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Despite the simplicity of the Duffing oscillator, its dynamical behaviour is extremely rich. Hence, the Duffing equations are used to describe the dynamic behaviour of many real-world nonlinear systems for a wide range of frequency bands and amplitude of the excitation signal in basic sciences and engineering. For example, the Duffing oscillator has been successfully used to model a variety of physical processes such as stiffening springs, beam buckling, nonlinear electronic circuits, superconducting Josephson parametric amplifiers, and ionisation waves in plasmas etc. Therefore, the identification of the Duffing oscillator model/parameters directly from the measured input–output data is a topic of active research in many scientific fields In this paper, we use the concept of stochastic differential equations (SDEs) to identify a model of the Duffing oscillator. SDE-based grey-box models allow us to capture the underlying mathematical structure describing the physics of the system (e.g. the original Duffing equations) using the drift term and explicitly handling of model uncertainty (or the process noise) using the diffusion term whereas the measurement uncertainty is modelled using the measurement noise term respectively. In this paper, we propose a slight variation of the maximum likelihood estimation framework used for the identification of SDEs based grey-box models yielding improved performance for long-term predictions. The proposed framework is combined with an iterative residual analysis to develop a grey-box model of the forced Duffing oscillator. The benchmark data from the so-called Brussels “Silverbox system”, which is an electrical circuit mimicking the forced Duffing oscillator dynamics is used for the identification purpose. Finally, the identified model performance (the simulation errors) is compared with the existing results available in the literature.

Flexible and responsive demand is key to the decarbonising of energy systems. In this paper, an economic dispatch model of a district heating system, modelled as a linear program, is soft-linked to a so-called flexibility function of end-consumer responses to time-varying heat prices, modelled generically as a set of ordinary differential equations. This linkage allows us to determine the cost savings potential of demand response in order to quantify its role in smart energy systems. Our key contribution is an optimal soft-linking framework for energy system and demand response models, named Frigg. Frigg finds the aforementioned economic potential under consideration of end-consumer behaviour. The dynamics of this behaviour lead to a significant problem complexity and pose computational challenges. Hence, the proposed method decomposes the problem based on backward dynamic programming and solves it efficiently. The framework is to be understood as a generic blue-print for soft-linking demand response and energy system models: It can be applied to a variety of energy systems and sources of demand response as well as other objectives than production cost minimisation. We compute the cost savings in a case study of the district heating system of Ejby, Denmark, under different degrees of demand flexibility. The results are compared with the alternative of investing in heat storage systems. Our results suggest substantial cost savings through demand response. Nevertheless, cost savings from heat storage that is cost-optimal in size exceed those achieved through demand response in the system configurations analysed.

Real-time fault monitoring and control of the Vacuum Assisted Resin Transfer Moulding production process requires knowledge of the position of the epoxy flow-front inside the mould. Therefore, a fast and accurate flow-front tracking system is highly prized. Physics-informed grey-box models deliver a good trade-off between high fidelity and data-driven black-box models for designing such a flow-front tracking system. In this paper, we propose stochastic differential equations (SDEs) based grey-box model of the flow-front dynamics in the case of missing sensor information. The proposed method uses the finite difference approximation of the spatial domain of the flow-front for estimating the spatial flow pattern of the epoxy. To accommodate for the missing sensor data, we utilize a modified version of the continuous-discrete extended Kalman filter based estimation framework for SDEs that takes into consideration the effective dimension of the measurement space during the identification process. The performance of the method is evaluated for common fault scenarios.

