This paper explores the detection and localization of cyber-attacks on time-series measurements data in power systems, focusing on comparing conventional machine learning (ML) like k-means, deep learning method like autoencoder, …
This paper explores the detection and localization of cyber-attacks on time-series measurements data in power systems, focusing on comparing conventional machine learning (ML) like k-means, deep learning method like autoencoder, and graph neural network (GNN)-based techniques. We assess the detection accuracy of these approaches and their potential to pinpoint the locations of specific sensor measurements under attack. Given the demonstrated success of GNNs in other time-series anomaly detection applications, we aim to evaluate their performance within the context of power systems cyber-attacks on sensor measurements. Utilizing the IEEE 68-bus system, we simulated four types of false data attacks, including scaling attacks, additive attacks, and their combinations, to test the selected approaches. Our results indicate that GNN-based methods outperform k-means and autoencoder in detection. Additionally, GNNs show promise in accurately localizing attacks for simple scenarios, although they still face challenges in more complex cases, especially ones that involve combinations of scaling and additive attacks.
With the emergence of low-inertia microgrids powered by inverter-based generation, there remains a concern about the operational resilience of these systems. Grid-forming inverters (GFMs), enabled by various device-level (primary) and …
With the emergence of low-inertia microgrids powered by inverter-based generation, there remains a concern about the operational resilience of these systems. Grid-forming inverters (GFMs), enabled by various device-level (primary) and system-level (secondary) control methods, are poised to play a significant role in achieving certain operational objectives, such as the effective utilization of clean energy resources while maintaining stability. However, despite the recent advances in GFMs, there is a lack of suitable controls that can ascertain resilience-constrained operations, like maintaining critical operational safety limits during transients under various cyber-physical disruptions. In this work, we develop decentralized autonomous controllers (DACs) that enforce resilience-constrained operation via local, minimally invasive adjustments (e.g., changes in set-points) while co-existing within the hierarchy of existing (primary and secondary) controls. The DACs work autonomously by sensing only local GFM measurements and act only when operational resilience constraints are violated. The proposed DAC scheme is computationally efficient (only algebraic computations), which enables fast, real-time execution and demonstrates the efficacy of the proposed control framework on GridLAB-D-HELICS-based control-grid co-simulations on the IEEE 123-node networked microgrid. Finally, we show how the developed DACs empower the grid by utilizing the available resources entirely to ensure resilience (maintain frequency safe limits).
Inverter-based storages are poised to play a prominent role in future power grids with massive renewable generation. Grid-forming inverters (GFMs) are emerging as a dominant technology with synchronous generators (SG)-like …
Inverter-based storages are poised to play a prominent role in future power grids with massive renewable generation. Grid-forming inverters (GFMs) are emerging as a dominant technology with synchronous generators (SG)-like characteristics through primary control loops. Advanced secondary control schemes, e.g., consensus algorithms, allow GFM-interfaced storage units to participate in frequency regulations and restore nominal frequency following grid disturbances. However, it is imperative to ensure transient frequency excursions do not violate critical safety limits while the grid transitions from pre- to post-disturbance operating point. This paper presents a hierarchical safety-enforced consensus method -- combining a device-layer (decentralized) transient safety filter with a secondary-layer (distributed) consensus coordination -- to achieve three distinct objectives: limiting transient frequency excursions to safe limits, minimizing frequency deviations from nominal, and ensuring coordinated power sharing among GFM-storage units. The proposed hierarchical (two-layered) safety-consensus technique is illustrated using a GFM-interfaced storage network on an IEEE 68-bus system under multiple grid transient scenarios.
The paper discusses fast frequency control in bulk power systems using embedded networks of grid-forming energy storage resources. Differing from their traditional roles of regulating reserves, the storage resources in …
The paper discusses fast frequency control in bulk power systems using embedded networks of grid-forming energy storage resources. Differing from their traditional roles of regulating reserves, the storage resources in this work operate as fast-acting grid assets shaping transient dynamics. The storage resources in the network are autonomously controlled using local measurements for distributed frequency support during disturbance events. Further, the grid-forming inverter systems interfacing with the storage resources, are augmented with fast-acting safety controls designed to contain frequency transients within a prescribed tolerance band. The control action, derived from the storage network, improves the frequency nadirs in the system and prevents the severity of a disturbance from propagating far from the source. The paper also presents sensitivity studies to evaluate the impacts of storage capacity and inverter controller parameters on the dynamic performance of frequency control and disturbance localization. The performance of the safety-constrained grid-forming control is also compared with the more common grid-following control. The results are illustrated through case studies on an IEEE test system.
The paper presents a theoretical study on small-signal stability and damping in bulk power systems with multiple grid-forming inverter-based storage resources. A detailed analysis is presented, characterizing the impacts of …
The paper presents a theoretical study on small-signal stability and damping in bulk power systems with multiple grid-forming inverter-based storage resources. A detailed analysis is presented, characterizing the impacts of inverter droop gains and storage size on the slower eigenvalues, particularly those concerning inter-area oscillation modes. From these parametric sensitivity studies, a set of necessary conditions are derived that the design of droop gain must satisfy to enhance damping performance. The analytical findings are structured into propositions highlighting potential design considerations for improving system stability. The findings are illustrated via numerical studies on an IEEE 68-bus grid-forming storage network.
In a controlled cyber-physical network, such as a power grid, any malicious data injection in the sensor measurements can lead to widespread impact due to the actions of the closed-loop …
In a controlled cyber-physical network, such as a power grid, any malicious data injection in the sensor measurements can lead to widespread impact due to the actions of the closed-loop controllers. While fast identification of the attack signatures is imperative for reliable operations, it is challenging to do so in a large dynamical network with tightly coupled nodes. A particularly challenging scenario arises when the cyberattacks are strategically launched during a grid stress condition, caused by non-malicious physical disturbances. In this work, we propose an algorithmic framework -- based on Koopman mode (KM) decomposition -- for online identification and visualization of the cyberattack signatures in streaming time-series measurements from a power network. The KMs are capable of capturing the spatial embedding of both natural and anomalous modes of oscillations in the sensor measurements and thus revealing the specific influences of cyberattacks, even under existing non-malicious grid stress events. Most importantly, it enables us to quantitatively compare the outcomes of different potential cyberattacks injected by an attacker. The performance of the proposed algorithmic framework is illustrated on the IEEE 68-bus test system using synthetic attack scenarios. Such knowledge regarding the detection of various cyberattacks will enable us to devise appropriate diagnostic scheme while considering varied constraints arising from different attacks.
Recent Odessa disturbance events have brought attention to the challenges associated with the interaction between Inverter-Based Resources (IBRs) and the transmission and distribution system. The NERC event diagnosis report has …
Recent Odessa disturbance events have brought attention to the challenges associated with the interaction between Inverter-Based Resources (IBRs) and the transmission and distribution system. The NERC event diagnosis report has highlighted several issues, emphasizing the need for continuous performance monitoring of these IBRs by system operators. Key areas of concern include the mismatch of control and protection performance of IBRs between the original equipment manufacturer (OEM)-provided models and field measurements. The inability to replicate the realistic response can result in incorrect reliability and resilience studies. In this paper, we developed an approach on how to emulate the behavior of an IBR using measurement data obtained for system operators to utilize in real-time and long-term planning. Two experiments are conducted in the phasor domain and electromagnetic transients (EMT) domain to emulate the behavior for grid forming and grid following inverters under various operating conditions and the effectiveness of the proposed model is demonstrated in terms of accuracy and ease of utilizing user-defined models (UDMs).
The modern power grid has seen a rise in the integration of non-linear loads, presenting a significant concern for operators. These loads introduce unwanted harmonics, leading to potential issues such …
The modern power grid has seen a rise in the integration of non-linear loads, presenting a significant concern for operators. These loads introduce unwanted harmonics, leading to potential issues such as overheating and improper functioning of circuit breakers. In pursuing a more sustainable grid, the adoption of electric vehicles (EVs) and photovoltaic (PV) systems in residential networks has increased. Understanding and examining the effects of high-order harmonic frequencies beyond $1.5$ kHz is crucial to understanding their impact on the operation and planning of electrical distribution systems under varying nonlinear loading conditions. This study investigates a diverse set of critical power electronic loads within a household modeled using PSCAD/EMTdc, analyzing their unique harmonic spectra. This information is utilized to run the time-series harmonic analysis program in OpenDSS on a modified IEEE 34 bus test system model. The impact of high-order harmonics is quantified using metrics that evaluate total harmonic distortion (THD), transformer harmonic-driven eddy current loss component, and propagation of harmonics from the source to the substation transformer.
Net load forecasting is crucial for energy planning and facilitating informed decision-making regarding trade and load distributions. However, evaluating forecasting models' performance against benchmark models remains challenging, thereby impeding experts' …
Net load forecasting is crucial for energy planning and facilitating informed decision-making regarding trade and load distributions. However, evaluating forecasting models' performance against benchmark models remains challenging, thereby impeding experts' trust in the model's performance. In this context, there is a demand for technological interventions that allow scientists to compare models across various timeframes and solar penetration levels. This paper introduces a visual analytics-based application designed to compare the performance of deep-learning-based net load forecasting models with other models for probabilistic net load forecasting. This application employs carefully selected visual analytic interventions, enabling users to discern differences in model performance across different solar penetration levels, dataset resolutions, and hours of the day over multiple months. We also present observations made using our application through a case study, demonstrating the effectiveness of visualizations in aiding scientists in making informed decisions and enhancing trust in net load forecasting models.
This paper addresses the following fundamental research question: how does the integration of grid-forming inverters (GFMs) replacing conventional synchronous generators (SGs) impact the slow coherent eigen-structure and the low-frequency oscillatory …
This paper addresses the following fundamental research question: how does the integration of grid-forming inverters (GFMs) replacing conventional synchronous generators (SGs) impact the slow coherent eigen-structure and the low-frequency oscillatory behavior of future power systems? Due to time-scale separated dynamics, generator states inside a coherent area synchronize over a fast time-scale due to stronger coupling, while the areas themselves synchronize over a slower time scale. Our mathematical analysis shows that due to the large-scale integration of GFMs, the weighted Laplacian structure of the frequency dynamics is preserved, however, the entries of the Laplacian may be significantly modified based on the location and penetration levels of the GFMs. This can impact and potentially significantly alter the coherency structure of the system. We have validated our findings with numerical results using the IEEE 68-bus test system.
