Distributed Control Nodes for Material Flow System Controls on the Example of Unit Load Conveyor and Sorter Facilities

The Chair of Transportation and Ware-housing at the University of Dortmund together with its industrial partner has developed and implemented a decentralized control system based on embedded technology and Internet standards. This innovative, highly flexible system uses autonomous software modules to control the flow of unit loads in real-time. The system is integrated into Chair’s test facility consisting of a wide range of conveying and sorting equipment. It is built for proof of concept purposes and will be used for further research in the fields of decentralized automation and embedded controls. This presentation describes the implementation of this decentralized control system.

Distributed control strategies for photovoltaic systems in urban environments

Este trabajo es una contribución a los sistemas fotovoltaicos (FV) con seguimiento distribuido del punto de máxima potencia (DMPPT), una topología que se caracteriza porque lleva a cabo el MPPT a nivel de módulo, al contrario de las topologías más tradicionales que llevan a cabo el MPPT para un número más elevado de módulos, pudiendo ser hasta cientos de módulos. Las dos tecnologías DMPPT que existen en el mercado son conocidos como microinversores y optimizadores de potencia, y ofrecen ciertas ventajas sobre sistemas de MPPT central como: mayor producción en situaciones de mismatch, monitorización individual de cada módulo, flexibilidad de diseño, mayor seguridad del sistema, etc. Aunque los sistemas DMPPT no están limitados a los entornos urbanos, se ha enfatizado en el título ya que es su mercado natural, siendo difícil una justificación de su sobrecoste en grandes huertas solares en suelo. Desde el año 2010 el mercado de estos sistemas ha incrementado notablemente y sigue creciendo de una forma continuada. Sin embargo, todavía falta un conocimiento profundo de cómo funcionan estos sistemas, especialmente en el caso de los optimizadores de potencia, de las ganancias energéticas esperables en condiciones de mismatch y de las posibilidades avanzadas de diagnóstico de fallos. El principal objetivo de esta tesis es presentar un estudio completo de cómo funcionan los sistemas DMPPT, sus límites y sus ventajas, así como experimentos varios que verifican la teoría y el desarrollo de herramientas para valorar las ventajas de utilizar DMPPT en cada instalación. Las ecuaciones que modelan el funcionamiento de los sistemas FVs con optimizadores de potencia se han desarrollado y utilizado para resaltar los límites de los mismos a la hora de resolver ciertas situaciones de mismatch. Se presenta un estudio profundo sobre el efecto de las sombras en los sistemas FVs: en la curva I-V y en los algoritmos MPPT. Se han llevado a cabo experimentos sobre el funcionamiento de los algoritmos MPPT en situaciones de sombreado, señalando su ineficiencia en estas situaciones. Un análisis de la ventaja del uso de DMPPT frente a los puntos calientes es presentado y verificado. También se presenta un análisis sobre las posibles ganancias en potencia y energía con el uso de DMPPT en condiciones de sombreado y este también es verificado experimentalmente, así como un breve estudio de su viabilidad económica. Para ayudar a llevar a cabo todos los análisis y experimentos descritos previamente se han desarrollado una serie de herramientas software. Una siendo un programa en LabView para controlar un simulador solar y almacenar las medidas. También se ha desarrollado un programa que simula curvas I-V de módulos y generador FVs afectados por sombras y este se ha verificado experimentalmente. Este