Optimal Power Flow (iOPF)

The software package, iOPF, is a set of short, concise routines for solving optimal power flow (OPF) with simple constraints. Its primary purpose is to illustrate much of the basic theory of Interior Point Method in as concise a code as possible, including an elegant vector computing method. The package also consists of some large cases which is from MATPOWER, a well-known package of MATLAB® M-files for solving OPF problems. The following tests show the iOPF can be competitive with MATPOWER and more robust for cases without line flow constraints. Furthermore, we also have a commercial version of OPF, written in ANSI C, with HVDC systems, more constraints and objective functions. The code is nearly two times faster than iOPF. For further information, you could contact me.

 We have carried out the following tests using Matlab R2014a, in a windows 10, with Intel Core i5-6500 CPU clocking at 3.20 GHz and 8GB of RAM memory. The MATPOWER 6.0 with MIPS solver is compared with iOPF. In order to keep all things being equal, the line flow constraints, before comparison, were removed from MATPOWER. However, the MATPOWER failed in calculating many big test cases. Therefore, we try to set the initialization of multipliers on inequality constraints (with the symbol “mu”) to zero and  it worked.

Case File Name	Time(sec)		Number of Iterations		Time for One Iteration(sec)
	MIPS	iOPF	MIPS	iOPF	MIPS	iOPF
Case300	0.21	0.13	19	19	0.011	0.007
Case1354pegase	1.05	0.80	29	26	0.036	0.031
Case2383wp	1.72	1.33	29	27	0.059	0.049
Case2746wop	1.81	1.18	27	21	0.067	0.056
Case2869pegase	2.47	1.85	30	27	0.082	0.068
Case3375sp	2.77	2.04	33	29	0.084	0.070
Case9241pegase	11.03	9.25	37	35	0.298	0.264
Case13659pegase	18.09	17.14	43	49	0.421	0.350

 

Closed-Form Eigenvalue Sensitivity (iCFES)

Eigenvalue Sensitivity is of great use for small-signal stability analysis, control and optimization, especially for online applications. It is superior to participation factor.
As for computing the eigenvalue sensitivities, the numerical differentiation method is widely used, which performs eigenvalue analysis to get the eigenvalues \(\lambda(\boldsymbol A)\) of the state matrix at the equilibrium point and then vary one variable \(x_i\) by a small quantity \(\varepsilon\) to get the perturbed state matrix \(A_\varepsilon\) and its eigenvalues \(\lambda( \boldsymbol A_\varepsilon)\).

The eigenvalue sensitivity with respect to \(x_i\) can be approximated by

\(\frac{\partial \lambda}{\partial  x_i} \approx \frac{\lambda( \boldsymbol A_\varepsilon)-\lambda(\boldsymbol A)}{ \varepsilon}\)

The numerical differentiation method is easy to implement, but for large systems its calculation burden can be heavy due to the repetitive procedure. Also, the sensitivity with respect to the power of the slack bus cannot be obtained.Alternatively, the eigenvalue sensitivity can be obtained by closed-form formulas.

Specifically, the \(j\)th eigenvalue sensitivity with respect to the \(i\)th variable \(x_i\) can be written as:

\(
\frac{\partial\lambda_j}{\partial x_i}=\frac{\boldsymbol\psi_j\frac{\partial \boldsymbol A }{\partial  x_i}\boldsymbol\phi_j}{\boldsymbol\psi_j \boldsymbol \phi_j},
\)

where \(\boldsymbol \psi_j \;\text{and}\; \boldsymbol\phi_j\) are, respectively, the left and right eigenvectors of the eigenvalue \(\lambda_j\);

So the derivation is required only once and has extremely high efficiency. However, from the above formula it is seen that the derivation of the eigenvalue sensitivities for all of the variables requires considerable work since the elements of state matrix can be different functions of several variables.  Although the closed-form eigenvalue sensitivity formula had been proposed many years ago, to our best knowledge, we have not found anyone who has provided the detailed results of closed-form eigenvalue sensitivity with respect to the power of generator. Therefore, I would like to publish the data to make available a common set of data that could be used by program developers and users to verify the correctness of their solutions.

• Dynamic data and results of closed-form eigenvalue sensitivity for WSCC 3 Machine 9 Bus System, IEEE 10 Machines 39 Bus System and IEEE 54 Machines 118 Bus System [Click link here]
• Please refer to the paper “Eigenvalue Sensitivity with Respect to Power Generation Revisited: A Vector-Bbased Closed-Fform Formulation” (to be reviewed)

Cascading Blackout Simulation Toolbox

A realistic cascading blackout simulation toolbox that considers wide-area monitoring, protection, and control. To be released in 2018.