Power system examples - Revision history

Schuetz2 at 12:27, 26 February 2026

2026-02-26T12:27:25Z

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	}}		}}

	~~__NUMBEREDHEADINGS__~~

	==Description==		==Description==

Saak: /* Test systems for small-signal stability analysis of large electric power system networks */

2023-11-24T09:38:55Z

Test systems for small-signal stability analysis of large electric power system networks

← Older revision		Revision as of 09:38, 24 November 2023
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	==Test systems for small-signal stability analysis of large electric power system networks==		==Test systems for small-signal stability analysis of large electric power system networks==
	All test '''power systems''' in [https://sites.google.com/site/rommes/software https://sites.google.com/site/rommes/software], except the PI sections example, are used for '''power system''' small-signal stability studies, and several of them are briefly described in <ref name="MarLP96"></ref>. Such studies include stability analysis, controller design, computation of MOR, etc. The dynamic phenomena of interest require the detailed modeling of the electrical energy generators (mainly the large sized ones) and other important devices such as flexible AC transmission systems (FACTS). The 3-phase alternating current (AC) network ~~is comprised~~ of nodes (electrical buses, which may be the representation of an entire electrical energy transmission (or distribution) substation, and branches (transmission lines, transformers, series capacitors) and need have only a static model in this application except for the occasional high-power electronic HVDC (High voltage direct current) transmission links and FACTS devices.		All test '''power systems''' in [https://sites.google.com/site/rommes/software https://sites.google.com/site/rommes/software], except the PI sections example, are used for '''power system''' small-signal stability studies, and several of them are briefly described in <ref name="MarLP96"></ref>. Such studies include stability analysis, controller design, computation of MOR, etc. The dynamic phenomena of interest require the detailed modeling of the electrical energy generators (mainly the large sized ones) and other important devices such as flexible AC transmission systems (FACTS). The 3-phase alternating current (AC) network consists of nodes (electrical buses), which may be the representation of an entire electrical energy transmission (or distribution) substation, and branches (transmission lines, transformers, series capacitors) and need to have only a static model in this application except for the occasional high-power electronic HVDC (High voltage direct current) transmission links and FACTS devices.

	The input <math>u(t)</math> depends on the study of interest, but the excitation system voltage reference is a frequent input variable in small-signal stability studies. Another reference is the active power setpoint of a rotor-speed regulator. The input signal used in all the above mentioned files is of Vref type. The output could be an entry of the generalized state <math>x(t)</math> (state or algebraic variable) or a linear combination of these variables. The generalized states <math>x(t)</math> are associated with equations of generators, controllers and network. The interconnected network has only algebraic variables, except when FACTS devices are connected.		The input <math>u(t)</math> depends on the study of interest, but the excitation system voltage reference is a frequent input variable in small-signal stability studies. Another reference is the active power setpoint of a rotor-speed regulator. The input signal used in all the above-mentioned files is of Vref type. The output could be an entry of the generalized state <math>x(t)</math> (state or algebraic variable) or a linear combination of these variables. The generalized states <math>x(t)</math> are associated with equations of generators, controllers, and network. The interconnected network has only algebraic variables, except when FACTS devices are connected.
	The test systems are represented by generalized state-space models, see ~~<math>(1)</math>~~, where the feed through matrix <math>D</math> is generally zero.		The test systems are represented by generalized state-space models, see above, where the feed through matrix <math>D</math> is generally zero.

	==Test systems for electromagnetic transients and harmonic distortion studies==		==Test systems for electromagnetic transients and harmonic distortion studies==

Saak at 09:35, 24 November 2023

2023-11-24T09:35:26Z

← Older revision		Revision as of 09:35, 24 November 2023
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	:<math>		:<math>
	E\dot{x}(t)=A x(t)+B u(t), \~~quad~~		\begin{array}{rcl}
	y(t)=Cx(t)+Du(t),\~~quad E,A\in\mathbb{R}^{n\times n},~B\in\mathbb{R}^{n\times m},~C\in\mathbb{R}^{p\times n},~D\in\mathbb~~{R}~~^{p\times m}\quad(1)~~		E\dot{x}(t)&=&A x(t)+B u(t), \\
			y(t)&=&Cx(t)+Du(t),
			\end{array}
	</math>		</math>
			where <math>E,A\in\mathbb{R}^{n\times n}</math>, <math>B\in\mathbb{R}^{n\times m}</math>, <math>C\in\mathbb{R}^{p\times n}</math>, <math>D\in\mathbb{R}^{p\times m}</math>,
	and originated at [http://www.cepel.br/ CEPEL] for simulating large '''power systems'''.		and originated at [http://www.cepel.br/ CEPEL] for simulating large '''power systems'''.

