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Research Papers

# A Distributed Predictive Control of Energy Resources in Radiant Floor BuildingsPUBLIC ACCESS

[+] Author and Article Information
Soroush Rastegarpour

Dipartimento di Elettronica,
Informazione e Bioingegneria of
Politecnico di Milano,
Piazza L. da Vinci, 32,
Milano 20133, Italy
e-mail: Soroush.Rastegarpour@polimi.it

Luca Ferrarini

Dipartimento di Elettronica,
Informazione e Bioingegneria of Politecnico di
Milano,
Piazza L. da Vinci, 32,
Milano 20133, Italy
e-mail: Luca.Ferrarini@polimi.it

Foivos Palaiogiannis

Smart Grids Research Unit,
School of Electrical and Computer Engineering,
National Technical University of Athens,
Zografou 15780, Greece
e-mail: Foivosp@mail.ntua.gr

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received November 12, 2018; final manuscript received May 23, 2019; published online June 27, 2019. Assoc. Editor: Alessandro Rizzo.

J. Dyn. Sys., Meas., Control 141(10), 101013 (Jun 27, 2019) (15 pages) Paper No: DS-18-1507; doi: 10.1115/1.4043935 History: Received November 12, 2018; Revised May 23, 2019

## Abstract

This paper studies the impact of using different types of energy storages integrated with a heat pump for energy efficiency in radiant-floor buildings. In particular, the performance of the building energy resources management system is improved through the application of distributed model predictive control (DMPC) to better anticipate the effects of disturbances and real-time pricing together with following the modular structure of the system under control. To this end, the load side and heating system are decoupled through a three-element mixing valve, which enforces a fixed water flow rate in the building pipelines. Hence, the building temperature control is executed by a linear model predictive control, which in turn is able to exchange the building information with the heating system controller. On the contrary, there is a variable action of the mixing valve, which enforces a variable circulated water flow rate within the tank. In this case, the optimization problem is more complex than in literature due to the variable circulation water flow rate within the tank layers, which gives rise to a nonlinear model. Therefore, an adaptive linear model predictive control is designed for the heating system to deal with the system nonlinearity trough a successive linearization method around the current operating point. A battery is also installed as a further storage, in addition to the thermal energy storage, in order to have the option between the charging and discharging of both storages based on the electricity price tariff and the building and thermal energy storage inertia. A qualitative comparative analysis has been also carried out with a rule-based heuristic logic and a centralized model predictive control (CMPC) algorithm. Finally, the proposed control algorithm has been experimentally validated in a well-equipped smart grid research laboratory belonging to the ERIGrid Research Infrastructure, funded by European Union's Horizon 2020 Research and Innovation Programme.

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## Introduction

Thanks to the pressure to reduce consumption and to the trend to implement smart grids paradigm, energy efficiency in buildings has been widely addressed in the recent years. According to Ref. [1], in the U.S. and other developed countries, about one-third of all energy consumption can be attributed to buildings. A variety of international regulations, including EN 15232, the European Energy Performance of Buildings Directive (EPBD 2010), ISO 50001, and sustainability and energy certificates such as LEED or energy star, have led to the creation of technical standards and design of specialized applications to optimize the energy efficiency of buildings.

Although there are numerous strategies to improve the energy consumption trend in building sector, recent uses of building energy resource management system as well as complex optimization techniques uncovered new ways to achieve thermal and electrical energy saving [2].

In Ref. [3], authors study the cost of battery and control systems, including financing, maintenance, and operating expenses, finding a point when this is smaller than electricity bill savings that the storage can enable over the lifetime of the storage system itself. It is investigated that if such break-even can be achieved, which storage technology and which dispatch strategy (i.e., when and how to discharge/charge the storage) creates the lowest overall cost to the residential consumer. Moreover, they developed an agent-based stochastic demand model to randomly generate demand profiles for a single, representative household in the U.S. Although these analyses are valuable in a demand side management point of view, the distributed predictive control strategies are still missing, where many benefits can be obtained through the modular structure of the system. Furthermore, the impacts of thermal storages instead of electrical ones is still challenging. Smart buildings with the microgrids (MGs) interaction are relatively new configurations, which allow to design and implement optimal control strategies in order to optimize energy consumption without compromising the comfort and safety [4]. The overall picture is also characterized by the presence of distributed energy resources, which can be dispatchable, such as gas engines, and also nondispatchable, such as renewable energy generation including wind, and solar photovoltaics (PV), which in turn can be estimated by available strategies and software tools [5,6]. In Refs. [7] and [8], authors developed a mathematical tool, which aims at optimizing the energy flows occurring between autonomous consumers. Thus, the constructed model amounts to be a useful tool for the definition of urban energy action plans centered on the exploitation of renewable sources and on the installation of energy efficient systems. However, the optimal energy resource management system together with exploitation of the modular structure of the system through distributed techniques is still a challenging issue.

