Chance constrained stochastic MPC for building climate control under combined parametric and additive uncertainty

Abstract

This paper presents a chance constrained stochastic model predictive control (SMPC) approach for building climate control under combined parametric and additive uncertainties. The proposed SMPC^ap approach enables the quantification, and manipulation, of both the mean and covariance of the stochastic system states and inputs. Its enhanced uncertainty anticipation is shown to induce improved thermal comfort in closed-loop simulations compared to the conventional deterministic MPC (DMPC) and the state-of-the-art SMPC^a only accounting for additive uncertainties, at the cost of a maximum relative increase in energy use of 21.6% and 4.2%, respectively. By incorporating the SMPC^ap strategy in an integrated optimal control and design (IOCD) approach, its additional added value for obtaining a more appropriate, yet robust, heat supply system sizing is illustrated. Via simulations, size reductions up to 33.3% are shown to be achievable for a terraced single-family dwelling without increasing thermal discomfort compared to an IOCD approach incorporating DMPC.

Keywords:

• stochastic model predictive control
• chance constraints
• parametric uncertainty
• additive uncertainty
• building climate control
• thermostatically controlled loads

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 MPC strategies that take into account uncertainty in a more implicit way, such as offset-free MPC, adaptive MPC and learning-based MPC, are not considered here. For a detailed review on these methods, see Drgoňa et al. (2020).

2 This is especially relevant in case of a large-scale roll-out of smart meters and controllers, also involving the older, existing building stock.

3 In a residential context, it is not realistic to run large, non-linear optimization models on a thermostat; although the computations could be done on a server, this may not be cost-efficient.

4 The uncertainty about the building model is limited to parametric uncertainty, i.e. uncertainty about the value of the model parameters. Model-form uncertainty, representing the discrepancy between the complex physical process and the simplified mathematical characterization, is out of scope of this work.

5 Since the covariance matrix ${\mathrm{\Sigma }}_{\mathbf{p}}$ is per definition a positive (semi-)definite matrix, this square root form can for example be obtained via the Cholesky factorization (Rasmussen and Williams 2006).

6 More particularly, by using Equation (12(12) $vec\left(\mathbf{A}\mathbf{X}{\mathbf{B}}^T\right)=(\mathbf{B}\otimes \mathbf{A})vec\left(\mathbf{X}\right)$(12) ), the product of a matrix $\mathbf{A}$ and a vector $\mathbf{x}$ can be rewritten as follows: $\mathbf{A}\mathbf{x}=vec\left(\mathbf{A}\mathbf{x}\right)=vec\left(\mathbf{I}\mathbf{A}\mathbf{x}\right)=({\mathbf{x}}^T\otimes \mathbf{I})vec\left(\mathbf{A}\right).$

7 The considered theoretical, reduced order white-box building model entails one important assumption that may limit practical implementation (not limiting in the simulations), being that direct measurements of the internal heat gains are available, which are in practice difficult to obtain. An alternative approach would be to use the domestic electricity use, available from digital/smart meter data, as an alternative input to construct a data-driven, grey-box building model, as is for example suggested by Reynders, Diriken, and Saelens (2014a).

8 Since only one heat input is considered, the index j is omitted from now on.

9 In Uytterhoeven et al. (2021), an RC model of order nine is considered. In this work, the model order is further reduced to an order of four, to ensure that the optimization problem remains tractable, given the repetition of the parametric uncertainty in the latent variable $\tilde{\mathbf{p}}$.

10 The uncertainty on the ground temperature is not considered in this work, since the forecasts and actual measurement data required for a proper uncertainty characterization are hard to obtain (Lambrichts 2020). In fact, the temporal variation of the ground temperature is not even considered, as argued in Reynders, Diriken, and Saelens (2014a), since it is unrealistic that local measurements can be actually used as an input to the building controller model; rather, a constant value of 10${}^{\circ }$C is used.

11 Note that in this work, an updated StROBe version is used, extended by Jelger Jansen, to include the metabolic heat rates in addition to the heat gains caused by appliances and lighting; see https://github.com/jelgerjansen/StROBe/tree/output_occupancy_profiles

12 In this work, thermal comfort is assessed based on the state of the air thermal capacity of the building model, instead of on the operative temperature (which is typically used in comfort standards (Peeters et al. 2009)), since the radiative temperatures of the surrounding building components are not explicitly considered in the chosen building model.

13 Because of the particular setting considered in this paper, focusing on optimal space heating at minimal energy use, the controller is compelled to stick to the lower temperature bound, implying that the upper temperature bound is of minor importance here.

14 The nominal heat demand is quantified following NBN EN 12831, considering an extremely cold day (according to the Belgian climate) with an outside temperature of −8${}^{\circ }$C, a ground temperature of 10${}^{\circ }$C, and an indoor temperature of 20${}^{\circ }$C (NBN 2017).

15 The point of view adopted in this paper differs from the one adopted in current practice, where one typically assumes that a sufficiently accurate building model is available. To account for the impact of the significant parametric uncertainty in this work, a high safety factor is used on top of the worst-case static design conditions.

16 The upper thermal comfort bound is of minor importance in this work, since the controller will try to stick to the lower temperature bound as it tries to minimize the energy use.

17 In contrast to the uncertainty characterization of the building model parameters and weather forecasts, wich are both based on a rigorous and extensive data-analysis.

18 Deterministic approaches do not hedge against uncertainty, making thermal discomfort due to closed-loop perturbations inevitable.

19 Similar thermal discomfort levels can be obtained for different combinations of system sizes and risk averseness levels (where the risk averseness level is rather a control preference characteristic to the user), illustrating the interchangeability between control and design.

20 This explains the need for larger system sizes in case of a DMPC strategy: to be able to correct for wrong control strategies, real-time flexibility is required in the form of spare capacity.

21 Also the DMPC strategy shows a certain degree of anticipation. However, this anticipation has nothing to do with uncertainties, but is merely due to the fact that the decreasing installed power becomes insufficient to deliver an instantaneous heat power peak.

22 With a variable energy price, a higher energy use could result in a lower energy cost.

Additional information

Funding

This work was supported by KU Leuven (C2 Research Project C24/16/018 ‘Energy Storage as a Disruptive Technology in the Energy System of the Future’) and the Research Foundation – Flanders (FWO) (postdoctoral mandate 12J3320N). Besides, the resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation – Flanders (FWO) and the Flemish Government.