Introduction

In this chapter the concept, the theory, and calculation of modal induced torque coefficients (MITCs) in multi-machine power systems are introduced. The concept of a modal induced torque coefficient is new [1], [2]. It forms the basis for calculation of the shifts in rotor modes when the stabilizer gains of one or more PSSs and/or FDSs are incremented by Δk (pu) on device base. Based on the concept of MITCs, the background theory of the rotor modes shifts, together with analysis of the effectiveness of, and interactions between, PSSs and FDSs in multi-machine systems are described in Chapter 13.

The theoretical development of MITCs in this chapter is fairly detailed and can be omitted if the practical applications of the analysis of rotor modes shifts are of primary interest. Where relevant, references are made in Chapter 13 to the results and equations that are developed in this chapter. A case study in the latter chapter demonstrates the significance of the MITCs and the insights that they provide into the dynamic performance of a multi-machine power system.

In essence, the concept of a modal induced torque coefficient is a further development of the concepts of damping and synchronising torque coefficients based on frequency response analysis (i.e. s = Jωf) [1]. In this chapter the torque coefficients are evaluated at the complex rotor modes (λh = α±jω). The frequency-analysis-based torque coefficients are introduced in Section 5.3 for a SMIB system and in Sections 9.5 and 10.6 for the multi-machine case. With shaft dynamics enabled, the modal induced torque coefficient, Tijh, for the complex rotor mode λh is defined as a complex torque coefficient which is induced on generator i due to a perturbation in the stabilizing signal of stabilizer j. The stabilizer in question may be a power system stabilizer (PSS) or a FACT device stabilizer (FDS) installed on a FACTS device. Such FACTS devices are static var compensators (SVCs), high voltage DC links, thyristor controlled series capacitors (TCSCs) among others. It will be shown that the effect of any such stabilizer on the damping of any mode of rotor oscillation can be quantified. The concept can be extended to other devices such as wind turbine generators, photovoltaics and any other power-converter based transmission or generation equipment.

Why analyse the small-signal dynamic performance of power systems?

We shall be concerned mainly with the analysis of the dynamic performance and control of large, interconnected electric power systems in the following chapters. The differential-algebraic equations which describe the behaviour of a power system are inherently non-linear. Among the non-linearities are functional types (e.g. sinδ), product types (e.g. voltage × current), limits on controller action, saturation in magnetic circuits, etc. The general method of assessing the performance of the system, with all its non-linearities, is through a time-domain simulation which reveals the response of the system to a specific disturbance, e.g. a fault, the loss of a generating unit, line switching. Typically, it may be necessary to conduct many such studies with disturbances applied in various locations in the system to ascertain its stability and dynamic characteristics. Even with many such studies, many of the characteristics of the dynamic behaviour may be missed and insights into system performance lost. In small-signal analysis of dynamic performance of multi-machine systems the stability and characteristics of the system are readily derived from eigenanalysis and other tools. Further-more, in such linear analysis the design of controllers and their integration into the dynamics of plant are facilitated.

Modern linear control system theory contains many powerful techniques, not only for determining the stability and dynamic characteristics of large linear systems, but also for tuning controllers that satisfy steady-state and dynamic performance specifications. Fortunately and importantly, Henri Poincaré [1] showed that if the linearized form of the non-linear system is stable, so is the non-linear system stable at the steady-state operating condition at which the system is linearized. Moreover, the dynamic characteristics of the system at the selected operating condition can be established from linear control system theory and, as long as the perturbations are small, the time-domain responses can be calculated. With such information the design of linear controllers may be undertaken and the resulting controls embedded in the non-linear system. In practice, if the modelling of the devices is adequate, small-signal tests involving generator controls, for example, have revealed close agreement between simulation and test results. Continuously-acting controllers of interest for synchronous generating units are Automatic Voltage Regulators (AVRs), Power System Stabilizers (PSSs) and speed governors.

Public policy measures can do much to accelerate deployment of carbon mitigation measures, and there are basically three options. One approach is to put a price on emissions, which can be done through a cap-and-trade system or an outright tax. The approach is motivated by the following premise: if the goal is to put a large dent in GHG emissions, a price tag must be put on the emissions. A second approach is to mandate reductions through the regulatory process. Forms of government regulation could include corporate average fuel economy (CAFE) standards for automotive transportation, renewable portfolio standards (RPS) for power production, and efficiency standards for buildings and home appliances. The third approach is to provide financial incentives to ease the cost of mitigation. Incentives can be provided as outright grants, tax credits for producing carbon-free energy, or preferential treatment for sale of the energy.

