab3549.png
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:alt: contingency_458
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contingency_458
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The figure shows the percentage of correct results using different
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tools. It can be seen that PSS/E has some chance to deliver incorrect
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results, and the chance increases with loading level. In contrast,
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PowerSAS.m still returns results all correctly.
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We also compared the computation speeds of PowerSAS.m and PSS/E. The
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figure below shows the average contingency analysis computation time of
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on the 458-bus system. The results show that SAS’s speed is comparable
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to and even faster than PSS/E’s.
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.. figure:: /img/comp_speed_458.png
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:alt: x
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x
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.. |accuracy_039_1| image:: https://user-images.githubusercontent.com/96191387/183999952-362734f7-d40c-4d27-aa79-eb48bdebcebf.png
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.. |accuracy_039_2| image:: https://user-images.githubusercontent.com/96191387/184000210-90382d81-06bb-4cf6-a423-b8588579e0fd.png
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