Data centers today consume between 2% and 4% of the electricity worldwide. Small data centers are responsible for half of this consumption. Data centers continue to expand rapidly and this presents a threat to their sustainable operation. Aquifer Thermal Energy Storage (ATES) is one of the promising technologies for cooling of data centers. ATES consists of two boreholes, which serve as cold and hot storage. One of our case studies in Denmark recently installed the ATES storage. During summertime, cold is extracted from the ground to provide cooling for data centers, while in winter it’s recharged with free coolers that can operate when the temperature outside falls below 8 oC. However, as a consequence of the climate change, summers are becoming hotter, and winters are becoming milder, creating an imbalance in the projected operation. In order to improve the operation of the ATES, an on site integration of heating and cooling systems is proposed. A mixed integer optimization model was developed for on site integration of heating and cooling systems. Themodel takes into account operational constraints of the system and also serves as a capacity expansion model. The results showed that two heat pumps, using ATES hot storage as a heating source, should be installed, with capacities of 200 kW and 240 kW, respectively. The payback period was calculated to be 6.51 years, while the internal rate of return was 13%. Gas consumption for district heating dropped by 76% and CO2 emissions reduced by 49.6%. The results showed that there are still low-hanging fruits regarding the data center energy efficiency. However, data centers need to be fully integrated to the energy systems.

This paper introduces a non-linear grey-box (GB) model based on stochastic differential equations that describes the heat dynamics of a school building in Denmark, equipped with a water-based heating system. The building is connected to a local district heating network through a heat exchanger. The heat is delivered to the rooms mainly through radiators and partially through a ventilation system. A monitoring system based on IoT sensors provides data on indoor climate in the rooms and on the heat load of the building. Using this data, we estimate unknown states and parameters of a model of the building’s heating system using the maximum likelihood method. Important novelties of this paper include models of the water flow in the circuit and the state of the valves in the radiator thermostats. The non-linear model accurately predicts the indoor air temperature, return water temperature and heat load. The ideas behind the model lay a foundation for GB models of buildings that use different kinds of water-based heating systems such as air-to-water/water-to-water heat pumps. Such GB models enable model predictive control to control e.g. the indoor air climate or provide flexibility services.

It is estimated that data centres consume between 2% to 4% of the total electricity consumption worldwide. Around half of this consumption is used for the cooling of data centres. As data centres continue to expand rapidly, increasing cooling efficiency presents a significant hurdle towards their sustainable operation. Recently, many data centres started using Aquifer Thermal Energy Storage (ATES) technology to cool their data centres. The ATES consists of two deep boreholes, one used as hot storage and one used for cold storage. During summertime, cold is extracted from the ground to cool data centres, while during winter, storage is recharged using cold ambient air by means of free coolers. However, due to climate change, summers are becoming hotter than expected, and winters milder than anticipated. As a consequence, ATES is continuously imbalanced, and its economics is becoming infeasible. In order to counteract this occurrence, an on-site integration with heating demand was proposed in order to increase the total efficiency of the energy system. A mixed-integer optimization model was developed in order to calculate the optimal investments and operational behaviour. The model was applied for the case of mediumsized data centres located at Naviair, Danish airspace control. The results showed that two heat pumps need to be installed. The project's payback period was calculated to be 6.51 years, and the internal rate of return was 13%. Initial gas consumption used for heating was reduced by 76%. CO2 emissions dropped from 658 to 331 tons (49.6%). The results showed that there are lowhanging fruits regarding data centres operation that are located within urban areas. However, silo-thinking is currently a significant obstacle to the broader adoption of the proposed solution.

With the aim of enabling effective flexible electricity demand, a machine learning algorithm is developed to forecast the CO2 emission intensities in European electrical power grids distinguishing between average and marginal emissions. The analysis focuses on Danish bidding zone DK2 and was done on a data set comprised of a large number (473) of explanatory variables such as power production, demand, import, weather conditions etc., collected from selected neighboring zones. The number of variables was reduced to less than 30 using both LASSO (a penalized linear regression analysis) and a forward feature selection algorithm. Three linear regression models that capture different aspects of the data (non-linearities and coupling of variables etc.), were created and combined into a final model using Softmax weighted average. Cross-validation is performed for debiasing and an autoregressive moving average model (ARIMA) implemented to correct the residuals, making the final model an ARIMA with exogenous inputs (ARIMAX). Forecast errors vary between 0.095 and 0.183 (NRMSE) for the average emissions and 0.029–0.160 for the marginals depending on the forecast horizon (1–24 h). The forecasts with the corresponding uncertainties are analyzed and performance on very short (below six hours) and longer horizons are discussed –. One interesting result is that the marginal emissions were shown to be highly independent of any variables in the DK2 zone, suggesting that the marginal generators are located in the neighboring zones. The developed methodology can be applied to any bidding zone in the European electricity network without requiring detailed knowledge about the zone and with very few manual interactions.