Accurate net load forecasting is vital for energy planning, aiding decisions on trade and load distribution. However, assessing the performance of forecasting models across diverse input variables, like temperature and …
Accurate net load forecasting is vital for energy planning, aiding decisions on trade and load distribution. However, assessing the performance of forecasting models across diverse input variables, like temperature and humidity, remains challenging, particularly for eliciting a high degree of trust in the model outcomes. In this context, there is a growing need for data-driven technological interventions to aid scientists in comprehending how models react to both noisy and clean input variables, thus shedding light on complex behaviors and fostering confidence in the outcomes. In this paper, we present Forte, a visual analytics-based application to explore deep probabilistic net load forecasting models across various input variables and understand the error rates for different scenarios. With carefully designed visual interventions, this web-based interface empowers scientists to derive insights about model performance by simulating diverse scenarios, facilitating an informed decision-making process. We discuss observations made using Forte and demonstrate the effectiveness of visualization techniques to provide valuable insights into the correlation between weather inputs and net load forecasts, ultimately advancing grid capabilities by improving trust in forecasting models.
The influx of non-linear power electronic loads into the distribution network has the potential to disrupt the existing distribution transformer operations. They were not designed to mediate the excessive heating …
The influx of non-linear power electronic loads into the distribution network has the potential to disrupt the existing distribution transformer operations. They were not designed to mediate the excessive heating losses generated from the harmonics. To have a good understanding of current standing challenges, a knowledge of the generation and load mix as well as the current harmonic estimations are essential for designing transformers and evaluating their performance. In this paper, we investigate a mixture of essential power electronic loads for a household designed in PSCAD/EMTdc and their potential impacts on transformer eddy current losses and derating using harmonic analysis. The various scenarios have been studied with increasing PV penetrations. The Peak load conditions are chosen for each scenario to perform a transformer derating analysis. Our findings reveal that in the presence of high power electronic loads (especially third harmonics), along with increasing PV generation may worsen transformer degradation. However, with a low amount of power electronic loads, additional PV generation helps to reduce the harmonic content in the current and improve transformer performance.
Accurate net load forecasting is vital for energy planning, aiding decisions on trade and load distribution. However, assessing the performance of forecasting models across diverse input variables, like temperature and …
Accurate net load forecasting is vital for energy planning, aiding decisions on trade and load distribution. However, assessing the performance of forecasting models across diverse input variables, like temperature and humidity, remains challenging, particularly for eliciting a high degree of trust in the model outcomes. In this context, there is a growing need for data-driven technological interventions to aid scientists in comprehending how models react to both noisy and clean input variables, thus shedding light on complex behaviors and fostering confidence in the outcomes. In this paper, we present Forte, a visual analytics-based application to explore deep probabilistic net load forecasting models across various input variables and understand the error rates for different scenarios. With carefully designed visual interventions, this web-based interface empowers scientists to derive insights about model performance by simulating diverse scenarios, facilitating an informed decision-making process. We discuss observations made using Forte and demonstrate the effectiveness of visualization techniques to provide valuable insights into the correlation between weather inputs and net load forecasts, ultimately advancing grid capabilities by improving trust in forecasting models.
Recent studies have demonstrated the potential of flexible loads in providing frequency response services. However, uncertainty and variability in various weather-related and end-use behavioral factors often affect the demand-side control …
Recent studies have demonstrated the potential of flexible loads in providing frequency response services. However, uncertainty and variability in various weather-related and end-use behavioral factors often affect the demand-side control performance. This work addresses this problem with the design of a demand-side control to achieve frequency response under load uncertainties. Our approach involves modeling the load uncertainties via stochastic processes that appear as both multiplicative and additive to the system states in closed-loop power system dynamics. Extending the recently developed mean square exponential stability (MSES) results for stochastic systems, we formulate multiobjective linear matrix inequality (LMI)-based optimal control synthesis problems to not only guarantee stochastic stability, but also promote sparsity, enhance closed-loop transient performance, and maximize allowable uncertainties. The fundamental trade-off between the maximum allowable (critical) uncertainty levels and the optimal stochastic stabilizing control efforts is established. Moreover, the sparse control synthesis problem is generalized to the realistic power systems scenario in which only partial-state measurements are available. Detailed numerical studies are carried out on the IEEE 39-bus system to demonstrate the closed-loop stochastic stabilizing performance of the sparse controllers in enhancing frequency response under load uncertainties; as well as illustrate the fundamental trade-off between the allowable uncertainties and optimal control efforts.
Critical energy infrastructures are increasingly relying on advanced sensing and control technologies for efficient and optimal utilization of flexible energy resources. Algorithmic procedures are needed to ensure that such systems …
Critical energy infrastructures are increasingly relying on advanced sensing and control technologies for efficient and optimal utilization of flexible energy resources. Algorithmic procedures are needed to ensure that such systems are designed to be resilient to a wide range of cyber-physical adversarial events. This paper provides a robust optimization framework to quantify the range of adversarial perturbations that a system can accommodate without violating pre-specified resiliency metrics. An inner-approximation of the set of adversarial events which can be mitigated by the availabe flexibility is constructed using an optimization based approach. The proposed algorithm is illustrated on an islanded microgrid example: a modified IEEE 123-node feeder with distributed energy resources. Simulations are carried out to validate that the resiliency metrics are met for any event sampled from the constructed adversarial set for varying levels of available flexibility (energy reserves).
Modern safety-critical energy infrastructures are operated in a hierarchical and modular control framework allowing only limited data exchange between modules. In this context, to assure system-wide safety each module must …
Modern safety-critical energy infrastructures are operated in a hierarchical and modular control framework allowing only limited data exchange between modules. In this context, to assure system-wide safety each module must synthesize and communicate constraints on the values of exchanged data. To ensure transient safety in inverter-based microgrids, we develop a set invariance-based distributed safety verification algorithm for each inverter module. Applying Nagumo's invariance condition, we construct a robust polynomial optimization problem to jointly search for safety-admissible set of control set-points and design parameters, under allowable disturbances from neighbors. We solve the verification problem with sum-of-squares programming and we perform numerical simulations using grid-forming inverters to illustrate our method.
Integration of electronics-based residential appliances and distributed energy resources in homes is expected to rise with grid decarbonization. These devices may introduce significant harmonics into power networks that need to …
Integration of electronics-based residential appliances and distributed energy resources in homes is expected to rise with grid decarbonization. These devices may introduce significant harmonics into power networks that need to be closely studied in order to accurately model and forecast load. However, it can be difficult to obtain harmonic-rich voltage and current data – necessary for identifying accurate load models – for residential electrical loads. Recognizing this need, first a set of electronics-based end-use loads is identified and modeled in an electromagnetic transients program tool for a residence. Second, an impedance-varying method is proposed to generate harmonic data that captures harmonic propagation to the supply voltage and harmonic interactions among end-use loads connected to the same supply voltage. Third, a harmonic-rich dataset produced via the proposed methodology is demonstrated to successfully identify frequency coupling matrix–based harmonic load models using the least-squares method. Numerical results demonstrate the accuracy of the model. The impact of limited data availability on model identification is also explored.
Continuous monitoring of the spatio-temporal dynamic behavior of critical infrastructure networks, such as the power systems, is a challenging but important task. In particular, accurate and timely prediction of the …
Continuous monitoring of the spatio-temporal dynamic behavior of critical infrastructure networks, such as the power systems, is a challenging but important task. In particular, accurate and timely prediction of the (electro-mechanical) transient dynamic trajectories of the power grid is necessary for early detection of any instability and prevention of catastrophic failures. Existing approaches for the prediction of dynamic trajectories either rely on the availability of accurate physical models of the system, use computationally expensive time-domain simulations, or are applicable only at local prediction problems (e.g., a single generator). In this paper, we report the application of two broad classes of data-driven learning models -- along with their algorithmic implementation and performance evaluation -- in predicting transient trajectories in power networks using only streaming measurements and the network topology as input. One class of models is based on the Koopman operator theory which allows for capturing the nonlinear dynamic behavior via an infinite-dimensional linear operator. The other class of models is based on the graph convolutional neural networks which are adept at capturing the inherent spatio-temporal correlations within the power network. Transient dynamic datasets for training and testing the models are synthesized by simulating a wide variety of load change events in the IEEE 68-bus system, categorized by the load change magnitudes, as well as by the degree of connectivity and the distance to nearest generator nodes. The results confirm that the proposed predictive models can successfully predict the post-disturbance transient evolution of the system with a high level of accuracy.
Critical energy infrastructures are increasingly relying on advanced sensing and control technologies for efficient and optimal utilization of flexible energy resources. Algorithmic procedures are needed to ensure that such systems …
Critical energy infrastructures are increasingly relying on advanced sensing and control technologies for efficient and optimal utilization of flexible energy resources. Algorithmic procedures are needed to ensure that such systems are designed to be resilient to a wide range of cyber-physical adversarial events. This paper provides a robust optimization framework to quantify the range of adversarial perturbations that a system can accommodate without violating pre-specified resiliency metrics. An inner-approximation of the set of adversarial events which can be mitigated by the available flexibility is constructed using an optimization based approach. The proposed algorithm is illustrated on an islanded microgrid example: a modified IEEE 123-node feeder with distributed energy resources. Simulations are carried out to validate that the resiliency metrics are met for any event sampled from the constructed adversarial set for varying levels of available flexibility (energy reserves).
Modern safety-critical energy infrastructures are increasingly operated in a hierarchical and modular control framework which allows for limited data exchange between the modules. In this context, it is important for …
Modern safety-critical energy infrastructures are increasingly operated in a hierarchical and modular control framework which allows for limited data exchange between the modules. In this context, it is important for each module to synthesize and communicate constraints on the values of exchanged information in order to assure system-wide safety. To ensure transient safety in inverter-based microgrids, we develop a set invariance-based distributed safety verification algorithm for each inverter module. Applying Nagumo's invariance condition, we construct a robust polynomial optimization problem to jointly search for safety-admissible set of control set-points and design parameters, under allowable disturbances from neighbors. We use sum-of-squares (SOS) programming to solve the verification problem and we perform numerical simulations using grid-forming inverters to illustrate the algorithm.
With the expected rise in behind-the-meter solar penetration within the distribution networks, there is a need to develop time-series forecasting methods that can reliably predict the net-load, accurately quantifying its …
With the expected rise in behind-the-meter solar penetration within the distribution networks, there is a need to develop time-series forecasting methods that can reliably predict the net-load, accurately quantifying its uncertainty and variability. This paper presents a deep learning method to generate probabilistic forecasts of day-ahead net-load at 15-min resolution, at various solar penetration levels. Our proposed deep-learning based architecture utilizes the dimensional reduction, from a higher-dimensional input to a lower-dimensional latent space, via a convolutional Autoencoder (AE). The extracted features from AE are then utilized to generate probability distributions across the latent space, by passing the features through a kernel-embedded Perron-Frobenius (kPF) operator. Finally, long short-term memory (LSTM) layers are used to synthesize time-series probability distributions of the forecasted net-load, from the latent space distributions. The models are shown to deliver superior forecast performance (as per several metrics), as well as maintain superior training efficiency, in comparison to existing benchmark models. Detailed analysis is carried out to evaluate the model performance across various solar penetration levels (up to 50\%), prediction horizons (e.g., 15\,min and 24\,hr ahead), and aggregation level of houses, as well as its robustness against missing measurements.
