This paper presents a distributed computational framework for stochastic convex optimization problems using the so-called scenario approach. Such a problem arises, for example, in a large-scale network of interconnected linear systems with local and common uncertainties. Due to the large number of required scenarios to approximate the stochasticity of these problems, the stochastic optimization involves formulating a large-scale scenario program, which is in general computationally demanding. We present two novel ideas in this paper to address this issue. We first develop a technique to decompose the large-scale scenario program into distributed scenario programs that exchange a certain number of scenarios with each other to compute local decisions using the alternating direction method of multipliers (ADMM). We show the exactness of the de-composition with a-priori probabilistic guarantees for the desired level of constraint fulfillment for both local and common uncertainty sources. As our second contribution, we develop a so-called soft communication scheme based on a set parametrization technique together with the notion of probabilistically reliable sets to reduce the required communication between the subproblems. We show how to incorporate the probabilistic reliability notion into existing results and provide new guarantees for the desired level of constraint violations. Two different simulation studies of two types of interconnected network, namely dynamically coupled and coupling constraints, are presented to illustrate advantages of the proposed distributed framework.

@en

In this article two limitations in current distributed model based approaches for anomaly detection in large-scale uncertain nonlinear systems are addressed. The first limitation regards the high conservativeness of deterministic detection thresholds, against which a novel family of set-based thresholds is proposed. Such set-based thresholds are defined in a way to guarantee robustness in a user-defined probabilistic sense, rather than a deterministic sense. They are obtained by solving a chance-constrained optimization problem, thanks to a randomization technique based on the Scenario Approach. The second limitation regards the requirement, in distributed anomaly detection architectures, for different parties to regularly communicate local measurements. In settings where these parties want to preserve their privacy, communication may be undesirable. In order to preserve privacy and still allow for distributed detection to be implemented, a novel privacy-preserving mechanism is proposed and a so-called privatized communication protocol is introduced. Theoretical guarantees on the achievable level of privacy, along with a characterization of the robustness properties of the proposed distributed threshold set design, taking into account the privatized communication scheme, are provided. Finally, simulation studies are included to illustrate our theoretical developments.

@en

This paper considers imbalance problems arising in Energy Management in Smart Grids (SG) as discrete-time stochastic linear systems subject to chance constraints, and proposes a Model Predictive Control (MPC) approach to solve them. It is well-known that handling the closed-loop constraint feasibility of such systems is in general difficult due to the presence of a potentially unbounded uncertainty source. To overcome such a difficulty, we propose two new ideas. We first reformulate the chance constraint using the so-called Conditional Value at Risk (CVaR), which is known to be the tightest convex approximation for chance constraints. We then relax the CVaR constraint using a penalty function depending on a coefficient parameter. An optimal solution is therefore obtained by solving a single unconstrained problem which, intuitively, takes into consideration a risk of the system trajectories in an undesirable state. A case study using an academic example is presented to estimate the a-posteriori probability of the coefficient parameter in order to show when such a penalty function is exact by means of probabilistic constraint fulfillment.@en

This work presents a solution method for a day-ahead stochastic reserve scheduling (RS) problem using an AC optimal power flow (OPF) formulation. Such a problem is known to be non-convex and in general hard to solve. Existing approaches follow either linearized (DC) power flow or iterative approximation of nonlinearities, which may lead to either infeasibility or computational intractability. In this work we present two new ideas to address this problem. We first develop an algorithm to determine the level of reserve requirements using vertex enumeration (VE) on the deviation of wind power scenarios from its forecasted value. We provide a theoretical result on the level of reliability of a solution obtained using VE. Such a solution is then incorporated in OPF-RS problem to determine up- and down-spinning reserves by distributing among generators, and relying on the structure of constraint functions with respect to the uncertain parameters. As a second contribution, we use the sparsity pattern of the power system to reduce computational time complexity. We then provide a novel recovery algorithm to find a feasible solution for the OPF-RS problem from the partial solution which is guaranteed to be rank-one. The IEEE 30 bus system is used to verify our theoretical developments together with a comparison with the DC counterpart using Monte Carlo simulations.@en

