John Bagterp Jørgensen

Economic Model Predictive Control for building climate control in a Smart Grid

Model Predictive Control (MPC) can be used to control a system of energy producers and consumers in a Smart Grid. In this paper, we use heat pumps for heating residential buildings with a floor heating system. We use the thermal capacity of the building to shift the energy consumption to periods with low electricity prices. In this way the heating system of the house becomes a flexible power consumer in the Smart Grid. This scenario is relevant for systems with a significant share of stochastic energy producers, e.g. wind turbines, where the ability to shift power consumption according to production is crucial. We present a model for a house with a ground source based heat pump used for supplying thermal energy to a water based floor heating system. The model is a linear state space model and the resulting controller is an Economic MPC formulated as a linear program. The model includes forecasts of both weather and electricity price. Simulation studies demonstrate the capabilities of the proposed model and algorithm. Compared to traditional operation of heat pumps with constant electricity prices, the optimized operating strategy saves 25-35% of the electricity cost.

Unreachable Setpoints in Model Predictive Control

In this work, a new model predictive controller is developed that handles unreachable setpoints better than traditional model predictive control methods. The new controller induces an interesting fast/slow asymmetry in the tracking response of the system. Nominal asymptotic stability of the optimal steady state is established for terminal constraint model predictive control (MPC). The region of attraction is the steerable set. Existing analysis methods for closed-loop properties of MPC are not applicable to this new formulation, and a new analysis method is developed. It is shown how to extend this analysis to terminal penalty MPC. Two examples are presented that demonstrate the advantages of the proposed setpoint-tracking MPC over the current target-tracking MPC.

The potential of Economic MPC for power management

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.

Nonconvex model predictive control for commercial refrigeration

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 model, with a full year simulation and 15-minute time periods, using historical electricity prices and weather data, as well as random variations in thermal load. These simulations show substantial cost savings, on the order of 30%, compared to a standard thermostat-based control system. Perhaps more important, we see that the method exhibits sophisticated response to real-time variations in electricity prices. This demand response is critical to help balance real-time uncertainties in generation capacity associated with large penetration of intermittent renewable energy sources in a future smart grid.

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

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.

Robust economic MPC for a power management scenario with uncertainties

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 e.g. 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.

An ESDIRK method with sensitivity analysis capabilities

Abstract A new algorithm for numerical sensitivity analysis of ordinary differential equations (ODEs) is presented. The underlying ODE solver belongs to the Runge–Kutta family. The algorithm calculates sensitivities with respect to problem parameters and initial conditions, exploiting the special structure of the sensitivity equations. A key feature is the reuse of information already computed for the state integration, hereby minimizing the extra effort required for sensitivity integration. Through case studies the new algorithm is compared to an extrapolation method and to the more established BDF based approaches. Several advantages of the new approach are demonstrated, especially when frequent discontinuities are present, which renders the new algorithm particularly suitable for dynamic optimization purposes.

Finite horizon MPC for systems in innovation form

System identification and model predictive control have largely developed as two separate disciplines. Nevertheless, the major part of industrial MPC commissioning is generation of data and identification of models. In this contribution we attempt to bridge this gap by contributing some of the missing links. Input-output models (FIR, ARX, ARMAX, Box-Jenkins) as well as subspace models can be represented as state space models in innovation form. These models have correlated process and measurement noise. The correct LQG control law for systems with correlated process and measurement noise is not well known. We provide the correct finite-horizon LQG controller for this system and use this to develop a state space representation of the closed-loop system. This representation is used for closed-loop frequency and covariance analysis. These measures are used in tuning of the unconstrained and constrained MPC. We demonstrate our results on a simulated industrial furnace.

Numerical Methods for Large Scale Moving Horizon Estimation and Control

Abstract Both the linear moving horizon estimator and controller may be solved by solving a linear quadratic optimal control problem. A primal active set, a dual active set, and an interior point algorithm for solution of the linear quadratic optimal control problem are presented. The major computational effort in all these algorithms reduces to solution of certain unconstrained linear quadratic optimal control problems. A Riccati recursion procedure for efficient solution of such unconstrained problems is stated

High-performance small-scale solvers for linear Model Predictive Control

In Model Predictive Control (MPC), an optimization problem needs to be solved at each sampling time, and this has traditionally limited use of MPC to systems with slow dynamic. In recent years, there has been an increasing interest in the area of fast small-scale solvers for linear MPC, with the two main research areas of explicit MPC and tailored on-line MPC. State-of-the-art solvers in this second class can outperform optimized linear-algebra libraries (BLAS) only for very small problems, and do not explicitly exploit the hardware capabilities, relying on compilers for that. This approach can attain only a small fraction of the peak performance on modern processors. In our paper, we combine high-performance computing techniques with tailored solvers for MPC, and use the specific instruction sets of the target architectures. The resulting software (called HPMPC) can solve linear MPC problems 2 to 8 times faster than the current state-of-the-art solver for this class of problems, and the high-performance is maintained for MPC problems with up to a few hundred states.