Time-varying Outputs Research Articles

Microgrid Formation and Real-Time Scheduling of Active Distribution Networks Considering Source-Load Stochasticity

The post-disruption microgrid (MG) formation and the subsequent scheduling are resilience-enhancing measures for active distribution networks (ADNs) against disastrous events. This paper proposes an integrated MG formation and scheduling solution, considering stochastic loads and distributed generators (DGs). Specifically, a first-stage MG formation model and a sec-ond-stage MG scheduling model are proposed to flexibly identify MG boundaries and restore as many critical loads as possible. The MG formation problem is formulated over multiple time steps to address time-varying generation outputs of intermittent DGs. Moreover, in the MG scheduling problem, the stochasticity of load demands and DG outputs is modeled as joint chance constraints (JCCs) to describe the possibility of security violations in the over-all system. An efficient approximation method is next developed to convexify the JCCs into solvable second-order cones. The interac-tion between the MG formation and scheduling by the model predictive control implementation achieves a responsive solution for enhancing the emergency support capability of ADNs by incor-porating power forecasts and real-time scheduling. The performance of the proposed method is verified in a modified real-world 136-node test feeder under catastrophic failures.

Probabilistic energy flow analysis of the integrated electricity and gas system considering multiform uncertainties

The coupling of electricity and gas systems has been ever-augmented with the wide deployment of gas-fired generators, which facilitates the conception of the integrated electricity and gas system (IEGS). Probabilistic energy flow (PEF) analysis is usually conducted to assess the operating status of the IEGS by calculating the probability distribution of state variables (e.g., gas pressure, gas flow, voltage, and power flow). However, multiform time-variant uncertainties can simultaneously reside in the IEGS, including discrete (e.g., the component failure or functioning) and continuous ones (e.g., renewable energy outputs). Existing PEF analysis works cannot completely deal with time-variant multiform uncertainties featured with different mathematical characteristics. This limitation hinders the estimation of potential operating risks of the IEGS. To address this, this paper proposes a generalized framework for analyzing the probabilistic energy flow of the IEGS considering multiform uncertainties. Firstly, both time-varying random working states and variable outputs of components are represented as probabilistic models utilizing the Lz- transform technique. The probabilistic model is composed of some representative states depicting possible realizations of the component's performance and corresponding probabilities. On this basis, the optimal energy flow (OEF) operator is defined to aggregate probabilistic models of different components to determine probabilistic models of energy flows in the IEGS. Furthermore, multidimensional indices are constructed to comprehensively explore the probabilistic features of energy flows and the impact of probabilistic energy flows on the system performance. In this paper, the system performance mainly refers to the energy-serving capability of the IEGS. Specifically, probabilistic distribution characteristics of energy flows are explicitly displayed by relevant expectations, standard deviations as well as skewnesses. Indices such as the nodal expected gas and electricity not supplied are adopted to evaluate the influence of the probabilistic energy flow on the system performance. Numerical studies reveal that energy flows through different pipelines or power lines present diversified statistical characteristics, which indicates that they are influenced by multiform uncertainties to different extents.

Mass and energy balance-assisted data-driven modeling and optimization of batch processes: The case of a batch polymerization process

In this paper, we use two data-driven modeling methodologies that we have been recently developed: the design of dynamic experiments (DoDE) (Georgakis 2013) and the dynamic response surface methodology (DRSM) (Klebanov and Georgakis 2016, Wang and Georgakis 2017). DoDE allows time-varying inputs, and DRSM models time-varying process outputs. We combine the above two data-driven tools and partial process knowledge to develop an integrated data and knowledge-driven model. We use this hybrid model to improve process productivity. The material and energy balances we use here lack a quantitative description of all rate phenomena. The optimization of the process is evolutionary and thus done in cycles. After three cycles, we have increased the productivity of the process, defined as the ratio of polymer produced over the batch duration, by 39.6% compared to the base case productivity. We also show how this productivity increase could have been as high as 81%.