This paper presents a framework to carry out multi-area stochastic reserve scheduling (RS) based on an AC optimal power flow (OPF) model with high penetration of wind power using distributed consensus and the alternating direction method of multipliers (ADMM). We first formulate the OPF-RS problem using semidefinite programming (SDP) in infinite dimensional spaces that are in general computationally intractable. Using a novel affine policy, we develop an approximation of the infinite dimensional SDP as a tractable finite dimensional SDP, and explicitly quantify the performance of the approximation. To this end, we adopt the recent developments in randomized optimization that allow a priori probabilistic feasibility guarantees to optimally schedule generating units while simultaneously determining the required reserve. We then use the geographical pattern of the power system to decompose the large-scale system into a multi-area power network, and provide a consensus ADMM algorithm to find a feasible solution for both local and overall multi-area network. Using our distributed stochastic framework, each area can use its own wind information to achieve local feasibility certificates, while ensuring overall feasibility of the multi-area power network under mild conditions. We provide numerical comparisons with a new benchmark formulation, the so-called converted DC (CDC) power flow model, using Monte Carlo simulations for two different IEEE case studies.

@en

Aquifer Thermal Energy Storage (ATES) is a building technology used to seasonally store thermal energy in the subsurface, which can reduce the energy use of larger buildings by more than half. The spatial layout of ATES systems is a key aspect for the technology, as thermal interactions between neighboring systems can degrade system performance. In light of this issue, current planning policies for ATES aim to avoid thermal interactions; however, under such policies, some urban areas already lack space for the further development of ATES, limiting achievable energy savings. We show how information exchange between ATES systems can support the dynamic management of thermal interactions, so that a significantly denser layout can be applied to increase energy savings in a given area without affecting system performance. To illustrate this approach, we simulate a distributed control framework across a range of scenarios for spatial planning and ATES operation in the city center of Utrecht, in The Netherlands. The results indicate that the dynamic management of thermal interactions can improve specific greenhouse gas savings by up to 40% per unit of allocated subsurface volume, for an equivalent level of ATES economic performance. However, taking advantage of this approach will require revised spatial planning policies to allow a denser development of ATES in urban areas.@en

This work presents a distributed stochastic energy management framework for a thermal grid with uncertainties in the consumer demand profiles. Using the model predictive control (MPC) paradigm, we formulate a finite-horizon chance-constrained mixed-integer linear optimization problem at each sampling time, which is in general non-convex and hard to solve. We then provide a unified framework to deal with production planning problems for uncertain systems, while providing a-priori probabilistic certificates for the robustness properties of the resulting solutions. Our methodology is based on solving a random convex program to compute the uncertainty bounds using the so-called scenario approach and then, solving a robust mixed-integer optimization problem with the computed randomized uncertainty bounds at each sampling time. Using a tractable approximation of uncertainty bounds, the proposed formulation retains the complexity of the problem without chance constraints. We also present two distributed approaches that are based on the alternating direction method of multipliers (ADMM) to solve the robust mixed-integer problem. The performance of the proposed methodology is illustrated using Monte Carlo simulations and employing two different problem formulations: optimization over input sequences (open-loop MPC) and optimization over affine feedback policies (closed-loop MPC).@en

Distributed fault diagnosis has been proposed as an effective technique for monitoring large scale, nonlinear and uncertain systems. It is based on the decomposition of the large scale system into a number of interconnected subsystems, each one monitored by a dedicated Local Fault Detector (LFD). Neighboring LFDs, in order to successfully account for subsystems interconnection, are thus required to communicate with each other some of the measurements from their subsystems. Anyway, such communication may expose private information of a given subsystem, such as its local input. To avoid this problem, we propose here to use differential privacy to pre-process data before transmission.

