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

Abstract: 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 original non-convex problems are convexified using a convex envelope approach. The approach is tested on two case studies: a benchmark building HVAC system control problem from the literature and control of a hybrid ground-coupled heat pump (HybGCHP) system. For the first application, convexified model predictive control was used and results were compared with fuzzy and adaptive control results. For the HybGCHP system, convexified optimal control was applied and the results were compared with dynamic programming based optimal control. In the first case superior performance was observed over the corresponding fuzzy and adaptive control results from the literature. For the HybGCHP system the associated convexified optimal control gave almost global optimal results in terms of responses and cost criteria. The suggested method is especially useful for optimal building HVAC control/design problems which include non-convex bilinear or fractional terms. Since a polynomial expression can be recursively expressed as a system of bilinear equations, the proposed method, in principle, can be applied to all systems where polynomial non-convexities exist.

## Summary (4 min read)

### 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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