mismo programa se ha utilizado para desarrollar un programa todavía más completo que estima las pérdidas anuales y las ganancias obtenidas con DMPPT en instalaciones FVs afectadas por sombras. Finalmente, se han desarrollado y verificado unos algoritmos para diagnosticar fallos en sistemas FVs con DMPPT. Esta herramienta puede diagnosticar los siguientes fallos: sombras debido a objetos fijos (con estimación de la distancia al objeto), suciedad localizada, suciedad general, posible punto caliente, degradación de módulos y pérdidas en el cableado de DC. Además, alerta al usuario de las pérdidas producidas por cada fallo y no requiere del uso de sensores de irradiancia y temperatura. ABSTRACT This work is a contribution to photovoltaic (PV) systems with distributed maximum power point tracking (DMPPT), a system topology characterized by performing the MPPT at module level, instead of the more traditional topologies which perform MPPT for a larger number of modules. The two DMPPT technologies available at the moment are known as microinverters and power optimizers, also known as module level power electronics (MLPE), and they provide certain advantages over central MPPT systems like: higher energy production in mismatch situations, monitoring of each individual module, system design flexibility, higher system safety, etc. Although DMPPT is not limited to urban environments, it has been emphasized in the title as it is their natural market, since in large ground-mounted PV plants the extra cost is difficult to justify. Since 2010 MLPE have increased their market share steadily and continuing to grow steadily. However, there still lacks a profound understanding of how they work, especially in the case of power optimizers, the achievable energy gains with their use and the possibilities in failure diagnosis. The main objective of this thesis is to provide a complete understanding of DMPPT technologies: how they function, their limitations and their advantages. A series of equations used to model PV arrays with power optimizers have been derived and used to point out limitations in solving certain mismatch situation. Because one of the most emphasized benefits of DMPPT is their ability to mitigate shading losses, an extensive study on the effects of shadows on PV systems is presented; both on the I-V curve and on MPPT algorithms. Experimental tests have been performed on the MPPT algorithms of central inverters and MLPE, highlighting their inefficiency in I-V curves with local maxima. An analysis of the possible mitigation of hot-spots with DMPPT is discussed and experimentally verified. And a theoretical analysis of the possible power and energy gains is presented as well as experiments in real PV systems. A short economic analysis of the benefits of DMPPT has also been performed. In order to aide in the previous task, a program which simulates I-V curves under shaded conditions has been developed and experimentally verified. This same program has been used to develop a software tool especially designed for PV systems affected by shading, which estimates the losses due to shading and the energy gains obtained with DMPPT. Finally, a set of algorithms for diagnosing system faults in PV systems with DMPPT has been developed and experimentally verified. The tool can diagnose the following failures: fixed object shading (with distance estimation), localized dirt, generalized dirt, possible hot-spots, module degradation and excessive losses in DC cables. In addition, it alerts the user of the power losses produced by each failure and classifies the failures by their severity and it does not require the use of irradiance or temperature sensors.

