Existing methods for nonlinear robust control often use scenario-based approaches to formulate the control problem as large nonlinear optimization problems. The optimization problems are challenging to solve due to their size, especially if the control problems include time-varying uncertainty. This paper draws from local reduction methods used in semi-infinite optimization to solve robust optimal control problems with parametric and time-varying uncertainty. By iteratively adding interim worst-case scenarios to the problem, methods based on local reduction provide a way to manage the total number of scenarios. We show that the local reduction method for optimal control problems consists of solving a series of simplified optimal control problems to find worst-case constraint violations. In particular, we present examples where local reduction methods find worst-case scenarios that are not on the boundary of the uncertainty set. We also provide bounds on the error if local solvers are used. The proposed approach is illustrated with two case studies with parametric and additive time-varying uncertainty. In the first case study, the number of scenarios obtained from local reduction is 101, smaller than in the case when all (Formula presented.) extreme scenarios are considered. In the second case study, the number of scenarios obtained from the local reduction is two compared to 512 extreme scenarios. Our approach was able to satisfy the constraints both for parametric uncertainty and time-varying disturbances, whereas approaches from literature either violated the constraints or became computationally expensive.

Online Feedback Optimization (OFO) controllers steer a system to its optimal operating point by treating optimization algorithms as auxiliary dynamic systems. Implementation of OFO controllers requires setting the parameters of the optimization algorithm that allows reaching convergence, posing a challenge because the convergence of the optimization algorithm is often decoupled from the performance of the controlled system. OFO controllers are also typically designed to ensure steady-state tracking by fixing the sampling time to be longer than the time constants of the system. In this paper, we first quantify the impact of OFO parameters and the sampling time on the tracking error and number of oscillations of the controlled system, showing that adjusting them without waiting for steady state allows good tracking. We then propose a tuning method for the sampling time of the OFO controller together with the parameters to allow tracking fast trajectories while reducing oscillations. We validate the proposed tuning approach in a pressure controller in a centrifugal compressor, tracking trajectories faster than the time needed to reach the steady state by the compressor. The results of the validation confirm that simultaneous tuning of the sampling time and the parameters of OFO yields up to 87% times better tracking performance than manual tuning based on steady state.

Industrial electrification provides a motivation to explore new methods for electrical power management. This paper addresses the situation where the available power is not enough to meet the full demand of a large production site, which can happen because of an electrical contingency. An industrial power management system typically handles electrical power deficits by disconnecting equipment, also known as load shedding. However, disconnections may cause an industrial process to shut down giving a very significant negative economic impact. Using a gas export plant as an example, this paper shows that flexible operation of the plant can offer an alternative to disconnection by making optimal use of any remaining available power, thus avoiding load shedding and enabling an electricity-intensive process to ride through a contingency without an expensive shut-down. Instead of load shedding, the electrical drives will enforce an immediate partial reduction of the power supplied to electrical motors and other machinery on a timescale similar to that of a circuit breaker. An innovation of the research is a framework for pre-computing feasible operating points of the process that match the available power. Technology suitable for industrial implementation is identified in the article and practical considerations are addressed. The findings show promise for the development of new systems for managing electrical power in electricity-intensive sites.

Existing methods for nonlinear robust control often use scenario-based approaches to formulate the control problem as nonlinear optimization problems. Increasing the number of scenarios improves robustness, while increasing the size of the optimization problems. Mitigating the size of the problem by reducing the number of scenarios requires knowledge about how the uncertainty affects the system. This paper draws from local reduction methods used in semi-infinite optimization to solve robust optimal control problems with parametric uncertainty. We show that nonlinear robust optimal control problems are equivalent to semi-infinite optimization problems and can be solved by local reduction. By iteratively adding interim globally worst-case scenarios to the problem, methods based on local reduction provide a way to manage the total number of scenarios. In particular, we show that local reduction methods find worst case scenarios that are not on the boundary of the uncertainty set. The proposed approach is illustrated with a case study with both parametric and additive time-varying uncertainty. The number of scenarios obtained from local reduction is 101, smaller than in the case when all 2 ¹⁴⁺³× ¹⁹² boundary scenarios are considered. A validation with randomly drawn scenarios shows that our proposed approach reduces the number of scenarios and ensures robustness even if local solvers are used.

Mitigating the energy use in buildings, together with satisfaction of comfort requirements are the main objectives of efficient building control systems. Augmenting building energy systems with batteries can improve the energy use of a building, while posing the challenge of considering battery degradation during control operation. We demonstrate the performance of a data-enabled predictive control (DeePC) approach applied to a single multizone building and an energy hub comprising an electric heat pump and a battery. In a comparison with a standard rule-based controller, results demonstrate that the performance of DeePC is superior in terms of satisfaction of comfort constraints without increasing grid power consumption. Moreover, DeePC achieved two-fold decrease in battery degradation over one year, as compared to a rule-based controller.