In each of the three approaches, a critical issue, both politically and economically, is cost management. If costs are too high, economic growth is stifled; if they are too low, emission reductions and innovation are stifled.

Cap-and-trade

Lawmakers are generally averse to increasing taxes. Hence, if they believe that a price should be placed on carbon emissions, they're inclined to favor a cap-and-trade system, even though, like a tax, the cost of implementation is ultimately borne by the consumer. In principle, the system works as follows. Governments impose caps (limits) on GHG emissions from large central sources such as power plants, oil refineries, natural gas producers, and manufacturers of energy-intensive products such as concrete, steel, and glass. Initially, the caps are high to allow time for adjustment but are gradually reduced until atmospheric GHG concentrations drop to desired levels. Commensurate with a prescribed cap, permits (allowances) are granted and/or auctioned to the emitting entities, with one permit corresponding to a unit of annual emissions such as 1 t-CO2eq.

Whether auctioned or granted, permits are traded on a market, which can be regional, national, or international. Permits are sold by those able to economically reduce emissions below their cap and bought by those finding such purchases to be a more cost-effective approach to compliance.

Environmental effects can be differentiated in terms of time scales – also called time constants. Pursuant to an input that alters the equilibrium of a system, a time scale provides an approximate measure of how long it takes for the input to achieve a significant portion (e.g., 67%) of its final effect and hence to reach a new equilibrium. The larger the time scale, the larger the inertia of the system.

Consider the infamous fogs that plagued London from the mid-1700s to the mid-1900s. They were not a natural phenomenon, but were anthropogenic and caused by the use of coal for everything from space heating in homes and businesses to, in the twentieth century, generation of electric power. During cold and still winter days, the soot and sulfur-laden gases produced by burning coal would hover over the city, reducing visibility to near zero and inducing serious illness and death among those with respiratory problems. Created by several million sources of coal combustion, the last London fog in 1952 resulted in 4,000 deaths. The problem could no longer be ignored. In 1956, Parliament passed a clean air act prohibiting coal combustion in the homes and businesses of England's largest cities, and the problem, if not eliminated, was significantly alleviated within a few years. This example illustrates an environmental problem for which the causal agents were well understood, the scope was local, there was no uncertainty concerning the nature and seriousness of the consequences, and remedies could be quickly implemented.

The ozone hole in the Earth's stratosphere provides a more recent example, one that's global in scope and for which time scales associated with remediation are much longer. Stratospheric ozone absorbs much of the ultraviolet (UV) components of solar radiation. With depletion of the ozone, more of the UV reaches the Earth's surface, with adverse effects on human health and the environment. The effects could not be ignored, and chemical reactions with the chlorofluorocarbons (CFCs) used in vapor-compression refrigeration and air-conditioning systems were determined to be the culprit.

Recognition of the problem spurred global cooperation in obtaining a solution, highlighted in 1987 by the Montreal Protocol, which committed the world's industrial nations to 50% and 85% reductions in CFC production by 2005 and 2007, respectively.

The atmosphere is our constant companion. It envelops us 24/7, providing life-sustaining oxygen. In school we learned that, excluding water vapor, nitrogen and oxygen comprise about 98% of the atmosphere's dry air, with argon and carbon dioxide providing much of the remainder. In urban and industrial surroundings, we may also sense the intrusion of atmospheric pollutants such as ozone and sulfur dioxide. In addressing global warming, we turn our attention to atmospheric constituents that qualify as greenhouse gases (GHGs).

The atmospheric concentration of some GHGs is determined by both natural and anthropogenic effects. Hence, the gases would still exist if humans did not inhabit the planet. Water vapor (H2O) and carbon dioxide (CO2) are the biggest contributors to the Earth's natural greenhouse effect, with smaller roles played by methane (CH4) and nitrous oxide (N2O). But due to human activities, the atmospheric concentrations of these species are increasing, providing an anthropogenic component to the greenhouse effect. The effect is amplified by the existence of other, strictly anthropogenic GHGs that number in the hundreds (Ramaswamy et al., 2001; Stine and Sturges, 2007).

