IN DEGREE PROJECT ELECTRICAL ENGINEERING,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2018

Model Predictive Control Applied to Ground Source Heat Pumps

PAUL VERRAX

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE

Model Predictive Control Applied to GroundSource Heat Pumps

Master Thesis

Paul Verraxverrax@kth.se

Automatic Control Department, EECSElectrical Engineering and Computer Science, KTH

Supervisor : Olle TrollbergExaminer : Elling Jacobsen

February 26, 2018

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Abstract

Building heating is one of the most important sources of energy consumption. GroundSource Heat Pumps (GSHP) are efficient heating systems, particularly popular in the Nordiccountries. However, the GSHPs available for the consumer market today typically only utilizebasic control schemes that are relatively inflexible. More advanced strategies such as ModelPredictive Control (MPC) appear as a promising approach to improve comfort while reducingconsumption. The present thesis considers a typical user case of a single family house heatedby a ground source heat pump willing to reduce its environmental impact. We design a MPCcontroller to be used on top of the existing heat pump system and with almost no additionalhardware needed. Specific attention is dedicated to the system’s efficiency in order to reflectthe real working performances of a ground source heat pump. The controller is evaluated insimulation on different scenarios using an identified model of a single family house. The resultsshow the MPC strategy becomes most beneficial when including time varying prices or reducedcomfort during certain hours of the day. When both are conjugated the economic savings areup to 8% despite the loss of efficiency of the heat pump. The controller was implemented andtested on a real system with promising results.

Sammanfattning

Uppvärmning av byggnader är en av samhällets största energiförbrukare. I Norden användsofta bergvärmepumpar som ett effektivt alternativ för uppvärmning. De flesta bergvärmepum-par styrs av väldigt enkla algoritmer. Model Predictive Control (MPC) är en lovande metodför att ta hänsyn till både inomhustemperatur och energiförbrukning, speciellt om man hartillgång till en väderprognos. Denna rapport studerar hur ett typiskt enfamiljshus kan minskasin energiåtgång och miljöpåverkan. Ansatsen är att lägga till ett yttre reglersystem till berg-värmepumpen. Speciell vikt läggs vid modellering av bergvärmepumpens effektivitet. Olikascenarion jämförs, bland annat att ta hänsyn till elprisets variation över dygnet med hjälpav ekonomisk MPC. Simuleringar visar att användning av MPC är mest fördelaktig i det fallbåde elpriset och krav på innetemperatur varierar över dygnet. I sådana fall kan energikost-naden minska med upp till 8%, trots att bergvärmepumpen stundtals arbetar i ett ogynnsamtdriftläge. Ett verkligt system har använts för systemidentifiering och experiment med en MPC-regulator.

Résumé

Le chauffage représente aujourd’hui l’un des plus importants postes de dépense énergé-tique. Les pompes à chaleur géothermiques sont une alternative efficace, particulièrementprisée dans les pays nordiques. Les systèmes actuellement disponibles sur le marché disposentd’un contrôle basique et peu flexible. Des stratégies de commande plus avancées telles que lacommande prédictive peuvent permettre de garantir une température intérieure confortabletout en diminuant la consommation. L’objectif de cette étude est d’évaluer le potentiel deréduction tant de la consommation énergétique que de la facture d’électricité. La prise encompte de l’efficacité de la pompe à chaleur s’avère alors primordiale. Différents scénarios sontprésentés, et incluent notamment des prix de l’électricité variables en utilisant la commandeprédictive économique (EMPC). Les résultats en simulation montrent que la commande parMPC devient bénéfique en intégrant à la fois un confort et des prix variables, avec des écono-mies allant jusqu’à 8% et ce malgré la perte d’efficacité de la pompe à chaleur. Les applicationspratiques sont traitées en procédant à une identification du système et avec l’implémentationet le test d’un contrôleur.

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Acknowledgements

I would first like to thank my examiner Pr. Elling Jacobsen and my supervisor Olle Trollberg whosupported me and especially helped me with the most challenging technical parts of the study. Iwould like to offer my special thanks to my employer at Dreik Ingenjörskonst, Paul Dreik, whohas been a constant source of inspiration during the project. In particular, without his passionateparticipation and input, the implementation could not have been successfully conducted.

I am also grateful to my opponents Ahmed Elfeky and Motoya Ohnishi whose pertinent remarksand comments helped me improved my work. I would also like to acknowledge Alberto Díaz Do-rado for his feedback and our passionate discussions on our respective thesis.

Finally, I must express my very profound gratitude to my parents and to my sister for provid-ing me with unfailing support and continuous encouragement through the process of researchingand writing this thesis. Thank you.

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Contents

1 Introduction 71.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Background 92.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Physical modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2 Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Modelling of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 Ground source heat pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Thermal modelling of buildings . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 System Identification 143.1 Setup and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.3 Input-signal design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.1 Pre-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.2 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.3 Model evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Results and adopted model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.1 Model comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3.2 Towards MPC : state space model . . . . . . . . . . . . . . . . . . . . . . . 233.3.3 Extending the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Controller design 254.1 Unconstrained MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.1 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1.2 Dense formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1.3 Cost function design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Hard constraints and feasibility handling . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Economic MPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Results 305.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.1.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.1.2 Scenario A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.1.3 Scenario B: varying comfort cost . . . . . . . . . . . . . . . . . . . . . . . . 355.1.4 Scenario C : electricity prices . . . . . . . . . . . . . . . . . . . . . . . . . . 375.1.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2.1 Dealing with uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2.2 Implementation aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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6 Conclusion and future work 46

Bibliography 47

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Chapter 1

Introduction

1.1 MotivationAccording to the European Commission, building heating and cooling accounts for half of theenergy consumption in Europe, of which 84% is still generated from fossil fuels [1]. In Sweden,the heating market represents 100TWh and 100 billion SEK, single house market being by far thelargest sub-market [2]. Economically speaking, district heating and electrical heat pumps are themost popular heating sources, representing each about 40% of the market while the rest mostlycomes from biomass. Finally when comparing the origins of the costs regarding, heat pumps,operating costs account for three quarters, the rest being mostly capital costs. With more than amillion heat pumps installed, the market is quite saturated and consists mostly of replacements ortap water heat pump systems.

Ground source heat pump is an efficient system for building heating, notably since the mainheat source is the ground and considered to be free and clean energy. Hence GSHPs have a lowimpact on the environment which is particularly true if the electricity used to power the heatpumps (mostly the compressor) originates from renewable energy sources. This is relatively validin Sweden where hydro and wind power account for more than 50% of the power mix, the restbeing mostly nuclear, therefore also carbon free energy. Hence GSHP is a technology that has aparticularly small ecological footprint. Apart from better building insulation, the energy consump-tion and cost can be cut off even further by using temperature control in smart buildings. Hencethe optimization of heating energy in buildings has appeared as a promising way to reduce energyconsumption. While renovation typically requires a significant investment and long constructionwork, use of Information and Communication Technology (ICT) generally has very low hardwarecost which makes it interesting for a very wide range of applications. In smart buildings, there ispotential to adapt the indoor temperature to the use of the building such as occupancy. Officebuildings heating can for instance adapt to office hours to reduce the consumption.

Moreover, such technologies appear as key components of smart grids in order not only to re-duce consumption but also to adapt load to production and cut off electricity demand peaks.Electric energy is particularly difficult and expensive to store, both in terms of economic cost andenvironmental impact. In order to continuously match the load to production one possibility is toencourage load shifting of equipments and devices that can be delayed. Heating is a good candidateto do load shifting since it allows to store heat energy in the building. This is beneficial at thedemand peaks such as in the morning when demand is the highest and becomes more and moreimportant when integrating high rate of renewable energy sources. Varying electricity prices are away to reflect, for the end user, the needs of the grid. Higher prices account for high demand andlow production (due to meteorological condition for renewable sources). Electricity prices are gen-erally forecast 24 hours in advance. Hence one can adapt his consumption to reduce its electricitybill and would correspond to a more manageable grid for the Transmission System Operator (TSO,Svenska kraftnät in Sweden). Hence in the future an increase in the general public awareness andeconomical interest would increase the grid flexibility, making the integration of renewable energysources easier.

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1.2 ObjectivesIn this thesis the main objective is to design a novel strategy for control of the indoor temperature,applicable to GSHP systems. In particular, we aim at utilizing model predictive control to enableoptimization of both the indoor comfort and the cost of operating the system. The heat pump’sefficiency is taken into account to match the real behaviour of the system. The use of EconomicMPC allows us to include electricity prices in the optimization problem. Different scenarios arecompared in simulation in order to determine the most promising approach for energy and eco-nomics savings. Data reflecting real life situations regarding weather and prices are used in thesimulations. The suggested controller is also tested in a real-world application. We perform asystem identification step in order to obtain a model of the plant suitable for the experimentaltesting. This model is also used in the different simulations.

In chapter 2 we present some background regarding previous studies on the subject and physics ofthe system, both for the thermal modelling of the house and the heat pump. The system identifi-cation is presented in Chapter 3 using a black box approach. Chapter 4 focuses on the design ofthe controller. The different simulated scenarios and their respective potential in terms of savingsare detailed in Chapter 5.

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Chapter 2

Background

In this chapter we provide some background regarding the existing literature on the differentrelated subjects. That includes both studies on the physical modelling of the system — mostlythe building and the heat pump — and studies focusing on using MPC to indoor climate control.Physical modelling of the heat pump and the building is then addressed in the second section.

2.1 Literature ReviewGeothermal ground source heat pump (GSHP) systems have become more and more popular world-wide in the past decades. The market has particularly grown in countries with a cold climate andwith environmental concerns such as Canada, Sweden and Switzerland. To this day, Sweden is thefirst country in terms of GSHP installed per capita which account for 23TWh of heating energy peryear [3]. The geological conditions are generally beneficial in Scandinavia while Swedish climatemakes the cooling mode mostly irrelevant. In Sweden, GSHP are being used both for individualheating and district heating. The systems and have become more powerful with capacities com-monly up to 25kW and borehole depths up to 200m. From this perspective, the system used forpractical experiments in this study, which consists of a single house heated by a 16kW GSHP, maybe considered a typical installation.

