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Voltage collapse in complex power grids.

Simpson-Porco JW, Dörfler F, Bullo F - Nat Commun (2016)

Bottom Line: A large-scale power grid's ability to transfer energy from producers to consumers is constrained by both the network structure and the nonlinear physics of power flow.Violations of these constraints have been observed to result in voltage collapse blackouts, where nodal voltages slowly decline before precipitously falling.We extensively test our predictions on large-scale systems, highlighting how our condition can be leveraged to increase grid stability margins.

View Article: PubMed Central - PubMed

Affiliation: Department of Electrical and Computer Engineering, Engineering Building 5, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1.

ABSTRACT
A large-scale power grid's ability to transfer energy from producers to consumers is constrained by both the network structure and the nonlinear physics of power flow. Violations of these constraints have been observed to result in voltage collapse blackouts, where nodal voltages slowly decline before precipitously falling. However, methods to test for voltage collapse are dominantly simulation-based, offering little theoretical insight into how grid structure influences stability margins. For a simplified power flow model, here we derive a closed-form condition under which a power network is safe from voltage collapse. The condition combines the complex structure of the network with the reactive power demands of loads to produce a node-by-node measure of grid stress, a prediction of the largest nodal voltage deviation, and an estimate of the distance to collapse. We extensively test our predictions on large-scale systems, highlighting how our condition can be leveraged to increase grid stability margins.

No MeSH data available.


Related in: MedlinePlus

Mechanical and energy interpretations of power flow.(a) An example power network with two generators (green) supplying power to three loads (red). Power demands (Q1, Q2, Q3) are placed on the load nodes; (b) a mechanical analogy: a linear spring network placed in a potential field. The generator voltages (green) are ‘pinned' at constant values, while the load voltages (red) are masses ‘hanging' off the generators, their equilibrium values being determined by their weights (the power demands QL=(Q1, Q2, Q3)), the heights of the fixed-generator voltages (V4, V5), and by the stiffness of the spring network (the susceptance matrix B). Voltage collapse can occur when one of the masses crosses an appropriate collapse boundary curve; (c) Contour plot of energy function when Q3=0 and node 3 is eliminated via Kron reduction13. Since E(VL) contains logarithms, it tends to −∞ as either axis is approached. In a normalized system of units, the stable high-voltage equilibrium rests in a local minimum at (0.94, 0.94), while an unstable low-voltage equilibrium sits at the saddle (0.68, 0.30). Voltage collapse occurs when these equilibria coalesce and the system trajectory diverges.
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f1: Mechanical and energy interpretations of power flow.(a) An example power network with two generators (green) supplying power to three loads (red). Power demands (Q1, Q2, Q3) are placed on the load nodes; (b) a mechanical analogy: a linear spring network placed in a potential field. The generator voltages (green) are ‘pinned' at constant values, while the load voltages (red) are masses ‘hanging' off the generators, their equilibrium values being determined by their weights (the power demands QL=(Q1, Q2, Q3)), the heights of the fixed-generator voltages (V4, V5), and by the stiffness of the spring network (the susceptance matrix B). Voltage collapse can occur when one of the masses crosses an appropriate collapse boundary curve; (c) Contour plot of energy function when Q3=0 and node 3 is eliminated via Kron reduction13. Since E(VL) contains logarithms, it tends to −∞ as either axis is approached. In a normalized system of units, the stable high-voltage equilibrium rests in a local minimum at (0.94, 0.94), while an unstable low-voltage equilibrium sits at the saddle (0.68, 0.30). Voltage collapse occurs when these equilibria coalesce and the system trajectory diverges.

Mentions: A novel mechanical analogy for the power flow (1) is shown in Fig. 1b. The equilibrium configuration of the spring network corresponds to the desirable high-voltage solution of (1), and can be interpreted as a local minimum (Fig. 1c) of the energy function31


Voltage collapse in complex power grids.

Simpson-Porco JW, Dörfler F, Bullo F - Nat Commun (2016)

Mechanical and energy interpretations of power flow.(a) An example power network with two generators (green) supplying power to three loads (red). Power demands (Q1, Q2, Q3) are placed on the load nodes; (b) a mechanical analogy: a linear spring network placed in a potential field. The generator voltages (green) are ‘pinned' at constant values, while the load voltages (red) are masses ‘hanging' off the generators, their equilibrium values being determined by their weights (the power demands QL=(Q1, Q2, Q3)), the heights of the fixed-generator voltages (V4, V5), and by the stiffness of the spring network (the susceptance matrix B). Voltage collapse can occur when one of the masses crosses an appropriate collapse boundary curve; (c) Contour plot of energy function when Q3=0 and node 3 is eliminated via Kron reduction13. Since E(VL) contains logarithms, it tends to −∞ as either axis is approached. In a normalized system of units, the stable high-voltage equilibrium rests in a local minimum at (0.94, 0.94), while an unstable low-voltage equilibrium sits at the saddle (0.68, 0.30). Voltage collapse occurs when these equilibria coalesce and the system trajectory diverges.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4759633&req=5

f1: Mechanical and energy interpretations of power flow.(a) An example power network with two generators (green) supplying power to three loads (red). Power demands (Q1, Q2, Q3) are placed on the load nodes; (b) a mechanical analogy: a linear spring network placed in a potential field. The generator voltages (green) are ‘pinned' at constant values, while the load voltages (red) are masses ‘hanging' off the generators, their equilibrium values being determined by their weights (the power demands QL=(Q1, Q2, Q3)), the heights of the fixed-generator voltages (V4, V5), and by the stiffness of the spring network (the susceptance matrix B). Voltage collapse can occur when one of the masses crosses an appropriate collapse boundary curve; (c) Contour plot of energy function when Q3=0 and node 3 is eliminated via Kron reduction13. Since E(VL) contains logarithms, it tends to −∞ as either axis is approached. In a normalized system of units, the stable high-voltage equilibrium rests in a local minimum at (0.94, 0.94), while an unstable low-voltage equilibrium sits at the saddle (0.68, 0.30). Voltage collapse occurs when these equilibria coalesce and the system trajectory diverges.
Mentions: A novel mechanical analogy for the power flow (1) is shown in Fig. 1b. The equilibrium configuration of the spring network corresponds to the desirable high-voltage solution of (1), and can be interpreted as a local minimum (Fig. 1c) of the energy function31

Bottom Line: A large-scale power grid's ability to transfer energy from producers to consumers is constrained by both the network structure and the nonlinear physics of power flow.Violations of these constraints have been observed to result in voltage collapse blackouts, where nodal voltages slowly decline before precipitously falling.We extensively test our predictions on large-scale systems, highlighting how our condition can be leveraged to increase grid stability margins.

View Article: PubMed Central - PubMed

Affiliation: Department of Electrical and Computer Engineering, Engineering Building 5, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1.

ABSTRACT
A large-scale power grid's ability to transfer energy from producers to consumers is constrained by both the network structure and the nonlinear physics of power flow. Violations of these constraints have been observed to result in voltage collapse blackouts, where nodal voltages slowly decline before precipitously falling. However, methods to test for voltage collapse are dominantly simulation-based, offering little theoretical insight into how grid structure influences stability margins. For a simplified power flow model, here we derive a closed-form condition under which a power network is safe from voltage collapse. The condition combines the complex structure of the network with the reactive power demands of loads to produce a node-by-node measure of grid stress, a prediction of the largest nodal voltage deviation, and an estimate of the distance to collapse. We extensively test our predictions on large-scale systems, highlighting how our condition can be leveraged to increase grid stability margins.

No MeSH data available.


Related in: MedlinePlus