Modern Control of DC-Based Power Systems

A Problem-Based Approach

Marco Cupelli, Antonino Riccobono, Markus Mirz, Mohsen Ferdowsi, Antonello Monti, marco Cupelli

• Publisher: Academic Press
• Year: 2018
• Print ISBN:
  9780128132203
• eBook ISBN:
  9780128132210
• Topic: Technology & Engineering
• Subtopic: Electrical Engineering & Telecommunications
• Index: Technology & Engineering

Modern Control of DC-Based Power Systems: A Problem-Based Approach addresses the future challenges of DC Grids in a problem-based context for practicing power engineers who are challenged with integrating DC grids in their existing architecture. This reference uses control theory to address the main concerns affecting these systems, things like generation capacity, limited maximum load demands and low installed inertia which are all set to increase as we move towards a full renewable model. Offering a new approach for a problem-based, practical approach, the book provides a coordinated view of the topic with MATLAB®, Simulink® files and additional ancillary material provided. - Includes Simulink® Files (of examples and for lab training classes) and MATLAB® files - Presents video slides to support the problem-based approach to understanding DC Power System control and application - Provides stability analysis of DC networks and examples of common stability problems

Chapter One

Overview—Voltage Stabilization of Constant Power Loads

Abstract

Throughout this chapter the reader will be introduced to the characteristics of the Constant Power Load (CPL) while being connected to a DC bus in Section 1.1. The Sections 1.2–1.4 present an overview on the previous research activities performed for stabilizing CPLs. Section 1.5 offers a summary of this chapter.

Keywords

Constant Power Load; DC bus; converter; actuator

Throughout this chapter the reader will be introduced to the characteristics of the Constant Power Load (CPL) while being connected to a DC bus in Section 1.1. The Sections 1.2–1.4 present an overview on the previous research activities performed for stabilizing CPLs. Section 1.5 offers a summary of this chapter.

1.1 Constant Power Load Connected to a DC Bus

The interaction of electrical subsystems around a DC bus can lead to the instability of the latter. Often in these cases, it is associated with the occurrence of instability when CPLs are connected on the DC bus [1–14]. The phenomenon of instability is considered then as the consequence of the connection of a CPL on a DC bus.

When there is an element present which regulates the power consumption of a load (e.g., inverter/actuator, converter/battery converter, drive), it is possible to assume the aforementioned CPL behavior. The control of the load will then set the load to compensate disturbances on the bus. The load does not take into account the electrical state of the network (including DC bus). Thus, it will be seen by the DC bus as a constant power consuming element as shown in Fig. 1.1. The output of the source converter is represented by the voltage source E, which is interfaced through a filter circuit consisting of the equivalent series resistor of the inductor r, inductor L and capacitor C with the nonlinear load, which is represented by a current source with the characteristic P/V, V denotes the voltage drop over the capacitor.

Figure 1.1 Electrical system containing a nonlinear load.

The current absorbed by CPL on the DC bus can be modeled as Eq. (1.1) where for a given operating point ($V_0,I_0)$

, the product of the load voltage and current is kept constant. The introduction of a CPL in the network implies the appearance of a nonlinearity of the $P/V$

type. The characteristic current–voltage curve across the CPL is represented in Fig. 1.2.

$I=\frac{P}{V}$ (1.1)

(1.1)

Figure 1.2 Characteristic I–V curve of a Constant Power Load.

The rate of change of the current can be obtained by a linear approximation, deriving from Eq. (1.1) in the small area around the operating point. This yields the linear equivalent impedance R_CPL at the given operating point Eq. (1.2).