Below, we provide brief overviews of the literature regarding the physical modelling of GSHPsystems, and for MPC, the main control strategy considered in this paper. However, we first re-mark that apart from the specific literature review some work covers a very wide range of topics,including physical modelling and control aspects. Notably [4] emphasizes very clearly the differentlevels of control, from the building to the borehole, and the corresponding required models forMPC though an air to water heat pump is studied. The study in [5] focuses on the modelling andoptimization of the variable speed heat pump. The readers interested in cooling applications ofGSHP may refer to [6]. Finally for Swedish speaking people [7] presents a more practical approachto GSHP including economic aspects.

2.1.1 Physical modellingIn this section, we provide an overview of the literature regarding the modelling of the system, fromthe borehole to the building, emphasizing the importance of the heat pump efficiency, often namedCoefficient of Performance (COP). The author of [8] proposes a numerical modelling strategy ofthe heat pump system taking into account the ground side: ground temperature, borehole lengthand refrigerant properties. [9] presents a physical analysis of the real efficiency of a heat pump andthe savings - both in term of energy and economics - a GSHP can offer. The COP modelling isalso the focus of [10] where an identification strategy using manufacturer’s data only is successfullydeveloped.

Regarding the building modelling and identification, a grey box approach using electrical circuitanalogy seems very popular, see [11] and [12]. [12] uses the output of solar cells to investigatesolar radiations. Building modelling papers often take into account the purpose of controlling thetemperature which is emphasized in [13] with a particular focus on MPC.

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2.1.2 Model Predictive ControlIn the past years, an increasing number of academic papers has considered applying MPC forbuilding climate control. A popular setup is to combine a central MPC controller with decen-tralized PI controllers for the different actuators (valves, heaters, . . . ), leading to a hierarchicalcontrol structure. Apart from the specific control strategy used, the different studies can also bedifferentiated with respect to certain aspects of the controlled system:

• Type of the heat pump (air to water, water to water, etc.)

• Control of the compressor (on/off or frequency control)

• Heating load (radiators, floor heating, tap water tank, . . . )

• Climate (heating and/or cooling mode)

An extensive literature review regarding MPC applied to air conditioning can for instance be foundin [14]. Some papers have a specific focus on the heat pump side, taking into consideration itsefficiency. For instance [15] tested an optimization strategy for GSHP in order to reduce electricityconsumption, in cooling mode though. [16] implemented a MPC controller for a large building withencouraging results in terms of energy savings. The OptiControl-II Project is an complete studywith long term experimental testing that has been carried out in an office building in Switzerland.It includes partnership with industrials but results presented in [17] suggest the savings are notworth the complete investment, which includes Thermally Activated Building System (TABS) andAir Handling Unit (AHU).

Regarding uncertainties, various ways of handling them have been studied, including stochasticMPC and robust MPC, respectively in [18], [19] and [20].

Recently Economic Model Predictive Control (EMPC) has been employed to optimize the costtaking into account varying electricity prices in the optimization. The study presented in [21]obtained particularly good results in simulation, while th authout in [22] combines Hybrid MPCand EMPC and discusses the importance of the heat pump efficiency. In [23] and [24] EMPC ispart of a more advanced smart building system that includes batteries, photovoltaic, electric waterheater or thermal energy storage (TES) with Air Source Heat Pumps (ASHP).

The literature is quite rich and diversified. However most of the mentioned studies lack exper-imental concern. Apart from actual testing which might be difficult to implement, experimentaldata such as actual outdoor temperature (or solar radiations) and varying electricity prices shouldbe used in order to reflect real life situations. In the same spirit simulations should take place ona rather long time scale, whereas results on a few days might be biased due to particular condi-tions (weather or electricity prices). Moreover the different papers assume high complexity controlin terms of sensors, actuators and computations is possible, which might not be possible in anindustrial context.

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2.2 Modelling of the systemIn this section we present some physical background regarding the modelling of the system. Under-standing of the heat pump’s behaviour is necessary to limit the consumption while the building’smodelling is used to maintain a comfortable indoor temperature. Hence the heat pump’s modellingis to be used in chapter 4 for the MPC design and the building modelling in chapter 3 for systemidentification. First we explain the ground source heat pump principle with a specific focus on theefficiency, we then go on to the modelling of the building itself.

2.2.1 Ground source heat pumpPrinciple

A ground source heat pump (GSHP) is a heat engine that, when working in heating mode, extractsgeothermal energy from a cold source (the ground) to heat a hot source (the building) while usingelectrical power. The system can be divided into three main parts, corresponding to three differentcircuits.

The ground side circuit consists of one or several boreholes connected to the heat pump. Whileshallow geothermal exists, deep boreholes (currently around 200m deep) require a smaller footprintand offer better performances due to higher ground temperatures. Heat extraction is carried out bya fluid circulating through a U tube, the use of alcohol based refrigerant is required to avoid freezing.

The building side is mainly composed of the heating actuators, which may include floor heat-ing system, radiators, tap water tanks, etc. Though extra heating systems such as electric heatersare sometimes included, they are not part of the GSHP itself and will be ignored. For the systemconsidered in this study radiators are the heating actuators.

The heat pump itself consists of an evaporator where the heat is extracted from the ground sideand the refrigerant evaporates, a compressor that increases the temperature, a condenser thattransfers the heat to the house side and a thermo-valve that makes the pressure drop before anew cycle begins. The compressor is the main controlled actuator through its frequency. Whileearly systems were equipped with on/off compressors, varying frequency systems are know quitecommon and allow better control.

Figure 2.1 – Diagram of a ground source heat pump (from [25])

Efficiency

The main parameter characterizing the behaviour of a heat pump is its efficiency, generally refereedto as Coefficient Of Performance (COP). For the heating mode of the system, it is defined as thethermodynamic efficiency i.e., the ration of the heating power coming out of the heat pump overthe consumed electric power:

COP =ΦhPel

(2.1)

When considering an ideal engine with reversible transformations, thermodynamics laws lead tothe Carnot efficiency:

COPcarnot =Th

Th − Tc=

TfTf − Tg

(2.2)

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where Th and Tc respectively refer to the hot and cold side temperatures of the engine. For aheat pump in heating mode, the hot side is the feed temperature Tf of the water going to theheaters while the cold side is the fluid from the borehole, named the ground temperature Tg. Eventhough Equation (2.2) does not correspond to the real efficiency (it’s an upper bound), it reflectsthat the cold and hot source temperatures should be as close as possible. Like in most studies,we will consider the ground temperature constant, which is valid over time horizons such as a dayor a week, but not over a month or year. When considering Tg constant, the hot side should beas low as possible. This explains why floor heating is more efficient than radiators, the thermalmass of the floor being more important than the one of heaters, the required temperature is lower,making the heat pump (or other devices) operate in a more efficient working region. The heat-pump manufacturer provides some values for the COP for specific cold and hot temperatures [26],see for example Table 2.1 which correspond to the particular heat pump used in this study.

Table 2.1 – Manufacturer’s data regarding COP at specific operating points.

cold side (◦C) hot side (◦C) ideal COP real COP rated output (kW) supplied power (kW)0 35 8.80 4.85 8.89 1.830 45 7.07 3.77 8.63 2.2910 35 12.3 6.11 11.22 1.8410 45 9.09 4.72 10.92 2.13

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2.2.2 Thermal modelling of buildingsA common way of modelling the heat dynamics of buildings is to use the electrical circuit analogywhere thermal resistances act as electrical resistance and thermal capacity as electrical capacitors.Temperature then corresponds to voltage and current to heat flow source. As presented in [11] theheat flow can be described by a set of differential equations such as

dTidt

=1

RiaCi(Ta − Ti) +

1

RihCi(Th − Ti) +

1

CiAwΦs

dThdt

=1

RihCh(Ti − Th) +

1

ChΦh (2.3)

where the system is modelled with two states corresponding to the indoor and the heaters tem-perature Ti and Th, knowing the outdoor temperature Ta. R accounts for the thermal resistancebetween two areas, e.g., between the indoor and outdoor temperatures. Heat capacities C arerelated to the inertia of the different subsystems, typically a floor heating system corresponds to agreater Ch than a radiator system. Two heat sources are present, the solar radiation Φs acting onthe indoor temperature through the effective window area Aw and the heating power Φh from theheating system.

Figure 2.2 – RC network model with states TiTe, from [11].

This framework can be extended by adding new states such as an envelope temperature Tebetween the indoor and outdoor temperature or a dedicated state for the temperature sensor Ts.One can also differentiate between indoor temperatures for different rooms, Ti1, Ti2, . . . with cor-responding thermal resistances between them. In this work however we will always consider thetemperature to be homogeneous throughout the house. Different inputs can also be added to ac-count for additional sources such as electrical equipment or living beings. However it is generallyhard to take into consideration all possible factors, and extra sources are typically modelled asdisturbances in a simpler model. For instance the solar radiation term Φs is often dropped becauseit is hard to model or measure. For the system under study we consider the temperature of thewater coming out of the heat pump to the radiators and the temperature of the water out ofthe radiators going back to the heat pump, respectively named feed temperature Tf and returntemperature Tr.

Finally, we precise that the presence of a tap water tank is not modelled here. The heatingsystem studied in this thesis includes one that is heated by the same heat pump as the radiators.Occasionally the heat pump switches from building heating to tap water generation, typically afterintensive tap water use such as showering. Such behaviour is hard to predict, but more importantlyfor the system at our disposal we do not control the switching to tap water generation. Hence weconsider it as disturbances.

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Chapter 3

System Identification

In this chapter we perform the system identification to obtain the model later used both forsimulations and experimental testing. The reference book used regarding system identification onboth theoretical and practical aspects is [27]. In the first section we precise the objectives of thesystem identification and the experimental setup. Section two explains the methodology used foridentifying the model and section three presents the experimental results and the final model.

3.1 Setup and objectives

3.1.1 ObjectivesOur objective is to identify a suitable model of the house for control purposes of the indoortemperature. A black box approach is first developed to identify the building dynamics with thefeed temperature coming out of the heat pump as the main input. The obtained model is thenextended to include the heat pump’s behaviour. Meteorological data such as outdoor temperatureare non controlled inputs of the system, or can be seen as measured disturbances. The returntemperature from the heaters is an extra output that has little interest itself but is measuredduring the identification process. The feed temperature going go to the heaters Tf can be seen asa physical input of the system, it is not directly manipulated. In fact several inputs are meaningful

• Tfeed is the physical input from the building perspective since it drives the temperature ofthe heaters.