Distinguishing features

Not all GHGs contribute to global warming in the same way. For one thing, spectral absorption bands corresponding to discrete regions of the electromagnetic spectrum in which the gases absorb terrestrial radiation differ according to their strength and wavelengths. For example, methane molecules absorb terrestrial radiation at different wavelengths and more strongly than carbon dioxide molecules. Also, once released to the atmosphere, gases differ according to the amount of time they remain in the atmosphere.

Distinguishing features of five comparatively long-lived and well-mixed GHGs that account for about 96% of the anthropogenic greenhouse effect are provided in Table 3.1, along with the cumulative effect of fifteen minor, halogenated gases that account for much of the remaining 4%. Containing chlorine, fluorine, and/or bromine, halogens are man-made chemicals used in many industrial processes and commercial products such as refrigerants, aerosols, and foaming agents. Examples include the families of chlorofluorocarbons (CFCs) and hydrochlorofluorocarbons (HCFCs). Halogens containing chlorine or bromine contribute to another environmental problem, namely depletion of stratospheric ozone. While the atmospheric concentrations of CO2, CH4, and N2O are influenced by both natural and anthropogenic effects, the halogens are strictly anthropogenic.

Basic elements of a complete end-to-end CCS system include CO2 capture and compression, transport from the source to a repository, and monitoring the status of the sequestered CO2 over time. Each element contributes to the total cost of CCS. Capture and compression costs increase with decreasing concentration and pressure of CO2 in the gas stream from which it is extracted, while transport costs increase with increasing distance from the source to the storage site. Costs of monitoring the sequestered CO2 are relatively small.

The cost of carbon capture for coal-fired power plants depends on the type of power plant, its efficiency, and whether the carbon is captured by pre- or post-combustion processes. To understand these options, it's helpful to first examine the makeup of a conventional plant. Consider the system shown in Figure E.1.

Pulverized coal (PC) and air enter the boiler, where the coal is burned to produce high-temperature products of combustion (POC). As the POC pass over tubes within the boiler (the steam generator), heat is transferred to pressurized water flowing through the tubes, converting it to steam. The steam is then routed through a set of steam turbines, where work is done on the turbine blades, enabling the turbine shaft to do work on an electric generator. To reduce losses in transmission by the grid, the voltage is increased by a transformer before the electricity enters the grid. The turbine has multiple stages to accommodate reheating in the boiler and to maximize the efficiency with which work is done by the steam. After leaving the last turbine stage, the steam is converted to water in a condenser and passed through a series of feedwater heaters and pumps before returning to the boiler to repeat the process.

Frank Incropera has done a masterful job of making the case for taking action now to reduce the impact of greenhouse gases attributable to human activity. He does it in a way that even the most ardent skeptics will have to acknowledge is a persuasive and balanced case that respects counterarguments but engages them thoroughly and convincingly.

I have had the pleasure of knowing Dr. Incropera for almost twenty years. I first met him when he agreed to leave Purdue University and move north to take the helm as Dean of the College of Engineering at the University of Notre Dame. As a member of the Advisory Council of the College, I watched as Dr. Incropera put in place the building blocks of a great engineering school. And while many of those building blocks – nanotechnology, bioengineering, advanced control systems, supersonic wind tunnels, and the like – were beyond the ken of this nuclear submarine engineer, I was delighted to discover that the dean and I shared a passion for all facets of the production and use of energy for the benefit of society.

Over the years, I and several other members of the Advisory Council were asked to speak to his classes about the practical aspects of the energy system in America and around the world. You see, Dr. Incropera is more than just a superb engineer. As is amply demonstrated in his book, he has a gift for taking the most complex of technical subjects and making them understandable to casual readers. As one who almost flunked quantum mechanics, I can attest that this is a real gift. Beyond the technical, Dr. Incropera is always interested in the whole range of considerations for any complex energy system. Obviously, a sustainable system must be founded on good science. That is just the starting point. Economics, the environment, history, politics, tax policy, and socioeconomic issues all have roles to play in understanding why we have the energy mix we have today. More importantly, he believes that unless you clearly understand the role each of these dimensions plays, it will be impossible to craft a solution to this “wicked problem.”

