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Journal Article•DOI•

A convex approach to a class of non-convex building HVAC control problems: Illustration by two case studies

Ercan Atam1, Lieve Helsen1•
Katholieke Universiteit Leuven1
15 Apr 2015-Energy and Buildings (Elsevier)-Vol. 93, pp 269-281
TL;DR: In this article, a convexification approach is presented for a class of non-convex optimal/model predictive control problems more specifically applied to building HVAC control problems.
About: This article is published in Energy and Buildings.The article was published on 2015-04-15 and is currently open access. It has received 28 citations till now. The article focuses on the topics: Optimal control & Adaptive control.

Summary (4 min read)

Jump to: [1. Introduction] – [2. PART I: Theoretical Foundations] – [2.1.2. Non-convex Model Predictive Control Problem Class] – [2.2.1. Overview of convex optimization] – [2.2.2. Convex Envelope] – [2.4. Recap of Dynamic Programming] – [3. PART II: Applications] – [3.1. Application to a Benchmark Problem] – [3.2.1. HybGCHP System Description] – [3.2.2. Heat & Cold Demand Satisfaction] – [3.2.3. Circulating Fluid Temperature Bounds] – [3.2.5. Borehole Dynamics] – [3.2.6. Non-convex Optimal Control Problem for Total EnergyCost Minimization] – [3.2.7. Convexified Optimal Control Problem and Comparison with Dynamic Programming] – [3.3. HybGCHP System Control Using Convexified MPC] and [4. Conclusion]

1. Introduction

  • In the context of energy-efficient buildings HVAC control has gained increasing attention in recent years.
  • This leads to the risk of designing a non-working controller on the real system or a working controller with suboptimal results.
  • De Ridder et al. and Verhelst [12, 5] used mathematical model-based control methods for HybGCHP systems, which allow global optimization.
  • There xist analytical formulas for a bilinear function or a rational function of two varibles.

2. PART I: Theoretical Foundations

  • First, the authors introduce a class of non-convex control p blems which include the optimal/model predictive control of HybGCHP systems.
  • Next, the proposed convex relaxation method for the given non-convexcontrol problems is detailed.
  • Since in Section 3.2 the proposed convex relaxation is applied to optimal control of HybGCHP systems and the results are compared with its dynamic programming-based control results, the authors also give a short oveview of the dynamic programming at the end of this part.

2.1.2. Non-convex Model Predictive Control Problem Class

  • The rest of descriptions of variables and functions are the same as in Section 2.1.1.
  • In MPC control, the control input over the prediction horizon is calculated at every time step and the first element of the control input vector is applied at the current time step.
  • At the next time step, the same calculation procedure is peated.
  • Due to this implementation scheme, MPC is sometimes called receding horizon control.

2.2.1. Overview of convex optimization

  • This means that if the authors take any two points inS and draw a line segment between these two points, then every point on that line segmentalso belongs toS. Definition 2.2 (Convex Function [13]).
  • This definition means that if the authors take any two points on the graph of a convex functionf and draw a straight line between them, then the portion of thefunction between these two points will lie below this straight line.
  • Armed with the above definitions of convex sets and functions, the authors are now ready to define a convex optimization problem.
  • Definition 2.3 (Convex Optimization Problem: the most general form [13]).
  • Given a convex functionf , a convex setS and the decision variable vectorx, the associated convex optimization problem is defined as minimizef(x) subject tox ∈ S. Definition 2.4 (Convex Optimization Problem: less general form [13]).

2.2.2. Convex Envelope

  • Next, the concept of convex envelope is defined.
  • Given a continuous functionk(x), its convex envelope, denoted byconvk(x), over a convex setS is defined as the pointwise supremum of all convex functions which are majorized byk(x): convk(x) = sup{r(x)| r convex andr(y) < k(y) ∀y ∈ S}.
  • The overall system of relaxed constraints consists of 18 scalar convex constraints, which can be compactly represented as Efrw ≥ gfr-c(x1, x2), (12) whereEfr is a constant vector andgfr-c(x1, x2) is a convex function with respect to the variablesx1, x2.
  • The derivation, although simple, is lengthy and hence was skipped.

2.4. Recap of Dynamic Programming

  • Dynamic programming is a closed-loop, global optimal contrl method (global optimal up to approximations due to state-input gridding and interpolations).
  • It is based on the “principle of optimality” [16] which simply says that in a multi-stage process whatever the previous states are, the remaining decisions must be optimal with regard to the state following from the current state.
  • This principle allows the optimal control problem of aK-stage process to be recursively formulated starting from the last stage.
  • For dynamic programming-based control methods, the most important issue is to have an accurate model with minimum number of states and inputs due to the famous curse of dimensionality problem [16, 17].
  • The reader is referred to references [16, 17] for details on dynamic programming.

