Design and Optimization of Power Delivery and Distribution Systems Using Evolutionary Computation Techniques

Nowadays computing platforms consist of a very large number of components that require to be supplied with diferent voltage levels and power requirements. Even a very small platform, like a handheld computer, may contain more than twenty diferent loads and voltage regulators. The power delivery designers of these systems are required to provide, in a very short time, the right power architecture that optimizes the performance, meets electrical specifications plus cost and size targets. The appropriate selection of the architecture and converters directly defines the performance of a given solution. Therefore, the designer needs to be able to evaluate a significant number of options in order to know with good certainty whether the selected solutions meet the size, energy eficiency and cost targets. The design dificulties of selecting the right solution arise due to the wide range of power conversion products provided by diferent manufacturers. These products range from discrete components (to build converters) to complete power conversion modules that employ diferent manufacturing technologies. Consequently, in most cases it is not possible to analyze all the alternatives (combinations of power architectures and converters) that can be built. The designer has to select a limited number of converters in order to simplify the analysis. In this thesis, in order to overcome the mentioned dificulties, a new design methodology for power supply systems is proposed. This methodology integrates evolutionary computation techniques in order to make possible analyzing a large number of possibilities. This exhaustive analysis helps the designer to quickly define a set of feasible solutions and select the best trade-off in performance according to each application. The proposed approach consists of two key steps, one for the automatic generation of architectures and other for the optimized selection of components. In this thesis are detailed the implementation of these two steps. The usefulness of the methodology is corroborated by contrasting the results using real problems and experiments designed to test the limits of the algorithms.

Pressure management in water distribution systems: current status, proposals, and future trends

Pressure management (PM) is commonly used in water distribution systems (WDSs). In the last decade, a strategic objective in the field has been the development of new scientific and technical methods for its implementation. However, due to a lack of systematic analysis of the results obtained in practical cases, progress has not always been reflected in practical actions. To address this problem, this paper provides a comprehensive analysis of the most innovative issues related to PM. The methodology proposed is based on a case-study comparison of qualitative concepts that involves published work from 140 sources. The results include a qualitative analysis covering four aspects: (1) the objectives yielded by PM; (2) types of regulation, including advanced control systems through electronic controllers; (3) new methods for designing districts; and (4) development of optimization models associated with PM. The evolution of the aforementioned four aspects is examined and discussed. Conclusions regarding the current status of each factor are drawn and proposals for future research outlined