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	There is a pressing need for better utilization of the transmission network and its cost-effective expansions to reliably carry electric power from the generating plants, driven by several primary energy sources, to the loads. This imposes stricter requirements on network design and '''power system''' control equipment, as well as on the adopted control laws, which, coupled with the continental dimensions of modern interconnected '''power systems''', must be studied with the help of simulators employing advanced NLA algorithms. This wiki page contains system models developed when dealing with these NLA issues, with emphasis on the analysis and control of small signal stability.		There is a pressing need for better utilization of the transmission network and its cost-effective expansions to reliably carry electric power from the generating plants, driven by several primary energy sources, to the loads. This imposes stricter requirements on network design and '''power system''' control equipment, as well as on the adopted control laws, which, coupled with the continental dimensions of modern interconnected '''power systems''', must be studied with the help of simulators employing advanced NLA algorithms. This wiki page contains system models developed when dealing with these NLA issues, with emphasis on the analysis and control of small signal stability.

	Modal analysis has been used for Model Order Reduction (MOR) of lightly damped systems like flexible mechanical structures and RLC networks from either '''power systems''' or microprocessor interconnectors. The advent of subspace accelerated dominant pole algorithms in <ref name="MarLP96"></ref><ref name="RomM06a"></ref><ref name="RomM06b"></ref>, made modal reduction effective for better damped systems, such as interconnected '''power systems''', which have eigenvalue clusters in the <math>0.2</math> to <math>2.0 Hz</math> range and damping ratios between <math>-0.05</math> and <math>0.25</math>. Important characteristics of model reduction methods include preserving the dominant dynamics and stability in the Reduced Order Models (ROMs).		Modal analysis has been used for Model Order Reduction (MOR) of lightly damped systems like flexible mechanical structures and RLC networks from either '''power systems''' or microprocessor interconnectors. The advent of subspace accelerated dominant pole algorithms in <ref name="MarLP96"></ref><ref name="RomM06a"></ref><ref name="RomM06b"></ref>, made modal reduction effective for better damped systems, such as interconnected '''power systems''', which have eigenvalue clusters in the <math>0.2</math> to <math>2.0 \mathrm{Hz}</math> range and damping ratios between <math>-0.05</math> and <math>0.25</math>. Important characteristics of model reduction methods include preserving the dominant dynamics and stability in the Reduced Order Models (ROMs).

	Model reduction is important to the linear control system analysis and design of '''power system''' dynamic models. Reduced-order '''power system''' transfer function models are effective in lowering the computational costs of controller design. Once the dominant transfer function poles have been found, the calculation of their associated residues can be obtained by the scalar product of the left and right eigenvectors by the input and output vectors of the transfer function. From the knowledge of the dominant pole-residue set, one can build a transfer function ROM whose accuracy varies with the number of retained poles. See, e.g., the SADPA, SAMDP papers <ref name="RomM06a"></ref><ref name="RomM06b"></ref>.		Model reduction is important to the linear control system analysis and design of '''power system''' dynamic models. Reduced-order '''power system''' transfer function models are effective in lowering the computational costs of controller design. Once the dominant transfer function poles have been found, the calculation of their associated residues can be obtained by the scalar product of the left and right eigenvectors by the input and output vectors of the transfer function. From the knowledge of the dominant pole-residue set, one can build a transfer function ROM whose accuracy varies with the number of retained poles. See, e.g., the SADPA, SAMDP papers <ref name="RomM06a"></ref><ref name="RomM06b"></ref>.

