There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Probability and Measure

Model predictive control (MPC) has demonstrated exceptional success for the high-performance control of complex systems [1], [2]. The conceptual simplicity of MPC as well as its ability to effectively cope with the complex dynamics of systems with multiple inputs and outputs, input and state/output constraints, and conflicting control objectives have made it an attractive multivariable constrained control approach [1]. MPC (a.k.a. receding-horizon control) solves an open-loop constrained optimal control problem (OCP) repeatedly in a receding-horizon manner [3]. The OCP is solved over a finite sequence of control actions {u0,u1,f,uN- 1} at every sampling time instant that the current state of the system is measured. The first element of the sequence of optimal control actions is applied to the system, and the computations are then repeated at the next sampling time. Thus, MPC replaces a feedback control law p(m), which can have formidable offline computation, with the repeated solution of an open-loop OCP [2]. In fact, repeated solution of the OCP confers an "implicit" feedback action to MPC to cope with system uncertainties and disturbances. Alternatively, explicit MPC approaches circumvent the need to solve an OCP online by deriving relationships for the optimal control actions in terms of an "explicit" function of the state and reference vectors. However, explicit MPC is not typically intended to replace standard MPC but, rather, to extend its area of application [4]-[6].

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Stochastic Model Predictive Control: An Overview and Perspectives for Future Research

A tutorial review of economic model predictive control methods

We consider a network of devices, such as generators, fixed loads, deferrable loads, and storage devices, each with its owndynamic constraints and objective, connected by AC and DC lines. The problem is to minimize the total network objective subject tothe device and line constraints, over a given time horizon. This is a large optimization problem, with variables for consumptionor generation for each device, power flow for each line, and voltage phase angles at AC buses, in each time period. In this paperwe develop a decentralized method for solving this problem called proximal message passing. The method is iterative: At each step,each device exchanges simple messages with its neighbors in the network and then solves its own optimization problem, minimizing itsown objective function, augmented by a term determined by the messages it has received. We show that this message passing methodconverges to a solution when the device objective and constraints are convex. The method is completely decentralized, and needs noglobal coordination other than synchronizing iterations; the problems to be solved by each device can typically be solved extremelyefficiently and in parallel. The method is fast enough that even a serial implementation can solve substantial problems inreasonable time frames. We report results for several numerical experiments, demonstrating the method's speed and scaling,including the solution of a problem instance with over 10 million variables in under 50 minutes for a serial implementation;with decentralized computing, the solve time would be less than one second.

/pdf/dynamic-network-energy-management-via-proximal-message-15sdzfsb0o.pdf

Dynamic Network Energy Management Via Proximal Message Passing

Model predictive control technologies for efficient and flexible power consumption in refrigeration systems

Economic Model Predictive Control is a receding horizon controller that minimizes an economic objective function rather than a weighted least squares objective function as in Model Predictive Control (MPC). We use Economic MPC to operate a portfolio of power generators and consumers such that the cost of producing the required power is minimized. The power generators are controllable power generators such as combined heat and power generators (CHP), coal and gas fired power generators, as well as a significant share of uncontrollable power generators such as parks of wind turbines. In addition, some of the power consumers are controllable. In this paper, the controllable power consumers are exemplified by large cold rooms or aggregations of super markets with refrigeration systems. We formulate the Economic MPC as a linear program. By simulation, we demonstrate the performance of Economic MPC for a small conceptual example.

/pdf/the-potential-of-economic-mpc-for-power-management-iorunl5m0k.pdf

The potential of Economic MPC for power management

We consider the control of a commercial multi-zone refrigeration system, consisting of several cooling units that share a common compressor, and is used to cool multiple areas or rooms. In each time period we choose cooling capacity to each unit and a common evaporation temperature. The goal is to minimise the total energy cost, using real-time electricity prices, while obeying temperature constraints on the zones. We propose a variation on model predictive control to achieve this goal. When the right variables are used, the dynamics of the system are linear, and the constraints are convex. The cost function, however, is nonconvex due to the temperature dependence of thermodynamic efficiency. To handle this nonconvexity we propose a sequential convex optimisation method, which typically converges in fewer than 5 or so iterations. We employ a fast convex quadratic programming solver to carry out the iterations, which is more than fast enough to run in real time. We demonstrate our method on a realistic mod...

/pdf/nonconvex-model-predictive-control-for-commercial-47tvvp847l.pdf

Nonconvex model predictive control for commercial refrigeration

Supermarket refrigeration consumes substantial amounts of energy. However due to the thermal capacity of the refrigerated goods, parts of the cooling capacity delivered can be shifted in time without deteriorating the food quality. In this paper we introduce a novel economic-optimizing MPC scheme that reduces operating costs by utilizing the thermal storage capabilities. In the study we specifically address advantages coming from daily variations in outdoor temperature and electricity prices but other aspects such as peak load reduction are also considered. An important contribution of this paper is also the formulation of a new cost function for our proposed power management system. This means the refrigeration system is enabled to contribute with ancillary services to the balancing power market. Since significant amounts of regulating power are needed for a higher penetration of intermittent renewable energy sources such as wind turbines, this feature will be in high demand in a future intelligent power grid (Smart Grid). Our perspective is seen from the refrigeration system but, as we demonstrate, the involvement in the balancing market can be economically beneficial for the system itself, while delivering crucial services to the Smart Grid. We simulate the system using models validated against data from real supermarkets as well as weather data and spot and regulating power prices from the Nordic power market.

/pdf/flexible-and-cost-efficient-power-consumption-using-economic-2ykgnxb6vi.pdf

Flexible and cost efficient power consumption using economic MPC a supermarket refrigeration benchmark

This paper presents a novel incorporation of probabilistic constraints and Second Order Cone Programming (SOCP) with economic Model Predictive Control (MPC) Hereby the performance of the controller is robustyfied in the presence of both model and forecast uncertainties Economic MPC is a receding horizon controller that minimizes an economic objective function and we have previously demonstrated its usage to include a refrigeration system as a controllable power consumer with a portfolio of power generators such that total cost is minimized The main focus for our work is power management of the refrigeration system Whereas our previous study was entirely deterministic, models of eg supermarket refrigeration systems are uncertain, as are forecasts of outdoor temperatures and electricity demand The linear program we have formulated does not cope with uncertainties and thus it is, liable to drive an optimal solution to an infeasible or very expensive solution The main contribution of this paper is the Finite Impulse Response (FIR) formulation of the system models, allowing us to describe and handle model uncertainties in the framework of probabilistic constraints Our new solution using this setup for robustifying the economic MPC is demonstrated by simulation of a small conceptual example The scenario is primarily chosen to illustrate the effect of our proposed method in that it can be compared with our previous deterministic simulations

/pdf/robust-economic-mpc-for-a-power-management-scenario-with-1syt4gb8h4.pdf

Tobias Gybel Hovgaard

Papers

Model predictive control technologies for efficient and flexible power consumption in refrigeration systems

The potential of Economic MPC for power management

Nonconvex model predictive control for commercial refrigeration

Flexible and cost efficient power consumption using economic MPC a supermarket refrigeration benchmark

Robust economic MPC for a power management scenario with uncertainties