Co-simulation Methodologies for Hybrid and Electric Vehicle Dynamics

In recent decades, full electric and hybrid electric vehicles have emerged as an alternative to conventional cars due to a range of factors, including environmental and economic aspects. These vehicles are the result of considerable efforts to seek ways of reducing the use of fossil fuel for vehicle propulsion. Sophisticated technologies such as hybrid and electric powertrains require careful study and optimization. Mathematical models play a key role at this point. Currently, many advanced mathematical analysis tools, as well as computer applications have been built for vehicle simulation purposes. Given the great interest of hybrid and electric powertrains, along with the increasing importance of reliable computer-based models, the author decided to integrate both aspects in the research purpose of this work. Furthermore, this is one of the first final degree projects held at the ETSII (Higher Technical School of Industrial Engineers) that covers the study of hybrid and electric propulsion systems. The present project is based on MBS3D 2.0, a specialized software for the dynamic simulation of multibody systems developed at the UPM Institute of Automobile Research (INSIA). Automobiles are a clear example of complex multibody systems, which are present in nearly every field of engineering. The work presented here benefits from the availability of MBS3D software. This program has proven to be a very efficient tool, with a highly developed underlying mathematical formulation. On this basis, the focus of this project is the extension of MBS3D features in order to be able to perform dynamic simulations of hybrid and electric vehicle models. This requires the joint simulation of the mechanical model of the vehicle, together with the model of the hybrid or electric powertrain. These sub-models belong to completely different physical domains. In fact the powertrain consists of energy storage systems, electrical machines and power electronics, connected to purely mechanical components (wheels, suspension, transmission, clutch…). The challenge today is to create a global vehicle model that is valid for computer simulation. Therefore, the main goal of this project is to apply co-simulation methodologies to a comprehensive model of an electric vehicle, where sub-models from different areas of engineering are coupled. The created electric vehicle (EV) model consists of a separately excited DC electric motor, a Li-ion battery pack, a DC/DC chopper converter and a multibody vehicle model. Co-simulation techniques allow car designers to simulate complex vehicle architectures and behaviors, which are usually difficult to implement in a real environment due to safety and/or economic reasons. In addition, multi-domain computational models help to detect the effects of different driving patterns and parameters and improve the models in a fast and effective way. Automotive designers can greatly benefit from a multidisciplinary approach of new hybrid and electric vehicles. In this case, the global electric vehicle model includes an electrical subsystem and a mechanical subsystem. The electrical subsystem consists of three basic components: electric motor, battery pack and power converter. A modular representation is used for building the dynamic model of the vehicle drivetrain. This means that every component of the drivetrain (submodule) is modeled separately and has its own general dynamic model, with clearly defined inputs and outputs. Then, all the particular submodules are assembled according to the drivetrain configuration and, in this way, the power flow across the components is completely determined. Dynamic models of electrical components are often based on equivalent circuits, where Kirchhoff’s voltage and current laws are applied to draw the algebraic and differential equations. Here, Randles circuit is used for dynamic modeling of the battery and the electric motor is modeled through the analysis of the equivalent circuit of a separately excited DC motor, where the power converter is included. The mechanical subsystem is defined by MBS3D equations. These equations consider the position, velocity and acceleration of all the bodies comprising the vehicle multibody system. MBS3D 2.0 is entirely written in MATLAB and the structure of the program has been thoroughly studied and understood by the author. MBS3D software is adapted according to the requirements of the applied co-simulation method. Some of the core functions are modified, such as integrator and graphics, and several auxiliary functions are added in order to compute the mathematical model of the electrical components. By coupling and co-simulating both subsystems, it is possible to evaluate the dynamic interaction among all the components of the drivetrain. ‘Tight-coupling’ method is used to cosimulate the sub-models. This approach integrates all subsystems simultaneously and the results of the integration are exchanged by function-call. This means that the integration is done jointly for the mechanical and the electrical subsystem, under a single integrator and then, the speed of integration is determined by the slower subsystem. Simulations are then used to show the performance of the developed EV model. However, this project focuses more on the validation of the computational and mathematical tool for electric and hybrid vehicle simulation. For this purpose, a detailed study and comparison of different integrators within the MATLAB environment is done. Consequently, the main efforts are directed towards the implementation of co-simulation techniques in MBS3D software. In this regard, it is not intended to create an extremely precise EV model in terms of real vehicle performance, although an acceptable level of accuracy is achieved. The gap between the EV model and the real system is filled, in a way, by introducing the gas and brake pedals input, which reflects the actual driver behavior. This input is included directly in the differential equations of the model, and determines the amount of current provided to the electric motor. For a separately excited DC motor, the rotor current is proportional to the traction torque delivered to the car wheels. Therefore, as it occurs in the case of real vehicle models, the propulsion torque in the mathematical model is controlled through acceleration and brake pedal commands. The designed transmission system also includes a reduction gear that adapts the torque coming for the motor drive and transfers it. The main contribution of this project is, therefore, the implementation of a new calculation path for the wheel torques, based on performance characteristics and outputs of the electric powertrain model. Originally, the wheel traction and braking torques were input to MBS3D through a vector directly computed by the user in a MATLAB script. Now, they are calculated as a function of the motor current which, in turn, depends on the current provided by the battery pack across the DC/DC chopper converter. The motor and battery currents and voltages are the solutions of the electrical ODE (Ordinary Differential Equation) system coupled to the multibody system. Simultaneously, the outputs of MBS3D model are the position, velocity and acceleration of the vehicle at all times. The motor shaft speed is computed from the output vehicle speed considering the wheel radius, the gear reduction ratio and the transmission efficiency. This motor shaft speed, somehow available from MBS3D model, is then introduced in the differential equations corresponding to the electrical subsystem. In this way, MBS3D and the electrical powertrain model are interconnected and both subsystems exchange values resulting as expected with tight-coupling approach.When programming mathematical models of complex systems, code optimization is a key step in the process. A way to improve the overall performance of the integration, making use of C/C++ as an alternative programming language, is described and implemented. Although this entails a higher computational burden, it leads to important advantages regarding cosimulation speed and stability. In order to do this, it is necessary to integrate MATLAB with another integrated development environment (IDE), where C/C++ code can be generated and executed. In this project, C/C++ files are programmed in Microsoft Visual Studio and the interface between both IDEs is created by building C/C++ MEX file functions. These programs contain functions or subroutines that can be dynamically linked and executed from MATLAB. This process achieves reductions in simulation time up to two orders of magnitude. The tests performed with different integrators, also reveal the stiff character of the differential equations corresponding to the electrical subsystem, and allow the improvement of the cosimulation process. When varying the parameters of the integration and/or the initial conditions of the problem, the solutions of the system of equations show better dynamic response and stability, depending on the integrator used. Several integrators, with variable and non-variable step-size, and for stiff and non-stiff problems are applied to the coupled ODE system. Then, the results are analyzed, compared and discussed. From all the above, the project can be divided into four main parts: 1. Creation of the equation-based electric vehicle model; 2. Programming, simulation and adjustment of the electric vehicle model; 3. Application of co-simulation methodologies to MBS3D and the electric powertrain subsystem; and 4. Code optimization and study of different integrators. Additionally, in order to deeply understand the context of the project, the first chapters include an introduction to basic vehicle dynamics, current classification of hybrid and electric vehicles and an explanation of the involved technologies such as brake energy regeneration, electric and non-electric propulsion systems for EVs and HEVs (hybrid electric vehicles) and their control strategies. Later, the problem of dynamic modeling of hybrid and electric vehicles is discussed. The integrated development environment and the simulation tool are also briefly described. The core chapters include an explanation of the major co-simulation methodologies and how they have been programmed and applied to the electric powertrain model together with the multibody system dynamic model. Finally, the last chapters summarize the main results and conclusions of the project and propose further research topics. In conclusion, co-simulation methodologies are applicable within the integrated development environments MATLAB and Visual Studio, and the simulation tool MBS3D 2.0, where equation-based models of multidisciplinary subsystems, consisting of mechanical and electrical components, are coupled and integrated in a very efficient way.