• Tfeed,ref is the input actually given to the system. Though it has to pass through the internallogic of the heat pump.

• The compressor speed or frequency is, in the end, the actual physical actuator controlled.Valves openings should also be considered but those data are unknown to us. The electricalpower consumed by the heat pump is directly related to the speed but is not measured.

• The heating power coming out of the pump is, as seen in section 2.2.2, the input for modellingusing electrical circuit analogy. However that quantity is also hard to measure.

The chosen input for system identification is Tfeed, knowing that the actual control signal sent isin fact Tfeed,ref . The input data used for the identification are then coming from internal measure-ments of the feed temperature and not from the manipulated signal. Deciding which weather datashould be included as additional inputs is part of the model selection process. Relevant weatherdata includes outdoor air temperature, solar radiation and wind speed.

3.1.2 Experimental setupThe system under study is a two floors individual house situated in Nacka, Stockholm. Measure-ments available mostly come from internal sensors furnished with the heat pump or, for weather,from open data available. Internal sensors at our disposal are the ones available with a classicalheat pump system:

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• Sensors inside the heat pump system measuring the temperatures of the water coming outof the heat pump to the radiators and back to the heat pump.

• One sensor in the house is located in the kitchen and measures the indoor temperature.

• One sensor is located outside, on a window sill, to measure the outdoor temperature.

After the first trials, the outdoor temperature measurements appeared to be flawed. Being locatedvery close to the house the measured temperature is a few degrees higher than the actual outdoortemperature, and this gap may depend on other conditions such as wind speed and direction.Therefore it was decided to use data from the Swedish Meteorological and Hydrological Institute(SMHI) station in Tullinge instead (located 20km from the house). Those data are more reliableand more robust to unexpected events such as snow but might be a bit off due to the differentlocation. The wind speed data are coming from the same station in Tullinge[28] while the solarirradiance can bound at[29].

Table 3.1 – Measured inputs and outputs for system identification. The original sampling timebefore any resampling is indicated.

Data Notation Kind Sources Sampling UnitIndoor temp. Ti Output sensor 5s KReturn temp. Tr Output sensor 5s K

Outdoor temp. Ta Input sensor or external 5s & 1h KFeed temp. Tf Input sensor 5s K

Solar radiation Φs Input external 1h W/m2

Wind speed ws Input external 1h m/s

Note that only one room temperature is measured, meaning the indoor temperature is assumeduniform throughout the house. As mentioned earlier, the actual electric consumption of the heatpump is unknown.

3.1.3 Input-signal designThe input signal that can be controlled is the required feed temperature that comes out of the heatpump to the radiators. However that signal passes through the internal logic of the heat pumpsystem, meaning it will be filtered before it is applied to the system. Since that internal logic ismostly unknown and that the actual feed temperature is measured, it is assumed for input signaldesign purposes, that the feed temperature is directly controlled. The measured feed temperaturewill hence be a filtered version of the input signal.

The purpose of a well designed input signal is to excite the system in the appropriate frequencyrange, where the input signal should then have a strong frequency content. For control, the mostimportant frequency range is typically the frequencies close to the closed loop bandwidth. A com-monly used input signal to do so is a Random Binary Signal (RBS) that takes only two values andswitches between them at pseudo - random times in order for the final signal to have its powerspectrum within a certain range. In [13] the following strategy is given to determine the spectrumto cover. Let τ and τ be the lowest and highest time constant of the system respectively, thespectrum to be covered by the input signal is [f ; f ] with

f =1

βτf =

α

τ

where typical values for the parameters are α = 2, β = 3. According to the TiTeTh model in [11]reasonable time constants can be evaluated: τmin = 0.13h and τmax = 18h. However, values fordesigning the input signal are τ = 1h and τ = 20h. Longer time constants are considered becausethe house to be identified is larger and therefore likely to be slower than the one in [11].

The amplitude of the signal should cover the normal operating range with the mean representing atypical operating point. However too large variations causes fast changes in the compressor speedand might not be a suitable from a mechanical perspective. As a comprise the signal range was

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set up at [33 45] degrees which was deemed large enough to yield sufficient information, but smallenough to avoid mechanical issues in the compressor.

The signal utilized for identification is illustrated in Figure 3.1 and Figure 3.2. The first shows thetime realizations of the signal while the second shows its frequency content.

Time (h)

0 10 20 30 40 50 60 70 80 90 100

Feed tem

pera

ture

(°C

)

32

34

36

38

40

42

44

46Designed input signal

Figure 3.1 – Realization of the PRBS signal designed as the input signal.

Frequency (mHz)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

|U(f

)|

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Power Spectrum of the input signal

Figure 3.2 – Power spectrum of the designed input signal.

16

3.2 Methodology

3.2.1 Pre-processingAll data are to be divided between an identification set and a validation set. The first is selected bychoosing the specific days during when the house is unoccupied in order to limit disturbances, theremaining data being used for validation. The full experiment lasts over a week, with respectively4 and 3 days for validation and identification. Since the data originates from different sources,different pre-processing is suitable according to the kind of data at hand. Low pass filtering isapplied to internal measurements in order to filter out the noise, knowing that the interestingfrequency content is quite far from noise and disturbance frequencies. The measurements fromexternal sources (weather data) are already handled. However they have to be up-sampled tothe common sampling time of 5s. This is achieved as follow : first a zero order hold is applied,then data is then smoothed using a moving average filter. The different data obtained after preprocessing are displayed in Figure 3.3.

3.2.2 IdentificationParametric identification strategies aim at explaining the measurements YN using a model char-acterized by a parameter set θ. That can be done by minimizing the error between predicted andmeasured outputs, which is known as Prediction Error Methods (PEM) and can be formulated as

θ = arg minθ

N∑k=1

ε[k, θ]2 (3.1)

where θ is the set of parameters to be identified and ε[k] = y[k] − y[k] the prediction error. Themodel structure chosen is an autoregressive with external input (ARX) model of order na, nb. Ifone considers only one output and one input the difference equation for ARX model is :

y[k] +

na∑i=1

aiy[k − i] =

nb−1∑i=0

biu[k − i] + ε[k] (3.2)

which yields to the following prediction error :

ε[k] = y[k] +

na∑i=1

aiy[k − i]−nb−1∑i=0

biu[k − i] (3.3)

When considering N � na, nb measurements one can re-write the problem as an overdeterminedlinear regression :

YN = ϕTNθ + εN (3.4)

where YN is the vector of the measured outputs and ϕN contains measurements of inputs andoutputs :

θ =

a1...ana

b1...bnb

and ϕTN =

−y[k − 1] . . . −y[k − na]...

−y[k +N − 1] . . . −y[k +N − na]u[k − 1] . . . u[k − nb]

...u[k +N − 1 . . . u[k +N − nb]

Hence the estimated parameters θ that minimizes the prediction error can be computed by usinga least squares approach :

θ = (ϕTNϕN )−1ϕTNYN (3.5)

Such an approach is however not suitable regarding numerical errors, typically due to poor con-ditioning of (ϕTNϕN ). For implementation, the control package from Octave [30] is used with theARX function which solves this in a robust manner and applies also to the multiple input multipleoutput case, which interests us.

17

50 100 150 2000

2

4

6

8

10

Time(h)

Speed (

m/s

)

validation data

identification data

Experimental data for the wind speed after smoothing

Wind speed

50 100 150 2000

50

100

150

200

250

300

Time(h)

Sola

r ra

dia

tion (

W/m

2)

validation data identification data

Experimental data for the solar radiation after smoothing

Solar radiation

Figure 3.3 – Plots of the different signals used as inputs and outputs after pre-processing. On eachplot dashed lines delimit the range for identification data and validation data.

18

3.2.3 Model evaluationTo evaluate the quality of models both identification and validation data were used. We presentfirst two ways of obtaining a prediction using the identified model. Afterwords, the residues - thedifference between the measurements and the predictions - can be used to compare and select themodels.

Output Error (OE) - N step ahead estimator

The output error (OE) model assumes that the errors are white and added to the output. Thatmeans that there are no additional information to be gained from looking at previous output. Thepredicted output is thus computed using only the past and present inputs. The predictor may beformulated as:

yOE [k] =B(q−1)

A(q−1)u[k] (3.6)

or in the temporal domain as

yOE [k] = −na∑i=1

aiyOE [k − i] +

nb∑i=0

biu[k − i] (3.7)

The error between the predicted and the measured output for a given set of parameters is then

εOE [k, θ] = yOE [k]− y[k] (3.8)

Prediction Error Minimization (PEM) - One step ahead predictor

The one step ahead predictor, knowing the past measurements, can be formulated using the ARXdifference equation as in eq. (3.2):

yPEM [k|k − 1] = −na∑i=1

aiy[k − i] +

nb∑i=0

biu[k − i] = y[k]− ε[k]

Where y[k− 1], y[k− 2], . . . denotes the actual measurements while yPEM [k|k− 1] is the predictedoutput at time k. The prediction error is thus

εPEM (k, θ) = y[k]− yPEM [k|k − 1] (3.9)

Or equivalently in the frequency domain

εPEM [k] = A(q−1)y[k]−B(z−1)u[k]

Hence one can express the PEM residues as the OE residues filtered through A(q−1):

εPEM [k] = A(q−1)

[y[k]− B(q−1)

A(q−1)u[k]

](3.10)

Even though that approach presents good results it has two important downsides:

• At the measurement sampling frequency h = 5s all the variables have very small variations(especially indoor temperature) which means that a simple model Ti[k] = Ti[k − 1] wouldalso give very convincing results even though it would not carry any useful model. To avoidthat the model was re-sampled at h = 120s before proceeding with the analysis of the systemidentification.

• This approach supposes measurements from step k are available to predict step k + 1, whileMPC must predict steps k, k + 1, . . . k + Nh using measurements only up to step k. This iswhy we also present the output error estimator that is the one to be applied in the MPC,even though the identification can be evaluated with PEM only.