Distributed control strategies for photovoltaic systems in urban environments : Versión definitiva

Este trabajo es una contribución a los sistemas fotovoltaicos (FV) con seguimiento distribuido del punto de máxima potencia (DMPPT), una topología que se caracteriza porque lleva a cabo el MPPT a nivel de módulo, al contrario de las topologías más tradicionales que llevan a cabo el MPPT para un número más elevado de módulos, pudiendo ser hasta cientos de módulos. Las dos tecnologías DMPPT que existen en el mercado son conocidos como microinversores y optimizadores de potencia, y ofrecen ciertas ventajas sobre sistemas de MPPT central como: mayor producción en situaciones de mismatch, monitorización individual de cada módulo, flexibilidad de diseño, mayor seguridad del sistema, etc. Aunque los sistemas DMPPT no están limitados a los entornos urbanos, se ha enfatizado en el título ya que es su mercado natural, siendo difícil una justificación de su sobrecoste en grandes huertas solares en suelo. Desde el año 2010 el mercado de estos sistemas ha incrementado notablemente y sigue creciendo de una forma continuada. Sin embargo, todavía falta un conocimiento profundo de cómo funcionan estos sistemas, especialmente en el caso de los optimizadores de potencia, de las ganancias energéticas esperables en condiciones de mismatch y de las posibilidades avanzadas de diagnóstico de fallos. El principal objetivo de esta tesis es presentar un estudio completo de cómo funcionan los sistemas DMPPT, sus límites y sus ventajas, así como experimentos varios que verifican la teoría y el desarrollo de herramientas para valorar las ventajas de utilizar DMPPT en cada instalación. Las ecuaciones que modelan el funcionamiento de los sistemas FVs con optimizadores de potencia se han desarrollado y utilizado para resaltar los límites de los mismos a la hora de resolver ciertas situaciones de mismatch. Se presenta un estudio profundo sobre el efecto de las sombras en los sistemas FVs: en la curva I-V y en los algoritmos MPPT. Se han llevado a cabo experimentos sobre el funcionamiento de los algoritmos MPPT en situaciones de sombreado, señalando su ineficiencia en estas situaciones. Un análisis de la ventaja del uso de DMPPT frente a los puntos calientes es presentado y verificado. También se presenta un análisis sobre las posibles ganancias en potencia y energía con el uso de DMPPT en condiciones de sombreado y este también es verificado experimentalmente, así como un breve estudio de su viabilidad económica. Para ayudar a llevar a cabo todos los análisis y experimentos descritos previamente se han desarrollado una serie de herramientas software. Una siendo un programa en LabView para controlar un simulador solar y almacenar las medidas. También se ha desarrollado un programa que simula curvas I-V de módulos y generador FVs afectados por sombras y este se ha verificado experimentalmente. Este mismo programa se ha utilizado para desarrollar un programa todavía más completo que estima las pérdidas anuales y las ganancias obtenidas con DMPPT en instalaciones FVs afectadas por sombras. Finalmente, se han desarrollado y verificado unos algoritmos para diagnosticar fallos en sistemas FVs con DMPPT. Esta herramienta puede diagnosticar los siguientes fallos: sombras debido a objetos fijos (con estimación de la distancia al objeto), suciedad localizada, suciedad general, posible punto caliente, degradación de módulos y pérdidas en el cableado de DC. Además, alerta al usuario de las pérdidas producidas por cada fallo y no requiere del uso de sensores de irradiancia y temperatura. ABSTRACT This work is a contribution to photovoltaic (PV) systems with distributed maximum power point tracking (DMPPT), a system topology characterized by performing the MPPT at module level, instead of the more traditional topologies which perform MPPT for a larger number of modules. The two DMPPT technologies available at the moment are known as microinverters and power optimizers, also known as module level power electronics (MLPE), and they provide certain advantages over central MPPT systems like: higher energy production in mismatch situations, monitoring of each individual module, system design flexibility, higher system safety, etc. Although DMPPT is not limited to urban environments, it has been emphasized in the title as it is their natural market, since in large ground-mounted PV plants the extra cost is difficult to justify. Since 2010 MLPE have increased their market share steadily and continuing to grow steadily. However, there still lacks a profound understanding of how they work, especially in the case of power optimizers, the achievable energy gains with their use and the possibilities in failure diagnosis. The main objective of this thesis is to provide a complete understanding of DMPPT technologies: how they function, their limitations and their advantages. A series of equations used to model PV arrays with power optimizers have been derived and used to point out limitations in solving certain mismatch situation. Because one of the most emphasized benefits of DMPPT is their ability to mitigate shading losses, an extensive study on the effects of shadows on PV systems is presented; both on the I-V curve and on MPPT algorithms. Experimental tests have been performed on the MPPT algorithms of central inverters and MLPE, highlighting their inefficiency in I-V curves with local maxima. An analysis of the possible mitigation of hot-spots with DMPPT is discussed and experimentally verified. And a theoretical analysis of the possible power and energy gains is presented as well as experiments in real PV systems. A short economic analysis of the benefits of DMPPT has also been performed. In order to aide in the previous task, a program which simulates I-V curves under shaded conditions has been developed and experimentally verified. This same program has been used to develop a software tool especially designed for PV systems affected by shading, which estimates the losses due to shading and the energy gains obtained with DMPPT. Finally, a set of algorithms for diagnosing system faults in PV systems with DMPPT has been developed and experimentally verified. The tool can diagnose the following failures: fixed object shading (with distance estimation), localized dirt, generalized dirt, possible hot-spots, module degradation and excessive losses in DC cables. In addition, it alerts the user of the power losses produced by each failure and classifies the failures by their severity and it does not require the use of irradiance or temperature sensors.