Signal analysis and feature generation for pattern identification of partial discharges in high-voltage equipment

This paper proposes a method for the identification of different partial discharges (PDs) sources through the analysis of a collection of PD signals acquired with a PD measurement system. This method, robust and sensitive enough to cope with noisy data and external interferences, combines the characterization of each signal from the collection, with a clustering procedure, the CLARA algorithm. Several features are proposed for the characterization of the signals, being the wavelet variances, the frequency estimated with the Prony method, and the energy, the most relevant for the performance of the clustering procedure. The result of the unsupervised classification is a set of clusters each containing those signals which are more similar to each other than to those in other clusters. The analysis of the classification results permits both the identification of different PD sources and the discrimination between original PD signals, reflections, noise and external interferences. The methods and graphical tools detailed in this paper have been coded and published as a contributed package of the R environment under a GNU/GPL license.

El papel de la hidroelectricidad en el contexto de las energías renovables en España

En el artículo se discute el papel de la energía hidroeléctrica en el marco del sistema eléctrico español, donde existe una elevada penetración de energías no gestionables con una tendencia clara a aumentar en los próximos años. El desarrollo de nuevas centrales hidroeléctricas se basará probablemente en centrales reversibles. La energía hidroeléctrica es una tecnología madura y eficiente para el almacenamiento de energía a gran escala y contribuye por tanto de manera decisiva a la integración de fuentes renovables no gestionables. Los beneficios obtenidos con la operación punta-valle pueden ser insuficientes para compensar el coste de una nueva central. Sin embargo, los ingresos obtenidos pueden incrementarse sustancialmente mediante su participación en los servicios de ajuste del sistema. Ello requeriría un diseño apropiado del mercado eléctrico. La contribución de las centrales hidráulicas reversibles al balance producción-consumo puede extenderse a las horas valle utilizando, bien bombeo en velocidad variable o bien una configuración de cortocircuito hidráulico. La necesidad de mitigar los efectos hidrológicos aguas abajo de las centrales hidroeléctricas puede introducir algunas restricciones en la operación que limitaría de algún modo los servicios descritos más arriba. Sin embargo, cabe esperar que los efectos ambientales provocados por las centrales hidráulicas reversibles sean significativamente menores. In this paper the role of hydropower in electric power systems is discussed, in the framework of the Spanish system, where a high penetration of intermittent power sources exists, showing a clear trend to increase in next years. The development of new hydro power facilities will be likely based on pumped storage hydro power plants. Hydropower is a mature and efficient technology for large-scale energy storage and therefore represents a key contribution for the integration of intermittent power sources, such as wind or photovoltaic. The benefits obtained from load shifting may be insufficient to compensate the costs of a new plant. However, the obtained revenues can significantly increase through its contribution to providing ancillary services. This would require an appropriate design of the electricity market. The contribution of pumped storage hydro power plants to balancing services can be extended to off-peak hours, using either variable speed pumping or the hydraulic shortcircuit configuration. The need to mitigate hydrological effects downstream of hydro plants may introduce some operational constraints which could limit to some extent the services described above. However environmental effects caused by pumped storage hydro power plants are expected to be significantly smaller.

