Multi-period, Multi-contingency and Interval Optimal Power Flow for Dimensioning and Allocation of Spinning Reserve

Borba, Ricardo Augusto; Fernandes, Thelma Solange Piazza

doi:10.1590/1678-4324-2025240973

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Article - Engineering, Technology and Techniques • Braz. arch. biol. technol. 68 • 2025 • https://doi.org/10.1590/1678-4324-2025240973 linkcopy

Multi-period, Multi-contingency and Interval Optimal Power Flow for Dimensioning and Allocation of Spinning Reserve

Authorship SCIMAGO INSTITUTIONS RANKINGS

Due to their low environmental impact and decreasing costs, there has been significant growth in the use of renewable sources worldwide. However, their power generation fluctuates randomly, necessitating reliable energy supply methods. Spinning reserve can mitigate the risk of energy deficits during generation interruptions and line contingencies. It is therefore necessary to develop specific techniques for dispatching and scheduling energy, power and reserve, with a focus on attending to emergency situations, congestion of transmission lines and spinning reserve. In this context, the present work proposes a linear multi-period, multi-contingency and interval optimal power flow scheme, which can schedule the power dispatch of a hydrothermal system (including electrical and energetic restrictions) for a horizon of one day ahead, and is also capable of dimensioning and allocating the amounts of spinning reserve necessary to supply different situations involving the outflow of lines and balancing levels of renewable energy not supplied. In addition, as a strategy to incorporate the uncertainties inherent in the problem, interval mathematics is applied to the post-optimization problem; that is, after the deterministic optimization obtained based on the optimal power flow, ranges of uncertainties are added to some network parameters so that an optimal range of network operation can be determined. Results are presented for a system of 291 buses representing the south-southeast network of Brazil. The proposed methodology adequately sizes and allocates the necessary reserve to meet the most important contingencies affecting transmission.

Keywords:
Spinning reserve; hydrothermal operation planning; interval mathematics; optimal power flow; Krawczyk method; load and generation uncertainty.

Allocation of the spinning reserve in a hydro-thermal-wind system.

Linear multi-period, multi-contingency and interval optimal power flow is used.

A high-voltage hydro-thermal-wind system with 291 buses is used to test the model.

Name	Operation
Addition	$X + Y = [\underline{x} + \underline{y}; \bar{x} + \bar{y}]$
Subtraction	$X + Y = [\underline{x} - \bar{y}; \bar{x} - \underline{y}]$
Multiplication	$X \cdot Y = [S; S]$ , $S = [\underline{x} \cdot \underline{y}; \underline{x} \cdot \bar{y}; \bar{x} \cdot \underline{y}; \bar{x} \cdot \bar{y}]$
Diameter	$d i a m (X) = \underline{X} - \bar{X}$

		MMPOF			MOPF
Generator	Period	*Pgh* [pu]	$R e s_{i n c r e s a e}$ [pu]	$R e s_{d e c r e a s e}$ [pu]	*Pgh* [pu]	*Rgh* [pu]
1	5	7.56	0.010	-2.877	7.30	0.281
	12	14.36	0.076	-1.614	14.09	0.537
	20	16.52	0	-0.772	16.62	0.549
3	5	9.30	1.341	-0.021	8.97	0.653
	12	12.59	0.580	-0.029	12.06	0.773
	20	13.48	0.183	-0.010	12.78	0.097
	Daily Energy (24 hour)	1374.6	49.58	54.25	1372.8	54.98

Method	Period	$P g h_1_{s u p}$ [pu]	Dif %	$P g h_3_{s u p}$ [pu]	Dif %	$P g h_1_{i n f}$ [pu]	Dif %	$P g h_3_{i n f}$ [pu]	Dif %
IMMOPF	5	7.79	1.54	9.47	0.32	7.32	0.13	9.11	0.44
Random		7.67	1.54	9.44	0.32	7.33	0.13	9.15	0.44
Mean		7.73 0.085		9.46 0.020		7.33 0.007		9.13 0.028
Std Dev.		7.73 0.085		9.46 0.020		7.33 0.007		9.13 0.028
IMMOPF	12	14.87	0.95	12.77	0.23	13.88	0.65	8.63	1.05
Random		14.73	0.95	12.74	0.23	13.97	0.65	8.54	1.05
Mean		14.80 0.09		12.76 0.020		13.93 0.064		8.58 0.036
Std Dev.		14.80 0.09		12.76 0.020		13.93 0.064		8.58 0.036
IMMOPF	20	16.91	0.35	13.70	0.73	16.12	0.75	13.28	0.07
Random		16.85	0.35	13.60	0.73	16.24	0.75	13.27	0.07
Mean		16.88 0.042		13.65 0.070		16.18 0.085		13.27 0.007
Std Dev		16.88 0.042		13.65 0.070		16.18 0.085		13.27 0.007