Operational Probabilistic Power Flow Analysis for Hybrid AC-DC Interconnected Power Systems with High Penetration of Offshore Wind Energy

With increasing penetration of intermittent renewable energy, power systems are facing great pressures in real-time operation and control. Minutes ahead short-term static risk assessment using the probabilistic power flow (PPF) can help system operators detect potential over-limit risks in advance and take preventive measures in time. However, current probability models of renewable power outputs lack enough adaption and efficiency in operational stage, and the PPF formulation also neglects the effects of control system and tie line power exchange of large power systems. This paper establishes a systematic operational PPF scheme for hybrid AC-DC interconnected power systems with offshore wind energy integration. An adaptive probabilistic modeling and fast update method is proposed to model multiple time-varying offshore wind power outputs. Considering different responses of control systems and AC/DC tie line power exchanges, a novel hybrid PPF is proposed to reflect power flow uncertainty more robustly and accurately. The above techniques are validated on a modified IEEE 30 bus system and the whole scheme is applied to the short-term static risk assessment of a real system in Southern China. The results show that our proposed scheme can well meet online calculation requirements, and can also achieve a more realistic characterization of operational PPF.

Data-Driven Process Modeling and Optimization Aided by Material and Energy Balances: The Case of a Batch Polymerization Process

Abstract Our group recently defined two novel data-driven modeling methodologies: The Design of Dynamic Experiments (DoDE) and the Dynamic Response Surface Methodology (DRSM). These two methods enable the quick and efficient data-driven modeling of processes with a partial understanding of their inner workings. They generalize the Design of Experiments (DoE) and the Response Surfaces Methodology (RSM). DoDE allows time-varying inputs, and DRSM models time-varying process outputs. In this paper, we combine the above data-driven tools and partial knowledge of a batch polymerization process to develop an integrated data and knowledge-driven model. The optimization objective is to minimize the process’s batch time while producing the same product quality, increasing productivity. The process knowledge incorporated into the model consists of material and energy balances in which we lack a quantitative description of the rate phenomena, such as reaction or mass/heat transfer rates. The optimization is evolutionary; initially, targeting small improvements through constrained extrapolations around the normal operating conditions. Then, we build the first models and use such models to design the next set of experiments that meet our specifications. This cycle of running experiments and updating the models is repeated until an optimum is reached. After three cycles, we succeeded in reducing the batch time by 26%, while producing acceptable product.

Multibeam Sonar Ray-Tracing Uncertainty Evaluation from a Hydrodynamic Model in a Highly Stratified Estuary

The Port of Saint John is a highly stratified, hyper-tidal estuary that requires frequent hydrographic surveys to ensure that minimum under-keel clearances are maintained and to monitor dredge operations. The potential achievable accuracy and efficiency of a multibeam sonar survey correlates to the knowledge of the physical oceanographic conditions of the harbour. A high-resolution baroclinic hydrodynamic model of the Port of Saint John was constructed to understand the dynamic physical oceanographic environment over four periods representing the limits in river flow conditions. The model provides time-varying outputs of temperature and salinity at predefined depth intervals over the domain. Sound speed is derived from the model output and is used for ray-tracing multibeam sonar data and survey planning to minimize refraction errors. Ray-tracing depth error statistics are calculated from Moving Vessel Profiler (MVP) temperature and salinity casts and used to evaluate the model skill. Results demonstrate that the model can be used to replace observed sound speed profiles and stay within depth accuracy specifications within a swath width of 60 degrees, or more, for most of the year throughout the harbour. Uncertainty increases for swath widths of more than 55 degrees during the most severe stratification of the spring freshet.

Adaptive finite-time fault-tolerant controller for a class of uncertain MIMO nonlinear switched systems subject to output constraints and unknown input nonlinearities

In this work, design of an adaptive finite-time fault-tolerant controller for a class of uncertain multi-input multi-output (MIMO) nonlinear switched systems with unmodeled dynamics subject to asymmetric time-varying output constraints and unknown faulty input nonlinearities has been addressed. The number of actuator faults can be infinite. In addition, the proposed control algorithm can cope with different unknown types of input nonlinearities namely, saturation, dead zone, backlash, and hysteresis. Actuator faults and input nonlinearities can be different in different modes. To estimate the system uncertainties, neural networks (NNs) have been employed and the unmodeled dynamics has been handled utilizing a dynamical signal. Finite-time convergence of the system outputs tracking errors to the neighborhood of the origin have been guaranteed, and the time-varying outputs constraints have been satisfied. By applying the backstepping control approach and using the common Lyapunov function (CLF) method, virtual and actual controllers and adaptive law for each subsystem have been constructed. The closed-loop stability and boundedness of all signals have been established. Through the simulation study, effectiveness of the proposed control scheme has been demonstrated.