$\frac{\partial I}{\partial V}=-\frac{P}{V_0^2}=-\frac{1}{R_{\mathit{CPL}}}$ (1.2)

(1.2)

From Eq. (1.2) it can be observed that R_CPL is dependent on the actual voltage and current. The curve representing the current versus voltage for a CPL in Eq. (1.3) can be approximated by a straight line that is tangent to the power curve at the operating point. The equation for this line is given as follows:

$I_{\mathit{CPL}}=2I_0-\frac{1}{R_{\mathit{CPL}}}V=2\frac{P}{V_0}-\frac{P}{V_0^2}V$ (1.3)

(1.3)

One can conclude that a CPL around its operating point has a behavior similar to that of a negative resistance in parallel with a constant current source. The diagram corresponding to the electrical behavior of the CPL around an operating point is depicted in Fig. 1.3. The output of the source converter is represented by the voltage source E, which is interfaced through a filter circuit consisting of the equivalent series resistor of the inductor r, the inductor L, and the capacitor C; where V denotes the voltage over the capacitor.

Figure 1.3 Electrical system containing a Constant Power Load, equivalent circuit around operating point.

In order to have a mathematical observation of the effect of the CPL and R_CPL on the network and to exclusively consider the variations of the system around the operating point lead toward omitting the DC component of Eq. (1.3), the resultant model is described in accordance to Eq. (1.4) and its characteristic polynomial is written in Eq. (1.5).

$\left[\begin{array}{c}\dot{V}\\ \dot{I}\end{array}\right]=\left[\begin{array}{cc}\frac{-1}{C{R}_{\mathit{CPL}}}& \frac{1}{C}\\ \frac{-1}{L}& \frac{-r}{L}\end{array}\right]\left[\begin{array}{c}V\\ I\end{array}\right]$ (1.4)

(1.4)

$P\left(\lambda \right)={\lambda }^2+\left(\frac{r}{L}+\frac{1}{C{R}_{\mathit{CPL}}}\right)\lambda +\frac{1}{\mathit{LC}}\left(1+\frac{r}{R_{\mathit{CPL}}}\right)$ (1.5)

(1.5)

From the characteristic polynomial, one can deduce the relationship Eq. (1.6) which states the criteria for the eigenvalues of the system to have a negative real part. The first inequality holds because of the negative sign of $R_{\mathit{CPL}}$

and usually the equivalent series resistor r is very small when comparing the absolute values, while the second will not always be respected since an increase in load power causes an increase in current which has as consequence an increase of $R_{\mathit{CPL}}$

(which goes to 0).

$\{\begin{array}{c}{R}_{\mathit{CPL}}<-r\\ R_{\mathit{CPL}}<-\frac{L}{\mathit{rC}}\end{array}$ (1.6)

(1.6)

Replacing $R_{\mathit{CPL}}$

by this inequality the relationship (1.7) is obtained, which gives the local stability condition for the system.

$P<\min \left(\frac{\mathit{rC}}{L}{V}_0^2\mathrm{;}\frac{V_0^2}{r}\right)$ (1.7)

(1.7)

The relation $P<V_0^2/r$

is less restrictive than the relationship $P<V_0^2/4r$

which ensures the existence of an operating point for the system while relation (1.8) gives the stability condition of the system operating points.

$\{\begin{array}{c}{R}_{\mathit{CPL}}<-r\\ R_{\mathit{CPL}}<-\frac{L}{\mathit{rC}}\end{array}$ (1.8)

(1.8)

It is seen from Eq. (1.8) that a relationship between the stability of system, the sizing of the filter (r, L, C, and indirectly E) and the power consumed by the load can be expressed. It is seen that as the capacity of the inverter increases the system becomes more stable and vice versa for the inductance of the filter. In addition, these observations show that the negative resistance is used to excit...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• List of Figures
• List of Tables
• List of Contributors
• List of Acronyms
• Symbols Used in this Work
• Preface
• Introduction
• Chapter One. Overview—Voltage Stabilization of Constant Power Loads
• Chapter Two. Small-Signal Analysis of Cascaded Systems
• Chapter Three. Background
• Chapter Four. Generation Side Control
• Chapter Five. Control Approaches for Parallel Source Converter Systems
• Chapter Six. Simulation
• Chapter Seven. Hardware In the Loop Implementation and Challenges
• Index