3. PART II: Applications

  • The purpose of this section is to test the proposed convexification-based control methods first on a benchmark HVAC building control system case study from the literature to which the authors apply the convexified MPC control method and compare these control results with the adaptive and fuzzy control results from the literature.
  • Next, the optimal control of a HybGCHP system is considered,which is the main application example for the developed methods.
  • For the HybGCHP system, the convexified optimal control results are compared with the dynamic programming control results of the same system to assess the performanceof th convexified optimal control method.

3.1. Application to a Benchmark Problem

  • The benchmark case study considered in this paper is the building heating control system, as described by Calvino et al. [18] and Chaudry and Das [19].
  • The model parameter values used in Eq.(14) are taken from Chaudhry and Das [19] and are given in Table 1.
  • Figure 3 shows the corresponding results when convexified MPC is usedwith Np = 10.

3.2.1. HybGCHP System Description

  • It is assumed that the heat demand (Q̇h) is provided by the heat pump and the boiler and the cold demand (Q̇c) is provided by the passive cooler (using a heat exchanger instead of an active chiller) and the chiller.
  • The coefficients of performance given by the above expressions depend on the temperatures of the source and the emission system, as expressed by COPhp = fhp(Tf , Tsw,h), COPpc = fpc(Tf , Tsw,c), COPch = fch(Ta, Tsw,c), whereTsw,h, Tsw,c represent the supply water temperature for heating and supply water temperature for cooling, respectively.
  • Next, the authors will discuss these constraints and present their expressions.

3.2.2. Heat & Cold Demand Satisfaction

  • The building heating and cooling demands should be satisfiedwith some acceptable violation margins: Q̇h(t)−.
  • Note that the margins are taken to be time-dependentto allow different degrees of flexibility over time.
  • During critical demand load periods these margins can be set very strictly.
  • It is assumed thatQ̇h andQ̇c are given and hence building modeling is not included in the optimization.

3.2.3. Circulating Fluid Temperature Bounds

  • The cooling of a building requires heat injection into the ground during summer.
  • This increases the ground temperature towards winter,which, in turn, increases COPhp.
  • The ground temperature, which is represented indirectly byTf , should be kept below the supply water temperature,Tsw,c, for passive cooling of the building.
  • Similarly, heating of a building requires heat extraction from the ground.

3.2.5. Borehole Dynamics

  • In the equivalent diameter approach, the heat transfer from the Utube is approximated by the heat transfer from a single pipe with a hypothetical diameter through which the heat exchanging fluid circulates.
  • T are soil nodal temperatures andp is the known parameter vector including thermal, physical and other parameters of the system (diffusivitiesαg, αs, conductivitieskg, ks, different radii rfg, rgs, discretization step sizes,etc.) andunet is the net heat power injected to the per borehole length.
  • POD is a flexible model-order reduction method compared to other model order reduction methods, also known as Remark 2.
  • Here, the authors consider optimal control which is an open-loop control method where control actions are c lculated off-line and the accuracy of these control actions is depending on the“simulation performance” of the model, not on its “prediction performance” because in classical optimal control measured values are not used and everythingis done off-line.
  • A too simple model cannot have a good simulation performance over a long period (like one year or multiple years), as shown in Figure 6.

3.2.6. Non-convex Optimal Control Problem for Total EnergyCost Minimization

  • (23j) With this formulation, the non-convexity of the optimal control problem comes from Eq.(23j): a bilinear term.
  • Note that the rational term includingTa in the cost function does not create any non-convexity becauseTa is not a decision variable or a function of decision variables.
  • In the next subsection, Eq.(23j) will be replaced by its convex approximation.

3.2.7. Convexified Optimal Control Problem and Comparison with Dynamic Programming

  • Note that all other constraints are already/or can be easilyput into the form given in Eq.(7).
  • To apply dynamic programming for assessing the performanceof onvexified optimal control, the optimization problem given by Eq.(23)is equivalently reformulated with two inputs instead of four by assuming zero violat n margins (see Eq.(17)) to alleviate the curse of dimensionality problem in dynamic programming.
  • This is done for allfe sible gridded states.
  • Figures 7(a)-7(b) show the time evolution of the mean temperature ofthe circulating fluid (Tf ), Figures 7(c)-7(d) show the heat pump power and heating demand (Q̇hp, Q̇h), Figures 8(a)-8(b) show the chiller power and cold demand (Q̇ch, Q̇c) and finally Figures 8(c)-8(d) show the evolution of accumulated cost (J).