19

Residuals analysis

Even though the identified model has two outputs, only the indoor temperature is considered forthe evaluation of the model. This is relevant since the control objectives mainly concern the indoortemperature. The analysis of the residuals allow to evaluate the quality of the model.

Typically the coefficient of determination is commonly used as a numerical parameter to com-pare the fitting of different models. It can be defined using the total sum of squares :

SStot =∑k

(y[k]− y)2

with y the mean value of the measurements, and the residuals sum of squares :

SSres =∑k

ε[k]2

The coefficient of determination is then defined as

R2 = 1− SSresSStot

(3.11)

The R2 coefficient can be interpreted as the ration of the explained variance by the model andshould then be as close as possible to one.

Moreover, the PEM residuals should approach the characteristics of a white noise. Otherwisethat means some kind of systemic variation in the data is not captured by the model. This isinvestigated by looking at the autocorrelation of the residuals and the cross correlation with theinput signal.

3.3 Results and adopted model

3.3.1 Model comparisonSeveral low order models are first tried to select the most appropriate one. When interpreting thedifferent models as continuous time systems using zero order hold, model with orders above twopresent very fast time constants (around a second) which is unlikely to correspond to a physical be-haviour and may on the contrary come from over-fitting, i.e., the model is fitting the noise.Keepingthe order low is also beneficial for computation time when applied to MPC. The remaining de-sign parameter is then the set of inputs to consider, from the most basic one that is feed andoutdoor temperature only to the most complete which adds solar radiation and wind speed. Thenext paragraphs investigate the results of the system identification using the previously explainedmethodology in order to select a suitable model. Since the fastest time constant of the identifiedmodels is around half an hour the model and data are down sampled accordingly with a samplingtime of two minutes.

Output Error predictor

In Figure 3.4 the different residuals according to the varying set of inputs are plotted. It suggeststhe improvements between the different models is not straightforward. Even though the differentcurves seem to be offset between each others one can notice their relative positions change. Forinstance the blue curve (representing the four inputs model) appears successively in a bottom(t = 10h), intermediate (t = 50h) and upper (t = 160h) position. As expected the fitting is muchbetter for the identification than the validation data. However that is partly due to the fact thatthe house was occupied when validation data was collected and empty when identification data wascollected. Occupancy can lead to unmodeled heat sources (from inhabitants and equipments) butalso unpredictable disturbances. The latter can be seen in the two downward peaks at t = 75 andt = 105 or in Figure 3.5 which corresponds to a rather long opening of the entrance door, makingthe immediate temperature around the sensor colder in a short time. The system then takes a fewhours to recover. Even though such an event is unfortunate and difficult to model, it should notbe filtered out since it does account for a physical behaviour of the system and the correspondingfrequency content is precisely within the interesting bandwidth. Hence such peaks are smoothedby the filter, but not cut.

Table 3.2 shows that the different models have comparable performances on the identificationdata but the one with four inputs can better predict the behaviour on the validation data.

20

0 50 100 150 200

-1

-0.5

0

0.5

1

1.5

Time(h)

Re

sd

idu

als

(K

)

validation data

identification data

Residuals of Output Error (OE) estimator

Outdoor, Feed, Wind and Sun

Outdoor, Feed and Sun

Outdoor and Feed

Outdoor, Feed and Wind

Figure 3.4 – Comparison of the residuals for validation and identification data with 4 different setsof inputs chosen within feed temperature, outdoor temperature, solar radiation and wind speed

Table 3.2 – Coefficient of determination for identification and validation data with the differentinput sets

Input set Identification data Validation dataOutdoor, feed temp., solar radiation and wind speed 0.89 -0.31

Outdoor, feed temperature and solar radiation 0.87 -0.71Outdoor and feed temperature 0.89 -0.67

Outdoor, feed temperature and wind speed 0.91 -0.77

Prediction Error Minimization predictor

When using the past measurements and PEM predictor the results improve a lot. The coefficientof determination is more than 0.999 and residues are very low, which can be seen in Figure 3.6.Correlation analysis is shown in Figure 3.7. One can notice some peaks for lags correspondingroughly to one and two days. This can be interpreted as unmodeled behaviour of daily periodic-ity. This can either come from bad or insufficient use of meteorological data or, more likely, thedisturbances from occupancy, since human activities can be considered to have be approximatelyperiodic with a period of 24 hours.

The PEM model shows convincing results, even though weather data still carry uncertaintiesand disturbances typically related to human behaviour is mostly difficult to tackle. However, itshould be kept in mind that even though such an approach allows to evaluate the system identi-fication, it can not be directly applied to MPC. Indeed the core idea of MPC is to predict on arather long time horizon the behaviour of the system whereas, as derived above, PEM gives onestep ahead predictions. Hence the model used for control would give results in between since it isstill possible to update the current state estimation with data from output measurements in orderto predict 24 hours ahead whereas the OE model never uses measurements to correct its estimate.

The final identified model utilizes all four inputs. However it is possible to use it with fewerinputs by dropping solar radiations and/or wind speed. This amounts to fixing those inputs at

21

0 50 100 150 20017

18

19

20

21

22

Time (h)

Ind

oo

r te

mp

era

ture

(°C

)

validation data identification data

Model fitting of the four input OE model

OE estimation

Measurement

Figure 3.5 – Comparison between OE estimator (four inputs) and measurements.

0 50 100 150 200-0.2

-0.15

-0.1

-0.05

0

0.05

Time (h)

Te

mp

era

ture

(K

)

Residuals of PEM estimator

Figure 3.6 – Residuals of the indoor temperature for the PEM estimator.

zero, which still has physical meaning (solar radiation is mostly zero during winter months forinstance).

22

0 50 100 150 200-0.2

0

0.2

0.4

0.6

0.8

1

lag (h)

au

toco

rr

Autocorrelation of the residuals

Figure 3.7 – Autocorrelation of the residuals for the PEM estimator

3.3.2 Towards MPC : state space modelSince MPC is conveniently formulated using a state space model, the identified model is presentedin that form. Since the identified model has two states and two outputs, one can perform a statetransformation — assuming the output matrix is invertible — so that the states matches theoutputs: x ≡ y. The states hence have direct physical interpretation as they correspond to theindoor and return temperature. The identified model is transformed into a continuous time modelusing zero order hold. This is consistent with the physical derivation presented in Section 2.2.2,though discrete time model is to be used in MPC. The resulting numerical model is:

˙(x1x2

)=

(−1.62e− 5 0

0 −5.98e− 4

)︸ ︷︷ ︸

=:Aid

(x1x2

)+

(1.12e− 5 4.57e− 6 6.79e− 8 −1.56e− 65.31e− 4 6.46e− 5 −8.19e− 7 −1.67e− 5

)︸ ︷︷ ︸

=:Bid

TfTaΦsws

(TiTr

)=

(1 00 1

)︸ ︷︷ ︸=:Cid

(x1x2

)(3.12)

Notice the decoupling between the states in the Aid matrix, tough the states are coupled throughthe inputs, especially Tf . In order to have physical intuition of the identified model one can lookat the time constants which respectively correspond to the dynamic of the heating system and ofthe house :

τ1 = 28min τ2 = 17h (3.13)

Those time constants are consistent with the one used for the input-signal design. The static gainsin Table 3.3 also carry physical meaning. For instance one would expect that, in the absence of sunand wind, identical outdoor and feed temperature would give the same indoor temperature. Thatmeans the static gains of the two first columns, for each row, should sum to one. This is almostthe case for the indoor temperature (0.97) and exactly for the return. However the negative gainassociated to solar radiation and return temperature appears odd.

23

Table 3.3 – Static gains of the identified model between the different inputs and outputs.

Tf (K) Ta(K) Φs(W ) ws(m/s)Ti(K) 0.69 0.28 0.0042 -0.096Tr(K) 0.89 0.11 -0.0014 -0.28

3.3.3 Extending the modelUsing the feed temperature as the main input for control is not deemed suitable as it does not easilytranslate into cost, so it was decided to switch to a heating power input, more in tune with thederivation in Section 2.2.2. As mentioned above that heating power coming out of the heat pumpdoes not have any physical interest itself but is the easiest to manipulate for control purposes. Anapproximation of the relationship between heating power, feed and return temperatures can befound in [11]:

Φhp = Cwf Tf +1

Rfr(Tf − Tr) (3.14)

where Cwf is the heat capacity of the water circulating in the pipes at the feed temperature andRfr the thermal resistance between the feed and return temperature. Using that relation the modelcan be extended with the feed temperature now as the third state and Φhp as the input :

˙TiTrTf

=

Aid(1, 1) 0 Bid(1, 1)0 Aid(2, 2) Bid(2, 1)0 1

CwfRfr− 1CwfRfr

︸ ︷︷ ︸

=:A

TiTrTf

+

0 Bid(1, 2) Bid(1, 3) Bid(1, 4)0 Bid(2, 2) Bid(2, 3) Bid(2, 4)1

Cwf0 0 0

︸ ︷︷ ︸

=:B

ΦhpTaΦsws

(3.15)

However, when interpreting the terms of the dynamic matrix, it appears the coefficients A(2, 3)and A(3, 2) should be the same. Physical modelling of the return temperature evolution indeedwould lead to :

Tr =1

CwrRrf(Tf − Tr) + . . .

Moreover, the water in the pipes can be divided between the water at temperature Tr and the waterat temperature Tf , the border corresponding to the radiators. Hence one can assume those twovolumes of water to be roughly the same (in the same spirit the heater’s temperature is generallyapproximated by the average of feed and return temperature). That implies the heat capacity arethe same, Cwr = Cwf . Since the resistance factors are also symmetrical, it holds

1

CwrRrf=

1

CwfRfr

So we can approximate the coefficient A(3,2) by the coefficient A(2,3) in the dynamic matrix.Finally Cwf only acts as a scaling factor on the input size and can be heuristically estimated usingthe water specific heat capacity Cwf ' 150cw.