In the classic model predictive control (MPC) techniques, the control action at each time-step is obtained by solving an online optimization problem; that is why it is usually named online MPC framework. Online MPC approaches have shown good performance for controlling large-scale systems, in particular in economic optimization of MG [9,10], energy management systems, and optimal real-time pricing applications [1113]. Moreover, meeting a peak demand during peak-price hours imposes a fluctuation on electricity grid and consequently more adverse impact on the users. In this case, energy storages play a crucial role on load shaping in demand side in order to shift demand from peak hours to off-peak periods. In Ref. [14], a chilled energy storage is used to find a cost-effective solution for heating, ventilation and air conditioning control system in buildings. The application of a thermal energy storage (TES) connected to a heat pump (HP) for the load shifting in demand side application is studied for a reference scenario in Northern Ireland (UK) in Ref. [15]. On the contrary, the role of electrical energy storages (batteries) in smoothing intermittent PV power flow, providing voltage peak shaving and shifting power generated by renewable resources to be more coincident with peaks time, is undeniable. Because of logical/algebraic models of battery charging/discharging in MGs, it makes a mixed logical dynamical (MLD) system, which is widely studied in MGs applications and energy management systems [12,16,17].

However, the application of different types of energy storages for cost minimization together with load and power shifting in demand side and power network is still a challenging and complex problem, which in turn can be coordinated by proper control strategies. A real-life scenario of residential microgrid including six apartments, a photovoltaic plant, a solar-based thermal energy plant, a geothermal heat pump, a thermal energy storage, and a battery is investigated in Ref. [18]. This study shows the effectiveness of using different type of storages in the plant. However, increasing the number of apartments and storages will result in a complex system which is computationally expensive to be solved in a centralized optimization problem. Hence, the modular structure of the system can be exploited in order to employ a distributed technique for the economic and optimal energy resources management. This paper will focus also on the distributed techniques to satisfy all constraints of local controllers, while they exchange information in order to converge to the optimal solution. It is also highlighted that the distributed solution is sufficiently close to the optimal solution obtained by centralized one. In Ref. [19], authors present an event-driven model predictive control approach for a local energy management system, where their main focus is on the economic energy resource management for the smart building including smart appliances in order to economically update the starting and stopping time of the appliances and charging and discharging of the batteries based on an event triggered by the user request. The idea is interesting, but the application of different types of storage in a modular structure is still an open issue in this contribution. Moreover, the nonlinear problem arises due to the variable water flowrate is not studied. The same holds for Ref. [20], where the authors addressed an almost similar idea considering a multi-apartment test case. The results show the advantages of using two types of storages, but the nonlinearity of the system due to variable water flow rate is not considered. Actually considering a fixed water flow rate oversimplifies the problem and also will deteriorate the advantages of using the thermal energy storage inertia. In this case, the distributed model predictive control is more complex, which is not studied in that paper.

The aim of this study is twofold. First, it proposes a distributed management and control scheme to exploit the advantages of using two different energy storages, i.e., TES and battery with slow and fast dynamics, respectively, to not only shift the demand away from peak hours, but also shift the power generated by renewable resources to be more coincident with peak times for optimal pricing concepts. Second, it guarantees the comfort level satisfaction for the building's occupants in presence of different disturbances. Similar to the modeling approach described in Refs. [21] and [22], a dynamical model of building, HP and TES is formulated. According to the proposed system structure in Fig. 1, the building pipelines are supplied through a three-element valve, which introduces more complexity due to variable position of valve.

Despite the fact that TES dynamical model is nonlinear due to variable water flow rate, variable valve action will provide the capability of using both water temperature and water flow rate as two control variables, which can have beneficial effects on load shifting and cost saving.

In order to deal with the previously-mentioned plant nonlinearity, the proposed control system composes of an adaptive structure based on successive linearization through which the plant model will be updated based on new operating point, according to the current valve position, at the beginning of each time interval. Moreover, a distributed model predictive control (DMPC) strategy is derived to better anticipate the effects of disturbances and real-time pricing together with following the modular structure of the system under control. In the proposed DMPC structure, similar to the concept described in Ref. [23], each MPC has different possibly conflicting objective function, and also information is transmitted and received once in each time interval. In this framework, according to the method studied in Ref. [24], the logic/algebraic and dynamical system model will be described in the form of an MLD model by which the logic constraints of system, such as charging/discharging of storages and power buying/selling from/to utility grid, can also be included in optimization problem.