	Modal analysis is also used in the harmonic distortion analysis of '''power system''' networks, mainly in subtransmission and distribution voltage levels. In this application, modal analysis allows for determining the main network parameters that impact distortion levels and suggests changes to their effective values among other alternatives. The linear RLC network is modeled unloaded, the conventional electrical generators can be entirely neglected and the nonlinear loads of various nature are modeled as current injections into the network buses -3rd, 5th, 7th, 11th, 13th harmonics of the nominal system frequency (<math>50 Hz</math> or <math>60 Hz</math>). An even more important application for the modal analysis of these large descriptor system RLC network models is the production of ROM for multiport electrical networks, generally in the form of a state-space system whose dimension is considerably smaller than that of the original system. When these ROMs take the form of RLC networks, they are referred to as RLC reduced equivalents. These ROMs, or equivalents, are used in real-time and off-line simulators for the study of '''power system''' high-frequency transient phenomena. They are produced in varying degrees of complexity by using advanced NLA algorithms, which is an area of intense research work.		Modal analysis is also used in the harmonic distortion analysis of '''power system''' networks, mainly in subtransmission and distribution voltage levels. In this application, modal analysis allows for determining the main network parameters that impact distortion levels and suggests changes to their effective values among other alternatives. The linear RLC network is modeled unloaded, the conventional electrical generators can be entirely neglected and the nonlinear loads of various nature are modeled as current injections into the network buses -3rd, 5th, 7th, 11th, 13th harmonics of the nominal system frequency (<math>50 \mathrm{Hz}</math> or <math>60 \mathrm{Hz}</math>). An even more important application for the modal analysis of these large descriptor system RLC network models is the production of ROM for multiport electrical networks, generally in the form of a state-space system whose dimension is considerably smaller than that of the original system. When these ROMs take the form of RLC networks, they are referred to as RLC reduced equivalents. These ROMs, or equivalents, are used in real-time and off-line simulators for the study of '''power system''' high-frequency transient phenomena. They are produced in varying degrees of complexity by using advanced NLA algorithms, which is an area of intense research work.

	==Test systems for small-signal stability analysis of large electric power system networks==		==Test systems for small-signal stability analysis of large electric power system networks==

Malhotra: Add correct infobox

2023-11-23T11:06:40Z

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Malhotra: Undo revision 3853 by Malhotra (talk)

2023-11-23T11:05:57Z

Undo revision 3853 by Malhotra (talk)

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Malhotra: Add infobox

2023-11-23T11:03:45Z

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Lund: /* Background */ small grammatical and formatting updates

2022-10-06T09:24:53Z

Background: small grammatical and formatting updates

← Older revision		Revision as of 09:24, 6 October 2022
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	==Background==		==Background==
	Electrical '''power systems''' experience several steady-state and dynamic phenomena that may hinder its reliable, stable operation if not properly designed and operated. Among the dynamic phenomena, '''power system''' angle stability is of major concern, its studies requiring both numerical integration of a stiff set of large, nonlinear differential-algebraic (DAE) equations (for transient stability simulations) and the eigensolution as well as the use of other ~~Numerical Linear Algebra~~ (NLA) algorithms applied to the linearized DAE equations of the same large stability models. '''Power systems''' with multiple electrical power plants, consumer and industrial loads experience complex electromechanical oscillations, much ~~alike~~ spring-mass mechanical systems experience mechanical oscillations. These oscillations, when the electric '''power system''' is under stressed conditions, may become poorly-damped or unstable. The analysis of these low-frequency oscillations, as well as their controller-induced damping control, is enhanced by the results from NLA algorithms, all ~~this constituting~~ the ~~study area~~ known as small signal stability.		Electrical '''power systems''' experience several steady-state and dynamic phenomena that may hinder its reliable, stable operation if not properly designed and operated. Among the dynamic phenomena, '''power system''' angle stability is of major concern, its studies requiring both numerical integration of a stiff set of large, nonlinear differential-algebraic (DAE) equations (for transient stability simulations) and the eigensolution as well as the use of other numerical linear algebra (NLA) algorithms applied to the linearized DAE equations of the same large stability models. '''Power systems''' with multiple electrical power plants, consumer loads, and industrial loads experience complex electromechanical oscillations, much as spring-mass mechanical systems experience mechanical oscillations. These oscillations, when the electric '''power system''' is under stressed conditions, may become poorly damped or unstable. The analysis of these low-frequency oscillations, as well as their controller-induced damping control, is enhanced by the results from NLA algorithms. The study of all these issues constitutes the field known as small signal stability.