@en

The objective of active power control in wind farms is to provide ancillary grid services. Improving this is vital for a smooth wind energy penetration in the energy market. One of these services is to track a power reference signal with a wind farm by dynamically de- and uprating the turbines. In this paper we present a computationally efficient model predictive controller (MPC) for computing optimal control signals for each time step. It is applied in the PArallelized Large-eddy simulation Model (PALM), which is considered as the real wind farm in this paper. By taking measurements from the PALM, we show that the closed-loop controller can provide power reference tracking while minimizing variations in the axial forces by solving a constrained optimization problem at each time step. A six turbine simulation case study is presented in which the controller operates with optimised turbine yaw settings. We show that with these optimized yaw settings, it is possible to track a power signal that temporarily exceeds the power harvested when operating under averaged greedy control turbine settings. Additionally, variations in the turbine's force signals are studied under different controller settings.

@en

This dissertation aims to develop a rigorous distributed approach to decision making using scenario-based techniques for large-scale networks of interconnected uncertain dynamical systems (called agents). A scenario program is a finite-dimensional optimization problem in which an objective function is minimized under constraints that are associated with finitely many, independently and identically distributed (i.i.d.), scenarios of a random parameter. Theoretical and practical interest in scenario programs originates from the fact that these problems are typically efficiently solvable while being closely related to robust and chance-constrained programs. In the former, the constraint is enforced for all admissible random parameters, whereas in the latter, the constraint is enforced up to a given level of probability. However, finding solutions of the resulting large-scale scenario optimization problem for uncertain networked systems poses several difficulties, e.g., computational cost for a central control unit. The main contribution of this dissertation is the design of a technique to decompose a large-scale scenario program into small-scale distributed scenario programs for each agent. Building on existing results in literature, we provide novel guarantees to quantify the robustness of the resulting solutions in a distributed framework. In this setting, each agent needs to exchange some information with its neighboring agents that is necessary due to the statistical learning features of the proposed setup. However, this inter-agent communication scheme might give rise to some concerns about the agents' private information. We therefore present a novel privatized distributed framework, based on the so-called differential privacy concept, such that each agent can share requested information while preserving its privacy. In addition, a soft communication scheme based on a set parameterization technique, along with the notion of probabilistically reliable set, is introduced to reduce the required communication burden. Such a reliability measure is incorporated into the feasibility guarantees of agent decisions in a probabilistic sense. The theoretical guarantees of the proposed distributed scenario-based decision making framework coincide with the centralized counterpart, however the scaling of the results with the number of agents remains an issue. @en

This paper presents an energy management framework for building climate comfort (BCC) systems interconnected in a grid via aquifer thermal energy storage (ATES) systems in the presence of two types of uncertainty (private and common). ATES can be used either as a heat source (hot well) or sink (cold well) depending on the season. We consider the uncertain thermal energy demand of individual buildings as private uncertainty source and the uncertain common resource pool (ATES) between neighbors as common uncertainty source. We develop a large-scale stochastic hybrid dynamical model to predict the thermal energy imbalance in a network of interconnected BCC systems together with mutual interactions between their local ATES. We formulate a finite-horizon mixed-integer quadratic optimization problem with multiple chance constraints at each sampling time, which is in general a non-convex problem and hard to solve. We then provide a computationally tractable framework by extending the so-called robust randomized approach and offering a less conservative solution for a problem with multiple chance constraints. A simulation study is provided to compare completely decoupled, centralized and move-blocking centralized solutions. We also present a numerical study using a geohydrological simulation environment (MODFLOW) to illustrate the advantages of our proposed framework.

