4. Numerical results

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Intensive Programme “Renewable Energy Sources”

Following stability algorithms have been investigated and compared to the original Newton-Raphson code (marked as No. 1). No. 2 contains the truncation algorithm with modified DXT limit values. One-Shot Fast-Decoupled procedure with V updates only is included in No. 3. One-Shot Fast-Decoupled and One-Shot Gauss-Seidel techniques (V updates only) combined with the truncation method are implemented in No. 4 and No. 5, respectively. Power mismatch minimization approach with α relaxation factors is realized in No. 6.

For evaluation of presented stability algorithms, following networks with convergence difficulties have been prepared (see Tab. 1). For transparency reasons, they are labelled with capital letters A – F.

Table 1 – Overview of analyzed test power systems

Networks	Description
A	Reduced Mato Grosso System (11-bus)
B	Simplified Czech Transmission System (56-bus)
C	Simplified 14-Gen SE Australian System (59-bus)
D	South England Power System (61-bus)
E	Power System of Iowa (145-bus)
F	Simplified Scottish System (629-bus)

Main emphasis of the evaluation process is placed on degree of numerical stability obtained, reduction of total number of iterations, CPU time minimization and computational burden of individual stability techniques. For each load flow case and numerical approach applied, total number of iterations and total CPU time are presented in Tab. 2 and Tab. 3, respectively.

Table 2 – Total number of iterations for individual load flow cases

						Examined stability algorithms
Networks		No. 1.	No. 2			No. 3		No. 4		No. 5			No. 6
A		5*	6			5		7		8			7*
B		11	6			5		5		5			6
C		50**	30*			18*		9		6			8
D		50**	50**			5		5		5			5
E		50**	8			50**		8		8			10
F		8*	7			4		4		7			10
	* convergence to non-physical solutions								** divergence
	Table 3 – Total CPU time per iteration [in ms]
					Examined stability algorithms
	Networks			No. 1.	No. 4		No. 5		No. 6
	A			14.6	13.4		11.7		19.4
	B			71.5	89.5		86.4		95.4
	C			73.2	84.5		90.4		102.9
	D			75.1	101.1		98.9		118.7
	E			425.2	480.5		472.6		592.2
	F			9791.6	11653.6		10114.6		12753.1

Note: No sparsity techniques applied.

It can be seen from results above, that the most reliable stability algorithms are procedures No. 4 and 5. Subsequent testings of various 42 real test power systems have been accomplished preferring approach No. 4 for smallest number of iterations needed. Algorithm No. 6 has been evaluated as

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