The rest of the paper is organized as follows: First of all, the control-oriented models of components considered in this paper are explained in Sec. 2. The main steps of the design procedure of the proposed DMPC together with centralized model predictive control (CMPC) and heuristic controller are described in Sec. 3. In Secs. 46, some simulation analysis and experimental results are summarized and discussed. In particular, the proposed control scheme has been experimentally validated in a well-equipped smart grid research laboratory, on a real MG equipment including PV panels, electrical storages, and controllable loads. The results of above-mentioned experiments are summarized at the end in Sec. 7. The concluding remarks are given in Sec. 8, respectively.

## Case Study

In this section, the building model, heating system, battery model, and PV panel estimator are briefly introduced. As it has been already mentioned, the main challenge of the paper is to employ suitably the distributed MPC algorithm in order to exploit inertia and storages of energy to decouple load request to the electrical grid from the user comfort control problem. Therefore, the main concepts of the plant under study are briefly discussed to better understand the complexities of the model, such as nonlinearity in TES, in the optimal control problem formulation.

The house considered in this paper is heated with a radiant floor technology. It is modeled based on single volume modeling approach as studied in Refs. [25] and [26]. This modeling approach takes into account the dynamic behavior of thermal variables and can be considered for the energy analysis and temperature profile estimation in buildings. The building dynamics include wall temperature ($Tw$), air temperature ($Tzone$), pavement temperature ($Tpav$), and pipeline temperature ($Tpi$ includes both water and pipeline) as depicted in Fig. 1.

This dynamical behavior is modeled by a set of ordinary differential equations as follows: Display Formula

(1)$Cw Ṫw=Uwce Toa−Tw+UwciTzone−TwCz Ṫzone=Uw Tw−Tzone+UpaviTpav−TzoneCp Ṫpav=Upav Tzone−Tpav+PTERMCwt Ṫpi=w Cpwt Tinlet−Tpi+(w Cpwte−α−1−e−αα+U1 Ap L 1−e−αα)(Tpav−Tinlet)$

where $PTERM$ defines the heat exchanges between inlet water temperature, as the main manipulated variable, and pavement temperature as follows:

$PTERM=U1 Ap L (1−e−α)αTinlet−Tpav$

Additionally, $U1, Ap, and L$ refer to the pipeline specification, which are pipeline thermal conductivity (W/mK), pipeline external perimeter (m) and pipeline length (m), respectively. $α$ is also the absorbance coefficient of the pipelines, which is experimentally tuned and considered fixed equal to 0.59 [27]. The rest of parameters used in the four-state model are defined in Table 1.

The return water temperature of the pipeline can be either considered equal to pipeline temperature $Tpi$, or formulated in a more accurate way based on pavement temperature ($Tpav(k)$) and inlet water temperature ($Tinlet(k)$) passing through pipelines in order to take into account the pipelines configuration and metal specifications as follows [27]: Display Formula

(2)$Tr(k)=(1−e−α)TPav(k)+e−αTinlet(k)$

This building is heated by hot water passing through pipelines, which is regulated by a valve at the entrance of the radiant panels to keep the water mass flow rate constant inside the panels and also limit the unexpected fluctuations of hot water temperature.

Due to the fact that in practice there is usually one sensor for measuring the building air temperature, here in this paper, the radiant floor building is modeled by one single thermal zone which is sufficiently good to represent the average building air temperature.

###### Heating System.

All the parameters used in this section are summarized in Table 2.

###### Air-to-Water Heat Pump.

The model of the heat pump in control purposes is defined by its coefficient of performance (COP). The COP of a heat pump can vary as a function of compressor frequency ($fHz$), the ambient temperature ($Toa$), humidity and the temperature of hot and cold sources, where condenser and evaporator are located, respectively.

Here, in this work, we use a model of heat pump with an increasing level of simplification. The objective of a heat pump cycle is to deliver heat transfer $Qout$ to the warm region, which is the space to be heated. In a given time interval, the rate at which energy is supplied to the warm region by heat transfer, $Qout$, is the sum of the energy supplied to the working fluid from the cold region, $Qin$, and the net rate of work input to the cycle, $W$. The coefficient of performance of a heat pump is defined as the ratio between the heating effect and the network required to achieve that effect. In practice, the COP could be approximated using the temperatures of the cold and hot zones, i.e., Display Formula

(3)$COP=QoutW≅THTH−TC$

where $TH$ is the temperature of the hot zone, which is the space to be heated, and $TC$ is the temperature of the cold zone, which is the surroundings space. As it is well known, the COP may vary depending on the temperature $TC$ of the cold region, as well as on the operating conditions. However, for a given prediction horizon of 5–6 h, considered in this paper, the outside temperature variation is negligible, and the operating conditions have a limited variation so that the COP can be assumed to be fixed [28] over the prediction horizon.

###### Thermal Energy Storage.

Thermal energy storage is a crucial part of energy management systems in buildings, which allows excess thermal energy to be stored and used later at different building scales. In this paper, a TES is considered to economically balance energy demand between day time and night time. As shown in Fig. 2, there are two different control variables, namely water flowrate and temperature, which in turn can be controlled by valve position and HP power, respectively.