@en

Aquifer Thermal Energy Storage (ATES) is an increasingly popular technology which can be used to seasonally store thermal energy in the subsurface. This can significantly reduce the consumption of energy for space heating and cooling in larger buildings, when combined with a heat pump. However, the adoption of ATES by building owners puts pressure on available subsurface space in dense urban areas. The spatial planning of ATES requires a minimum clearance between neighboring systems to avoid thermal interactions; in the Netherlands, some cities already have a scarcity of space for new systems, limiting the energy savings which could be delivered. This situation is in part caused by the overallocation of subsurface space under current methods for ATES planning and operation. In this context, previous work using an idealized case study has shown that cooperative ATES operation could support more efficient spatial planning, by dynamically managing thermal interactions between neighboring systems in response to operating conditions. This work extends this approach for a realistic case study of ATES operation in the city of Utrecht, in the Netherlands. This case is simulated using a coupled agentbased/ geohydrological environment, under 5 different scenarios for spatial planning (representing different layout guidelines for ATES systems). In addition, a distributed stochastic model predictive control (DSMPC) approach is used to impose coupling constraints on the stored thermal volumes of neighboring ATES systems. Compared to a case without coordination, the dynamic management of thermal interactions with DSMPC can significantly improve the energy savings obtained per unit of subsurface volume allocated for ATES, without penalizing the thermal performance of individual systems. Furthermore, the DSMPC controller provides comparable computational performance, with runtimes scaling linearly with the number of simulated systems. @en

This paper presents an approach to distributed stochastic model predictive control (SMPC) of large-scale uncertain linear systems with additive disturbances. Typical SMPC approaches for such problems involve formulating a large-scale finite-horizon chance-constrained optimization problem at each sampling time, which is in general non-convex and difficult to solve. Using an approximation, the so-called scenario approach, we formulate a large-scale scenario program and provide a theoretical guarantee to quantify the robustness of the obtained solution. However, such a reformulation leads to a computational tractability issue, due to the large number of required scenarios. To this end, we present two novel ideas in this paper to address this issue. We first provide a technique to decompose the large-scale scenario program into distributed scenario programs that exchange a certain number of scenarios with each other in order to compute local decisions. We show the exactness of the decomposition with a-priori probabilistic guarantees for the desired level of constraint fulfillment. As our second contribution, we develop an inter-agent soft communication scheme based on a set parametrization technique together with the notion of probabilistically reliable sets to reduce the required communication between each subproblem. We show how to incorporate the probabilistic reliability notion into existing results and provide new guarantees for the desired level of constraint violations. A simulation study is presented to illustrate the advantages of our proposed framework.

@en

In this paper, we present an implementation of a model predictive controller (MPC) for wind farm power tracking problem. The controller is evaluated in the high-fidelity PAral-lelized Large-eddy simulation Model (PALM). By taking measurements from PALM, we show that the closed-loop MPC can provide power reference tracking while reducing force variations on a farm level by solving a constrained optimization problem at each time step. A six turbine wind farm case study is presented in which the controller operates with yawed turbines that increases the potential power that can be harvested with the wind farm, and we show that it is possible to track a reference power signal that temporarily exceeds the power harvested when operating under the so-called greedy control settings.

@en

In this paper, we present an energy management framework for building climate comfort systems that are interconnected in a grid via aquifer thermal energy storage (ATES) systems in the presence of two types of uncertainty namely private and common uncertainty sources. The ATES system is considered as a large-scale storage system that can be a heat source or sink, or a storage for thermal energy While the private uncertainty source refers to uncertain thermal energy demand of individual buildings, the common uncertainty source describes the uncertain common resource pool (ATES) between neighbors. To this end, we develop a large-scale uncertain coupled dynamical model to predict the thermal energy imbalance in a network of interconnected building climate comfort systems together with mutual interactions between the local ATES systems. A finite-horizon mixed-integer quadratic optimization problem with multiple chance constraints is formulated at each sampling time, which is in general a non-convex problem and hard to solve. We then provide a computationally tractable framework based on an extension to the so-called robust randomized approach which offers a less conservative solution for a problem with multiple chance constraints. A simulation study is provided to compare two different configurations, namely: completely decoupled, and centralized solutions.