3.3. HybGCHP System Control Using Convexified MPC

  • The non-convex model predictive control (NMPC) problem forHybGCHP system is shortly described as follows.
  • Here, the authors do not present the results for convexified MPC, they just wanted to show how the proposed idea can be used in the context of NMPC.
  • In practice this will cause the following problems.
  • Firstly, the large-scale model with 506 states is not observable since there are 506 states and only one input-output pair.

4. Conclusion

  • In this paper a convexification approach using convex envelopes for hard-tosolve non-convex optimization problems involving rational and/or bilinear terms of decision variables was proposed.
  • For the first case study of an HVAC building control system, the performance of convexified MPC was compared to the performance of fuzzy and adaptive control from the literature.
  • For the second application, the total energy costminimization of buildings with a HybGCHP system, convexified optimal control was used and its results were compared to dynamic programming based control results, which is a closedloop, global optimal control method (global optimal up to approximations in the gridding of states/inputs and used interpolations).
  • The overall message of this paper is that given a non-convex optimization/control problem of thermal systems, the first step should be to analyse the given system in terms of the non-convex terms and then investigate whether convex envelopes for the associated terms exist or not.
  • The accuracy degree of the approximation of the original non-convex optimization problem by a convex one may be very case dependent.

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Ján Drgoňa1, Ján Drgoňa2, Javier Arroyo2, Iago Cupeiro Figueroa2, David Blum3, Krzysztof Arendt4, Donghun Kim3, Donghun Kim5, Enric Perarnau Ollé2, Juraj Oravec6, Michael Wetter3, Draguna Vrabie1, Lieve Helsen2 •
Pacific Northwest National Laboratory1, Katholieke Universiteit Leuven2, Lawrence Berkeley National Laboratory3, University of Southern Denmark4, Purdue University5, Slovak University of Technology in Bratislava6
01 Jan 2020-Annual Reviews in Control
TL;DR: This paper provides a unified framework for model predictive building control technology with focus on the real-world applications and presents the essential components of a practical implementation of MPC such as different control architectures and nuances of communication infrastructures within supervisory control and data acquisition (SCADA) systems.

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TL;DR: In this paper, a compact overview of the state-of-the-art in modeling of ground-coupled heat pump (GCHP) systems and an in-depth review of their optimal control along with the associated research challenges are given.
Abstract: In this paper, a compact overview of the state-of-the-art in modeling of ground-coupled heat pump (GCHP) systems and an in-depth review of their optimal control along with the associated research challenges are given. The main focus is on optimal control but since design of an optimal controller may require a model, a relatively short literature review of modeling approaches is also discussed. Adopting the adage “a picture is worth a thousand words”, we tried to include a minimal number of representative schematics and result figures for some of the reviewed studies for clarity and a better understanding of the presented material. In addition to the literature review, we included our comments, points of view, alternative solutions and some potential future directions. This review paper is useful both for engineers and researchers involved in modeling and optimal control of GCHP systems. The second part of the paper, “Ground-Coupled Heat Pumps: Part 2 – Literature Review and Research Challenges in Optimal Design”, focuses on the literature review on optimal design and the associated design challenges for GCHP systems.

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Giuseppe Tommaso Costanzo1, Sandro Iacovella2, Frederik Ruelens2, T. Leurs2, Bert Claessens3 •
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TL;DR: In this article, a data-driven control approach for building climate control is presented based on reinforcement learning, where the underlying sequential decision making problem is cast into a Markov decision problem, after which the control algorithm is detailed.

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Experimental analysis of data-driven control for a building heating system

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Giuseppe Tommaso Costanzo1, Sandro Iacovella2, Frederik Ruelens2, T. Leurs2, Bert Claessens3 •
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TL;DR: Model-assisted batch reinforcement learning is applied to the setting of building climate control subjected to dynamic pricing and it is found that within 10 to 20 days sensible policies are obtained that can be used for different outside temperature regimes.
Abstract: Driven by the opportunity to harvest the flexibility related to building climate control for demand response applications, this work presents a data-driven control approach building upon recent advancements in reinforcement learning. More specifically, model assisted batch reinforcement learning is applied to the setting of building climate control subjected to a dynamic pricing. The underlying sequential decision making problem is cast on a markov decision problem, after which the control algorithm is detailed. In this work, fitted Q-iteration is used to construct a policy from a batch of experimental tuples. In those regions of the state space where the experimental sample density is low, virtual support samples are added using an artificial neural network. Finally, the resulting policy is shaped using domain knowledge. The control approach has been evaluated quantitatively using a simulation and qualitatively in a living lab. From the quantitative analysis it has been found that the control approach converges in approximately 20 days to obtain a control policy with a performance within 90% of the mathematical optimum. The experimental analysis confirms that within 10 to 20 days sensible policies are obtained that can be used for different outside temperature regimes.