New Dynamic Response Surface Methodology for Modeling Nonlinear Processes over Semi-infinite Time Horizons

We follow up on the recently proposed Dynamic Response Surface Methodology (DRSM) [Klebanov and Georgakis Ind. Eng. Chem. Res. 2016, 55(14), 4022] as an effective data-driven approach for modeling time-varying outputs of batch processes with finite time durations. The present new DRSM methodology, DRSM-2, is capable of accurately modeling nonlinear continuous processes over both finite and semi-infinite time horizons as easily and accurately as modeling batch processes, as DRSM-1 did in the initial publication, cited above. The key innovation here is the introduction of an exponential transformation of time, converting the semi-infinite time duration into a finite one. We also propose a systematic model selection procedure to determine the optimal values of the decision parameters affecting the accuracy of the obtained DRSM-2 model. We demonstrate the power of the DRSM-2 approach in two representative processes, a continuous propylene polymerization and a semibatch penicillin fermentation. It is shown that the obtained data-driven models accurately represent the time-varying process outputs in both of the examined, and quite different, case studies.

A Stochastic Stability Enhancement Method of Grid-Connected Distributed Energy Storage Systems

Integrating distributed energy storage systems (DESSs) into the distribution system can facilitate the high-level penetration of renewable energy source-based distributed generations (RES-DGs). To mitigate irregularly time-varying power outputs from RES-DGs, supervisory controllers of DESSs need to allocate corresponding power set points for DESS inverter primary controllers. Consequently, operating conditions of DESS inverters determined by allocated power set points could suddenly change over a wide range and seriously threaten the stability of DESSs. To guarantee DESSs operate stably over a wide range of conditions, this paper proposes a stochastic stability enhancement method embedded in the DESS grid-connected dc-ac inverter controllers. Specifically, by considering the stochastic change of DESS operating conditions as a Markov chain process, the small-signal expression of the distribution system is modeled as a Markov jump linear system (MJLS). Based on this MJLS model, a mode-dependent supplementary controller is developed for the DESS grid-connected dc-ac inverter by solving a set of linear matrix inequalities. This controller can adjust its control gains according to the measured real-time operating mode to enhance the stochastic stability of the distribution system. Studies of the Canadian urban benchmark distribution system have shown the proposed controller can enable the distribution system operate in a stable manner when its operating condition stochastically varies.

Dynamic Response Surface Models: A Data-Driven Approach for the Analysis of Time-Varying Process Outputs

In a recent publication (Ind. Eng. Chem. Res. 2013, 52 (35), 12369) we generalized the classic design of experiments (DoE) methodology by introducing the Design of Dynamic Experiments (DoDE), allowing for the systematic design of experiments involving time-varying inputs. Here, we expand the response surface model (RSM) methodology, used in DoE and DoDE problems, so that it describes the time-evolution of the process, not just the results at the end of the experiment. We apply this generalized type of RSM model, to be denoted by DRSM, to three example processes; a nonisothermal batch reactor with a simple reaction, an isothermal semibatch reactor with several reactions, and a semibatch penicillin fermentation process. Using a limited number of online measurements at prespecified equidistant time instants, we are able to quickly and accurately represent the time evolution of the process output through these simple interpolative data-driven models. The ever-increasing availability of time-resolved measureme...

Reduced order models for assessing CO2 impacts in shallow unconfined aquifers

Risk assessment studies of potential CO2 sequestration projects consider many factors, including the possibility of brine and/or CO2 leakage from the storage reservoir. Detailed multiphase reactive transport simulations have been developed to predict the impact of such leaks on shallow groundwater quality; however, these simulations are computationally expensive and thus difficult to directly embed in a probabilistic risk assessment analysis. Here we present a process for developing computationally fast reduced-order models which emulate key features of the more detailed reactive transport simulations. A large ensemble of simulations that take into account uncertainty in aquifer characteristics and CO2/brine leakage scenarios were performed. Twelve simulation outputs of interest were used to develop response surfaces (RSs) using a MARS (multivariate adaptive regression splines) algorithm (Milborrow, 2015).A key part of this study is to compare different measures of ROM accuracy. We show that for some computed outputs, MARS performs very well in matching the simulation data. The capability of the RS to predict simulation outputs for parameter combinations not used in RS development was tested using cross-validation. Again, for some outputs, these results were quite good. For other outputs, however, the method performs relatively poorly. Performance was best for predicting the volume of depressed-pH-plumes, and was relatively poor for predicting organic and trace metal plume volumes. We believe several factors, including the non-linearity of the problem, complexity of the geochemistry, and granularity in the simulation results, contribute to this varied performance.The reduced order models were developed principally to be used in probabilistic performance analysis where a large range of scenarios are considered and ensemble performance is calculated. We demonstrate that they effectively predict the ensemble behavior. However, the performance of the RSs is much less accurate when used to predict time-varying outputs from a single simulation. If an analysis requires only a small number of scenarios to be investigated, computationally expensive physics-based simulations would likely provide more reliable results. If the aggregate behavior of a large number of realizations is the focus, as will be the case in probabilistic quantitative risk assessment, the methodology presented here is relatively robust.