Continuous monitoring of the spatio-temporal dynamic behavior of critical infrastructure networks, such as the power systems, is a challenging but important task. In particular, accurate and timely prediction of the …
Continuous monitoring of the spatio-temporal dynamic behavior of critical infrastructure networks, such as the power systems, is a challenging but important task. In particular, accurate and timely prediction of the (electro-mechanical) transient dynamic trajectories of the power grid is necessary for early detection of any instability and prevention of catastrophic failures. Existing approaches for prediction of dynamic trajectories either rely on the availability of accurate physical models of the system, use computationally expensive time-domain simulations, or are applicable only at local prediction problems (e.g., a single generator). In this paper, we report the application of two broad classes of data-driven learning models – along with their algorithmic implementation and performance evaluation – in predicting transient trajectories in power networks using only streaming measurements and the network topology as input. One class of models is based on the <i>Koopman operator theory</i> which allows for capturing the nonlinear dynamic behavior via an infinite-dimensional linear operator. The other class of models is based on the <i>graph convolutional neural networks</i> which are adept at capturing the inherent spatio-temporal correlations within the power network. Transient dynamic datasets for training and testing the models are synthesized by simulating a wide variety of load change events in the IEEE 68-bus system, categorized by the load change magnitudes, as well as by the degree of connectivity and the distance to nearest generator nodes. The results confirm that the proposed predictive models can successfully predict the post-disturbance transient evolution of the system with a high level of accuracy.
Critical energy infrastructure are constantly understress due to the ever increasing disruptions caused by wildfires, hurricanes, other weather related extreme events and cyber-attacks. Hence it becomes important to make critical …
Critical energy infrastructure are constantly understress due to the ever increasing disruptions caused by wildfires, hurricanes, other weather related extreme events and cyber-attacks. Hence it becomes important to make critical infrastructure resilient to threats from such cyber-physical events. Such events are however hard to predict and numerous in nature and type, making it infeasible to become resilient to all possible cyber-physical event as such an approach would make the system operation overly conservative. Furthermore, distributions of such events are hard to predict and historical data available on such events is sparse. To deal with these issues, we present a policy-mode framework that enumerates and predicts the probability of various cyber-physical events on top of a distributionally robust optimization (DRO) that is robust to the sparsity of the available historical data. The proposed algorithm is illustrated on an islanded microgrid example: a modified IEEE 123-node feeder with distributed energy resources (DERs) and energy storage.
The influx of non-linear power electronic loads into the distribution network has the potential to disrupt the existing distribution transformer operations. They were not designed to mediate the excessive heating …
The influx of non-linear power electronic loads into the distribution network has the potential to disrupt the existing distribution transformer operations. They were not designed to mediate the excessive heating losses generated from the harmonics. To have a good understanding of current standing challenges, a knowledge of the generation and load mix as well as the current harmonic estimations are essential for designing transformers and evaluating their performance. In this paper, we investigate a mixture of essential power electronic loads for a household designed in PSCAD/EMTdc and their potential impacts on transformer eddy current losses and derating using harmonic analysis. The various scenarios have been studied with increasing PV penetrations. The peak load conditions are chosen for each scenario to perform a transformer derating analysis. Our findings reveal that in the presence of high power electronic loads (especially third harmonics), along with increasing PV generation may worsen transformer degradation. However, with a low amount of power electronic loads, additional PV generation helps to reduce the harmonic content in the current and improve transformer performance.
Koopman operator theory provides a model-free and purely data-driven technique for studying nonlinear dynamical systems. Since the Koopman operator is infinite-dimensional, researchers have developed several methods that provide a finite-dimensional …
Koopman operator theory provides a model-free and purely data-driven technique for studying nonlinear dynamical systems. Since the Koopman operator is infinite-dimensional, researchers have developed several methods that provide a finite-dimensional approximation of the Koopman operator so that it can be applied for practical use cases. One common thing with most of the methods is that their solutions are obtained by solving a centralized minimization problem. In this work, we treat the dynamical system to be a multi-agent system and propose an algorithm to compute the finite-dimensional approximation of the Koopman operator in a distributed manner using the knowledge of the topology of the underlying multi-agent system. The proposed distributed approach results in a sparse Koopman whose block structure mimics the Laplacian of the multi-agent system. Extensive simulation studies illustrate the proposed framework on the network of oscillators and the IEEE 68 bus system.
Recent studies have demonstrated the potential of flexible loads in providing frequency response services. However, uncertainty and variability in various weather-related and end-use behavioral factors often affect the demand-side control …
Recent studies have demonstrated the potential of flexible loads in providing frequency response services. However, uncertainty and variability in various weather-related and end-use behavioral factors often affect the demand-side control performance. This work addresses this problem with the design of a demand-side control to achieve frequency response under load uncertainties. Our approach involves modeling the load uncertainties via stochastic processes that appear as both multiplicative and additive to the system states in closed-loop power system dynamics. Extending the recently developed mean square exponential stability (MSES) results for stochastic systems, we formulate multi-objective linear matrix inequality (LMI)-based optimal control synthesis problems to not only guarantee stochastic stability, but also promote sparsity, enhance closed-loop transient performance, and maximize allowable uncertainties. The fundamental trade-off between the maximum allowable ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">critical</i> ) uncertainty levels and the optimal stochastic stabilizing control efforts is established. Moreover, the sparse control synthesis problem is generalized to the realistic power systems scenario in which only partial-state measurements are available. Detailed numerical studies are carried out on IEEE 39-bus system to demonstrate the closed-loop stochastic stabilizing performance of the sparse controllers in enhancing frequency response under load uncertainties; as well as illustrate the fundamental trade-off between the allowable uncertainties and optimal control efforts.
The proliferation of inverter-based generation and advanced sensing, controls, and communication infrastructure has facilitated the accelerated deployment of microgrids. A coordinated network of microgrids can maintain reliable power delivery to …
The proliferation of inverter-based generation and advanced sensing, controls, and communication infrastructure has facilitated the accelerated deployment of microgrids. A coordinated network of microgrids can maintain reliable power delivery to critical facilities during extreme events. Low-inertia offered by the power-electronics-interfaced energy resources, however, can present significant challenges to ensuring stable operation of the microgrids. In this work, distributed small-signal stability conditions for inverter-based microgrids are developed that involve the droop-controller parameters and the network parameters (e.g. line impedances, loads). The distributed closed- form parametric stability conditions derived in this work can be verified in a computationally efficient manner, facilitating the reliable design and operations of networks of microgrids. Dynamic phasor models have been used to capture the effects of electromagnetic transients. Numerical results are presented, along with PSCAD simulations, to validate the analytical stability conditions. Effects of design choices, such as the conductor types, and inverter sizes, on the small-signal stability of inverter- based microgrids are investigated to identify interpretable stable/unstable region estimates.
For effective integration of building operations into the evolving demand response programs of the power grid, real-time decisions concerning the use of building appliances for grid services must excel on …
For effective integration of building operations into the evolving demand response programs of the power grid, real-time decisions concerning the use of building appliances for grid services must excel on multiple criteria, ranging from the added value to occupants' comfort to the quality of the grid services. In this paper, we present a data-driven decision-support framework to dynamically rank load control alternatives in a commercial building, addressing the needs of multiple decision criteria (e.g. occupant comfort, grid service quality) under uncertainties in occupancy patterns. We adopt a stochastic multi-criteria decision algorithm recently applied to prioritize residential on/off loads, and extend it to i) complex load control decisions (e.g. dimming of lights, changing zone temperature set-points) in a commercial building; and ii) systematic integration of zonal occupancy patterns to accurately identify short-term grid service opportunities. We evaluate the performance of the framework for curtailment of air-conditioning, lighting, and plug-loads in a multi-zone office building for a range of design choices. With the help of a prototype system that integrates an interactive Data Analytics and Visualization frontend, we demonstrate a way for the building operators to monitor the flexibility in energy consumption and to develop trust in the decision recommendations by interpreting the rationale behind the ranking.
For effective integration of building operations into the evolving demand response programs of the power grid, real-time decisions concerning the use of building appliances for grid services must excel on …
For effective integration of building operations into the evolving demand response programs of the power grid, real-time decisions concerning the use of building appliances for grid services must excel on multiple criteria, ranging from the added value to occupants' comfort to the quality of the grid services. In this paper, we present a data-driven decision-support framework to dynamically rank load control alternatives in a commercial building, addressing the needs of multiple decision criteria (e.g. occupant comfort, grid service quality) under uncertainties in occupancy patterns. We adopt a stochastic multi-criteria decision algorithm recently applied to prioritize residential on/off loads, and extend it to i) complex load control decisions (e.g. dimming of lights, changing zone temperature set-points) in a commercial building; and ii) systematic integration of zonal occupancy patterns to accurately identify short-term grid service opportunities. We evaluate the performance of the framework for curtailment of air-conditioning, lighting, and plug-loads in a multi-zone office building for a range of design choices. With the help of a prototype system that integrates an interactive \textit{Data Analytics and Visualization} frontend, we demonstrate a way for the building operators to monitor the flexibility in energy consumption and to develop trust in the decision recommendations by interpreting the rationale behind the ranking.
Integration of electronics-based residential appliances and distributed energy resources in homes is expected to rise with grid decarbonization. These devices may introduce significant harmonics into power networks that need to …
Integration of electronics-based residential appliances and distributed energy resources in homes is expected to rise with grid decarbonization. These devices may introduce significant harmonics into power networks that need to be closely studied in order to accurately model and forecast load. However, it can be difficult to obtain harmonic-rich voltage and current data -- necessary for identifying accurate load models -- for residential electrical loads. Recognizing this need, first a set of electronics-based end-use loads is identified and modeled in an electromagnetic transients program tool for a residence. Second, an impedance-varying method is proposed to generate harmonic data that captures harmonic propagation to the supply voltage and harmonic interactions among end-use loads connected to the same supply voltage. Third, a harmonic-rich dataset produced via the proposed methodology is demonstrated to successfully identify frequency coupling matrix-based harmonic load models using the least-squares method. Numerical results demonstrate the accuracy of the model. The impact of limited data availability on model identification is also explored.
This report documents recent technical work on developing and validating stochastic occupancy models in commercial buildings, performed by the Pacific Northwest National Laboratory (PNNL) as part of the Sensor Impact …
This report documents recent technical work on developing and validating stochastic occupancy models in commercial buildings, performed by the Pacific Northwest National Laboratory (PNNL) as part of the Sensor Impact Evaluation and Verification project under the U.S. Department of Energy (DOE) Building Technologies Office (BTO). In this report, we present our work on developing and validating inhomogeneous semi-Markov chain models for generating sequences of zone-level occupancy presence and occupancy counts in a commercial building. Real datasets are used to learn and validate the generative occupancy models. Relevant metrics such as normalized Jensen-Shannon distance (NJSD) are used to demonstrate the ability of the models to express realistic occupancy behavioral patterns.