Validation of a novel approach to subjective quality evaluation of conventional and 3D broadcasted video services

Transmission errors are the main cause of degradation of the quality of real broadcasted video services. Therefore, knowing their impact on the quality of experience of the end users is a crucial issue. For instance, it would help to improve the performance of the distribution systems, and to develop monitoring tools to automatically estimate the quality perceived by the end users. In this paper we validate a subjective evaluation approach specifically designed to obtain meaningful results of the effects of degradations caused by transmission errors. This methodology has been already used in our previous works with monoscopic and stereoscopic videos. The validation is done by comparing the subjective ratings obtained for typical transmission errors with the proposed methodology and with the standard method Absolute Category Rating. The results show that the proposed approach could provide more representative evaluations of the quality of experience perceived by end users of conventional and 3D broadcasted video services.

Signal analysis and feature generation for pattern identification of partial discharges in high-voltage equipment

This paper proposes a method for the identification of different partial discharges (PDs) sources through the analysis of a collection of PD signals acquired with a PD measurement system. This method, robust and sensitive enough to cope with noisy data and external interferences, combines the characterization of each signal from the collection, with a clustering procedure, the CLARA algorithm. Several features are proposed for the characterization of the signals, being the wavelet variances, the frequency estimated with the Prony method, and the energy, the most relevant for the performance of the clustering procedure. The result of the unsupervised classification is a set of clusters each containing those signals which are more similar to each other than to those in other clusters. The analysis of the classification results permits both the identification of different PD sources and the discrimination between original PD signals, reflections, noise and external interferences. The methods and graphical tools detailed in this paper have been coded and published as a contributed package of the R environment under a GNU/GPL license.

Tratamientos avanzados de agua potable para eliminación de materia orgánica disuelta : aplicación del BAC