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18 May 2016-IEEE Control Systems Magazine
TL;DR: The residential and commercial building sector is known to use around 40% of the total end-use energy and is considered to be the largest energy consumer sector in the world as mentioned in this paper.
Abstract: The residential and commercial building sector is known to use around 40% of the total end-use energy and, hence, is considered to be the largest energy consumer sector in the world [1]. Approximately half of this energy is used for heating/cooling, ventilation, and air-conditioning (HVAC), and this usage is increasing 0.5?5% per year in developed countries [2]. The distribution of energy use percentages within the building for the United States is shown in Figure 1. This trend is similar for the rest of the world.

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The last requirement is relevant for the synthesis of (real-time) dynamic optimization. In this thesis, mainly models for heat conduction of conductive and convective processes are considered, with a focus on applications in glass furnaces. A technique based on the orthogonal decomposition of a collection of measure- ments of physical quantities (such as temperature) in position and time (sig- nals) is used to reduce the complexity of models. Following ideas from Fourier series expansions, signals are represented as series of orthonormal functions. These so-called basis functions approximate the spatial distribution of the signal while the coe±cients of the basis functions represent the time-varying dynamics. The basis functions are derived from measured or simulated data and are physically relevant. The reduced order model is obtained by applying a Galerkin projection of the equations of the original model onto the space spanned by a ¯nite set of well selected basis functions. In this PhD thesis, this technique is applied and implemented to heat conduc- tion models and a Computational Fluid Dynamics model of an industrial glass melt feeder. From all applications described in this PhD thesis, the reduced order model consists of less than 1% of the number of equations of the original model. The examples show that with such a reduction a maximum error of 1This factor is based on experience for achieving high performance (real-time) dynamic optimization. 1% in the variation of the physical variables (signals) is achievable. The low complexity of the reduced order model enables the design and synthe- sis of optimal model-based controllers. Using the reduced order models, var- ious linear controllers have been synthesized to optimally control both linear and nonlinear heat conduction processes. The controller minimizes a quadratic criterion function in deviation between the desired and actual state and the input variables of the system. The application of such a controller to heat con- duction processes shows that the tracking of an arbitrary desired temperature pro¯le can be achieved in an optimal manner. Despite the drastic reduction of the number of equations, the computational gain for the reduction of nonlinear processes is low. To reduce the computation time, a method of 'missing point estimation' (MPE) is proposed in this PhD thesis and combined with the POD reduction technique. The reduced model is then based on a selected set of points in the spatial domain. On the basis of two selection criteria, an ordening of the relevant points in the spatial domain is proposed. The most relevant points are selected for describ- ing the process dynamics, while the dynamical features of the process in the remaining points are estimated. In combination with the reduction technique of orthogonal basis functions, this leads to a reduction of computation time. For nonlinear heat conduction model of a heated plate, a reduced order model that is 100 times faster than real-time can be achieved. In particular, these reduction and acceleration techniques (POD and MPE) are applied to simulate a transition of operating point in a glass melt feeder. This transition concerns a color change of glass in a glass melt feeder and leads to a drastic change of many physical quantities in the glass melt. For this transition, an optimal set of basis functions is determined from simulation data. A reduced order model for the temperature distribution in the feeder has been constructed from the nonlinear model by applying the POD and MPE technique. It is shown that a reduced model of order 18 attains a resolution with a maximum error of 1% in the variations of the physical quantities, while achieving a computational speed that is about 8 times faster than real-time. The attained accuracy and acceleration are adequate for the anticipation of the process dynamics and for process monitoring. However, the computational speed is not su±cient for on-line control design. If, apart from temperature, also the dynamics of the velocity and pressure ¯eld of the nonlinear model are incorporated in the reduction procedure, then a computational speed of about 30 times faster than real-time is feasible. For control design, the desired computational speed is minimally 50 times faster than real-time. The reduc- tion techniques presented in this thesis therefore have su±cient potential and perspective to enable model-based control system design in the near future, for example by further improvement of the MPE method, or by improving the speed of convergence in the numerical simulations. Further enhancement of the computational gain can be achieved by exploiting the advantages of par- allel computing. Parallel computing is already enabled in the new generation software for glass furnaces but it is not used in this PhD thesis.

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157 citations

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  • ...A challenging HVAC control application where bilinear/fractional terms exist is the control of ground-coupled heat pumps (GCHP) and hybrid groundcoupled heat pump systems combined with low-exergy heat emission systems [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]....

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Frequently Asked Questions (1)
Q1. What are the contributions in "A convex approach to a class of non-convex building hvac control problems: illustration by two case studies" ?

In this paper, a convexification approach is presented for a class of non-convex optimal/model predictive control problems more specifically applied to building HVAC control problems. The suggested method is especially useful for optimal building HVAC control/design problems which include non-convex bilinear or fractional terms.