Recent studies have demonstrated the potential of flexible loads in providing frequency response services. However, uncertainty and variability in various weather-related and end-use behavioral factors often affect the demand-side control …
Recent studies have demonstrated the potential of flexible loads in providing frequency response services. However, uncertainty and variability in various weather-related and end-use behavioral factors often affect the demand-side control performance. This work addresses this problem with the design of a demand-side control to achieve frequency response under load uncertainties. Our approach involves modeling the load uncertainties via stochastic processes that appear as both multiplicative and additive to the system states in closed-loop power system dynamics. Extending the recently developed mean square exponential stability (MSES) results for stochastic systems, we formulate multi-objective linear matrix inequality (LMI)-based optimal control synthesis problems to not only guarantee stochastic stability, but also promote sparsity, enhance closed-loop transient performance, and maximize allowable uncertainties. The fundamental trade-off between the maximum allowable (\textit{critical}) uncertainty levels and the optimal stochastic stabilizing control efforts is established. Moreover, the sparse control synthesis problem is generalized to the realistic power systems scenario in which only partial-state measurements are available. Detailed numerical studies are carried out on IEEE 39-bus system to demonstrate the closed-loop stochastic stabilizing performance of the sparse controllers in enhancing frequency response under load uncertainties; as well as illustrate the fundamental trade-off between the allowable uncertainties and optimal control efforts.
For effective integration of building operations into the evolving demand response programs of the power grid, real-time decisions concerning the use of building appliances for grid services must excel on …
For effective integration of building operations into the evolving demand response programs of the power grid, real-time decisions concerning the use of building appliances for grid services must excel on multiple criteria, ranging from the added value to occupants' comfort to the quality of the grid services. In this paper, we present a data-driven decision-support framework to dynamically rank load control alternatives in a commercial building, addressing the needs of multiple decision criteria (e.g. occupant comfort, grid service quality) under uncertainties in occupancy patterns. We adopt a stochastic multi-criteria decision algorithm recently applied to prioritize residential on/off loads, and extend it to i) complex load control decisions (e.g. dimming of lights, changing zone temperature set-points) in a commercial building; and ii) systematic integration of zonal occupancy patterns to accurately identify short-term grid service opportunities. We evaluate the performance of the framework for curtailment of air-conditioning, lighting, and plug-loads in a multi-zone office building for a range of design choices. With the help of a prototype system that integrates an interactive \textit{Data Analytics and Visualization} frontend, we demonstrate a way for the building operators to monitor the flexibility in energy consumption and to develop trust in the decision recommendations by interpreting the rationale behind the ranking.
Malicious activities on measurements from sensors like Phasor Measurement Units (PMUs) can mislead the control center operator into taking wrong control actions resulting in disruption of operation, financial losses, and …
Malicious activities on measurements from sensors like Phasor Measurement Units (PMUs) can mislead the control center operator into taking wrong control actions resulting in disruption of operation, financial losses, and equipment damage. In particular, false data attacks initiated during power systems transients caused due to abrupt changes in load and generation can fool the conventional model-based detection methods relying on thresholds comparison to trigger an anomaly. In this paper, we propose a Koopman mode decomposition (KMD) based algorithm to detect and identify false data attacks in real-time. The Koopman modes (KMs) are capable of capturing the nonlinear modes of oscillation in the transient dynamics of the power networks and reveal the spatial embedding of both natural and anomalous modes of oscillations in the sensor measurements. The Koopman-based spatio-temporal nonlinear modal analysis is used to filter out the false data injected by an attacker. The performance of the algorithm is illustrated on the IEEE 68-bus test system using synthetic attack scenarios generated on GridSTAGE, a recently developed multivariate spatio-temporal data generation framework for simulation of adversarial scenarios in cyber-physical power systems.
Effective utilization of flexible loads for grid services, while satisfying end-user preferences and constraints, requires an accurate estimation of the aggregated predictive flexibility offered by the electrical loads. Virtual battery …
Effective utilization of flexible loads for grid services, while satisfying end-user preferences and constraints, requires an accurate estimation of the aggregated predictive flexibility offered by the electrical loads. Virtual battery (VB) models are often used to quantify the predictive flexibility in thermostatic loads (e.g. residential air-conditioners, electric water-heaters), which model the temporal evolution of a (virtual) energy state via a first order dynamics including self-dissipation rate, and power and energy capacities as parameters. Uncertainties and lack of information regarding end-usage and equipment models render deterministic VB models impractical. In this paper, we introduce the notion of stochastic VB models, and propose a variational autoencoder-based deep learning algorithm to identify the probability distribution of the VB model parameters. Using available sensors and meters data, the proposed algorithm generates not only point estimates of the VB parameters, but also confidence intervals around those values. Effectiveness of the proposed frameworks is demonstrated on a collection of electric water-heater loads, whose operation is driven by uncertain water usage profiles.
Unlike conventional generators, inverter-based generation do not possess any rotational inertia. While grid-forming inverters can synthesize small (virtual) inertia via advanced feedback control loops, additional control mechanisms are needed to …
Unlike conventional generators, inverter-based generation do not possess any rotational inertia. While grid-forming inverters can synthesize small (virtual) inertia via advanced feedback control loops, additional control mechanisms are needed to ensure safety and security of the power grid during transients. In this paper, we propose novel real-time safety-constrained feedback controllers ("safety filters") for droop-based (grid-forming) inverters to ensure transient security of the grid. The safety filter acts as a buffer between the network operational layer and the inverter-control layer, and only lets those dispatch control signals pass to the inverter droop-controller, which are guaranteed to not violate the safety specifications (frequency, voltage, current limits). Using a distributed barrier certificates method, we construct state-inclusive bounds on the allowable control inputs, which guarantee the satisfaction of transient safety specifications. Sum-of-square programming is used to synthesize the safety filters. Numerical simulation results are provided to illustrate the performance of the proposed safety filter in inverter- based microgrids.
The proliferation of inverter-based generation and advanced sensing, controls, and communication infrastructure have facilitated the accelerated deployment of microgrids. A coordinated network of microgrids can maintain reliable power delivery to …
The proliferation of inverter-based generation and advanced sensing, controls, and communication infrastructure have facilitated the accelerated deployment of microgrids. A coordinated network of microgrids can maintain reliable power delivery to critical facilities during extreme events. Low-inertia offered by the power-electronics interfaced energy resources, however, can present significant challenges to ensuring stable operation of the microgrids. In this work, distributed small-signal stability conditions for inverter-based microgrids are developed that involve the droop-controller parameters and the network parameters (e.g. line impedances, loads). The distributed closed-form parametric stability conditions derived in this paper can be verified in a computationally efficient manner, facilitating the reliable design and operations of networks of microgrids. Dynamic phasor models have been used to capture the effects of electromagnetic transients. Numerical results are presented, along with PSCAD simulations, to validate the analytical stability conditions. Effects of design choices, such as the conductor types, and inverter sizes, on the small-signal stability of inverter-based microgrids are investigated to identify interpretable stable/unstable region estimates.
Unlike conventional generators, inverter-based generation do not possess any rotational inertia. While grid-forming inverters can synthesize small (virtual) inertia via advanced feedback control loops, additional control mechanisms are needed to …
Unlike conventional generators, inverter-based generation do not possess any rotational inertia. While grid-forming inverters can synthesize small (virtual) inertia via advanced feedback control loops, additional control mechanisms are needed to ensure safety and security of the power grid during transients. In this paper, we propose novel real-time safety-constrained feedback controllers (safety filters) for droop-based (grid-forming) inverters to ensure transient security of the grid. The safety filter acts as a buffer between the network operational layer and the inverter-control layer, and only lets those dispatch control signals pass to the inverter droop-controller, which are guaranteed to not violate the safety specifications (frequency, voltage, current limits). Using a distributed barrier certificates method, we construct state-inclusive bounds on the allowable control inputs, which guarantee the satisfaction of transient safety specifications. Sum-of-square programming is used to synthesize the safety filters. Numerical simulation results are provided to illustrate the performance of the proposed filter in inverter-based microgrids.
Effective utilization of flexible loads for grid services, while satisfying end-user preferences and constraints, requires an accurate estimation of the aggregated predictive flexibility offered by the electrical loads. Virtual battery …
Effective utilization of flexible loads for grid services, while satisfying end-user preferences and constraints, requires an accurate estimation of the aggregated predictive flexibility offered by the electrical loads. Virtual battery (VB) models are often used to quantify the predictive flexibility in thermostatic loads (e.g. residential air-conditioners, electric water-heaters), which model the temporal evolution of a (virtual) energy state via a first order dynamics including self-dissipation rate, and power and energy capacities as parameters. Uncertainties and lack of information regarding end-usage and equipment models render deterministic VB models impractical. In this paper, we introduce the notion of stochastic VB models, and propose a \textit{variational autoencoder}-based deep learning algorithm to identify the probability distribution of the VB model parameters. Using available sensors and meters data, the proposed algorithm generates not only point estimates of the VB parameters, but also confidence intervals around those values. Effectiveness of the proposed frameworks is demonstrated on a collection of electric water-heater loads, whose operation is driven by uncertain water usage profiles.
Protection equipment is used to prevent damage to induction motor loads by isolating them from power systems in the event of severe faults. Modeling the response of induction motor loads …
Protection equipment is used to prevent damage to induction motor loads by isolating them from power systems in the event of severe faults. Modeling the response of induction motor loads and their protection is vital for power system planning and operation, especially in understanding system's dynamic performance and stability after a fault occurs. Induction motors are usually equipped with several types of protection with different operation mechanisms, making it challenging to develop adequate yet not overly complex protection models and determine their parameters for aggregate induction motor models. This paper proposes an optimization-based nonlinear regression framework to determine protection model parameters for aggregate induction motor loads in commercial buildings. Using a mathematical abstraction, the task of determining a suitable set of parameters for the protection model in composite load models is formulated as a nonlinear regression problem. Numerical examples are provided to illustrate the application of the framework. Sensitivity studies are presented to demonstrate the impact of lack of available motor load information on the accuracy of the protection models.
Malicious activities on measurements from sensors like Phasor Measurement Units (PMUs) can mislead the control center operator into taking wrong control actions resulting in disruption of operation, financial losses, and …
Malicious activities on measurements from sensors like Phasor Measurement Units (PMUs) can mislead the control center operator into taking wrong control actions resulting in disruption of operation, financial losses, and equipment damage. In particular, false data attacks initiated during power systems transients caused due to abrupt changes in load and generation can fool the conventional model-based detection methods relying on thresholds comparison to trigger an anomaly. In this paper, we propose a Koopman mode decomposition (KMD) based algorithm to detect and identify false data attacks in real-time. The Koopman modes (KMs) are capable of capturing the nonlinear modes of oscillation in the transient dynamics of the power networks and reveal the spatial embedding of both natural and anomalous modes of oscillations in the sensor measurements. The Koopman-based spatio-temporal nonlinear modal analysis is used to filter out the false data injected by an attacker. The performance of the algorithm is illustrated on the IEEE 68-bus test system using synthetic attack scenarios generated on GridSTAGE, a recently developed multivariate spatio-temporal data generation framework for simulation of adversarial scenarios in cyber-physical power systems.