A class of data-driven control methods has recently emerged based on Willems' fundamental lemma. Such methods can ease the modeling burden in control design but can be sensitive to disturbances acting on the system under control. In this article, we propose a restructuring of the problem to incorporate segmented prediction trajectories. The proposed segmentation leads to reduced tracking error for longer prediction horizons in the presence of unmeasured disturbance and noise when compared with an unsegmented formulation. The performance characteristics are illustrated in a set-point tracking case study in which the segmented formulation enables more consistent performance over a wide range of prediction horizons. The method is then applied to a building energy management problem using a detailed simulation environment. The case studies show that good tracking performance is achieved for a range of horizon choices, whereas performance degrades with longer horizons without segmentation.

Ensuring safety in optimization is challenging if the underlying functional forms of either the constraints or the objective function are unknown. The challenge can be addressed by using Gaussian processes to provide confidence intervals used to find solutions that can be considered safe. To iteratively find a trade-off between finding the solution and ensuring safety, the SafeOpt algorithm builds on algorithms using only the upper bounds (UCB-type algorithms) by performing an exhaustive search on the entire search space to find a safe iterate. That approach can quickly become computationally expensive. We reformulate the exhaustive search as a series of optimization problems to find the next recommended points. We show that the proposed reformulation allows using a wide range of available optimization solvers, such as derivative-free methods. We show that by exploiting the properties of the solver, we enable the introduction of new stopping criteria into safe learning methods and increase flexibility in trading off solver accuracy and computational time. The results from a non-convex optimization problem and an application for controller tuning confirm the flexibility and the performance of the proposed reformulation.

Online Feedback Optimization is a method used to steer the operation of a process plant to its optimal operating point without explicitly solving a nonlinear constrained optimization problem. This is achieved by leveraging a linear plant model and feedback from measurements. However the presence of plant-model mismatch leads to suboptimal results when using this approach. Learning the plant-model mismatch enables Online Feedback Optimization to overcome this shortcoming. In this work we present a novel application of Online Feedback Optimization with online model adaptation using Gaussian process regression. We demonstrate our approach with a realistic load sharing problem in a compressor station with parametric and structural plant-model mismatch. We assume imperfect knowledge of the compressor maps and design an Online Feedback Optimization controller that minimizes the compressor station power consumption. In the evaluated scenario, imperfect knowledge of the plant leads to a 5% increase in power consumption compared to the case with perfect knowledge. We demonstrate that Online Feedback Optimization with model adaptation reduces this increase to only 0.8%, closely approximating the case of perfect knowledge of the plant, regardless of the type of mismatch.

Real-time optimization can play a key role for improving the performance of industrial processes, but it becomes challenging in cases where the process characteristics are uncertain, particularly when the uncertain characteristics impact safety-related constraints. In this study, we present an adaptive and explorative real-time optimization framework that can effectively learn the characteristics of an industrial refrigeration plant through externally driven changes in cooling load targets and through self-exploration. We utilize Gaussian processes (GP) to facilitate learning unknown compressor characteristics of the plant and we leverage the uncertainty quantification of the GP to drive exploration using a weighted sum term in the objective function for real-time optimization. Furthermore, the uncertainty information is used to probabilistically enforce the maximum total power consumption constraint for the compressors with high confidence at all times. Our simulated experiments demonstrate that the proposed approach safely enhances the energy efficiency of the refrigeration process, closely approximating the performance of a best-case solution that has complete information about the plant performance characteristics. We also demonstrate the impact of varying the exploration term in the solution and how the uncertainty of plant behaviour is reduced even in the absence of cooling load target changes.

Process and energy industries have been recognised as adopters of high levels of automation compared to other sectors. Nonetheless, human cognitive input still plays a critical role in the operation of process plants and replication of these cognitive capabilities remains a key challenge for advancing automation levels. In this paper, we provide an analysis of process and energy industries based on a scenario of reduced availability of skilled labour and increased demands for safety, sustainability, and resilience. We consider the different mechanical, sensing, situational awareness, and decision-making tasks involved in the operation of plants and map them to possible realisations of unmanned and autonomous systems. We discuss the implications of current technology capabilities and future technology development perspectives, the factors influencing the complexity of operation in process plants, and the importance of human-machine collaboration. As part of autonomous system capabilities, we consider adaptation as a key capability and we make a connection to adaptation of model-based solutions. We argue that reaching higher and wider levels of autonomy requires a rethink of the design processes for both the physical plants as well as the way automation, control, and safety solutions are conceptualised.