	There is a pressing need for better utilization of the transmission network and its cost-effective expansions, to reliably carry electric power from the generating plants, driven by several primary energy sources, to the loads. This imposes stricter requirements on network design and '''power system''' control equipment, as well as in the adopted control laws, which coupled with the continental dimensions of modern interconnected '''power systems''', must be studied with the help of simulators employing advanced NLA algorithms. This wiki page contains system models developed when dealing with these NLA issues, with emphasis on the analysis and control of small signal stability.		There is a pressing need for better utilization of the transmission network and its cost-effective expansions to reliably carry electric power from the generating plants, driven by several primary energy sources, to the loads. This imposes stricter requirements on network design and '''power system''' control equipment, as well as on the adopted control laws, which, coupled with the continental dimensions of modern interconnected '''power systems''', must be studied with the help of simulators employing advanced NLA algorithms. This wiki page contains system models developed when dealing with these NLA issues, with emphasis on the analysis and control of small signal stability.
	Modal analysis has been used for Model Order Reduction (MOR) of lightly damped systems like flexible mechanical structures and RLC networks from either '''power systems''' or microprocessor interconnectors. The advent of subspace accelerated dominant pole algorithms in <ref name="MarLP96"></ref><ref name="RomM06a"></ref><ref name="RomM06b"></ref>, made modal reduction effective for better damped systems, such as interconnected '''power systems''', which have eigenvalue clusters in the <math>0.2</math> to <math>2.0 Hz</math> range and damping ratios between <math>-0.05</math> and <math>0.25</math>. Important characteristics of model reduction methods include preserving the dominant dynamics and stability in the Reduced Order Models (ROMs).
	Model reduction is important to the linear control system analysis and design of '''power system''' dynamic models. Reduced-order '''power system''' transfer function models are effective in lowering the computational costs of controller design. Once the dominant transfer function poles have been found, the calculation of their associated residues can be obtained by the scalar product of the left and right eigenvectors by the input and output vectors of the transfer function. From the knowledge of the dominant pole-residue set, one can build a transfer function ROM whose accuracy varies with the number of retained poles. See SADPA, SAMDP papers <ref name="RomM06a"></ref><ref name="RomM06b"></ref>.		Modal analysis has been used for Model Order Reduction (MOR) of lightly damped systems like flexible mechanical structures and RLC networks from either '''power systems''' or microprocessor interconnectors. The advent of subspace accelerated dominant pole algorithms in <ref name="MarLP96"></ref><ref name="RomM06a"></ref><ref name="RomM06b"></ref>, made modal reduction effective for better damped systems, such as interconnected '''power systems''', which have eigenvalue clusters in the <math>0.2</math> to <math>2.0 Hz</math> range and damping ratios between <math>-0.05</math> and <math>0.25</math>. Important characteristics of model reduction methods include preserving the dominant dynamics and stability in the Reduced Order Models (ROMs).
	Modal analysis is also used in the harmonic distortion analysis of '''power system''' networks, mainly in subtransmission and distribution voltage levels. In this application, modal analysis allows determining the main network parameters that impact distortion levels and ~~suggest~~ changes to their effective values among other alternatives. The linear RLC network is modeled unloaded, the conventional electrical generators can be entirely neglected and the nonlinear loads of various nature are modeled as current injections into the network buses -3rd, 5th, 7th, 11th, 13th harmonics of the nominal system frequency (<math>50 Hz</math> or <math>60 Hz</math>). An even more important application for the modal analysis of these large descriptor system RLC network models is the production of ROM for multiport electrical networks, generally in the form of a state-space system whose dimension is considerably smaller than the original system. When these ROMs take the form of RLC networks, they are referred to as RLC reduced equivalents. These ROMs, or equivalents, are used in real-time and off-line simulators for the study of '''power system''' high-frequency transient phenomena. They are produced in varying degrees of complexity by using advanced NLA algorithms, ~~this being~~ an area of intense research work.
			Model reduction is important to the linear control system analysis and design of '''power system''' dynamic models. Reduced-order '''power system''' transfer function models are effective in lowering the computational costs of controller design. Once the dominant transfer function poles have been found, the calculation of their associated residues can be obtained by the scalar product of the left and right eigenvectors by the input and output vectors of the transfer function. From the knowledge of the dominant pole-residue set, one can build a transfer function ROM whose accuracy varies with the number of retained poles. See, e.g., the SADPA, SAMDP papers <ref name="RomM06a"></ref><ref name="RomM06b"></ref>.