“Climate change is occurring, is very likely caused primarily by human activities, and poses significant risks to humans and the environment.” This statement was made in a report prepared by the National Research Council (NRC, 2011), which is an agent of the U.S. National Academies of Science and Engineering and the Institute of Medicine. The view is shared by governmental science agencies across the world, as well as by an overwhelming majority of the global scientific community. We know that atmospheric GHG concentrations are rising, that the gases contribute to warming, and that warming is enhanced by feedback effects. We also know that atmosphere and ocean temperatures are rising, along with sea levels, as Arctic sea ice and glaciers continue to melt. The trends are due to the use of fossil fuels, and we know they will continue, even if atmospheric GHG concentrations were to remain at today's levels. And we are aware of the consequences, such as the impact of rising sea levels on low-lying coastal regions and the increased frequency of extreme weather events.

Yes, there are gaps in the scientific knowledge base. They've been addressed in previous chapters and are acknowledged in the NRC report. But the science underpinning a human impact on climate change is strong and getting stronger with each new study. This part of the debate is all but over. Most uncertainties are associated with partitioning of energy between the Earth's atmosphere and oceans and with understanding the consequences of climate change. Where and to what extent will the frequency and intensity of major storms increase; where will chronic drought and desertification be most pronounced; where and to what extent will ecosystems, biodiversity and agricultural production be adversely affected; and what will be the scope of attendant diasporas? It is not acceptable to focus on these uncertainties and to simply call for more research without addressing root causes of global warming and preparing to deal with the adverse effects of climate change. Because the potential consequences of global warming are profound, the prudent and morally responsible approach is to take action now. The NRC report states unequivocally that “uncertainty is not a reason for inaction.”

Consider the atmosphere from any point on the Earth's surface. Move east, west, north, or south; the atmosphere has no boundaries. If a CO2 molecule is discharged from a point source in the United States, it can move in any direction, and one year hence it could be over India, Brazil, or anywhere else in the world. Likewise, GHG emissions in India and other nations make their way to the United States and to every other nation. All nations contribute to GHG emissions, some more than others, and all nations are recipients of the emissions. Geographically, the atmosphere is nondiscriminatory.

To the extent that GHG emissions contribute to global warming and, in turn, warming contributes to climate change, inherent problems are not local, regional, or national. They are transnational. The problems cannot be solved by any one nation – although some can contribute more than others – but must be addressed by collaboration among all nations. Even the smallest of nations that contribute little to GHG emissions must be involved, for they may well be among those most adversely impacted by climate change and least able to adapt.

Whether the problem is perceived as potential or omnipresent, the fact that it is international is indisputable. And the international community has been paying attention. In this chapter, we will examine efforts to address the problem and the mine fields that have impeded substantive progress. As applied to climate change, the chapter is a history lesson in geopolitics, highlighting how difficult it is to achieve consensus among nations of widely disparate cultures and economic conditions.

The Intergovernmental Panel on Climate Change

International agreements have long been used to address environmental issues for segments of the planet as diverse as the atmosphere, the open seas, and Antarctica. A relatively recent agreement is the Montreal Protocol of 1987, which calls for protection of the stratosphere's ozone layer through elimination of industrial chemicals termed chlorofluorocarbons (CFCs) and hydrochlorofluorocarbons (HCFCs).

Although interest in global warming can be traced to the late nineteenth century (Weart, 2003), it was not until 1988 that the issue began to receive serious attention.

The foregoing results provide strong evidence that global warming is real and attributable in no small measure to human activities, a conclusion strengthened by other indicators that are updated annually (Blunden and Arndt, 2014; EPA, 2014b). But why should we be concerned? What is the impact of warming? How will it affect the natural environment? The built environment? Food production? Human health and security? And, although we've been careful to differentiate between climate and weather, could warming affect the frequency and intensity of extreme weather events?

The ramifications of global climate change (GCC) and their likelihood have been discussed extensively (IPCC, 2007d, 2014a; Cullen, 2010). Cullen extrapolates past and current warming trends into a prediction of circa-2050 climate. Droughts are projected to be more severe in some regions and floods in others. Depletion of ground water and food sources is predicted, as is deterioration of infrastructure in low-lying regions vulnerable to rising sea levels and storm surges of growing frequency and intensity. And the world would also have to contend with growing migrations of climate refugees.

The potential effects on ecological and socioeconomic systems are numerous and may be positive or negative, with significant differences from one region of the globe to another. Let's begin with the region in which GCC and its ramifications are most evident.