Non-invasive Detection of Cortical Control Signals for Brain-Computer Interfaces

Thesis (Ph.D.)--University of Washington, 2016-05

A structural method for the design of fault tolerant distributed control systems

Distributed digital control systems provide alternatives to conventional, centralised digital control systems. Typically, a modern distributed control system will comprise a multi-processor or network of processors, a communications network, an associated set of sensors and actuators, and the systems and applications software. This thesis addresses the problem of how to design robust decentralised control systems, such as those used to control event-driven, real-time processes in time-critical environments. Emphasis is placed on studying the dynamical behaviour of a system and identifying ways of partitioning the system so that it may be controlled in a distributed manner. A structural partitioning technique is adopted which makes use of natural physical sub-processes in the system, which are then mapped into the software processes to control the system. However, communications are required between the processes because of the disjoint nature of the distributed (i.e. partitioned) state of the physical system. The structural partitioning technique, and recent developments in the theory of potential controllability and observability of a system, are the basis for the design of controllers. In particular, the method is used to derive a decentralised estimate of the state vector for a continuous-time system. The work is also extended to derive a distributed estimate for a discrete-time system. Emphasis is also given to the role of communications in the distributed control of processes and to the partitioning technique necessary to design distributed and decentralised systems with resilient structures. A method is presented for the systematic identification of necessary communications for distributed control. It is also shwon that the structural partitions can be used directly in the design of software fault tolerant concurrent controllers. In particular, the structural partition can be used to identify the boundary of the conversation which can be used to protect a specific part of the system. In addition, for certain classes of system, the partitions can be used to identify processes which may be dynamically reconfigured in the event of a fault. These methods should be of use in the design of robust distributed systems.

The Complex Unified Evolutionary Approach to the Creation of the Multilevel Distributed Control System of a Gas-transport Company

The objects of a large-scale gas-transport company (GTC) suggest a complex unified evolutionary approach, which covers basic building concepts, up-to-date technologies, models, methods and means that are used in the phases of design, adoption, maintenance and development of the multilevel automated distributed control systems (ADCS).. As a single methodological basis of the suggested approach three basic Concepts, which contain the basic methodological principles and conceptual provisions on the creation of distributed control systems, were worked out: systems of the lower level (ACS of the technological processes based on up-to-date SCADA), of the middle level (ACS of the operative-dispatch production control based on MES-systems) and of the high level (business process control on the basis of complex automated systems ERP).

ADAM: A Decentralized Parallel Computer Architecture Featuring Fast Thread and Data Migration and a Uniform Hardware Abstraction

The furious pace of Moore's Law is driving computer architecture into a realm where the the speed of light is the dominant factor in system latencies. The number of clock cycles to span a chip are increasing, while the number of bits that can be accessed within a clock cycle is decreasing. Hence, it is becoming more difficult to hide latency. One alternative solution is to reduce latency by migrating threads and data, but the overhead of existing implementations has previously made migration an unserviceable solution so far. I present an architecture, implementation, and mechanisms that reduces the overhead of migration to the point where migration is a viable supplement to other latency hiding mechanisms, such as multithreading. The architecture is abstract, and presents programmers with a simple, uniform fine-grained multithreaded parallel programming model with implicit memory management. In other words, the spatial nature and implementation details (such as the number of processors) of a parallel machine are entirely hidden from the programmer. Compiler writers are encouraged to devise programming languages for the machine that guide a programmer to express their ideas in terms of objects, since objects exhibit an inherent physical locality of data and code. The machine implementation can then leverage this locality to automatically distribute data and threads across the physical machine by using a set of high performance migration mechanisms. An implementation of this architecture could migrate a null thread in 66 cycles -- over a factor of 1000 improvement over previous work. Performance also scales well; the time required to move a typical thread is only 4 to 5 times that of a null thread. Data migration performance is similar, and scales linearly with data block size. Since the performance of the migration mechanism is on par with that of an L2 cache, the implementation simulated in my work has no data caches and relies instead on multithreading and the migration mechanism to hide and reduce access latencies.