@en

The application of seasonal Aquifer Thermal Energy Storage (ATES) contributes to meet goals for energy savings and greenhouse gas (GHG) emission reductions. Heat pumps have a crucial position in ATES systems because they dictate the operation scheme of the ATES wells and therefore play an important role in utilizing the storage potential of the subsurface. In the Netherlands, suitable climatic and geohydrological conditions in combination with progressive building energy efficiency regulation have caused the adoption of ATES to take off, resulting in a situation where demand for ATES exceeds the available subsurface space in many urban areas. The most important aspects in this problem are A) the permanent and often unused claim resulting from static permits for ATES operation, and B) excessive safety zones around wells to prevent interaction between wells. Both aspects result in an artificial reduction of subsurface space for potential new ATES systems. Recent research has shown that ATES systems could be placed much closer to each other, and that a controlled/limited degree of interaction between them can actually benefit the overall energy savings of an entire area. Two different simulation experiments were carried out to evaluate the effect of an adaptive permit capacity policy, as well as revised layout guidelines for ATES wells. Our solution provides a framework in which smaller distances between wells and adaptability of the permit volume plays a key role, to allow for optimal utilization of subsurface space for ATES and maximize GHG emission reduction. This paper shows how the total GHG emission reduction of an area can be increased by intensifying the use of the aquifer by allowing (some) interaction between ATES wells, which opens up unused but claimed subsurface space, and increase the number of heat pumps and ATES systems installed.@en

This paper presents a solution method for a day-ahead stochastic reserve scheduling (RS) problem using an AC optimal power flow (OPF) formulation. Such a problem is known to be non-convex and in general hard to solve. Existing approaches follow either linearized (DC) power flow or iterative approximation of nonlinearities, which may lead to either infeasibility or computational intractability. In this paper we present two new ideas to address this problem. We first develop an algorithm to determine the level of reserve requirements using vertex enumeration (VE) on the deviation of wind power scenarios from its forecasted value. We provide a theoretical result on the level of reliability of a solution obtained using VE. Such a solution is then incorporated in RS-OPF problem to determine up-and down-spinning reserves by distributing among generators, and relying on the structure of constraint functions with respect to the uncertain parameters. As a second contribution, we use the sparsity pattern of the power system to reduce computational time complexity. We then provide a novel recovery algorithm to find a feasible solution for the RS-OPF problem from the partial solution which is guaranteed to be rank-one. The IEEE 30 bus system is used to verify theoretical developments together with a comparison with DC counterpart using Monte Carlo simulations.

@en

Traditional deterministic robust fault detection threshold designs, such as the norm-based or limit-checking method, are plagued by high conservativeness, which leads to poor fault detection performance. On one side they are ill-suited at tightly bounding the healthy residuals of uncertain nonlinear systems, as such residuals can take values in arbitrarily shaped, possibly non-convex regions. On the other hand, they must be robust even to worst-case, rare values of the modeling and measurement uncertainties. In order to maximize performance of detection, we propose two innovative ideas. First, we introduce threshold sets, parametrized in a way to bound arbitrarily well the residuals produced in healthy condition by an observer-based residual generator. Secondly, we formulate a chance-constrained cascade optimization problem to determine such a set, leading to optimal detection performance of a given class of faults, while guaranteeing robustness in a probabilistic sense. We then provide a computationally tractable framework by using randomization techniques, and a simulation analysis where a well-known three-tank benchmark system is considered.

@en

Aquifer Thermal Energy Storage (ATES) is an innovative building technology that can be used to store thermal energy in natural subsurface formations [1, 4, 10]. In combination with a heat pump, ATES can reduce the energy demand of larger buildings by more than half, which has made the technology increasingly popular in northern Europe (see Figure 1). Furthermore, the climate and subsurface conditions required for ATES use can be found in areas across Europe, Asia and North America. By the middle of the century, roughly half of the world’s urban population is therefore expected to live in areas technically suitable for ATES [2].@en