With all inputs but the outdoor temperature at zero, one would expect the indoor temperatureto follow the variations of the outdoor temperature, with less amplitude as a result of the inertiaof the system. But mostly the two quantities should have the same average temperature sinceall additional sources are shut off. However it was observed the indoor temperature was too low,which is linked to the static gains displayed above. In order to correct that behaviour one can addconstant heat sources. Intuitively those sources can correspond to electric equipment always turnedon such as servers and fridge or human occupancy. A static heating power of Φ0 = 800W showedto be sufficient to correct the observed bias. Hence the model has to be corrected, a constant offsetof Φ0 being added to the heat pump. For readability reasons, this is not included in the derivationof the controller.

An alternative approach would have been to exploit more the physical modelling as explainedin section 2.2.2 and carry out a grey box identification. However, this fall outside the scope if thisthesis.

24

Chapter 4

Controller design

In this chapter we explain the design the MPC controller. In the first section the unconstrainedcase is studied focusing on the cost function design. Section 2 addresses hard constraints andfeasibility issues. In the last section we explain the use of Economic MPC. The reference bookused regarding model predictive control theory and practice is [31].

4.1 Unconstrained MPC

4.1.1 PrincipleMPC is a control algorithm that uses the principle of finite time optimization over a receding timehorizon. At each step the algorithm predicts the behaviour of the plant over the next time horizonusing the system model and, in our case, forecasts of disturbances. Using a previously designedcost function, the algorithm solves the optimization problem over the time horizon. From theoptimal control vector that is obtained, only the first step is applied and the process is repeatedafter each sampling time. Though numerous variations exist, the standard MPC algorithm can beformulated as follow:

1. Estimate current state x[t] using measurements and previous state. In our case the estimationproblem is separated from the control one which means that in this chapter, we assume statesare perfectly known throughout the process.

2. Predict future states over the receding time horizon as a function of the control inputs.

3. Find the optimal control vector with respect to some cost function.

4. Apply the first input of optimal control sequence.

5. Wait according to sampling time and repeat from 1.

4.1.2 Dense formulationConsider the MPC problem as explained above with sampling time h and a receding horizon ofNh samples. The minimization of the cost function can be written as

minu[t],...,u[t+Nh−1]

J(u[t], . . . , u[t+Nh − 1], x[t+ 1], . . . , x[t+Nh]) (4.1)

Consider the model we obtained from the system identification. For simplicity reasons, we assumethat among the weather related disturbances only the outdoor temperature Ta is used while thesolar radiation and wind speed are considered as unmeasured disturbances. In all simulations theoutdoor temperature over the upcoming time horizon is considered to be perfectly known usingweather forecasts. This is reasonable for short term predictions such as the one of 24 hours usedin simulation. The model can then be written as:

x[k + 1] = Ax[k] +B

(ΦhTa

)(4.2)

25

One can then separate the actual control input u = Φh and weather data w = Ta:

x[k + 1] = Ax[k] +Buu[k] +Bww[k] (4.3)

In the MPC condensed formulation, the cost function ,Equation (4.1), is to be written as a functionof the inputs only. This can be done by using the system dynamics to ei=eliminate the states inthe cost function. In our case upcoming outdoor temperatures are considered known using weatherforecasts. Let’s introduce the states, input and weather vectors over the time horizon :

Xt =

x[t+ 1]...

x[t+Nh]

Ut =

u[t]...

u[t+Nh − 1]

Wt =

w[t]...

w[t+Nh − 1]

The aim is to express future states knowing the present state and future inputs, which can be doneas:

Xt = GuUt +GwWt +Hxt (4.4)

where

H =

A...

ANh

Gu =

Bu 0 . . . 0ABu Bu 0 . . . 0...

. . .ANh−1Bu ANh−2Bu . . . Bu 0

and Gw similarly as Gu with Bw acting as Bu.

An alternative formulation of MPC is to keep the states as optimization variables. Equalityconstraints such as Aeqx = beq then have to be added to account for the dynamics of the system.The pros and cons of the condensed and sparse approaches from the computational perspectiveare investigated in [32]. It is showed that taking full advantage of the matrices structure, a sparseapproach is faster than a condensed one whereas a more basic solver benefits from the reducednumber of variables of the condensed formulation. The latter was observed in our case, which iswhy we selected the condensed formulation.

4.1.3 Cost function designLQ problem

Above, no assumptions were made on the form taken by the cost function J . Linear Quadratic(LQ) cost functions are widely used as they allow to use quadratic programming solvers whichare numerically efficient. Hence the cost function, expressed only as a function of Ut, has to beformulated as

J =1

2UTt HLQUt + UTt gLQ (4.5)

where HLQ is the Hessian of the problem and should be positive semi-definite in order to ensurethat the optimization is convex.

The cost function is designed to reflect the two main objectives of the control: achieve a goodlevel of comfort regarding indoor temperature while keeping the electric consumption as low aspossible. Hence the cost function can be split into two parts

J = ccomfJcomf + celecJelec (4.6)

where celec, ccomf respectively represent scalar weights on comfort and consumption.

Comfort-cost design

One can express the user comfort using a reference indoor temperature Tref , usually constant.The cost function then penalizes the squared difference between the reference and actual indoortemperature.

Jcomf =

Nh+t∑k=t+1

(Ti[k]− Tref )2 =

Nh+t∑k=t+1

Ti[k]2 − 2Ti[k]Tref + T 2ref (4.7)

26

Dropping the constant term, it can then be formulated as a linear quadratic cost :

Jcomf =

Nh+t∑k=t+1

x[k]TQcomfx[k] + x[k]T qcomf

with

Qcomf =

1 0 00 0 00 0 0

qcomf =

−2Tref00

The sum can be removed when concatenating the matrices :

Jcomf = XT QcomfX +XT qcomf (4.8)

where

Qcomf =

Qcomf 0 . . .

0. . .

0 . . . Qcomf

qcomf =

qcomf...qcomf

(4.9)

Hence the comfort cost does not depend directly on the control inputs. In order to eliminate thestate variables one can use Equation (4.4) :

Jcomf = (GuUt +GwWt +Hxt)T Qcomf (GuUt +GwWt +Hxt) + (GuUt +GwWt +Hxt)

T qcomf

Jcomf = UTt [GTu QcomfGu]︸ ︷︷ ︸Rcomf

Ut + UTt [2GTu Qcomf (GwWt +Hxt) +GTu qcomf ]︸ ︷︷ ︸rcomf

(4.10)

Electric consumption cost design

The consumption cost is computed using the electrical power consumed at each step:

Jelec =

t+Nh∑k=t+1

Pel[k] (4.11)

The electrical power can be expressed using heating power and the COP

Pel =ΦhCOP

(4.12)

where Φh = u is the input of the model and the COP reflects the varying efficiency of the heatpump, as explained in section 2.2.1. It is then possible to write the consumption cost as a QPproblem with matrices Relec and relec in the same fashion as for the comfort cost,i.e.,

Jelec = UTt RelecUt + UTt relec

Final cost function

The global cost function is obtained by

J = UTt (celecRelec + ccomfRcomf )︸ ︷︷ ︸=:0.5HLQ

Ut + UTt (celecrelec + ccomfrcomf )︸ ︷︷ ︸=:gLQ

(4.13)

Moreover, one can recall the physical meaning of the cost function is

J =∑k

ccomf (Ti[k]− Tref )2 + celecPel[k]

where the weighting coefficients and electrical power are positive (or zero). This means the costfunction is positive semi-definite. Assuming the problem is well conditioned, an analytical solutionto the optimization exists :

Uopt = H−1LQgLQ (4.14)

27

Such an approach is not efficient nor robust to numerical errors so the use of a solver is recom-mended. However it emphasizes the importance of having a good conditioning of the HLQ matrix.However the obtained HLQ matrix had poor conditioning, which makes the problem harder tosolve and for instance very sensitive to initial conditions. To avoid such a behaviour the problemis regularized (see [33]) by adding an a extra matrix Rstable to HLQ to penalize strong changes inthe input :

Rstable = 2ε

1 −1 0 . . . 0−1 2 −1 0 . . . 00 −1 2 −1 . . . 0...

. . ....

0 . . . −1 2 −10 . . . 0 −1 1

So that

1

2UTt RstableUt = ε

∑k

(u[k + 1]− u[k])2

where ε� ccomf , celec (assuming ccomf , celec ∼ 1) is a coefficient negligible compared to other costsso that the other objectives of the optimization are almost not affected by this term. Simulationsshow an order of magnitude such as ε ∼ 10−6 is enough to obtain a sufficient conditioning andhave very small influence on the performances.

4.2 Hard constraints and feasibility handlingIntroducing hard constraints reflects physical limitations of the system such as saturations. Itcan also ensure that some minimal requirements are met. For the system at hand, inequalityconstraints can be formulated on the input u regarding minimum and maximum heating power:

Pmin ≤ u ≤ Pmax (4.15)

The assumption that the system is not working in cooling mode means Pmin = 0. A reasonablevalue for the maximum power is given by the nominal maximum heating power from the manu-facturer, namely Pmax = 16kW .

Inequality constraints on the indoor temperature can be formulated in order to ensure that theindoor temperature remains within some reasonable bounds in addition to the energy considera-tions:

Ti,min ≤ Ti ≤ Ti,max (4.16)

Note that those two constraints - heating power and indoor temperature - differ since the first onedirectly addresses the input of the system, while the latter concerns a state.

The constrained quadratic programming problem can be formulated as :

minUt

1

2UTt HLQUt + UTt gLQ (4.17)

subject to: (4.18)AeqUt = beq (4.19)lb ≤ Ut ≤ ub (4.20)Alb ≤ AinUt ≤ Aub (4.21)

Equality constraints usually represent the dynamics of the plant but are not needed in the con-densed formulation. Lower and upper bonds lb, ub on the input are then constant vectors containingthe minimum and maximum powers. To transform inequality constraints from Ti to Ut one has touse the condensed formulation from eq. (4.4). Due to this, vectors Alb and Aub have to be updatedonline whereas constraints on the input are formulated offline. From the computational point ofview, it should be emphasized that the computational cost of adding constraints on the states issignificantly more important than the one of adding constraints on the inputs.

28

When adding hard constraints the feasibility of the solution may become an issue. The qp solverfrom Octave control package [30] uses active-set method and thus requires a feasible first guess toinitialize the search. In order to provide the solver with a suitable feasible first guess, a two-stepstrategy is implemented:

1. Formulate and solve the optimization problem with zero weight on consumption and hardconstraints only on input size. The initial guess for the solver can typically be a constantvector between Pmin and Pmax. The result of the optimization is thus feasible with respectto power hard constraints but also to indoor temperature hard constraints since the costfunction used has only the "comfort" term.