DSF1305-00027B 8090-00069B EP/S001670/1 If CO2-emissions are to be reduced, the shares of renewable energy sources will have to be significantly increased. However, energy flexibility is required to cope with the increased share of renewable energy. Utilising it necessitates mathematical models of the operational response of energy flexible consumers. In this paper we present an accurate and general dynamic model of energy flexibility based on stochastic differential equations. The intuitive interpretation of the parameters is explained, to show the generality of the proposed model. To validate the approach, the parameters are estimated for three water towers and three buildings controlled by economic model predictive controllers. The model is then used to offer the energy flexibility on the current electricity market of Scandinavia, Nord Pool, using the so called “flexi orders”. Finally, the energy flexibility is used by controlling the demand of the water towers indirectly, through price signals designed based on the proposed model. Compared to having perfect foresight of electricity prices and future demand, between 63% and 98% of the potential savings were obtained in for these case studies. This shows that even without direct control of energy flexible systems, most of the potential can be reached under the current market conditions. publishersversion published

The energy flexibility associated with energy consumption must be exploited to accommodate more fluctuating renewable energy. The only solution that enables this without violating privacy concerns is penalty-based control, where penalty signals are designed to give incentives for the consumers to adjust their demand according to the needs of the grid. Designing the penalty signals is a challenging task due to different flexibility potential offered by various energy consuming systems. In this paper, it is shown that the best utilisation of energy flexibility requires individual penalty signals tuned towards the energy flexibility of each consumer. Here, we present a very simple yet novel approach for designing such individual penalty signals for each consumer, such that the value of the energy flexibility is increased, for both the grid operators and the consumers. (C) 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

The large penetration rate of renewable energy sources leads to challenges in planning and controlling the energy production, transmission, and distribution in power systems. A potential solution is found in a paradigm shift from traditional supply control to demand control. To address such changes, a first step lays in a formal and robust characterization of the energy flexibility on the demand side. The most common way to characterize the energy flexibility is by considering it as a static function at every time instant. The validity of this approach is questionable because energy-based systems are never at steady-state. Therefore, in this paper, a novel methodology to characterize the energy flexibility as a dynamic function is proposed, which is titled as the Flexibility Function. The Flexibility Function brings new possibilities for enabling the grid operators or other operators to control the demand through the use of penalty signals (e.g., price, CO2, etc.). For instance, CO2-based controllers can be used to accelerate the transition to a fossil-free society. Contrary to previous static approaches to quantify Energy Flexibility, the dynamic nature of the Flexibility Function enables a Flexibility Index, which describes to which extent a building is able to respond to the grid’s need for flexibility. In order to validate the proposed methodologies, a case study is presented, demonstrating how different Flexibility Functions enable the utilization of the flexibility in different types of buildings, which are integrated with renewable energies. © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/BY/4.0/)

The operation of electric and heat grids alike is complicated due to the dynamic demand, with the increasing penetration of renewable energy sources adding to the problem. In order to improve the integration of variable renewable energy sources, the flexibility of the system needs to be improved. This paper proposed a novel characterization of the short-term energy flexibility, which was further utilized for the district heating capacity extension. The soft-linking of the models includes feedback, but the added computational complexity is kept at a minimum. Compared to the other literature in the field, due to the accurate characterization of the dynamics of the energy flexibility, flexibility is utilized much more frequently. The method was demonstrated for the case of the district heating of Zagreb. Results showed that both capital and operational savings can be achieved by adopting the proposed method. In the best performing scenario, which included the capacity extension planning, the savings of the district heating system were 5.4%. The extensive power exchange in the best performing scenario meant that the flexibility was used to help balancing the power grid as well. Implementing flexibility into energy planning models : Soft-linking of a high- level energy planning model and a short-term operational model. © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/BY/4.0/).