Unlike conventional generators, inverter-based generation do not possess any rotational inertia. While grid-forming inverters can synthesize small (virtual) inertia via advanced feedback control loops, additional control mechanisms are needed to …
Unlike conventional generators, inverter-based generation do not possess any rotational inertia. While grid-forming inverters can synthesize small (virtual) inertia via advanced feedback control loops, additional control mechanisms are needed to ensure safety and security of the power grid during transients. In this paper, we propose novel real-time safety-constrained feedback controllers ("safety filters") for droop-based (grid-forming) inverters to ensure transient security of the grid. The safety filter acts as a buffer between the network operational layer and the inverter-control layer, and only lets those dispatch control signals pass to the inverter droop-controller, which are guaranteed to not violate the safety specifications (frequency, voltage, current limits). Using a distributed barrier certificates method, we construct state-inclusive bounds on the allowable control inputs, which guarantee the satisfaction of transient safety specifications. Sum-of-square programming is used to synthesize the safety filters. Numerical simulation results are provided to illustrate the performance of the proposed filter in inverter-based microgrids.
Effective utilization of flexible loads for grid services, while satisfying end-user preferences and constraints, requires an accurate estimation of the aggregated predictive flexibility offered by the electrical loads. Virtual battery …
Effective utilization of flexible loads for grid services, while satisfying end-user preferences and constraints, requires an accurate estimation of the aggregated predictive flexibility offered by the electrical loads. Virtual battery (VB) models are often used to quantify the predictive flexibility in thermostatic loads (e.g. residential air-conditioners, electric water-heaters), which model the temporal evolution of a (virtual) energy state via a first order dynamics including self-dissipation rate, and power and energy capacities as parameters. Uncertainties and lack of information regarding end-usage and equipment models render deterministic VB models impractical. In this paper, we introduce the notion of stochastic VB models, and propose a \textit{variational autoencoder}-based deep learning algorithm to identify the probability distribution of the VB model parameters. Using available sensors and meters data, the proposed algorithm generates not only point estimates of the VB parameters, but also confidence intervals around those values. Effectiveness of the proposed frameworks is demonstrated on a collection of electric water-heater loads, whose operation is driven by uncertain water usage profiles.
The proliferation of inverter-based generation and advanced sensing, controls, and communication infrastructure have facilitated the accelerated deployment of microgrids. A coordinated network of microgrids can maintain reliable power delivery to …
The proliferation of inverter-based generation and advanced sensing, controls, and communication infrastructure have facilitated the accelerated deployment of microgrids. A coordinated network of microgrids can maintain reliable power delivery to critical facilities during extreme events. Low inertia offered by the power electronics interfaced energy resources however, can present significant challenges to ensuring stable operation of the microgrids. In this work, distributed small-signal stability conditions for inverter-based microgrids are developed that involve the droop controller parameters and the network parameters such as line impedances, loads, etc. The distributed closed-form parametric stability conditions derived in this paper can be verified in a computationally efficient manner, facilitating the reliable design and operations of networks of microgrids. Dynamic phasor models have been used to capture the effects of electromagnetic transients. Numerical results are presented, along with PSCAD simulations, to validate the analytical stability conditions. Effects of design choices, such as the conductor types, and inverter sizes, on the small-signal stability of inverter-based microgrids are investigated to identify interpretable stable or unstable region estimates.
Protection equipment is used to prevent damage to induction motor loads by isolating them from power systems in the event of severe faults. Modeling the response of induction motor loads …
Protection equipment is used to prevent damage to induction motor loads by isolating them from power systems in the event of severe faults. Modeling the response of induction motor loads and their protection is vital for power system planning and operation, especially in understanding system's dynamic performance and stability after a fault occurs. Induction motors are usually equipped with several types of protection with different operation mechanisms, making it challenging to develop adequate yet not overly complex protection models and determine their parameters for aggregate induction motor models. This paper proposes an optimization-based nonlinear regression framework to determine protection model parameters for aggregate induction motor loads in commercial buildings. Using a mathematical abstraction, the task of determining a suitable set of parameters for the protection model in composite load models is formulated as a nonlinear regression problem. Numerical examples are provided to illustrate the application of the framework. Sensitivity studies are presented to demonstrate the impact of lack of available motor load information on the accuracy of the protection models.
The potential of distributed energy resources in providing grid services can be maximized with the recent advancements in demand side control. Effective utilization of this control strategy requires the knowledge …
The potential of distributed energy resources in providing grid services can be maximized with the recent advancements in demand side control. Effective utilization of this control strategy requires the knowledge of aggregate flexibility of the distributed energy resources (DERs). Recent works have shown that the aggregate flexibility of DERs can be modeled as a virtual battery (VB) whose state evolution is governed by a first order system including self dissipation. The VB parameters (self dissipation rate, energy capacity) are obtained by solving an optimization problem which minimizes the tracking performance of the ensemble and the proposed first order model. For the identified first order model, time varying power limits are calculated using binary search algorithms. Finally, this proposed framework is demonstrated for different homogeneous and heterogeneous ensembles consisting of air conditioners (ACs) and electric water heaters (EWHs).
This paper investigates the use of Infrastructure-To-Vehicle (I2V) communication to generate routing suggestions for drivers in transportation systems, with the goal of optimizing a measure of overall network congestion. We …
This paper investigates the use of Infrastructure-To-Vehicle (I2V) communication to generate routing suggestions for drivers in transportation systems, with the goal of optimizing a measure of overall network congestion. We define link-wise levels of trust to tolerate the non-cooperative behavior of part of the driver population, and we propose a realtime optimization mechanism that adapts to the instantaneous network conditions and to sudden changes in the levels of trust. Our framework allows us to quantify the improvement in travel time in relation to the degree at which drivers follow the routing suggestions. We then study the resilience of the system, measured as the smallest change in routing choices that results in roads reaching their maximum capacity. Interestingly, our findings suggest that fluctuations in the extent to which drivers follow the provided routing suggestions can cause failures of certain links. These results imply that the benefits of using Infrastructure-To-Vehicle communication come at the cost of new fragilities, that should be appropriately addressed in order to guarantee the reliable operation of the infrastructure.
Low-to-medium voltage distribution networks are experiencing rising levels of distributed energy resources, including renewable generation, along with improved sensing, communication, and automation infrastructure. As such, state estimation methods for distribution …
Low-to-medium voltage distribution networks are experiencing rising levels of distributed energy resources, including renewable generation, along with improved sensing, communication, and automation infrastructure. As such, state estimation methods for distribution systems are becoming increasingly relevant as a means to enable better control strategies that can both leverage the benefits and mitigate the risks associated with high penetration of variable and uncertain distributed generation resources. The primary challenges of this problem include modeling complexities (nonlinear, nonconvex power-flow equations), limited availability of sensor measurements, and high penetration of uncertain renewable generation. This paper formulates the distribution system state estimation as a nonlinear, weighted, least squares problem, based on sensor measurements as well as forecast data (both load and generation). We investigate the sensitivity of state estimator accuracy to (load/generation) forecast uncertainties, sensor accuracy, and sensor coverage levels.
The Koopman operator has recently garnered much attention for its value in dynamical systems analysis and data-driven model discovery. However, its application has been hindered by the computational complexity of …
The Koopman operator has recently garnered much attention for its value in dynamical systems analysis and data-driven model discovery. However, its application has been hindered by the computational complexity of extended dynamic mode decomposition; this requires a combinatorially large basis set to adequately describe many nonlinear systems of interest, e.g. cyber-physical infrastructure systems, biological networks, social systems, and fluid dynamics. Often the dictionaries generated for these problems are manually curated, requiring domain-specific knowledge and painstaking tuning. In this paper we introduce a computational framework for learning Koopman operators of nonlinear dynamical systems using deep learning. We show that this novel method automatically selects efficient deep dictionaries, requiring much lower dimensional dictionaries while outperforming state-of-the-art methods. We benchmark this method on partially observed nonlinear systems, including the glycolytic oscillator and show it is able to predict on test data quantitatively 100 steps into the future, using only a single timepoint as an initial condition, and quantitative oscillatory behavior 400 steps into the future.
As modern engineering systems grow in complexity, attitudes toward a modular design approach become increasingly more favorable. A key challenge to a modular design approach is the certification of robust …
As modern engineering systems grow in complexity, attitudes toward a modular design approach become increasingly more favorable. A key challenge to a modular design approach is the certification of robust stability under uncertainties in the rest of the network. In this paper, we consider the problem of identifying the parametric region, which guarantees stability of the connected module in the robust sense under uncertainties. We derive the conditions under which the robust stability of the connected module is guaranteed for some values of the design parameters, and present a sum-of-squares (SOS) optimization-based algorithm to identify such a parametric region for polynomial systems. Using the example of an inverter-based microgrid, we show how this parametric region changes with variations in the level of uncertainties in the network.
Abstract SeDuMi is an add-on for MATLAB, which lets you solve optimization problems with linear, quadratic and semidefiniteness constraints. It is possible to have complex valued data and variables in …
Abstract SeDuMi is an add-on for MATLAB, which lets you solve optimization problems with linear, quadratic and semidefiniteness constraints. It is possible to have complex valued data and variables in SeDuMi. Moreover, large scale optimization problems are solved efficiently, by exploiting sparsity. This paper describes how to work with this toolbox. Keywords: Symmetric conesemidefinite programmingsecond order cone programmingself-dualityMATLABSeDuMi *MATLAB is a registered trademark of the The Mathworks, Inc. *MATLAB is a registered trademark of the The Mathworks, Inc. Notes *MATLAB is a registered trademark of the The Mathworks, Inc.
. From a theoretical viewpoint, the GPM has developments and impact in var-ious area of Mathematics like algebra, Fourier analysis, functional analysis, operator theory, probabilityand statistics, to cite a few. …
. From a theoretical viewpoint, the GPM has developments and impact in var-ious area of Mathematics like algebra, Fourier analysis, functional analysis, operator theory, probabilityand statistics, to cite a few. In addition, and despite its rather simple and short formulation, the GPMhas a large number of important applications in various fields like optimization, probability, mathematicalfinance, optimal control, control and signal processing, chemistry, cristallography, tomography, quantumcomputing, etc.In its full generality, the GPM is untractable numerically. However when K is a compact basic semi-algebraic set, and the functions involved are polynomials (and in some cases piecewise polynomials orrational functions), then the situation is much nicer. Indeed, one can define a systematic numerical schemebased on a hierarchy of semidefinite programs, which provides a monotone sequence that converges tothe optimal value of the GPM. (A semidefinite program is a convex optimization problem which up toarbitrary fixed precision, can be solved in polynomial time.) Sometimes finite convergence may evenocccur.In the talk, we will present the semidefinite programming methodology to solve the GPM and describein detail several applications of the GPM (notably in optimization, probability, optimal control andmathematical finance).R´ef´erences[1] J.B. Lasserre, Moments, Positive Polynomials and their Applications, Imperial College Press, inpress.[2] J.B. Lasserre, A Semidefinite programming approach to the generalized problem of moments,Math. Prog. 112 (2008), pp. 65–92.