Designing computer architecture research workloads

MinneSPEC proposes reduced input sets that microprocessor designers can use to model representative short-running workloads. A four-step methodology verifies the program behavior similarity of these input sets to reference sets.

The effects of J-gate potential and interfaces on donor exchange coupling in the Kane quantum computer architecture

We calculate the electron exchange coupling for a phosphorus donor pair in silicon perturbed by a J-gate potential and the boundary effects of the silicon host geometry. In addition to the electron-electron exchange interaction we also calculate the contact hyperfine interaction between the donor nucleus and electron as a function of the varying experimental conditions. Donor separation, depth of the P nuclei below the silicon oxide layer and J-gate voltage become decisive factors in determining the strength of both the exchange coupling and hyperfine interaction-both crucial components for qubit operations in the Kane quantum computer. These calculations were performed using an anisotropic effective-mass Hamiltonian approach. The behaviour of the donor exchange coupling as a function of the parameters varied in this work provides relevant information for the experimental design of these devices.

Programming Robots with an Agent-Oriented BDI-based Control Architecture: Explorations using the JaCa and Webots Platforms

La crescente disponibilità di dispositivi meccanici e -soprattutto - elettronici le cui performance aumentano mentre il loro costo diminuisce, ha permesso al campo della robotica di compiere notevoli progressi. Tali progressi non sono stati fatti unicamente per ciò che riguarda la robotica per uso industriale, nelle catene di montaggio per esempio, ma anche per quella branca della robotica che comprende i robot autonomi domestici. Questi sistemi autonomi stanno diventando, per i suddetti motivi, sempre più pervasivi, ovvero sono immersi nello stesso ambiente nel quale vivono gli essere umani, e interagiscono con questi in maniera proattiva. Essi stanno compiendo quindi lo stesso percorso che hanno attraversato i personal computer all'incirca 30 anni fa, passando dall'essere costosi ed ingombranti mainframe a disposizione unicamente di enti di ricerca ed università, ad essere presenti all'interno di ogni abitazione, per un utilizzo non solo professionale ma anche di assistenza alle attività quotidiane o anche di intrattenimento. Per questi motivi la robotica è un campo dell'Information Technology che interessa sempre più tutti i tipi di programmatori software. Questa tesi analizza per prima cosa gli aspetti salienti della programmazione di controllori per robot autonomi (ovvero senza essere guidati da un utente), quindi, come l'approccio basato su agenti sia appropriato per la programmazione di questi sistemi. In particolare si mostrerà come un approccio ad agenti, utilizzando il linguaggio di programmazione Jason e quindi l'architettura BDI, sia una scelta significativa, dal momento che il modello sottostante a questo tipo di linguaggio è basato sul ragionamento pratico degli esseri umani (Human Practical Reasoning) e quindi è adatto alla implementazione di sistemi che agiscono in maniera autonoma. Dato che le possibilità di utilizzare un vero e proprio sistema autonomo per poter testare i controllori sono ridotte, per motivi pratici, economici e temporali, mostreremo come è facile e performante arrivare in maniera rapida ad un primo prototipo del robot tramite l'utilizzo del simulatore commerciale Webots. Il contributo portato da questa tesi include la possibilità di poter programmare un robot in maniera modulare e rapida per mezzo di poche linee di codice, in modo tale che l'aumento delle funzionalità di questo risulti un collo di bottiglia, come si verifica nella programmazione di questi sistemi tramite i classici linguaggi di programmazione imperativi. L'organizzazione di questa tesi prevede un capitolo di background nel quale vengono riportare le basi della robotica, della sua programmazione e degli strumenti atti allo scopo, un capitolo che riporta le nozioni di programmazione ad agenti, tramite il linguaggio Jason -quindi l'architettura BDI - e perché tale approccio è adatto alla programmazione di sistemi di controllo per la robotica. Successivamente viene presentata quella che è la struttura completa del nostro ambiente di lavoro software che comprende l'ambiente ad agenti e il simulatore, quindi nel successivo capitolo vengono mostrate quelle che sono le esplorazioni effettuate utilizzando Jason e un approccio classico (per mezzo di linguaggi classici), attraverso diversi casi di studio di crescente complessità; dopodiché, verrà effettuata una valutazione tra i due approcci analizzando i problemi e i vantaggi che comportano questi. Infine, la tesi terminerà con un capitolo di conclusioni e di riflessioni sulle possibili estensioni e lavori futuri.