2. Use previously found optimal control into the complete constrained optimal problem withnon zero weight on consumption and hard constraints on input and indoor temperature.

Note that the computational complexity of the first step is negligible compared to the last one.This is due to the high cost of adding constraints on the states.

Another approach to handle feasibility issues is to soften the constraints by adding slack vari-ables. Though such an approach is promising, it was not carried out by lack of time and wouldhave led to similar results as the one obtained.

4.3 Economic MPCEconomic MPC (EMPC) is a more and more popular approach to include economic costs directlyin the cost function instead of at a higher hierarchical level in the control strategy [34]. In ourcase, we use a broad interpretation where economic cost refers both to consumption and comfortwhereas EMPC sometimes refers only to economic costs. Including (non constant) electricity pricesallows to optimize for the electricity bill instead of the power consumption. Similarly the comfortrequired by the user is not constant over time. Typically the comfort level is irrelevant as soonas the house is unoccupied and a varying cost on comfort can reflect if cost on comfort should beenabled or not. Hence the cost function can be modified :

J =

t+Nh∑k=t+1

cellel[k]Pel[k] + ccomf lcomf [k](Ti[k]− Tref )2 (4.22)

where lel[k] is the electricity price at time k and lcomf [k] reflects how relevant comfort is at time k.Those quantities are time dependant whereas cel and ccomf are tuning scalar parameters regardingthe trade-off between comfort and consumption. Time varying prices should be non negative toguarantee convergence of the optimization problem, which might be restrictive when consideringhighly volatile electricity prices. High production peaks of renewable energy may indeed lead tonegative electricity prices, which has for instance already been observed in Denmark. Electricitymarket prices can generally be known 24 hours ahead through the Transmission System Operator(TSO) or electricity market operator such as NordPool in the Nordic countries. Including electric-ity prices then relates to smart grids and the ability for the grid to shift and attenuate consumptionpeaks. Indoor climate and tap water have been seen as good candidates for load shifting strategydue to their slow dynamics and large energy storage capacity.

Implementation wise the methodology is very similar to the one developed in section 4.1.3 forthe cost function derivation. The different cost matrices and vector are now time dependant whichmeans they have to be computed online

Regarding the comfort cost, that can for instance be done as:

Qcomfort,var[k] =

lcomf [k] 0 00 0 00 0 0

qcomfort,var[k] =

−2Tref lcomf [k]00

Which allows to use afterwords the same derivation as before for concatenated matrices.

29

Chapter 5

Results

In this chapter we detail the difference results obtained. Different scenarios are first investigatedand compared in simulations. A first experimental implementation is then presented in section 2.Possibilities for improvement are also discussed.

5.1 SimulationsThis section presents the different scenarios that are simulated and their relative benefits. To doso, we first explain the comparison methodology used for evaluation. The first scenario focuses onclassical MPC trade-off between consumption and comfort. Scenario B investigates the case wherethe user agrees on a reduced comfort level during specific hours of the day while scenario C usesvarying electricity prices. In all the following simulations, perfect model and forecasts are assumedso that only the savings strategy is evaluated. Robustness and practical implementation aspectsare addressed in section 5.2.

5.1.1 MethodologyThe evaluation of the strategy focuses on the two objectives of the optimization : comfort andconsumption. As a reference case for comparisons, we use a scenario with constant indoor temper-ature Ti(t) ≡ Ti,0 as reference case. Such a scenario corresponds to perfect control of the indoortemperature and can be achieved with a feed forward controller or using MPC with zero cost onconsumption. However if one wants to compare savings related to utilization of the dynamics of thesystem only, the constant indoor temperature reference should not equal the Tref used in MPC.Instead we use the average of the indoor temperature achieved by the MPC controller we want tocompare to. For instance if the results of the MPC with Tref = 21◦C is a varying indoor tem-perature of average 20.5◦ then it is compared to a constant indoor temperature of 20.5◦C. Thatis in fact very important because part of the savings achieved by MPC are only due to a reducedindoor temperature compared to the reference case and hence could be achieved with a simplercontroller such as feed-forward. Here we aim at evaluating the savings achieved by using the dy-namics cleverly, independently from reducing the indoor temperature. Though for completenessand as it is generally used in the literature, the feed-forward control at Ti(t) ≡ Tref is indicatedon the performance tables but not used when computing the savings.

The time horizon is fixed at 24 hours and the sampling time at 15min. That allows a goodcompromise between the complexity of the optimization (related to the number of decision vari-ables i.e., 96 inputs here). It also corresponds to a time horizon where weather prediction arereliable and electricity prices are known in advance.

Different MPC tunings are tried for each scenario. Though the reference temperature Tref isa tuning parameter, it will mostly be fixed at 20◦C or 21◦C. Hard constraints on the heatingpower are set at Pmin = 0 and Pmax = 16kW according to the technical specifications of the heatpump while hard constraints on the indoor temperature may vary depending on the scenario. Themain tuning parameters are the respective weights on comfort ccomf and consumption cel. Hencethe performance tables indicate different possible trade-offs between comfort and consumption rep-

30

resented by the fraction ccomf/cel.

Finally the performances regarding comfort and consumption can be accounted for using the fol-lowing indicators

• Consumption is represented by the average daily electricity consumption, denotedW el (kWh/day).When considering electricity prices (scenario C) the average daily electricity bill (e/day) isadded as an economic indicator. Those indicators are compared to the consumption andcost of the reference case feed-forward controller as explained above. Performances of thereference case are indicated with a subscript ref .

• Comfort is represented using the average indoor temperature T i (◦C). A confidence intervalat T i ± 0.3 (◦C) indicates the variance of the indoor temperature around the mean, the per-centage of indoor temperature within the interval is indicated under CI (confidence interval).When considering varying comfort zone (mostly scenario B) those values are computed usingonly for the time when comfort cost is enabled. If for instance the user agreed upon reducedindoor temperature during office hours, the corresponding temperatures are not taken intoaccount for comfort evaluation.

5.1.2 Scenario AIn this scenario we investigate how much standard MPC can save energy and at what cost interms of comfort. Intuitively one could imagine the MPC strategy leads to overheating when it’sbeneficial (regarding efficiency) and heat less otherwise. One of the main results is that this isnever the case : excluding comfort aspects, the most beneficial in terms of consumption is to heatas little as possible, all the time. One way of simulating this is to consider a zero cost on comfort,with only a rather high minimum indoor temperature (for instance Tmin = 19.5 ◦C) to account forreasonable indoor temperature. When doing so, the result is a flat indoor temperature constant: Ti(t) ≡ Tmin. From a computational point of view, the solution sticks to the boundaries of theproblem, resulting in longer computational times and sometimes poor convergence. Knowing thisis the most efficient solution with respect to electrical consumption, it is possible to compute andachieve such a control with zero cost on consumption but with the minimum temperature actingas the reference one : Tref := Tmin.

The other approach to this is when considering the trade-off done by MPC between comfortand consumption (both costs being now non-zero). It appears that almost all savings achieved bythe controller are due to the reduction of the average temperature instead of optimization of theheat pump’s efficiency. That can be seen on Figure 5.1 and mostly Figure 5.2 where trajectoriesof an MPC controller are compared with a feed-forward control with identical average indoor tem-peratures. While the MPC trajectories show more variations, the efficiency plotted in Figure 5.2suggests the use of MPC does not improve the global COP of the heat pump. Numerical resultsare displayed in Table 5.1 and show that the energy savings are negligible when compared to a ref-erence case with identical average indoor temperature. For completeness the reference case usuallyused for comparison in the literature is shown under feed-forward controller with 21◦C average.Hence one could argue MPC(1) (the most restrictive when it comes to comfort) saves more than3% energy compare to the feed-forward controller whereas almost all those savings are due to areduction, on average, of the indoor temperature. Hence such savings could identically be achievedby a feed-forward controller with a lower reference indoor temperature.

Table 5.1 – Performances of classical trade-off scenario, simulated over December 2017

Feed-forward MPC(1) MPC(2) MPC(3)Tuning ratio ccomf/cel 4/1 3/1 2/1Average indoor T i (◦C) 21 20.62 20.49 20.25

% of Ti within confidence interval 100% 100% 100 % 98 %Consumption W el (kWh/day) 6.42 6.22 6.15 6.02

Reference consumption W el,ref (kWh/day) 6.24 6.17 6.04Energy savings 0.3% 0.3% 0.3%

31

50 100 150

0

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40

50

Time (h)

Te

mp

era

ture

(C

)

Comparison of temperatures between reference and MPC controllers

Indoor (MPC)

Return (MPC)

Feed (MPC)

Indoor (Ref.)

Return (Ref.)

Feed (Ref.)

Outdoor

Figure 5.1 – Comparison of temperatures between feed forward and MPC controllers for identicalaverage indoor temperatures.

50 100 150

0

1

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3

4

5

6

Time (h)

Po

we

r (k

W)

Comparison of powers between reference and MPC controllers

Electrical power (MPC)

Efficiency (MPC)

Heating power (MPC)

Electrical power (Ref.)

Efficiency (Ref.)

Heating power (Ref.)

Figure 5.2 – Comparison of powers between feed forward and MPC controllers for identical averageindoor temperatures.

32

There is however a special case where the use of MPC only leads to savings. During summer,outdoor temperature is above the reference temperature during the day. Then the feed-forwardcontroller is unable to keep the indoor at the reference temperature due to impossible cooling.The model predictive approach can however foreseen those hours and hence lower the indoortemperature to benefit from "free" heating which can be seen on Figure 5.3. The resulting savingsdisplayed in Table 5.2 reach 3% to 5% depending on the reference case and tuning used (comparisonframework is here a bit difficult to adapt because constant indoor temperature is impossible toachieve). Those days of course are rare over the year and the corresponding consumption is alwayslow due to low required heating power and hence high efficiency.