			Modal analysis is also used in the harmonic distortion analysis of '''power system''' networks, mainly in subtransmission and distribution voltage levels. In this application, modal analysis allows for determining the main network parameters that impact distortion levels and suggests changes to their effective values among other alternatives. The linear RLC network is modeled unloaded, the conventional electrical generators can be entirely neglected and the nonlinear loads of various nature are modeled as current injections into the network buses -3rd, 5th, 7th, 11th, 13th harmonics of the nominal system frequency (<math>50 Hz</math> or <math>60 Hz</math>). An even more important application for the modal analysis of these large descriptor system RLC network models is the production of ROM for multiport electrical networks, generally in the form of a state-space system whose dimension is considerably smaller than that of the original system. When these ROMs take the form of RLC networks, they are referred to as RLC reduced equivalents. These ROMs, or equivalents, are used in real-time and off-line simulators for the study of '''power system''' high-frequency transient phenomena. They are produced in varying degrees of complexity by using advanced NLA algorithms, which is an area of intense research work.

	==Test systems for small-signal stability analysis of large electric power system networks==		==Test systems for small-signal stability analysis of large electric power system networks==

Lund: /* Data */ correct dimensions for BIPS/2007

2022-09-01T14:39:48Z

Data: correct dimensions for BIPS/2007

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	\|21476		\|21476
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	\|1		\|32
	\|DAE		\|DAE
	\|-		\|-

Himpe at 12:53, 25 August 2022

2022-08-25T12:53:39Z

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	[[Category:time invariant]]		[[Category:time invariant]]
	[[Category:first differential order]]		[[Category:first differential order]]
	[[Category:~~DAE index one‏‎~~]]		[[Category:Differential_algebraic_system]]
	~~[[Category:DAE index two‏‎]]~~
	__NUMBEREDHEADINGS__		__NUMBEREDHEADINGS__

Lund: remove dead link

2022-06-29T11:10:40Z

remove dead link

← Older revision		Revision as of 11:10, 29 June 2022
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	All transmission lines in the network are modeled by RLC ladder networks, of cascaded RLC PI-circuits, having fixed parameters <ref name="Wat03"></ref>. The transformers are modeled by series RL circuits. Loads and shunt elements are represented by series-connected RL (or RC) branch, or just an L (or C). As a consequence, the states <math>x(t)</math> of the dynamic system are either the current through an inductor or the voltage across a capacitor. So matrices <math>A, B, C, D, E</math> describe how each circuit element equation, node and voltage are connected, according to the Kirchhoff’s law.		All transmission lines in the network are modeled by RLC ladder networks, of cascaded RLC PI-circuits, having fixed parameters <ref name="Wat03"></ref>. The transformers are modeled by series RL circuits. Loads and shunt elements are represented by series-connected RL (or RC) branch, or just an L (or C). As a consequence, the states <math>x(t)</math> of the dynamic system are either the current through an inductor or the voltage across a capacitor. So matrices <math>A, B, C, D, E</math> describe how each circuit element equation, node and voltage are connected, according to the Kirchhoff’s law.

	On https://sites.google.com/site/rommes/software the systems labeled by PI Sections 20--80 are of index-2 ([https://drive.google.com/file/d/1ttJgCZeNEspA3Ty7vNPtsuPqo_-Qa-6g/view SISO_PI_n.zip and MIMO_PI_n.zip]). The data of these test systems can be converted into index-0 systems by applying a procedure for symbolic math elimination of algebraic variables and redundant state variables. This procedure is detailed in the Appendix C of <ref name="FreRM11"></ref> ~~([https://drive.google.com/file/d/1ttJgCZeNEspA3Ty7vNPtsuPqo_-Qa-6g/view SISO_PI_n.zip and MIMO_PI_n.zip])~~.		On https://sites.google.com/site/rommes/software the systems labeled by PI Sections 20--80 are of index-2 ([https://drive.google.com/file/d/1ttJgCZeNEspA3Ty7vNPtsuPqo_-Qa-6g/view SISO_PI_n.zip and MIMO_PI_n.zip]). The data of these test systems can be converted into index-0 systems by applying a procedure for symbolic math elimination of algebraic variables and redundant state variables. This procedure is detailed in the Appendix C of <ref name="FreRM11"></ref>.

	==References==		==References==