# A Network Graph-based Framework for Modeling, Calculating and Controlling Feasible AC Electric Power Delivery

thesis

posted on 01.10.2015, 00:00 by Andrew HsuAn influx of technology has changed the traditional, top-down approach to electric power systems,

into an interconnected electrical and informations network with large generators, intermittent renewables, storage, and demand response. The transmission network has seen its fair share of innovation, as new hardware is introduced for both monitoring and control purposes. For example, distributed line rating units (DLRs) are used to determine a more accurate thermal rating for transmission lines. Flexible AC Transmission Systems (FACTS) devices are capable of adjusting the reactance of lines using power electronics switching. Even newer technologies, like distributed series reactances (DSRs) are offering similar adjustment capabilities at lower cost. How can these devices be coordinated to run the electric power system more efficiently and reliably?

This thesis proposes a framework for modeling power systems using a graph representation.

This localizes each individual component's behavior and only interconnects the system through the

power and voltages of the components, over a network graph that obeys Kirchhoff's laws. Using

this framework, a number of new formulations for the power

flow problem are introduced.

The first formulation poses the conventional power flow problem, in terms of nodal voltage magnitude and phase angle, as a complex-value domain problem in terms of complex-valued nodal

voltages, for systems with only a slack bus and PQ buses.

Secondly, the complex-value domain power flow problem has been reposed in terms of the branch

voltages, or voltage difference across the lines of the network. By combining this with the S-E graph

model, a pi model of transmission lines, a set of power flow equations was formulated in terms of

the power transferred through each line, and the voltage across each line.

However, the branch-based formulation of power flow is more difficult to solve using the conventional numerical methods, especially Jacobi method. Therefore, a new optimization-based formulation of power

ow is developed, in terms of branch variables. This thesis shows how distributed

power flow calculation, performed by smart wires and buses, can be performed through only communication with neighbors. This new formulation is also instrumental in deriving methods for ensuring feasible power delivery.

If there is no power flow solution, there is no equilibrium value for the voltages of the network, and will lead to voltage collapse. To ensure that power is successfully delivered across the network, the components of the network must adjust so that there is a valid power flow solution. This thesis proposes two methods of ensuring feasible power delivery; targeted load shedding, and adjustment

of line reactances.

By using Lagrangian relaxation to solve the new optimization-based power flow problem, the

set of possible solutions is expanded beyond just the possibly power flow solutions of the network. If

there is no valid power flow solution, then the optimization problem can still converge to a solution.

The Lagrange multipliers, which correspond to physical constraints, such as nodal power balance,

therefore becomes a measure of power mismatch at each bus. Using the numerical results of the

Lagrange multipliers, adjustments to load can be made, allowing targeted load shedding.

Another method of reaching a feasible power flow solution is by adjusting the parameters of

the lines connecting the generators and loads. It may be less desirable to shed load, which would

disconnect customers from power, than to use devices such as DSRs and FACTS devices to change

the reactance of lines. By using the mathematical closed form power flow solution of a two-bus

power system, the conditions for feasible power delivery across one line can be derived. Extending

this to the multi-bus network allows each line to determine whether it should adjust, to reach a

valid power flow solution.

Finally, proof of concept simulations show the distributed calculation of power flow, targeted

load adjustment and line reactance adjustment on 3 bus, 14 bus, and 45 bus systems. An object-

oriented programming platform is used to simulate power systems in an actual distributed environment, with separate Matlab processes communicating between one another. The distributed power flow algorithm has been implemented for the 3 and 14 bus networks on this platform.

## History

## Date

01/10/2015## Degree Type

Dissertation## Department

Electrical and Computer Engineering## Degree Name

- Doctor of Philosophy (PhD)