0 50 100 150 200

10

15

20

25

30

35

40

Time (h)

Te

mp

era

ture

(C

)

Simulated temperatures of the system

Indoor

Return

Outdoor

Feed

Reference

Figure 5.3 – Simulated temperatures for the warm days scenario using 9 days data beginning2017-07-20 with controller MPC(1).

Table 5.2 – Comparison of different feed-forward and MPC controllers with warm days case.

Feed-forward(1) MPC(1) Feed-forward(2) MPC(2)Tref (◦C) 19.76 20 19.6 20

Tuning ratio ccomf/cel 4/1 2/1Average indoor T i (◦C) 19.82 19.82 19.66 19.66

% of Ti within confidence interval 95 % 97% 95 % 74%Consumption W el (kWh/day) 1.10 1.06 1.05 1

33

0 50 100 150 200

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3

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5

6

Time (h)

Po

we

r (k

W)

Simulated powers and efficiency of the heat pump

Electrical power

Efficiency

Heating power

Figure 5.4 – Simulated powers for the warm days scenario using 9 days data beginning 2017-07-20with controller MPC(1).

34

5.1.3 Scenario B: varying comfort costIn scenario B , we investigate the potential gains of having a varying cost on comfort. This reflectthe idea that the user agrees upon a reduced indoor temperature during specific hours of the day.Two sub-cases will be studied and compared :

• Office hours : Most of houses are unoccupied during office hours such as 8am - 4pm andhence do not need to be heated during that time.

• Night hours : As the activity in the house is reduced at night the indoor temperature can belowered between for instance 10pm - 6am. This is also a common recommendation for healthreasons and sleep quality.

In both cases, the underlying assumption is that a good level of comfort is restored in advanceso that the changes are "barely noticed" by the user. The indoor reference temperature used is21 ◦C while hard constraints on the indoor temperature are intentionally lowered at 17 ◦C togive more freedom to the controller. A varying comfort penalty is used such that it enables thecost on comfort only when needed. The system is simulated with weather from December 2017.The resulting temperatures and powers are displayed in Figure 5.5 and Figure 5.6 respectively.Performances are shown in Table 5.3 and Table 5.4. In order to make a fair comparison of thecontrol strategy, the reference case is a flat indoor temperature whose value corresponds to theaverage of the MPC taken only when the comfort constraints are active. For instance when theindoor temperature is reduced at night the reference case is fixed at the average indoor temperatureduring the day.

0 20 40 60 80 100 120 140

0

10

20

30

40

50

Time (h)

Te

mp

era

ture

(C

)

Simulated temperatures of the system

Indoor

Return

Outdoor

Feed

Reference

Figure 5.5 – Simulated temperatures for the reduced night comfort scenario. Dashed line representswhether the cost on comfort is enabled or not

It can be observed in the plots that the resulting signals are fairly smooth. There is for instanceno on/off control of the heat pump corresponding to night hours. First, this is due to the timehorizon of 24 hours that always foresee the morning where the indoor temperature has to catchup with the reference. Second, note that it is never beneficial to run the heat pump at full powerbecause that leads to important loss off COP. Hence it can be seen on Figure 5.6 that the periodicincrease in electrical and heating power also corresponds to a drop of the efficiency. This combinedwith the important inertia of the house makes the temperature drop around only about a degree.

35

0 20 40 60 80 100 120 140

0

1

2

3

4

5

6

Time (h)

Po

we

r (k

W)

Simulated powers and efficiency of the heat pump

Electrical power

Efficiency

Heating power

Figure 5.6 – Varying comfort scenario with reduced comfort at night

Table 5.3 – Performances of the reduced comfort at night scenario, simulated over December 2017

Feed-forward MPC(1) MPC(2) MPC(3)Tuning ratio ccomf/cel 4/1 3/1 2/1Average indoor T i (◦C) 21 20.84 20.72 20.39

% of Ti within confidence interval 100% 99% 96 % 90%Consumption W el (kWh/day) 6.42 6.25 6.18 5.99

Reference consumption W el,ref (kWh/day) 6.34 6.28 6.11Energy savings 1.4% 1.6% 2%

Table 5.4 – Performances of the reduced comfort during office hours scenario, (December 2017)

Feed-forward MPC(1) MPC(2) MPC(3)Tuning ratio ccomf/cel 4/1 3/1 2/1Average indoor T i (◦C) 21 20.84 20.72 20.39

% of Ti within confidence interval 100% 98% 95 % 89%Consumption W el (kWh/day) 6.42 6.26 6.18 6.00

Reference consumption W el,ref (kWh/day) 6.34 6.28 6.11Energy savings 1.3% 1.6% 1.8%

Note that in order to be able to compare the two strategies the reduced comfort periods bothlast 8 hours. From the performance tables it appears the two different strategies are equivalentin terms of savings and comfort. That means outdoor temperature does not have a significantinfluence on the performances. Though this is not included in the present simulations one couldexpect the sun to have an influence during fall or spring months. In both cases the savings arearound 1.5% compared to a constant indoor temperature, while when comparing to the flat indoorat Tref it can go from 2.7% for MPC(1) to 7.2% for MPC(3). When decreasing the cost on comfortthe average indoor temperature decreases and the confidence interval (CI) becomes smaller whilehaving only small benefits on energy savings.

36

5.1.4 Scenario C : electricity pricesPeak - Off peak prices

When integrating the electricity prices into the cost function, one has to look for the end userprices and not the market prices. The latter typically are available at Nord Pool [35] with bothdata and 24 hours ahead forecast. However the end user prices should also include taxes, gridrelated costs and likely renewable energy related prices such as Electricity Certificate [36]. Hencethat final cost is unknown for Sweden and it was thus chosen to consider a peak/off peak hourssystem. In such a set up the customer has the choice between a constant price over the month or apeak and off peak hours system. Hours that usually correspond to low demand are assigned lowerprices while the rest of the hours are higher. The number of off peak hours are fixed and limited toeight per day. Though different offers exist, we’ll consider the typical case where electricity pricesare lowered at night between 10pm and 6am. Logically, the prices when choosing constant pricesare in between the peaks and off peaks prices.

Table 5.5 – Examples of end user electricity prices (e/kWh) with base option and peak/off-peakhour option for two electricity providers

Supplier base option peak hours (06h-22h) off peak hours (22h-06h)A 0.14660 0.15930 0.12520B 0.13686 0.14844 0.11742

Prices displayed in Table 5.5 are representative of the offers available[37] on the French mar-ket1. Such a set up assumes that electricity prices are mostly determined by the demand whereasthe offer is quite constant, so that electricity is to be cheaper when demand is low (typically atnight). Hence that does not apply to electricity mix with a high rate of volatile renewable sources.This is valid for both France and Sweden where nuclear and hydro-power — two highly predictablesources — represent more than 80% of the electric production.

Simulations using prices from supplier B are done in order to evaluate the potential for economicsavings. Simulations in Figure 5.7 and especially Figure 5.8 show the variations in the prices areimportant enough to drive the behaviour of the heat pump. Not that lower prices induce moreimportant work from the heat pump and hence a reduction of the efficiency. Figure 5.8 shows theCOP goes below four after the low price hours. This means that, to some extent, the variationsin the prices are large enough to compensate the efficiency drop. When looking at the indoortemperature, the low price period allow to catch back on the reference temperature. That canbe seen as the the system storing heat when beneficial. Observe the indoor temperature nevergoes above the reference temperature (at 21) even during low prices period, possibly due to thesymmetric penalty on comfort (Ti − Tref )2.

1Prices are displayed for information and simulation purposes only and do not have any legal basis

37

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Time (h)

Te

mp

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ture

(C

)

Simulated temperatures of the system

Indoor

Return

Outdoor

Feed

Reference

Figure 5.7 – Simulated temperatures of the system for varying electricity prices over the first weekof December, controlled with MPC(2)

0 50 100 150

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Time (h)

Po

we

r (k

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Simulated powers and efficiency compared with electricity prices

0 50 100 1500

0.05

0.1

0.15

0.2

0.25P

rice

s (

eu

r/kW

h)

Electrical power

Electricity prices

Heating power

Efficiency

Figure 5.8 – Simulated powers of the system for varying electricity prices over the first week ofDecember. Electricity prices are also displayed, controlled with MPC(2)

38

To evaluate the savings potential the MPC is compared as before with a constant indoortemperature with constant electricity prices and tested over December with different tuning, seeTable 5.6. An important observation is that the global consumption increases slightly compared tothe reference cases, which confirms that this scenario does not make the heat pump work in a moreefficient region. However the economic savings range from 2.3% for MPC(1) to 2.8% for MPC(3).The indoor temperature appears to vary more than before when considering the confidence interval(CI). However around half of the temperatures outside of the CI are above the mean temperature,which is not that damageable for comfort.

Table 5.6 – Performances of the varying electricity prices scenario

Feed-forward MPC(1) MPC(2) MPC(3)Tuning ratio ccomf/cel 4/1 3/1 2/1Average indoor T i (◦C) 21 20.62 20.50 20.26

% of Ti within confidence interval 100% 75% 67 % 58%Consumption W el (kWh/day) 6.43 6.25 6.19 6.07

Reference consumption W el,ref (kWh/day) 6.23 6.17 6.05Cost (e/day) 0.882 0.835 0.824 0.804

Reference cost (e/day) 0.854 0.845 0.827Economic savings 2.2% 2.5% 2.8%

Scenario B+C : varying prices and comfort

It is possible to combine scenario B and C by applying varying electricity prices and varyingcomfort. One can notice on Figure 5.6 that in the case of reduced comfort at night in scenarioB the consumed power is already higher in the off peak hours. That means adding the varyingprices would push the optimization in the same direction whereas with reduced comfort duringthe office hours the two optimums would be the opposite direction. Compared with scenario B,the trajectories are slightly different. On Section 5.1.4 the indoor temperature is not symmetricanymore with respect to the reference temperature. Indoor temperature rises before comfort isenabled because that corresponds to night hours heating which is cheap whereas the temperaturedecreases before the comfort is disabled due to higher prices during the day. Hence the variationsboth in terms of temperatures and powers (Figure 5.10) are more important than in the previousscenarios. When focusing on comfort that means a greater risk of low indoor temperature at theboundaries of the comfort zone (e.g., just before 10pm).