Los procesos de biofiltración por carbón activo biológico se han utilizado desde hace décadas, primeramente en Europa y después en Norte América, sin embargo no hay parámetros de diseño y operación específicos que se puedan utilizar de guía para la biofiltración. Además, el factor coste a la hora de elegir el carbón activo como medio de filtración impacta en el presupuesto, debido al elevado coste de inversión y de regeneración. A la hora de diseñar y operar filtros de carbón activo los requisitos que comúnmente se buscan son eliminar materia orgánica, olor, y sabor de agua. Dentro de la eliminación de materia orgánica se precisa la eliminación necesaria para evitar subproductos en la desinfección no deseados, y reducir los niveles de carbono orgánico disuelto biodegradable y asimilable a valores que consigan la bioestabilidad del agua producto, a fin de evitar recrecimiento de biofilm en las redes de distribución. El ozono se ha utilizado durante años como un oxidante previo a la biofiltración para reducir el olor, sabor, y color del agua, oxidando la materia orgánica convirtiendo los compuestos no biodegradables y lentamente biodegradables en biodegradables, consiguiendo que puedan ser posteriormente eliminados biológicamente en los filtros de carbón activo. Sin embargo la inestabilidad del ozono en el agua hace que se produzcan ácidos carboxilos, alcoholes y aldehídos, conocidos como subproductos de la desinfección. Con esta tesis se pretende dar respuesta principalmente a los siguientes objetivos: análisis de parámetros requeridos para el diseño de los filtros de carbón activo biológicos, necesidades de ozonización previa a la filtración, y comportamiento de la biofiltración en un sistema compuesto de coagulación sobre un filtro de carbón activo biológico. Los resultados obtenidos muestran que la biofiltración es un proceso que encaja perfectamente con los parámetros de diseño de plantas con filtración convencional. Aunque la capacidad de eliminación de materia orgánica se reduce a medida que el filtro se satura y entra en la fase biológica, la biodegradación en esta fase se mantienen estable y perdura a lo lago de los meses sin preocupaciones por la regeneración del carbón. Los valores de carbono orgánico disuelto biodegradable se mantienen por debajo de los marcados en la literatura existente para agua bioestable, lo que hace innecesaria la dosificación de ozono previa a la biofiltración. La adición de la coagulación con la corrección de pH sobre el carbón activo consigue una mejora en la reducción de la materia orgánica, sin afectar a la biodegradación del carbón activo, cumpliendo también con los requerimientos de turbidez a la salida de filtración. Lo que plantea importantes ventajas para el proceso. Granular activated carbon filters have been used for many years to treat and produce drinking water using the adsorption capacity of carbon, replacing it once the carbon lost its adsorption capacity and became saturated. On the other hand, biological activated carbon filters have been studied for decades, firstly in Europe and subsequently in North America, nevertheless are no generally accepted design and operational parameters documented to be used as design guidance for biofiltration. Perhaps this is because of the cost factor; to choose activated carbon as a filtration media requires a significant investment due to the high capital and regeneration costs. When activated carbon filters are typically required it is for the reduction of an organic load, removal of colour, taste and / or odour. In terms of organic matter reduction, the primary aim is to achieve as much removal as possible to reduce or avoid the introduction of disinfection by products, the required removal in biodegradable dissolved organic carbon and assimilable organic carbon to produce a biologically stable potable water which prohibits the regrowth of biofilm in the distribution systems. The ozone has historically been used as an oxidant to reduce colour, taste and odour by oxidizing the organic matter and increasing the biodegradability of the organic matter, enhancing the effectiveness of organic removal in downstream biological activated carbon filters. Unfortunately, ozone is unstable in water and reacts with organic matter producing carboxylic acids, alcohols, and aldehydes, known as disinfection by products. This thesis has the following objectives: determination of the required parameters for the design of the biological activated filters, the requirement of ozonization as a pre-treatment for the biological activated filters, and a performance assessment of biofiltration when coagulation is applied as a pretreatment for biological activated carbon filters. The results show that the process design parameters of biofiltration are compatible with those of conventional filtration. The organic matter removal reduces its effectiveness as soon as the filter is saturated and the biological stage starts, but the biodegradation continues steadily and lasts for a long period of time without the need of carbon regeneration. The removal of the biodegradable dissolved organic carbon is enough to produce a biostable water according to the values shown on the existing literature; therefore ozone is not required prior to the filtration. Furthermore, the addition of coagulant and pH control before the biological activated carbon filter achieves a additional removal of organic matter, without affecting the biodegradation that occurs in the activated carbon whilst also complying with the required turbidity removal.

Response threshold models, stochastic learning automata and ant colony optimization-based decentralized self-coordination algorithms for heterogeneous multi-tasks distribution in multi-robot systems

