1. A Differential Transformation Toolbox for Solving Power System Differential Equations
1Peyton Spencer, 2Yang Liu, 2Dr. Kai Sun
1Auburn University 2The University of Tennessee, Knoxville
ABSTRACT
By modeling the grid with differential equations, utilities can predict the outcome of a fault and determine if the system remains resilient.
In modern times, the grid is pushed closer to its limits to save money on infrastructure
Faster simulation tools are needed to allow utilities to respond to faults proactively
The differential transform method is one possible solution, but it has never been automated
CONTRIBUTIONS
Designed a storage method for polynomial differential equations
Developed automatic transform code that can approximate solutions
Tested a single machine infinite bus system
METHOD
The tool relies on the Differential Transform (DT) which provides a set of rules to explicitly transform a function to its Taylor series coefficients.
The program is comprised of three large operations:
Read the input and convert equations to MTM cell array format, allowing simple indexing
𝑑 𝑥1 𝑑𝑡 =𝐴+𝐵+𝐶
𝑑 𝑥2 𝑑𝑡 =𝐷−𝐸
𝑑 𝑥3 𝑑𝑡 =𝐹+𝐺−𝐻
Monomial Term Matrix (MTM)
𝐴 .∗ 𝑥1 .∗ 𝑥2 .∗ 𝑥3 .∗….∗ 𝑥𝑚 = 𝐴1 𝑥11 𝑥12 … 𝑥1𝑚 𝐴2 𝑥21 𝑥22 … 𝑥2𝑚 ⋮ ⋮ ⋮ ⋱ ⋮ 𝐴𝑛 𝑥𝑛1 𝑥𝑛2 … 𝑥𝑛𝑚
MTM cell array Sign cell array
[𝐴] [𝐵] [𝐶] [𝐷] [𝐸] [𝐹] [𝐺] [𝐻]
[1 1] [2] [1 2]
Monomials
Construct a Taylor series approximation using initial values and the DT
Taylor Approximation
𝑥 𝑛 𝑡 = 𝑘=0 ∞ 𝑋 𝑛 𝑘 𝑡 𝑘
X(N,K) Transformed Matrix
𝑋 1 (0) 𝑋 … 𝑋 1 (𝐾) 𝑋 2 (0) 𝑋 … 𝑋 2 (𝐾) ⋮ ⋮ ⋱ ⋮ 𝑋 𝑁 (0) 𝑋 𝑁 (1) … 𝑋 𝑁 (𝐾)
MTM cell array
Sign cell array
Initial Values
Transform
Functions
Automatically repeat the second process across multiple time windows
The X(N,K) matrix is used to build a Taylor expansion for each differentiable variable. These approximations are accurate within a small time window. At the end of this window, the Taylor output becomes the new initial values and a new X(N,K) matrix is calculated. The next Taylor expansion is constructed with the new coefficient matrix, and the Taylor output is approximated for a time window of equal length. This process is repeated until the end of the time range specified.
RESULTS
This SMIB equation was tested against MATLAB ode45 solver:
𝑑𝑥 𝑑𝑡 =𝑊𝑠∗𝑦, 𝑥=𝑟𝑜𝑡𝑜𝑟 𝑎𝑛𝑔𝑙𝑒, 𝑦=𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
𝑑𝑦 𝑑𝑡 = 1 2𝐻 (𝑃𝑚 −𝑃𝑚𝑎𝑥∗ sin 𝑥 −𝑄∗𝑦)
Rotor Angle and Frequency of SMIB equation
sin 𝑥 was converted to a 9th order Taylor polynomial
The solvers graphed 𝑡= 0, 5 with 4000 equal time steps
DTsolver took 1.32 s while ode45 took .081 s
Transforming high powers took the longest
Increasing number of time steps affects accuracy more than increasing K order
Can increase new method’s speed by allowing a varying time step that is small at points of inflection and large when the slope is changing less
Acknowledgements
This work was supported primarily by the ERC Program of the National Science Foundation and DOE under NSF Award Number EEC and the CURENT Industry Partnership Program.