Performances are summarized in Table 5.7, recall that the mean indoor temperature onlytakes into account the period where comfort is enabled (that is not during the night). Contraryto scenario C, the electricity consumption itself is reduced. The economic savings are way moreimportant, going from 6.2% to 7.7%, and increases significantly when fostering consumption insteadof comfort. This is also different from the previous scenarios where lower comfort weight had littleinfluence on savings and can be explained by the bigger variations already observed on Section 5.1.4.Notice that the confidence interval (CI) ratio has improved compared to previous scenario C. Thismight be counter intuitive but is due to the fact the period previously associated with lower indoortemperatures are now concentrated when comfort cost is disabled and hence not taken into accountin the CI since it does not impact comfort perception. Finally the percentage of savings is greaterthan the sum of respective savings in scenario B and C, which is due to the two effects acting inthe same direction.

39

0 50 100 150

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40

50

Time (h)

Te

mp

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ture

(C

)

Simulated temperatures of the system

Indoor

Return

Outdoor

Feed

Reference

Figure 5.9 – Simulated temperatures of the system for varying electricity prices and reduced comfortat night over the first week of December, controlled with MPC(2).

0 50 100 150

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Simulated powers and efficiency compared with electricity prices

0 50 100 1500

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0.1

0.15

0.2

0.25

Price

s (

eu

r/kW

h)

Electrical power

Electricity prices

Heating power

Efficiency

Figure 5.10 – Simulated powers of the system for varying electricity prices and reduced comfortat night over the first week of December. Electricity prices are also displayed, controlled withMPC(2).

40

Table 5.7 – Performances when conjugating varying electricity prices and reduced comfort at night,simulated over December 2017.

Feed-forward MPC(1) MPC(2) MPC(3)Tuning ratio ccomf/cel 4/1 3/1 2/1Average indoor T i (◦C) 21 20.84 20.73 20.43

% of Ti within confidence interval 100% 92% 88 % 82%Consumption W el (kWh/day) 6.43 6.24 6.17 6.02

Reference consumption W el,ref (kWh/day) 6.34 6.29 6.13Cost (e/day) 0.882 0.818 0.805 0.778

Reference cost (e/day) 0.870 0.862 0.840Economic savings 6.2% 6.8% 7.7%

5.1.5 DiscussionWe would like to emphasize two key lessons from the simulations.

• The comparison methodology used allows to evaluate the MPC savings only compared toflat indoor temperature. That diminishes significantly the results since and simplest way tooreduce consumption is still to lower the indoor temperature. To us this is a truer comparisonbetween what can be achieved with a flat control (feed-forward or others) and dynamic MPCstrategy.

• The efficiency of the heat pump is taken into account which implies minimum heating isalways the most efficient solution. Each time the heat pump is over heating — compared tothe reference case — its efficiency drops. Hence the benefits of overheating at some momentmust do more that compensate the corresponding loss of efficiency. That is beneficial toscenario B where reduced comfort is allowed and also prevent from important variations inthe heating power that would lead to drop of efficiency. Varying electricity prices in scenarioC are important enough to make an more inefficient use of the heat pump when economicallybeneficial.

41

5.2 TestA first version of the MPC could be tested on the real system during a week, from January 26thto February 2nd. As a first test, the consumption aspects are dropped so only the ability to keepthe indoor temperature constant is considered. According to the previous framework that meansthe weight on consumption is zero cel = 0.

5.2.1 Dealing with uncertaintiesIn order to run, the MPC algorithm needs to have an estimate of the actual state. To do so,a Kalman filter is used taking as measurements the indoor temperature. Design of the Kalmanfilter is done using the built in Kalman function in Octave [30], with the tuning parameters q andr, respectively the covariances of the additive white noise process and of the additive white noisemeasurements. Since q and r generally are unknown, they may be considered to represent the trustone has in the plant model and the sensors, respectively. The Kalman filter can be characterizedby the Kalman gain matrix Lobs, the estimator then is computed with a prediction and correctionstep :

x[k + 1|k] = x[k] +Buu[k] +BwTa[k]︸ ︷︷ ︸prediction

+Lobs(y[k]− y[k])︸ ︷︷ ︸correction

(5.1)

where the estimated output is y[k] = Cx[k]. Matrices A, Bu Bw and C corresponds to the nom-inal model of the plant. Ta[k] and y[k] are measurements of respectively the outdoor and indoortemperature.

To tune the Kalman filter, one can simulate the difference between the true system and the model.One possibility to do so is by considering the four inputs model from system identification as thetrue system and the two inputs model used in MPC as the nominal model used for MPC. Suchan approach typically accounts for model errors, whereas other approaches could represent outputerrors. The indoor temperature measurements are considered to be fairly trustworthy, comparedto the uncertainties associated with the model. The simulations showed that for the estimatorto actually track the indoor temperature it was required a large covariance on the model noise,for instance q = 200000 and r = 1. (Figure 5.11) shows a one week simulation with this tuningwhere the objective of the MPC is only to keep the indoor at the reference temperature. Hencethe Kalman filter favours the measurements data over the nominal model.

5.2.2 Implementation aspectsThe implementation was carried out at Dreik Ingenjörskonst. From the MPC point of view, thecontrol signal sent is the feed temperature while in practice, the signal sent is the reference of theinternal feed temperature controller in heat pump. Hence the model input u is different from theactual input sent to the plant. The steps of the control algorithm are

1. Knowing x[k] from previous step and the weather forecast for the next 24 hours, solve theoptimization problem and hence obtain u[k].

2. Using current indoor measurement y[k], compute next step estimator x[k + 1].

3. Send control signal as the first element of x[k + 1](3) which corresponds to the feed temper-ature. Other variables x[k] and u[k] are stored.

This signal is then filtered to give the reference feed temperature Tfeed,ref going into the heatpump’s internal control. Weather forecast updated three times per day was included using SMHIopen data. As for the simulations the sampling time is 15min and the time horizon is 24 hours.

42

0 20 40 60 80 100 120 14020.6

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Time (h)

Te

mp

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)

Simulation of the indoor temperature estimation with model errors

Estimated indoor

"Measured" indoor

Modelled indoor

Reference

Figure 5.11 – Simulation of the Kalman filter estimator using a more complete model to simulatethe real plant

43

5.2.3 ResultsThe results to be analysed consist mostly of the indoor temperature and the control signals.

Let’s recall the only objective is to track the reference indoor temperature Tref = 21◦C. OnFigure 5.12 one can see the indoor temperature generally lies within 20.2◦ and 20.8◦ whereas theestimated indoor is always above 20.7◦. Hence most of the error between the actual indoor andthe reference is due to a bad estimate of the temperature. That suggests the tuning of the filterperformed in the previous step is too optimistic on the model accuracy.

50 100 15020

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Te

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MPC test : indoor temperature control

Estimated indoor

Measured indoor

Reference

Figure 5.12 – Difference between estimate, measured and reference indoor temperatures during thetest.

The accuracy of the feed temperatures can be studied in Figure 5.13 where the estimated andmeasured feed temperatures are plotted along the reference feed which is an internal control signalof the heat pump. Here the estimated feed temperature should "lead" the actual feed because itacts as the control signal. The sharp peaks correspond to turning on and turning off instants ofthe heat pump where huge variations of the flow makes the feed temperature vary a lot around thesensor but not in the entire circuit. The signal is filtered so that effect is reduced. However theestimated and measured feed still differ and the estimation seems in particular unable to capturereasonably fast variations. That can also be interpreted as the difficulty to drive completely thebehaviour of the heat pump due to internal control. It is however positive that the feed referenceand estimated feed are close.

44

Figure 5.13 – Difference between estimate, measured and reference feed temperatures during thetest.

5.2.4 DiscussionThe experimental results obtained seem promising though insufficient to guarantee a good level ofcomfort. When analysing the results, different sources of errors and improvements can be found.

• Separate the prediction and correction steps of the Kalman filter. Correction can be done atthe beginning of the loop using current measurements. This improves the current estimatedstate x[k]. Once the optimal control is computed, perform the prediction step at the end ofthe loop and use the third state of x[k + 1] as feed temperature for the actual control of theplant. This would improve the estimated state the MPC is working with.

• Improve the uncertainties handling. Apart from the tuning of the Kalman filter, it is possibleto include the disturbance as an extra state of the model so that the optimization takes itinto account when solving the optimal control problem.

• Improve the internal control of the heat pump, since the control strategy assumes the heatpump is able to apply the required feed temperature.

One could also investigate the weather forecast to see how trustworthy the predicted outdoortemperatures are compared to the measurements.

45

Chapter 6

Conclusion and future work

In this thesis we showed the application of MPC to control ground source heat pumps has a po-tential for economic and energy savings. Three main scenarios were investigated, the first witha basic reference temperature. The second scenario presents the idea of having a reduced indoortemperature during certain hours of the day such as office hours and the final scenario adds varyingelectricity prices to the optimization problem. Due to the way a eat pump’s efficiency varies withvarying temperature differences between the hot and cold side, a constant indoor temperature stillappears beneficial in a basic scenario since peaks in temperature reduces the efficiency of the heatpump. However when considering variable level of comfort according to different hours of the day,an MPC based strategy becomes relevant. Varying electricity prices makes the MPC even morebeneficial with economic savings up to 8% observed in this study.

Implementation aspects has been a limiting factor in this thesis. System identification was per-formed to obtain a suitable model of the plant for the MPC used both in simulation and a firstimplementation of an MPC controller was made. However, the results showed the that the un-certainties and disturbances aspects must be tackled further, but this was not possible within thetime-frame of this thesis-project. Though it would be possible to obtain a more reliable model fromthe system identification, we think a more robust approach to the MPC would be more beneficialto deal with disturbances difficult to model. Future work could also explore the idea of on-lineparameter estimation to allow the model to improve while running. A better control over the heatpump internal working would also improve the system’s behaviour.

In the future, including the tap water generation within the control system has great potential.Tap water tanks are generally large enough to allow flexibility in control without decreasing com-fort. There is then a huge potential for energy storage that could for instance benefit from lowelectricity price hours. From the control perspective this would lead to hybrid control and couldintegrate other elements such as solar cells.

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TRITA EECS-EX-2018:47

ISSN 1653-5146

ISRN 1653-5146

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