The Arctic: canary in a mine shaft

The Arctic occupies the region north of the Arctic Circle and includes northern portions of eight nations – Canada, Denmark (Greenland), Finland, Iceland, Norway, Russia, Sweden, and the United States (Alaska) – as well as the Arctic Ocean. Because warming is occurring much faster in the Arctic than in any other region of the world, it has become the proverbial canary in the mine shaft, a testimonial to the reality of global warming.

Since the 1950s, Arctic temperatures have increased at a rate more than twice the global average. Using Alaska as an example, its temperature increased by about 2°C, more than twice that of the lower forty-eight states. Were the trend to continue, an estimate of 2°C for the twenty-first-century rise in the global average temperature would translate to a 4°C increase for the Arctic.

By now, you would probably agree that the issue of climate change is wrapped in science, technology, economics, and politics. But is there yet another dimension? Many would say “yes,” maintaining the existence of a moral imperative. To address climate change from such a perspective, we'll examine some of the philosophical pillars of ethics, as well as foundations of ethical behavior derived from religious traditions. From both philosophical and theological perspectives, how do ethical considerations inform the debate on climate change?

Ethics involves reflection on human behavior and how to channel it in appropriate ways. A central question involves life and how it should be lived. Another involves the nature of good and standards by which an action is judged good or not. Such questions have been addressed by philosophers for millennia in attempts to delineate between right and wrong. But in applying these standards, difficulties often arise because many issues are multifaceted, complex, and nuanced.

Ethical dimensions of climate change

Technology has allowed humans to conquer space and time. Modern transportation systems provide movement of goods and people from one location to any other; modern communication systems enable ideas and knowledge to flow almost instantaneously across the world. Globalization has had an enormous impact on raising living standards throughout the world. But underpinning it all has been rising energy consumption, particularly from fossil fuels, and attendant environmental degradation. Some environmental damage is local, such as mining coal by mountaintop removal, or regional, such as acid rain. And most degradation is manifested over relatively short time scales, from immediate to months or years. But climate change is global and manifested over decades to millennia. Today's GHG emissions have consequences for all, anywhere on Earth, and for the unborn as well as the living. When I burn one gallon of gasoline, I discharge almost 9 kg-CO2 to the atmosphere, putting the entire planet at greater risk to the effects of climate change. When the U.S. transportation sector consumes 215 billion gallons of fuel, as it did in 2012 (Davis et al., 2014), it adds about 2 Gt-CO2 to the atmosphere, with Americans enjoying the benefits of consumption while calling upon the world to share the burdens.

We can't engage in a serious examination of climate change without considering its strong ties to energy. More than any other factor, anthropogenic contributions to climate depend on how energy is produced and used.

Over his illustrious career, Richard Smalley (1943–2005), a Nobel Laureate and pioneer in the field of nanoscience and technology, was invited to give many lectures on his work. However, in the last few years of his life, he felt compelled to use the lectures as a vehicle for sharing his concerns about the world's energy future. In one of his slides he presented his views on humanity's top ten problems of the next fifty years. His list included food, water, the environment, poverty, war, disease, education, democracy, and population. While we might attach different weights to the significance of each concern, we would probably agree that all are to be taken seriously. However, for Smalley, there was no equivocation on what belonged at the top of the list. Meeting the world's energy needs was paramount and linked, to varying degree, with the other nine.

Energy: an indispensable resource

It would be difficult to overstate the importance of energy to the well-being of humankind. It is the resource that sustains all life and economic activity. It enables the production and distribution of all manner of goods and services, as well as human mobility on the ground and in the air. It is absolutely essential to achieving an acceptable standard of living, and in the words of Paul Roberts (2004, p. 6), “Access to energy has emerged as the overwhelming imperative of the twenty-first century.”

While preindustrial societies functioned entirely on energy derived from the Sun, the Industrial Revolution marked a transition to the use of fossil fuels and by the mid-twentieth century to nuclear energy. It is difficult to appreciate the enormity of today's global energy supply chain. In 2013, humankind consumed approximately 505 quadrillion (505,000 trillion) British thermal units (Btu) of energy, or simply 505 quads (BP, 2014a). The amount is staggering, and trillions of dollars are spent annually to produce and distribute this energy.