Smart and flexible operation of components in district heating systems can play a crucial role in integrating larger shares of renewable energy sources in energy systems. Buildings are one of the crucial components that will enable flexibility in the district heating by using intelligent operation. Recent work suggests that such improved operation at the same time can increase thermal comfort and lower economic costs. We have digitalised the heating system in a Danish school by adding IoT devices, such as smart thermostats and temperature sensors to demonstrate the possibilities of making buildings smart. Based on experimental data, this paper introduces a non-linear grey-box model of the thermal dynamics of the building. A non-linear model predictive control method is presented for the thermostatic set-point control of the building's radiators. Based on the building model and the control algorithm, simulation studies are carried out to show the flexibility potential of the building. When used for lowering the return temperature the results suggest that operational costs can be lowered by around 10% using predictive control.

Renewable energy integration in power systems and increasing electrification of energy demand create new challenges to which energy flexibility can provide effective solutions. Trough an innovative use of cumulative energy consumption curves, which represent the maximum and minimum energy limits, as well as the associated flexible energy consumption, this paper presents a methodology to characterise and use the energy flexibility provided by water pumping and storage systems (WPSS) in order to achieve specific objectives at different levels of power systems. The methodology is applied to a case study considering a real WPSS where energy flexibility is used to reduce electricity costs and support the operation of the power system during a wind generation curtailment event. Collected results show that savings around 16% can be achieved while reducing pumping cycles by 57%. Furthermore, the WPSS operation can be modified according to the needs of the power system using the available energy flexibility.

Rapid increase of the renewable energy share in electricity production requires optimization and flexibility of the power consumption side. Thermostatically controlled loads (TCLs) have a large potential for regulation service provision. Economic model predictive control (MPC) is an advanced control method which can be used to syncronize the power consumption with undispatchable renewable electricity production. Thermal behavior of TCLs can be described by linear models based on energy balance of the system. In some cases, parameters of the model may be time-varying. In this work, we present a modified economic MPC based on linear parameter-varying model. In particular, we provide an exact transformation from a standard economic MPC formulation to a linear program. We assume that the variables influencing the model parameters are known (predictable) for the prediction horizon of the controller. As a case study, we present control system that minimizes operational cost of swimming pool heating system, where parameters of the model depend on the weather forecast. Simulation results demonstrate that the proposed method is able to deal with this kind of systems.

The future energy system is weather-driven. To take full and effective advantage of the renewable energy production, we need to make the demand flexible, such that it better coincides with the weather-driven energy production. We argue that this disruption of the energy system implies a need for new planning and control methodologies for the energy demand. This chapter briefly introduces the grey-box modelling principles (which facilitate the need for real-time applications) and illustrates how to use the grey-box models for multi-level control systems of smart buildings and neighbourhoods. Utilising the inherent energy flexibility of buildings, we describe how to use a dynamic price signal to make buildings behave in a desired way for better grid and energy balancing. Each building individually plans its own energy consumption based on the price signal, but it also needs to consider the local weather. We introduce a novel stochastic dynamical model for an integrated setup of grey-box models that are able to take advantage of the local weather disturbances. Finally, we show how to control a building given weather-driven disturbances using model predictive control and integrated grey-box model formulations.

Most district heating systems are operated in an inflexible way, mainly due to legacy operational software, business-as-usual state of mind and lack of awareness of new possibilities brought by digitalization. Moreover, shares of variable renewable energy generation are continually increasing, which causes large oscillations in electricity prices in day-ahead markets. Electric boilers are already common in district heating systems, and it is expected that heat pumps will gain a stronger ground in the future, resulting in variation and uncertainty in the cost of heat production. This calls for a more dynamic representation of the district heating system, including the implementation of demand response that could react to the cost of heat production. In order to check the potential of increasing the flexibility in district heating systems based on a price signal, a high-level energy planning model was soft-linked with the flexibility algorithm, including feedback between them. The case study was carried out for a district heating grid in Zagreb, Croatia. Flexibility was utilized by varying the forward temperature in district heating systems by 3.5 o C compared to the baseline operation. Compared to the relevant literature, flexibility was used much more in our case often due to the accurate representation of the flexibility dynamics. Results showed that both capital and operational savings were achieved. The best performing scenario resulted in socio-economic savings of 5.4%.