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# Graph Theory Helps Synchronize Power Generator Groups

May 17, 2018 | 1 Comment

A joint research team from Tokyo Institute of Technology (Tokyo Tech) and North Carolina State University has clarified the fundamental principles for achieving the synchronization of power generator groups in power networks. Synchronization is vital for the stable supply of alternating current electric power.

Synchronization of generator groups means the phase angles (timing of the alternations) of the rotors such as the turbines of multiple generators must be the same or reasonably close. Each generator’s rotor rotates according to the standard of a specific frequency (50 Hz or 60 Hz) in order to maintain its frequency. A difference in the frequency of each generator creates a difference in phase angle.

The synchronization phenomenon of generator groups such as at multiple thermal power plants is closely related to the stable supply of electric power. Specifically, if a generator becomes out of synchronization, that generator and its surrounding generators will not be able to operate stably, and in worst cases, serious accidents resulting in power outages can occur.

Added now is the energy problems driven by global warming fears and the concerns about depletion of fossil fuels. These feelings have become more serious and on a global scale. Therefore, from the viewpoint of reducing carbon dioxide and systemic use of energy, high expectations have been placed on renewable energy such as wind power and photovoltaic (WP-PV) generation.

When large-scale WP-PV generation equipment and power storage equipment are introduced, in addition to power generation such as thermal power, hydro power, and nuclear power that are commonly used today, it is necessary to consider power charge and discharge by WP-PV generated output and storage batteries in order to maintain equilibrium between supply and demand.

But the amount of power from WP-PV generation fluctuates since there is uncertainty related to changes in weather and changes in solar radiation volume according to the time zone. This makes it much more difficult to maintain the synchronization of generator groups. The need to analyze synchronization is greater than ever.

With conventional analysis, a major approach is based on numerical simulation. There are no studies that theoretically clarify the basic principles for how to properly synchronize generator groups according to the network structure of power transmission. There is an urgent need to build a power supply and demand framework that efficiently utilizes power storage equipment to allow for the uncertainty of WP-PV generation and demand predictions.

Assistant Professor Takayuki Ishizaki, Professor Jun-ichi Imura of Tokyo Tech, and Associate Professor Aranya Chakrabortty of the NSF ERC FREEDM System Center at North Carolina State University worked on multiple studies including power network modeling, stability analysis, and stabilization control from the perspective of graph theory. They have clarified that the symmetry of the network in graph theory is the fundamental principle for realizing the synchronization of generator groups at thermal power plants integrated with power grids (connected to a network).

Here graph theory is a mathematical theory related to graphs (network structure) composed of sets of vertices (nodes) and sets of edges. The power grid network is interpreted as a graph in which the connection point is the vertex and the transmission line linking the connection points is the edge.

The team’s research paper has been published in the journal Proceedings of the IEEE.

The behavior of generators connected through a network in a power grid is represented by complex equations (differential algebraic equations) that combine differential equations and algebraic equations. The differential equations express “behavior of generators” derived from Newton’s second law of motion, and the algebraic equations express “power balance at power grid connection points” derived from Ohm’s law and Kirchhoff’s law.

Ohm’s Law and Kirchhoff’s Law are physical laws that express the relationship between physical quantities such as voltage and current in an electric circuit. Ohm’s law indicates that the voltage difference between two points in a circuit is proportional to the current flowing between them. Kirchhoff’s law indicates that at the branch point in the circuit, the sum of the currents flowing to that point is equal to the sum of the currents flowing from that point.

Analysis of these differential algebraic equations was generally performed by transformation into a mathematically equivalent differential equation through a simplification method called the Kron reduction. However, the problems were that with the existing approach, since the algebraic equation representing the power grid is eliminated by deleting the redundant variable representing the connection point voltage, it was not very suitable for analyzing the relationship between the network structure of the power grid and the behavior of the generator.

To resolve this issue, they analyzed the network structure of the power grid contained in the algebraic equations from the viewpoint of symmetry based on an understanding of graph theory. Specifically, by analyzing the behavior of the generator without eliminating the algebraic equations, they discovered that the symmetry of the power grid is the basic principle for realizing synchronization of generator groups. In addition, based on a new idea of simultaneously integrating generator groups that show synchronous behavior and the power grid that couples these, it became possible to mathematically and physically construct a feasible aggregated model.

Its expected that the team’s graph theory will result in a basis for developing analysis and control methods for realizing stable power supply to large and complex electric power systems. In the future, Professor Imura says that the theory aims to develop more complex electric power systems including converters, and to establish a theory to approximate the synchronization of generator groups.

So far grid operators and power companies have, if the vernacular is forgiven, “been winging it” with impressive results. But its been expensive and for the operators and companies more than a bit exciting in a much less than positive way. These firms are to be commended for an exemplary performance in difficult situations.

To simplify, alternating current electrons are all going one way for a 50th or 60th of a second and then reversing. 50 or 60 times every second. A generator out of time with the others is a crisis. Its a bit like one of the cylinders in your car’s engine going out of time. Not good. At all.

With WP-PV the generators have massively increased in number and have made a mountain sized problem out of a tiny small hill. And the problem is growing into a mountain range of immense complexity and costs are sure to go up. One day in the future people will ask “why did they build so many little generators?” Its a pretty good question today.

With the green renewable enthusiasm well underway, the anxiety producing expensive and high risk production of lots of little power generators is expected to increase for the foreseeable future. Until the “big crises” accident happens. We’ll see if the blame goes to the grid operators and power companies or the hoard of little generators policy. Today we at least know we’re building into a problem with political policy.

# Comments

1 Comment so far

There is an urgent need to build a power supply and demand framework that efficiently utilizes power storage equipment to allow for the uncertainty of WP-PV generation and demand predictions.