Method			$R e s 1_{i n c r e a s e} s u p$	Dif %			$R e s 3_{i n c r e a s e} s u p$	Dif %			$R e s 1_{i n c r e a s e} i n f$	Dif %			$R e s 3_{i n c r e a s e} s u p$		Dif %
			[pu]	Dif %			[pu]	Dif %			[pu]	Dif %			[pu]		Dif %
MMOPF	5	0.380			2.64	1.700			11.7	0.0			0	0.940		8.51
Random		0.370			2.64	1.500			11.7	0.0			0	1.020		8.51
Mean		0.375 0.0071				1.600 0.140				0.0 0.0				0.980 0.057
Std Dev		0.375 0.0071				1.600 0.140				0.0 0.0				0.980 0.057
MMOPF	12	0.390			17.9	0.770			1.29	0.0			0	0.380		13.16
Random		0.320			17.9	0.760			1.29	0.0			0	0.430		13.16
Mean		0.360 0.050				0.765 0.0071				0.0 0.0				0.410 0.035
Std Dev		0.360 0.050				0.765 0.0071				0.0 0.0				0.410 0.035
MMOPF	20	0.130			8.33	0.300			6.67	0.0			0	0.050		0
Random		0.120			8.33	0.320			6.67	0.0			0	0.050		0
Mean		0.125 0.0071				0.310 0.014				0.0 0.0				0.050 0.000
Std Dev		0.125 0.0071				0.310 0.014				0.0 0.0				0.050 0.000

Method			$R e s 1_{d e c r e s}_s u p$	Dif %			$R e s 3_{d e c r e s}_s u p$	Dif %			$R e s 1_{d e c r e s}_i n f$		Dif %		$R e s 3_{d e c r e s}_i n f$		Dif %
			[pu]	Dif %			[pu]	Dif %			[pu]		Dif %		[pu]		Dif %
MMOPF	5	-2.70			0.37	0.0			0	-3.01		1.33		-0.41		9.75
Random		-2.71			0.37	0.0			0	-2.962		1.33		-0.37		9.75
Mean		-2.71 0.007				0.0 0.0				-2.99 0.029				-0.39 0.028
Std Dev		-2.71 0.007				0.0 0.0				-2.99 0.029				-0.39 0.028
MMOPF	12	-1.54			3.25	0.0			0	-2.50		2.80		-0.32		9.37
Random		-1.59			3.25	0.0			0	-2.43		2.80		-0.29		9.37
Mean		-1.57 0.035				0.0 0.0				-2.47				-0.31 0.021
Std Dev		-1.57 0.035				0.0 0.0				0.049				-0.31 0.021
MMOPF	20	-0.57			1.75	0.0			0	-0.97		1.03		-0.16		18.75
Random		-0.58			1.75	0.0			0	-0.96		1.03		-0.13		18.75
Mean		-0.58 0.007				0.0 0.0				-0.97 0.007				-0.15 0.021
Std Dev		-0.58 0.007				0.0 0.0				-0.97 0.007				-0.15 0.021

Variable	Generator 1	Generator 3
Phg	95.22%	95.33%
$R e s_{i n c r e s}$	99.74%	98.12%
$R e s_{d e c r e s}$	92.99%	96.00%

Period	A - Pgh	A - $R e s_{I n c r e a s e}$	A - $R e s_{d e c r e a s e}$
5	78.01%	73.17%	74.38%
12	81.01%	72.41%	62.81%
20	79.20%	83.14%	69.06%

Period	$A (P g h 1)$	$A (P g h 3)$	$A (R e s 1_{i n c r e a s e}$ )	$A (R e s 3_{i n c r e a s e})$	$A (R e s 1 d e c r e a s e$ )	$A (R e s 3 d e c r e a s e)$
5	71.66%	94.51%	77.91%	61.79%	73.39%	85.10%
12	76.93%	85.94%	79.76%	83.53%	84.30%	93.66%
20	77.99%	77.87%	90.39%	88.01%	98.57%	81.15%

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Instituto de Tecnologia do Paraná - Tecpar Rua Prof. Algacyr Munhoz Mader, 3775 - CIC, 81350-010 , Tel: +55 41 3316-3054 - Curitiba - PR - Brazil
E-mail: babt@tecpar.br

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