We present a methodology for the algorithmic construction of Lyapunov functions for the transient stability analysis of classical power system models. The proposed methodology uses recent advances in the theory …
We present a methodology for the algorithmic construction of Lyapunov functions for the transient stability analysis of classical power system models. The proposed methodology uses recent advances in the theory of positive polynomials, semidefinite programming, and sum of squares decomposition, which have been powerful tools for the analysis of systems with polynomial vector fields. In order to apply these techniques to power grid systems described by trigonometric nonlinearities we use an algebraic reformulation technique to recast the system's dynamics into a set of polynomial differential algebraic equations. We demonstrate the application of these techniques to the transient stability analysis of power systems by estimating the region of attraction of the stable operating point. An algorithm to compute the local stability Lyapunov function is described together with an optimization algorithm designed to improve this estimate.
Higher penetration of renewable generation will increase the demand for adequate (and cost-effective) controllable resources on the grid that can mitigate and contain the contingencies locally before it can cause …
Higher penetration of renewable generation will increase the demand for adequate (and cost-effective) controllable resources on the grid that can mitigate and contain the contingencies locally before it can cause a network-wide collapse. However, end-use constraints can potentially lead to load unavailability when an event occurs, leading to unreliable demand response services. Sensors measurements and knowledge of the local load dynamics could be leveraged to improve the performance of load control algorithms. In the context of hierarchical frequency response using ensemble of switching loads, we present a metric to evaluate the fitness of each device in successfully providing the ancillary service. Furthermore a fitness-based assignment of control set-points is formulated which achieves reliable performance under different operating conditions. Monte Carlo simulations of ensembles of electric water heaters and residential air-conditioners are performed to evaluate the proposed control algorithm.
As the penetration of intermittent energy sources grows substantially, loads will be required to play an increasingly important role in compensating the fast time-scale fluctuations in generated power. Recent numerical …
As the penetration of intermittent energy sources grows substantially, loads will be required to play an increasingly important role in compensating the fast time-scale fluctuations in generated power. Recent numerical modeling of thermostatically controlled loads (TCLs) has demonstrated that such load following is feasible, but analytical models that satisfactorily quantify the aggregate power consumption of a group of TCLs are desired to enable controller design. We develop such a model for the aggregate power response of a homogeneous population of TCLs to uniform variation of all TCL setpoints. A linearized model of the response is derived, and a linear quadratic regulator (LQR) has been designed. Using the TCL setpoint as the control input, the LQR enables aggregate power to track reference signals that exhibit step, ramp and sinusoidal variations. Although much of the work assumes a homogeneous population of TCLs with deterministic dynamics, we also propose a method for probing the dynamics of systems where load characteristics are not well known.
Inverter-interfaced microgrids differ from the traditional power systems due to their lack of inertia. Vanishing timescale separation between voltage and frequency dynamics makes it critical that faster-timescale stabilizing control laws …
Inverter-interfaced microgrids differ from the traditional power systems due to their lack of inertia. Vanishing timescale separation between voltage and frequency dynamics makes it critical that faster-timescale stabilizing control laws also guarantee by-construction the satisfaction of voltage limits during transients. In this article, we apply a barrier functions method to compute distributed active and reactive power setpoint control laws that certify satisfaction of voltage limits during transients. Using sum-of-squares optimization tools, we propose an algorithmic construction of these control laws. Numerical simulations are provided to illustrate the proposed method.
A method for analyzing large-scale nonlinear dynamical systems by decomposing them into coupled lower order subsystems that are sufficiently simple for computational analysis is presented. It is shown that the …
A method for analyzing large-scale nonlinear dynamical systems by decomposing them into coupled lower order subsystems that are sufficiently simple for computational analysis is presented. It is shown that the decomposition approach can be used to scale the Sum of Squares programming framework for nonlinear systems analysis. The method constructs subsystem Lyapunov functions which are used to form a composite Lyapunov function for the whole system. Further computational savings are achieved if a method based on sparsity maximization is used to obtain the subsystem Lyapunov functions.
With increasing availability of communication and control infrastructure at the distribution systems, it is expected that the distributed energy resources (DERs) will take an active part in future power systems …
With increasing availability of communication and control infrastructure at the distribution systems, it is expected that the distributed energy resources (DERs) will take an active part in future power systems operations. One of the main challenges associated with integration of DERs in grid planning and control is in estimating the available flexibility in a collection of (heterogeneous) DERs, each of which may have local constraints that vary over time. In this work, we present a geometric approach for approximating the flexibility of a DER in modulating its active and reactive power consumption. The proposed method is agnostic about the type and model of the DERs, thereby facilitating a plug-and-play approach, and allows scalable aggregation of the flexibility of a collection of (heterogeneous) DERs at the distributed system level. Simulation results are presented to demonstrate the performance of the proposed method.
Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD) and its generalization, the extended-DMD …
Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD) and its generalization, the extended-DMD (EDMD), are becoming increasingly popular in practical applications. The EDMD improves upon the classical DMD by the inclusion of a flexible choice of dictionary of observables that spans a finite dimensional subspace on which the Koopman operator can be approximated. This enhances the accuracy of the solution reconstruction and broadens the applicability of the Koopman formalism. Although the convergence of the EDMD has been established, applying the method in practice requires a careful choice of the observables to improve convergence with just a finite number of terms. This is especially difficult for high dimensional and highly nonlinear systems. In this paper, we employ ideas from machine learning to improve upon the EDMD method. We develop an iterative approximation algorithm which couples the EDMD with a trainable dictionary represented by an artificial neural network. Using the Duffing oscillator and the Kuramoto Sivashinsky PDE as examples, we show that our algorithm can effectively and efficiently adapt the trainable dictionary to the problem at hand to achieve good reconstruction accuracy without the need to choose a fixed dictionary a priori. Furthermore, to obtain a given accuracy we require fewer dictionary terms than EDMD with fixed dictionaries. This alleviates an important shortcoming of the EDMD algorithm and enhances the applicability of the Koopman framework to practical problems.
Recent studies have shown that the aggregated dynamic flexibility of an ensemble of thermostatic loads can be modeled in the form of a virtual battery. The existing methods for computing …
Recent studies have shown that the aggregated dynamic flexibility of an ensemble of thermostatic loads can be modeled in the form of a virtual battery. The existing methods for computing the virtual battery parameters require the knowledge of the first-principle models and parameter values of the loads in the ensemble. In real-world applications, however, it is likely that the only available information are end-use measurements such as power consumption, room temperature, device on/off status, etc., while very little about the individual load models and parameters are known. We propose a transfer learning based deep network framework for calculating virtual battery state of a given ensemble of flexible thermostatic loads, from the available end-use measurements. This proposed framework extracts first order virtual battery model parameters for the given ensemble. We illustrate the effectiveness of this novel framework on different ensembles of ACs and WHs.
As modern engineering systems grow in complexity, attitudes toward a modular design approach become increasingly more favorable. A key challenge to a modular design approach is the certification of robust …
As modern engineering systems grow in complexity, attitudes toward a modular design approach become increasingly more favorable. A key challenge to a modular design approach is the certification of robust stability under uncertainties in the rest of the network. In this paper, we consider the problem of identifying the parametric region, which guarantees stability of the connected module in the robust sense under uncertainties. We derive the conditions under which the robust stability of the connected module is guaranteed for some values of the design parameters, and present a sum-of-squares (SOS) optimization-based algorithm to identify such a parametric region for polynomial systems. Using the example of an inverter-based microgrid, we show how this parametric region changes with variations in the level of uncertainties in the network.
We develop a new method which extends dynamic mode decomposition (DMD) to incorporate the effect of control to extract low-order models from high-dimensional, complex systems. DMD finds spatial-temporal coherent modes, …
We develop a new method which extends dynamic mode decomposition (DMD) to incorporate the effect of control to extract low-order models from high-dimensional, complex systems. DMD finds spatial-temporal coherent modes, connects local-linear analysis to nonlinear operator theory, and provides an equation-free architecture which is compatible with compressive sensing. In actuated systems, DMD is incapable of producing an input-output model; moreover, the dynamics and the modes will be corrupted by external forcing. Our new method, dynamic mode decomposition with control (DMDc), capitalizes on all of the advantages of DMD and provides the additional innovation of being able to disambiguate between the underlying dynamics and the effects of actuation, resulting in accurate input-output models. The method is data-driven in that it does not require knowledge of the underlying governing equations---only snapshots in time of observables and actuation data from historical, experimental, or black-box simulations. We demonstrate the method on high-dimensional dynamical systems, including a model with relevance to the analysis of infectious disease data with mass vaccination (actuation).
Coordinated aggregation of a large population of thermostatically controlled loads (TCLs) presents a great potential to provide various ancillary services to the grid. One of the key challenges of integrating …
Coordinated aggregation of a large population of thermostatically controlled loads (TCLs) presents a great potential to provide various ancillary services to the grid. One of the key challenges of integrating TCLs into system-level operation and control is developing a simple and portable model to accurately capture their aggregate flexibility. In this paper, we propose a geometric approach to model the aggregate flexibility of TCLs. We show that the set of admissible power profiles of an individual TCL is a polytope, and their aggregate flexibility is the Minkowski sum of the individual polytopes. In order to represent their aggregate flexibility in an intuitive way and achieve a tractable approximation, we develop optimization-based algorithms to approximate the polytopes by the homothets of a given convex set. As a special application, this set is chosen as a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">virtual battery model</i> , and the corresponding optimal approximations are solved efficiently by equivalent linear programming problems. Numerical results show that our algorithms yield significant improvement in characterizing the aggregate flexibility over existing modeling methods. We also conduct case studies to demonstrate the efficacy of our approaches by coordinating TCLs to track a frequency regulation signal from the Pennsylvania–New Jersey–Maryland Interconnection.
Proper modeling of inverter-based microgrids is crucial for accurate assessment of stability boundaries. It has been recently realized that the stability conditions for such microgrids are significantly different from those …
Proper modeling of inverter-based microgrids is crucial for accurate assessment of stability boundaries. It has been recently realized that the stability conditions for such microgrids are significantly different from those known for large-scale power systems. In particular, the network dynamics, despite its fast nature, appears to have major influence on stability of slower modes. While detailed models are available, they are both computationally expensive and not transparent enough to provide an insight into the instability mechanisms and factors. In this paper, a computationally efficient and accurate reduced-order model is proposed for modeling inverter-based microgrids. The developed model has a structure similar to quasi-stationary model and at the same time properly accounts for the effects of network dynamics. The main factors affecting microgrid stability are analyzed using the developed reduced-order model and shown to be unique for microgrids, having no direct analogy in large-scale power systems. Particularly, it has been discovered that the stability limits for the conventional droop-based system are determined by the ratio of inverter rating to network capacity, leading to a smaller stability region for microgrids with shorter lines. Finally, the results are verified with different models based on both frequency and time domain analyses.