Opening the Black Box: Low-Dimensional Dynamics in High-Dimensional Recurrent Neural Networks

Recurrent neural networks (RNNs) are useful tools for learning nonlinear relationships between time-varying inputs and outputs with complex temporal dependencies. Recently developed algorithms have been successful at training RNNs to perform a wide variety of tasks, but the resulting networks have been treated as black boxes: their mechanism of operation remains unknown. Here we explore the hypothesis that fixed points, both stable and unstable, and the linearized dynamics around them, can reveal crucial aspects of how RNNs implement their computations. Further, we explore the utility of linearization in areas of phase space that are not true fixed points but merely points of very slow movement. We present a simple optimization technique that is applied to trained RNNs to find the fixed and slow points of their dynamics. Linearization around these slow regions can be used to explore, or reverse-engineer, the behavior of the RNN. We describe the technique, illustrate it using simple examples, and finally showcase it on three high-dimensional RNN examples: a 3-bit flip-flop device, an input-dependent sine wave generator, and a two-point moving average. In all cases, the mechanisms of trained networks could be inferred from the sets of fixed and slow points and the linearized dynamics around them.

Maximin distance optimal designs for computer experiments with time-varying inputs and outputs

Computer models of dynamic systems produce outputs that are functions of time; models that solve systems of differential equations often have this character. Time-indexed inputs, such as the functions that describe time-varying boundary conditions, are also common with such models. Morris (2012) described a generalization of the Gaussian process often used to produce “meta-models” when inputs are finite-dimensional vectors, that can be used in the functional input setting, and showed how the maximin distance design optimality criterion (Johnson et al., 1990) can also be extended to this case. This paper describes an upper bound on the maximin distance criterion for functional inputs. A class of designs that are optimal under certain conditions is also presented; while these designs are of limited practical value, they show that the derived bound cannot be improved in the general case.

Gaussian Surrogates for Computer Models With Time-Varying Inputs and Outputs

Computer models of dynamic systems produce outputs that are functions of time; models that solve systems of differential equations often have this character. In many cases, time series output can be usefully reduced via principal components to simplify analysis. Time-indexed inputs, such as the functions that describe time-varying boundary conditions, are also common with such models. However, inputs that are functions of time often do not have one or a few “characteristic shapes” that are more common with output functions, and so, principal component representation has less potential for reducing the dimension of input functions. In this article, Gaussian process surrogates are described for models with inputs and outputs that are both functions of time. The focus is on construction of an appropriate covariance structure for such surrogates, some experimental design issues, and an application to a model of marrow cell dynamics.

Sensitivity and uncertainty analyses for a SARS model with time-varying inputs and outputs

This paper presents a statistical study of a deterministic model for the transmission dynamics and control of severe acute respiratory syndrome (SARS). The effect of the model parameters on the dynamics of the disease is analyzed using sensitivity and uncertainty analyses. The response (or output) of interest is the control reproduction number, which is an epidemiological threshold governing the persistence or elimination of SARS in a given population. The compartmental model includes parameters associated with control measures such as quarantine and isolation of asymptomatic and symptomatic individuals. One feature of our analysis is the incorporation of time-dependent functions into the model to reflect the progressive refinement of these SARS control measures over time. Consequently, the model contains continuous time-varying inputs and outputs. In this setting, sensitivity and uncertainty analytical techniques are used in order to analyze the impact of the uncertainty in the parameter estimates on the results obtained and to determine which parameters have the largest impact on driving the disease dynamics.