From robots to animals : Virtual fences for controlling cattle

We consider the problem of monitoring and controlling the position of herd animals, and view animals as networked agents with natural mobility but not strictly controllable. By exploiting knowledge of individual and herd behavior we would like to apply a vast body of theory in robotics and motion planning to achieving the constrained motion of a herd. In this paper we describe the concept of a virtual fence which applies a stimulus to an animal as a function of its pose with respect to the fenceline. Multiple fence lines can define a region, and the fences can be static or dynamic. The fence algorithm is implemented by a small position-aware computer device worn by the animal, which we refer to as a Smart Collar.We describe a herd-animal simulator, the Smart Collar hardware and algorithms for tracking and controlling animals as well as the results of on-farm experiments with up to ten Smart Collars.

MATLAB toolboxes : robotics and vision for students and teachers

In this column, Dr. Peter Corke of CSIRO, Australia, gives us a description of MATLAB Toolboxes he has developed. He has been passionately developing tools to enable students and teachers to better understand the theoretical concepts behind classical robotics and computer vision through easy and intuitive simulation and visualization. The results of this labor of love have been packaged as MATLAB Toolboxes: the Robotics Toolbox and the Vision Toolbox. –Daniela Rus, RAS Education Cochair

Scaffolding children’s robot building and programming activities

Since 2001 the School of Information Technology and Electrical Engineering (ITEE) at the University of Queensland has been involved in RoboCupJunior activities aimed at providing children with the Robot building and programming knowledge they need to succeed in RoboCupJunior competitions. These activities include robotics workshops, the organization of the State-wide RoboCupJunior competition, and consultation on all matters robotic with schools and government organizations. The activities initiated by ITEE have succeeded in providing children with the scaffolding necessary to become competent, independent robot builders and programmers. Results from state, national and international competitions suggest that many of the children who participate in the activities supported by ITEE are subsequently able to purpose- build robots to effectively compete in RoboCupJunior competitions. As a result of the scaffolding received within workshops children are able to think deeply and creatively about their designs, and to critique their designs in order to make the best possible creation in an effort to win.

Multi-robot control in highly dynamic, competitive environments

The control and coordination of multiple mobile robots is a challenging task; particularly in environments with multiple, rapidly moving obstacles and agents. This paper describes a robust approach to multi-robot control, where robustness is gained from competency at every layer of robot control. The layers are: (i) a central coordination system (MAPS), (ii) an action system (AES), (iii) a navigation module, and (iv) a low level dynamic motion control system. The multi-robot coordination system assigns each robot a role and a sub-goal. Each robots action execution system then assumes the assigned role and attempts to achieve the specified sub-goal. The robots navigation system directs the robot to specific goal locations while ensuring that the robot avoids any obstacles. The motion system maps the heading and speed information from the navigation system to force-constrained motion. This multi-robot system has been extensively tested and applied in the robot soccer domain using both centralized and distributed coordination.