Potential of electrical loads in providing grid ancillary services is often limited due to the uncertainties associated with the load behavior. A knowledge of the expected uncertainties with a load …
Potential of electrical loads in providing grid ancillary services is often limited due to the uncertainties associated with the load behavior. A knowledge of the expected uncertainties with a load control program would invariably yield to better informed control policies, opening up the possibility of extracting the maximal load control potential without affecting grid operations. In the context of frequency responsive load control, a probabilistic uncertainty analysis framework is presented to quantify the expected error between the target and actual load response, under uncertainties in the load dynamics. A closed-form expression of an optimal demand flexibility, minimizing the expected error in actual and committed flexibility, is provided. Analytical results are validated through Monte Carlo simulations of ensembles of electric water heaters.
Recently, sum-of-squares (SOS) based methods have been used for the stability analysis and control synthesis of polynomial dynamical systems. This analysis framework was also extended to non-polynomial dynamical systems, including …
Recently, sum-of-squares (SOS) based methods have been used for the stability analysis and control synthesis of polynomial dynamical systems. This analysis framework was also extended to non-polynomial dynamical systems, including power systems, using an algebraic reformulation technique that recasts the system's dynamics into a set of polynomial differential algebraic equations. Nevertheless, for large scale dynamical systems this method becomes inapplicable due to its computational complexity. For this reason we develop a subsystem based stability analysis approach using vector Lyapunov functions and introduce a parallel and scalable algorithm to infer the stability of the interconnected system with the help of the subsystem Lyapunov functions. Furthermore, we design adaptive and distributed control laws that guarantee asymptotic stability under a given external disturbance. Finally, we apply this algorithm for the stability analysis and control synthesis of a network preserving power system.
We propose a novel operator-theoretic framework to study global stability of nonlinear systems. Based on the spectral properties of the so-called Koopman operator, our approach can be regarded as a …
We propose a novel operator-theoretic framework to study global stability of nonlinear systems. Based on the spectral properties of the so-called Koopman operator, our approach can be regarded as a natural extension of classic linear stability analysis to nonlinear systems. The main results establish the (necessary and sufficient) relationship between the existence of specific eigenfunctions of the Koopman operator and the global stability property of hyperbolic fixed points and limit cycles. These results are complemented with numerical methods which are used to estimate the region of attraction of the fixed point or to prove in a systematic way global stability of the attractor within a given region of the state space.
The proliferation of inverter-based generation and advanced sensing, controls, and communication infrastructure have facilitated the accelerated deployment of microgrids. A coordinated network of microgrids can maintain reliable power delivery to …
The proliferation of inverter-based generation and advanced sensing, controls, and communication infrastructure have facilitated the accelerated deployment of microgrids. A coordinated network of microgrids can maintain reliable power delivery to critical facilities during extreme events. Low-inertia offered by the power-electronics interfaced energy resources, however, can present significant challenges to ensuring stable operation of the microgrids. In this work, distributed small-signal stability conditions for inverter-based microgrids are developed that involve the droop-controller parameters and the network parameters (e.g. line impedances, loads). The distributed closed-form parametric stability conditions derived in this paper can be verified in a computationally efficient manner, facilitating the reliable design and operations of networks of microgrids. Dynamic phasor models have been used to capture the effects of electromagnetic transients. Numerical results are presented, along with PSCAD simulations, to validate the analytical stability conditions. Effects of design choices, such as the conductor types, and inverter sizes, on the small-signal stability of inverter-based microgrids are investigated to identify interpretable stable/unstable region estimates.
Spectral decomposition of the Koopman operator is attracting attention as a tool for the analysis of nonlinear dynamical systems. Dynamic mode decomposition is a popular numerical algorithm for Koopman spectral …
Spectral decomposition of the Koopman operator is attracting attention as a tool for the analysis of nonlinear dynamical systems. Dynamic mode decomposition is a popular numerical algorithm for Koopman spectral analysis; however, we often need to prepare nonlinear observables manually according to the underlying dynamics, which is not always possible since we may not have any a priori knowledge about them. In this paper, we propose a fully data-driven method for Koopman spectral analysis based on the principle of learning Koopman invariant subspaces from observed data. To this end, we propose minimization of the residual sum of squares of linear least-squares regression to estimate a set of functions that transforms data into a form in which the linear regression fits well. We introduce an implementation with neural networks and evaluate performance empirically using nonlinear dynamical systems and applications.
Sum-of-squares (SOS) methods have been shown to be very useful in computing polynomial Lyapunov functions for systems of reasonably small size. However for large scale systems it is necessary to …
Sum-of-squares (SOS) methods have been shown to be very useful in computing polynomial Lyapunov functions for systems of reasonably small size. However for large scale systems it is necessary to use a scalable alternative using vector Lyapunov functions. Earlier works have shown that under certain conditions the stability of an interconnected system can be studied through suitable comparison equations. However finding such comparison equations can be non-trivial. In this work we propose an SOS based systematic procedure to directly compute the comparison equations for interconnected system with polynomial dynamics. With an example of interacting Van der Pol systems, we illustrate how this facilitates a scalable and parallel approach to stability analysis.
Koopman operator is a composition operator defined for a dynamical system described by nonlinear differential or difference equation. Although the original system is nonlinear and evolves on a finite-dimensional state …
Koopman operator is a composition operator defined for a dynamical system described by nonlinear differential or difference equation. Although the original system is nonlinear and evolves on a finite-dimensional state space, the Koopman operator itself is linear but infinite-dimensional (evolves on a function space). This linear operator captures the full information of the dynamics described by the original nonlinear system. In particular, spectral properties of the Koopman operator play a crucial role in analyzing the original system. In the first part of this paper, we review the so-called Koopman operator theory for nonlinear dynamical systems, with emphasis on modal decomposition and computation that are direct to wide applications. Then, in the second part, we present a series of applications of the Koopman operator theory to power systems technology. The applications are established as data-centric methods, namely, how to use massive quantities of data obtained numerically and experimentally, through spectral analysis of the Koopman operator: coherency identification of swings in coupled synchronous generators, precursor diagnostic of instabilities in the coupled swing dynamics, and stability assessment of power systems without any use of mathematical models. Future problems of this research direction are identified in the last concluding part of this paper.
This article considers power networks governed by swing nonlinear dynamics and subject to disturbances. We develop a bilayered control strategy for a subset of buses that simultaneously guarantees transient frequency …
This article considers power networks governed by swing nonlinear dynamics and subject to disturbances. We develop a bilayered control strategy for a subset of buses that simultaneously guarantees transient frequency safety of each individual bus and asymptotic stability of the entire network. The bottom layer is a model predictive controller that, based on periodically sampled system information, optimizes control resources to have transient frequency evolve close to a safe desired interval. The top layer is a real-time controller assisting the bottom-layer controller to guarantee that transient frequency safety is actually achieved. We show that control signals at both layers are Lipschitz in the state and do not jeopardize network stability. Furthermore, we carefully characterize the information requirements at each bus necessary to implement the controller and employ saddle-point dynamics to introduce a distributed implementation that only requires information exchange with up to two-hop neighbors in the power network. Simulations on the IEEE 39-bus power network illustrate our results.
Safety critical systems involve the tight coupling between potentially conflicting control objectives and safety constraints. As a means of creating a formal framework for controlling systems of this form, and …
Safety critical systems involve the tight coupling between potentially conflicting control objectives and safety constraints. As a means of creating a formal framework for controlling systems of this form, and with a view toward automotive applications, this paper develops a methodology that allows safety conditions-expressed as control barrier functions-to be unified with performance objectives-expressed as control Lyapunov functions-in the context of real-time optimization-based controllers. Safety conditions are specified in terms of forward invariance of a set, and are verified via two novel generalizations of barrier functions; in each case, the existence of a barrier function satisfying Lyapunov-like conditions implies forward invariance of the set, and the relationship between these two classes of barrier functions is characterized. In addition, each of these formulations yields a notion of control barrier function (CBF), providing inequality constraints in the control input that, when satisfied, again imply forward invariance of the set. Through these constructions, CBFs can naturally be unified with control Lyapunov functions (CLFs) in the context of a quadratic program (QP); this allows for the achievement of control objectives (represented by CLFs) subject to conditions on the admissible states of the system (represented by CBFs). The mediation of safety and performance through a QP is demonstrated on adaptive cruise control and lane keeping, two automotive control problems that present both safety and performance considerations coupled with actuator bounds.
While depth tends to improve network performances, it also makes gradient-based training more difficult since deeper networks tend to be more non-linear. The recently proposed knowledge distillation approach is aimed …
While depth tends to improve network performances, it also makes gradient-based training more difficult since deeper networks tend to be more non-linear. The recently proposed knowledge distillation approach is aimed at obtaining small and fast-to-execute models, and it has shown that a student network could imitate the soft output of a larger teacher network or ensemble of networks. In this paper, we extend this idea to allow the training of a student that is deeper and thinner than the teacher, using not only the outputs but also the intermediate representations learned by the teacher as hints to improve the training process and final performance of the student. Because the student intermediate hidden layer will generally be smaller than the teacher's intermediate hidden layer, additional parameters are introduced to map the student hidden layer to the prediction of the teacher hidden layer. This allows one to train deeper students that can generalize better or run faster, a trade-off that is controlled by the chosen student capacity. For example, on CIFAR-10, a deep student network with almost 10.4 times less parameters outperforms a larger, state-of-the-art teacher network.
Stability analysis tools are essential to understanding and controlling any engineering system. Recently, sum-of-squares (SOS) based methods have been used to compute Lyapunov based estimates for the region-of-attraction (ROA) of …
Stability analysis tools are essential to understanding and controlling any engineering system. Recently, sum-of-squares (SOS) based methods have been used to compute Lyapunov based estimates for the region-of-attraction (ROA) of polynomial dynamical systems. But for a real-life large scale dynamical system this method becomes inapplicable because of growing computational burden. In such a case, it is important to develop a subsystem based stability analysis approach which is the focus of the work presented here. A parallel and scalable algorithm is used to infer stability of an interconnected system, with the help of the subsystem Lyapunov functions. Locally computable control laws are proposed to guarantee asymptotic stability under a given disturbance.