In recent decades, there has been an increasing interest in systems comprised of several autonomous mobile robots, and as a result, there has been a substantial amount of development in the eld of Articial Intelligence, especially in Robotics. There are several studies in the literature by some researchers from the scientic community that focus on the creation of intelligent machines and devices capable to imitate the functions and movements of living beings. Multi-Robot Systems (MRS) can often deal with tasks that are dicult, if not impossible, to be accomplished by a single robot. In the context of MRS, one of the main challenges is the need to control, coordinate and synchronize the operation of multiple robots to perform a specic task. This requires the development of new strategies and methods which allow us to obtain the desired system behavior in a formal and concise way. This PhD thesis aims to study the coordination of multi-robot systems, in particular, addresses the problem of the distribution of heterogeneous multi-tasks. The main interest in these systems is to understand how from simple rules inspired by the division of labor in social insects, a group of robots can perform tasks in an organized and coordinated way. We are mainly interested on truly distributed or decentralized solutions in which the robots themselves, autonomously and in an individual manner, select a particular task so that all tasks are optimally distributed. In general, to perform the multi-tasks distribution among a team of robots, they have to synchronize their actions and exchange information. Under this approach we can speak of multi-tasks selection instead of multi-tasks assignment, which means, that the agents or robots select the tasks instead of being assigned a task by a central controller. The key element in these algorithms is the estimation ix of the stimuli and the adaptive update of the thresholds. This means that each robot performs this estimate locally depending on the load or the number of pending tasks to be performed. In addition, it is very interesting the evaluation of the results in function in each approach, comparing the results obtained by the introducing noise in the number of pending loads, with the purpose of simulate the robot's error in estimating the real number of pending tasks. The main contribution of this thesis can be found in the approach based on self-organization and division of labor in social insects. An experimental scenario for the coordination problem among multiple robots, the robustness of the approaches and the generation of dynamic tasks have been presented and discussed. The particular issues studied are: Threshold models: It presents the experiments conducted to test the response threshold model with the objective to analyze the system performance index, for the problem of the distribution of heterogeneous multitasks in multi-robot systems; also has been introduced additive noise in the number of pending loads and has been generated dynamic tasks over time. Learning automata methods: It describes the experiments to test the learning automata-based probabilistic algorithms. The approach was tested to evaluate the system performance index with additive noise and with dynamic tasks generation for the same problem of the distribution of heterogeneous multi-tasks in multi-robot systems. Ant colony optimization: The goal of the experiments presented is to test the ant colony optimization-based deterministic algorithms, to achieve the distribution of heterogeneous multi-tasks in multi-robot systems. In the experiments performed, the system performance index is evaluated by introducing additive noise and dynamic tasks generation over time.

Application of self-organizing techniques for the distribution of heterogeneous multi-tasks in multi-robot systems

This paper focuses on the general problem of coordinating of multi-robot systems, more specifically, it addresses the self-election of heterogeneous and specialized tasks by autonomous robots. In this regard, it has proposed experimenting with two different techniques based chiefly on selforganization and emergence biologically inspired, by applying response threshold models as well as ant colony optimization. Under this approach it can speak of multi-tasks selection instead of multi-tasks allocation, that means, as the agents or robots select the tasks instead of being assigned a task by a central controller. The key element in these algorithms is the estimation of the stimuli and the adaptive update of the thresholds. This means that each robot performs this estimate locally depending on the load or the number of pending tasks to be performed. It has evaluated the robustness of the algorithms, perturbing the number of pending loads to simulate the robot’s error in estimating the real number of pending tasks and also the dynamic generation of loads through time. The paper ends with a critical discussion of experimental results.

Response threshold models and stochastic learning automata for self-coordination of heterogeneous multi-task distribution in multi-robot systems.

This paper focuses on the general problem of coordinating multiple robots. More specifically, it addresses the self-selection of heterogeneous specialized tasks by autonomous robots. In this paper we focus on a specifically distributed or decentralized approach as we are particularly interested in a decentralized solution where the robots themselves autonomously and in an individual manner, are responsible for selecting a particular task so that all the existing tasks are optimally distributed and executed. In this regard, we have established an experimental scenario to solve the corresponding multi-task distribution problem and we propose a solution using two different approaches by applying Response Threshold Models as well as Learning Automata-based probabilistic algorithms. We have evaluated the robustness of the algorithms, perturbing the number of pending loads to simulate the robot’s error in estimating the real number of pending tasks and also the dynamic generation of loads through time. The paper ends with a critical discussion of experimental results.

Energy Storage Systems for Electric Vehicles: Performance Comparison based on a Simple Equivalent Circuit and Experimental Tests

The decision to select the most suitable type of energy storage system for an electric vehicle is always difficult, since many conditionings must be taken into account. Sometimes, this study can be made by means of complex mathematical models which represent the behavior of a battery, ultracapacitor or some other devices. However, these models are usually too dependent on parameters that are not easily available, which usually results in nonrealistic results. Besides, the more accurate the model, the more specific it needs to be, which becomes an issue when comparing systems of different nature. This paper proposes a practical methodology to compare different energy storage technologies. This is done by means of a linear approach of an equivalent circuit based on laboratory tests. Via these tests, the internal resistance and the self-discharge rate are evaluated, making it possible to compare different energy storage systems regardless their technology. Rather simple testing equipment is sufficient to give a comparative idea of the differences between each system, concerning issues such as efficiency, heating and self-discharge, when operating under a certain scenario. The proposed methodology is applied to four energy storage systems of different nature for the sake of illustration.