Introducing flexibility in the time-discretisation mesh can improve convergence and computational time when solving differential equations numerically, particularly when the solutions are discontinuous, as commonly found in control problems with constraints. State-of-the-art methods use fixed mesh schemes, which cannot achieve superlinear convergence in the presence of non-smooth solutions. In this paper, we propose using a flexible mesh in an integrated residual method. The locations of the mesh nodes are introduced as decision variables, and constraints are added to set upper and lower bounds on the size of the mesh intervals. We compare our approach to a uniform fixed mesh on a real-world satellite reorientation example. This example demonstrates that the flexible mesh enables the solver to automatically locate the discontinuities in the solution, has superlinear convergence and faster solve times, while achieving the same accuracy as a fixed mesh.

Devising optimal operating strategies for a compressor station relies on the knowledge of compressor characteristics. As the compressor characteristics change with time and use, it is necessary to provide accurate models of the characteristics that can be used in optimization of the operating strategy. This paper proposes a new algorithm for online learning of the characteristics of the compressors using Gaussian Processes. The performance of the new approximation is shown in a case study with three compressors. The case study shows that Gaussian Processes accurately capture the characteristics of compressors even if no knowledge about the characteristics is initially available. The results show that the flexible nature of Gaussian Processes allows them to adapt to the data online making them amenable for use in real-time optimization problems.

Buildings are responsible for about a quarter of global energy-related CO ₂ emissions. Consequently, the decarbonisation of the housing stock is essential in achieving net-zero carbon emissions. Global decarbonisation targets can be achieved through increased efficiency in using energy generated by intermittent resources. The paper presents a co-design framework for simultaneous optimal design and operation of residential buildings using Model Predictive Control (MPC). The framework is capable of explicitly taking into account operational constraints and pushing the system to its efficiency and performance limits in an integrated fashion. The optimality criterion minimises system cost considering time-varying electricity prices and battery degradation. A case study illustrates the potential of co-design in enhancing flexibility and self-sufficiency of a system operating under different conditions. Specifically, numerical results from a low-fidelity model show substantial carbon emission reduction and bill savings compared to an a-priori sizing approach.

Two main approaches to optimisation of load allocation for a system of compressors working in parallel rely on either minimisation of power or maximisation of efficiency. This paper shows that the two approaches are not equivalent. An example with two compressors confirms that the two approaches do not give the same solution.

Efficient maintenance of industrial equipment requires degradation monitoring and prediction. Currently used prediction models are mostly deterministic and cannot consider uncertainty inherent to degradation measurements. In this paper we propose using time series models obtained using Facebook Prophet algorithm to predict the evolution of degradation of turbomachinery. We illustrate our considerations with data from large scale industrial centrifugal compressors. Our predictions are promising and confidence intervals cover the predictions well.

Normal practice in a compressor station with compressors in parallel is to allocate the mass flows equally. However, this strategy is not optimal if the compressors are not identical. A common reason why compressors become non-identical is because their performance degrades over time. Degradation increases the power necessary to run the compressor station and changes the optimal allocation of mass flows. This paper presents a framework for optimal operation in a compressor station with degrading compressors. The optimisation framework proposed in this work explicitly includes a model of degradation in the optimisation problem and analyses how the optimal load-sharing changes when the compressors are degrading. The optimisation framework was applied in an industrial case study of a compressor station in which three parallel compressors are subject to degradation. The case study confirms that it is possible to minimise the extra power consumption due to degradation by adjusting the operating conditions of the compressor station. The analysis also gives insights into the impact of degradation on the optimal solution when compressors work at their limits.

The analysis of equipment degradation has traditionally developed in two main directions. One approach, of great interest for control system design, has been to consider that degradation causes fundamental changes to the behaviour of a system. Another approach, used in optimal maintenance planning and production scheduling, considers degradation as a separate process that affects performance but does not necessarily change the behaviour. This article provides a review of mathematical models of degradation that will facilitate the integration of degradation modelling into control and optimisation schemes. To this end, a new unified classification is proposed. It takes into account the influence of degradation on the behaviour of the system, as well as the factors influencing degradation. Understanding these mutual influences will enable improved optimization, design and operation of control systems. The flexibility of the proposed classification is demonstrated in an industrial application to a multi-product batch scheduling process.

Despite a large body of research, the widespread application of Model Predictive Control (MPC) to residential buildings has yet to be realised. The modelling challenge is often cited as a significant obstacle. This chapter establishes a systematic workflow, from detailed simulation model development to control-oriented model generation to act as a guide for practitioners in the residential sector. The workflow begins with physics-based modelling methods for analysis and evaluation. Following this, model-based and data-driven techniques for developing low-complexity, control-oriented models are outlined. Through sections detailing these different stages, a case study is constructed, concluding with a final section in which MPC strategies based on the proposed methods are evaluated, with a price-aware formulation producing a reduction in operational space-heating cost of 11%. The combination of simulation model development, control design and analysis in a single workflow can encourage a more rapid uptake of MPC in the sector.