Motivated by the need to simultaneously guarantee safety and stability of safety-critical dynamical systems, we construct permissive barrier certificates in this paper that explicitly maximize the region where the system …
Motivated by the need to simultaneously guarantee safety and stability of safety-critical dynamical systems, we construct permissive barrier certificates in this paper that explicitly maximize the region where the system can be stabilized without violating safety constraints. An iterative search algorithm is developed to search for the maximum volume barrier certified region of safe stabilization. The barrier certified region, which is allowed to take any arbitrary shape, is proved to be strictly larger than safe regions generated with Lyapunov sublevel set based methods. The proposed approach effectively unites a Lyapunov function with multiple barrier functions that might not be compatible with each other. Simulation results of the iterative search algorithm demonstrate the effectiveness of the proposed method.
The Koopman operator has recently garnered much attention for its value in dynamical systems analysis and data-driven model discovery. However, its application has been hindered by the computational complexity of …
The Koopman operator has recently garnered much attention for its value in dynamical systems analysis and data-driven model discovery. However, its application has been hindered by the computational complexity of extended dynamic mode decomposition; this requires a combinatorially large basis set to adequately describe many nonlinear systems of interest, e.g. cyber-physical infrastructure systems, biological networks, social systems, and fluid dynamics. Often the dictionaries generated for these problems are manually curated, requiring domain-specific knowledge and painstaking tuning. In this paper we introduce a computational framework for learning Koopman operators of nonlinear dynamical systems using deep learning. We show that this novel method automatically selects efficient deep dictionaries, requiring much lower dimensional dictionaries while outperforming state-of-the-art methods. We benchmark this method on partially observed nonlinear systems, including the glycolytic oscillator and show it is able to predict on test data quantitatively 100 steps into the future, using only a single timepoint as an initial condition, and quantitative oscillatory behavior 400 steps into the future.
Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques …
Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems. This survey aims to provide the reader with a significant overview of the LMI techniques that are used in control systems for tackling optimization problems over polynomials, describing approaches such as decomposition in sum of squares, Positivstellensatz, theory of moments, Pólya's theorem, and matrix dilation. Moreover, it aims to provide a collection of the essential problems in control systems where these LMI techniques are used, such as stability and performance investigations in nonlinear systems, uncertain systems, time-delay systems, and genetic regulatory networks. It is expected that this survey may be a concise useful reference for all readers.
SOSTOOLS v3.00 is the latest release of the freely available MATLAB toolbox for formulating and solving sum of squares (SOS) optimization problems. Such problems arise naturally in the analysis and …
SOSTOOLS v3.00 is the latest release of the freely available MATLAB toolbox for formulating and solving sum of squares (SOS) optimization problems. Such problems arise naturally in the analysis and control of nonlinear dynamical systems, but also in other areas such as combinatorial optimization. Highlights of the new release include the ability to create polynomial matrices and formulate polynomial matrix inequalities, compatibility with MuPAD, the new MATLAB symbolic engine, as well as the multipoly toolbox v2.01. SOSTOOLS v3.00 can interface with five semidefinite programming solvers, and includes ten demonstration examples.
The flexibility of active ($p$) and reactive power ($q$) consumption in distributed energy resources (DERs) can be represented as a (potentially non-convex) set of points in the $p$-$q$ plane. Modeling …
The flexibility of active ($p$) and reactive power ($q$) consumption in distributed energy resources (DERs) can be represented as a (potentially non-convex) set of points in the $p$-$q$ plane. Modeling of the aggregated flexibility in a heterogeneous ensemble of DERs as a Minkowski sum (M-sum) is computationally intractable even for moderately sized populations. In this article, we propose a scalable method of computing the M-sum of the flexibility domains of a heterogeneous ensemble of DERs, which are allowed to be non-convex, non-compact. In particular, the proposed algorithm computes a guaranteed superset of the true M-sum, with desired accuracy. The worst-case complexity of the algorithm is computed. Special cases are considered, and it is shown that under certain scenarios, it is possible to achieve a complexity that is linear with the size of the ensemble. Numerical examples are provided by computing the aggregated flexibility of different mix of DERs under varying scenarios.
Protection strategies for transmission and distribution systems have been extensively investigated to facilitate better coordination of physical protection devices. A diverse range of functional motors with dedicated protection schemes are …
Protection strategies for transmission and distribution systems have been extensively investigated to facilitate better coordination of physical protection devices. A diverse range of functional motors with dedicated protection schemes are being used more and more in commercial, residential, and industrial buildings. This paper focuses on simulating several of the most popular protection schemes using the Electro-Magnetic Transient Program (EMTP) model for three-phase and single-phase induction motors in existing commercial buildings connected to typical distribution feeders. To investigate the behaviors of single-phase motors stalling, the actions of motor protection and reconnections, and the impacts of device-level protection on system-level dynamics, we imposed voltage depressions at the head of a feeder fully loaded with functional induction motors. Several distribution feeders are represented in a standard IEEE 39-bus transmission system to simulate fault-induced delayed voltage recovery (FIDVR) and explore mitigation strategies by optimally configuring the building-level motor protection settings.
Advantages of distributed control have been extensively discussed, while its impacts on microgrid performance and stability, especially in the case of communication latency, have not been explicitly studied or fully …
Advantages of distributed control have been extensively discussed, while its impacts on microgrid performance and stability, especially in the case of communication latency, have not been explicitly studied or fully understood. This paper addresses this gap by proposing a generalized theoretical framework for small-signal stability analysis and performance evaluation for microgrids using distributed control. The proposed framework synthesizes generator and load frequency-domain characteristics, primary and secondary control loops, as well as the communication latency into a frequency-domain representation which is further evaluated by the generalized Nyquist theorem. In addition, various parameters and their impacts on microgrid dynamic performance are investigated and summarized into guidelines to help better design the system. Case studies demonstrate the effectiveness of the proposed approach.
In this paper, we consider the problem of quantifying controllability and observability of a nonlinear discrete time dynamical system. We introduce the Koopman operator as a canonical representation of the …
In this paper, we consider the problem of quantifying controllability and observability of a nonlinear discrete time dynamical system. We introduce the Koopman operator as a canonical representation of the system and apply a lifting technique to compute gramians in the space of full-state observables. We illustrate the properties of these gramians and identify several relationships with canonical results on local controllability and observability. Once defined, we show that these gramians can be balanced through a change of coordinates on the observables space, which in turn allows for direct application of balanced truncation. Throughout the paper, we highlight the aspects of our approach with an example nonlinear system.
The stability of an equilibrium point of a nonlinear dynamical system is typically determined using Lyapunov theory. This requires the construction of an energy-like function, termed a Lyapunov function, which …
The stability of an equilibrium point of a nonlinear dynamical system is typically determined using Lyapunov theory. This requires the construction of an energy-like function, termed a Lyapunov function, which satisfies certain positivity conditions. Unlike linear dynamical systems, there is no algorithmic method for constructing Lyapunov functions for general nonlinear systems. However, if the systems of interest evolve according to polynomial vector fields and the Lyapunov functions are constrained to be sum-of-squares polynomials then stability verification can be cast as a semidefinite (convex) optimization programme. In this paper we describe recent advances in sum-of-squares programming that facilitate advanced stability analysis and control design.
The nonlinearity of dynamics in systems biology makes it hard to infer them from experimental data. Simple linear models are computationally efficient, but cannot incorporate these important nonlinearities. An adaptive …
The nonlinearity of dynamics in systems biology makes it hard to infer them from experimental data. Simple linear models are computationally efficient, but cannot incorporate these important nonlinearities. An adaptive method based on the S-system formalism, which is a sensible representation of nonlinear mass-action kinetics typically found in cellular dynamics, maintains the efficiency of linear regression. We combine this approach with adaptive model selection to obtain efficient and parsimonious representations of cellular dynamics. The approach is tested by inferring the dynamics of yeast glycolysis from simulated data. With little computing time, it produces dynamical models with high predictive power and with structural complexity adapted to the difficulty of the inference problem.
Identifying coordinate transformations that make strongly nonlinear dynamics approximately linear is a central challenge in modern dynamical systems. These transformations have the potential to enable prediction, estimation, and control of …
Identifying coordinate transformations that make strongly nonlinear dynamics approximately linear is a central challenge in modern dynamical systems. These transformations have the potential to enable prediction, estimation, and control of nonlinear systems using standard linear theory. The Koopman operator has emerged as a leading data-driven embedding, as eigenfunctions of this operator provide intrinsic coordinates that globally linearize the dynamics. However, identifying and representing these eigenfunctions has proven to be mathematically and computationally challenging. This work leverages the power of deep learning to discover representations of Koopman eigenfunctions from trajectory data of dynamical systems. Our network is parsimonious and interpretable by construction, embedding the dynamics on a low-dimensional manifold that is of the intrinsic rank of the dynamics and parameterized by the Koopman eigenfunctions. In particular, we identify nonlinear coordinates on which the dynamics are globally linear using a modified auto-encoder. We also generalize Koopman representations to include a ubiquitous class of systems that exhibit continuous spectra, ranging from the simple pendulum to nonlinear optics and broadband turbulence. Our framework parametrizes the continuous frequency using an auxiliary network, enabling a compact and efficient embedding at the intrinsic rank, while connecting our models to half a century of asymptotics. In this way, we benefit from the power and generality of deep learning, while retaining the physical interpretability of Koopman embeddings.
In this paper, we demonstrate the fragility of decentralized load-side frequency algorithm proposed in [1] against stochastic parametric uncertainty in power network model. The stochastic parametric uncertainty is motivated through …
In this paper, we demonstrate the fragility of decentralized load-side frequency algorithm proposed in [1] against stochastic parametric uncertainty in power network model. The stochastic parametric uncertainty is motivated through the presence of renewable energy resources in power system model. We show that relatively small variance value of the parametric uncertainty affecting the system bus voltages cause the decentralized load-side frequency regulation algorithm to become stochastically unstable. The critical variance value of the stochastic bus voltages above which the decentralized control algorithm become mean square unstable is computed using an analytical framework developed in [2], [3]. Furthermore, the critical variance value is shown to decrease with the increase in the cost of the controllable loads and with the increase in penetration of renewable energy resources. Finally, simulation results on IEEE 68 bus system are presented to verify the main findings of the paper.
In the electrical grid, the distribution system is themost vulnerable to severe weather events. Well-placed and coordinatedupgrades, such as the combination of microgrids, systemhardening and additional line redundancy, can greatly …
In the electrical grid, the distribution system is themost vulnerable to severe weather events. Well-placed and coordinatedupgrades, such as the combination of microgrids, systemhardening and additional line redundancy, can greatly reduce thenumber of electrical outages during extreme events. Indeed, ithas been suggested that resilience is one of the primary benefitsof networked microgrids. We formulate a resilient distributiongrid design problem as a two-stage stochastic program andmake use of decomposition-based heuristic algorithms to scaleto problems of practical size. We demonstrate the feasibilityof a resilient distribution design tool on a model of an actualdistribution network. We vary the study parameters, i.e., thecapital cost of microgrid generation relative to system hardeningand target system resilience metrics, and find regions in thisparametric space corresponding to different distribution systemarchitectures, such as individual microgrids, hardened networks,and a transition region that suggests the benefits of microgridsnetworked via hardened circuit segments.