Distributed Load Control in Multiphase Radial Networks

The current power grid is on the cusp of modernization due to the emergence of distributed generation and controllable loads, as well as renewable energy. On one hand, distributed and renewable generation is volatile and difficult to dispatch. On the other hand, controllable loads provide significant potential for compensating for the uncertainties. In a future grid where there are thousands or millions of controllable loads and a large portion of the generation comes from volatile sources like wind and solar, distributed control that shifts or reduces the power consumption of electric loads in a reliable and economic way would be highly valuable.

Load control needs to be conducted with network awareness. Otherwise, voltage violations and overloading of circuit devices are likely. To model these effects, network power flows and voltages have to be considered explicitly. However, the physical laws that determine power flows and voltages are nonlinear. Furthermore, while distributed generation and controllable loads are mostly located in distribution networks that are multiphase and radial, most of the power flow studies focus on single-phase networks.

This thesis focuses on distributed load control in multiphase radial distribution networks. In particular, we first study distributed load control without considering network constraints, and then consider network-aware distributed load control.

Distributed implementation of load control is the main challenge if network constraints can be ignored. In this case, we first ignore the uncertainties in renewable generation and load arrivals, and propose a distributed load control algorithm, Algorithm 1, that optimally schedules the deferrable loads to shape the net electricity demand. Deferrable loads refer to loads whose total energy consumption is fixed, but energy usage can be shifted over time in response to network conditions. Algorithm 1 is a distributed gradient decent algorithm, and empirically converges to optimal deferrable load schedules within 15 iterations.

We then extend Algorithm 1 to a real-time setup where deferrable loads arrive over time, and only imprecise predictions about future renewable generation and load are available at the time of decision making. The real-time algorithm Algorithm 2 is based on model-predictive control: Algorithm 2 uses updated predictions on renewable generation as the true values, and computes a pseudo load to simulate future deferrable load. The pseudo load consumes 0 power at the current time step, and its total energy consumption equals the expectation of future deferrable load total energy request.

Network constraints, e.g., transformer loading constraints and voltage regulation constraints, bring significant challenge to the load control problem since power flows and voltages are governed by nonlinear physical laws. Remarkably, distribution networks are usually multiphase and radial. Two approaches are explored to overcome this challenge: one based on convex relaxation and the other that seeks a locally optimal load schedule.

To explore the convex relaxation approach, a novel but equivalent power flow model, the branch flow model, is developed, and a semidefinite programming relaxation, called BFM-SDP, is obtained using the branch flow model. BFM-SDP is mathematically equivalent to a standard convex relaxation proposed in the literature, but numerically is much more stable. Empirical studies show that BFM-SDP is numerically exact for the IEEE 13-, 34-, 37-, 123-bus networks and a real-world 2065-bus network, while the standard convex relaxation is numerically exact for only two of these networks.

Theoretical guarantees on the exactness of convex relaxations are provided for two types of networks: single-phase radial alternative-current (AC) networks, and single-phase mesh direct-current (DC) networks. In particular, for single-phase radial AC networks, we prove that a second-order cone program (SOCP) relaxation is exact if voltage upper bounds are not binding; we also modify the optimal load control problem so that its SOCP relaxation is always exact. For single-phase mesh DC networks, we prove that an SOCP relaxation is exact if 1) voltage upper bounds are not binding, or 2) voltage upper bounds are uniform and power injection lower bounds are strictly negative; we also modify the optimal load control problem so that its SOCP relaxation is always exact.

To seek a locally optimal load schedule, a distributed gradient-decent algorithm, Algorithm 9, is proposed. The suboptimality gap of the algorithm is rigorously characterized and close to 0 for practical networks. Furthermore, unlike the convex relaxation approach, Algorithm 9 ensures a feasible solution. The gradients used in Algorithm 9 are estimated based on a linear approximation of the power flow, which is derived with the following assumptions: 1) line losses are negligible; and 2) voltages are reasonably balanced. Both assumptions are satisfied in practical distribution networks. Empirical results show that Algorithm 9 obtains 70+ times speed up over the convex relaxation approach, at the cost of a suboptimality within numerical precision.