The dynamical model of TES can be formulated based on convection and conduction heat transfer equations between different tank layers; similar to the approaches described in Refs. [22] and [29]. In these models, tank is discretized vertically into several nodes such that partial differential equation model of the tank, described in Ref. [29], is converted into a set of $n$ ordinary differential equations where $n$ is the number of nodes and, consequently, the order of TES model.

The energy balance equation for a sample node (Fig. 3) of the tank is shown below: Display Formula

(4)$mjcvjdTjdt=Q̇coil,j−Q̇wall,j+Q̇j+1−Q̇j−1+Q̇convj$

where $mj$ is the mass of water in the node $j$ and $cvj$ is the respective thermal capacity. This equation shows how the temperature of each layer will change due to heat exchange with ambient temperature ($Q̇wall,j$), lamped heat transfer rate of HP ($Q̇coil,j$), and heat exchange with surrounding nodes ($Q̇j+1, Q̇j−1$). Moreover, the convection heat transfer due to circulation water flow rate $Fs$ can be addressed as follows: Display Formula

(5)$Q̇conv=Fs cv(Tj+1−Tj)$

The model dynamics behaves linearly as long as circulation mass flow rate $Fs$ is fixed, i.e., the valve position is kept constant and nonzero. On the contrary, there is a more complex nonlinear dynamical behavior due to a variable circulated water flow rate, obtained by changing the valve position. Actually, that is one of the main contributions of this paper on the TES modeling, which is not studied in the other papers, for example, Refs. [22] and [29]. Accordingly, the valve position $xv(k)$ is a function of the system states, which is defined in steady-state through the mass and energy conservation equations Display Formula

(6)$Finlet=Fsk+Frk$
Display Formula
(7)$Finlet Tinletk=FskT1k+FrkTr(k)$

where $Finlet$, $Fs$, and $Fr$ are the inlet mass flow rate to the room, the outlet mass flow rate of the thermal storage, and the return mass flow rate to the valve, respectively, and $Tinletk$, $Tsk$, and $Tr(k)$ are the respective water temperatures (As depicted in Figs. 2 and 4). Then, at each time instant $k$, $xvk$ can be formulated as follows: Display Formula

(8)$xvk=FskFsk+Frk=Tinletk−Tr(k)Tsk−Tr(k)$
Display Formula
(9)$Fsk= xvk Fnominal$

where $Fnominal$ is the maximum water flow rate which, by definition, is equal to $Finlet$.

Here, in this paper, the proposed control scheme will take into account this nonlinear behavior of the tank (Eqs. (6)(9)) by using an adaptive structure based on successive linearization through which the plant model will be updated with the new operating point, based on the current valve position.

###### Battery Predictive Model.

By considering the battery power exchange as the input ($PBatt$) and by defining the binary variable $∂Batt$ and auxiliary variable $ψBatt(k)=∂Batt PBatt(k)$, the energy stored in battery at each time-step $k$, for a sampling time of $Ts$, can be represented with the internal state $EBatt$ representing the normalized state of the charge (SOC) and with different charging $ξch$ and discharging $ξdch$ efficiencies Display Formula

(10)$EBatt(k+1)=EBatt(k)+TsEnominalBatt (ξch−ξdch) ψBatt(k)+ξdchPBatt(k)$

where $EnominalBatt$ is the maximum capacity of the battery. According to the binary variable $∂Batt$, the value of power absorbed and released can be distinguished as follows: Display Formula

(11)$PBatt >0⇔∂Batt=1; charging mode PBatt <0⇔∂Batt=0; discharging mode$

Following the conditions mentioned in Eq. (11), the battery model (Eq. (10)) is a piecewise linear system, where the charging and discharging power commands are bounded ($−500W). Moreover, to reduce the risk of damage and based on battery limitation in the laboratory test bed, mainly because of battery aging, the battery SOC during the experiments has been limited by

$0.65
Furthermore, changing in battery efficiencies $(ξch,ξdch$) due to battery aging will result in major problems, which will be experimentally analyzed later on in this paper.

###### Photovoltaics-Panel Estimator.

Photovoltaics production forecasting is a challenging problem that has been widely studied in literature. The most prominent methods include a combination of the data available from different online forecasting website or private companies. Here, we considered a consistent PV estimator regarding the weather condition and PV production in a day before such that the PV estimator, at the beginning of each day, will be derived based on new package of recorded data. To be more precise, the PV production prediction is strongly dependent on the weather forecasting. As for the weather forecasting, the outside solar radiation is considered equal to the weighted average of the last days, i.e., the less weight for the past days, while higher weight for the nearer days. Based on the weather forecasting and the configuration of the PV panels, the PV production predictions are computed.

## Methodology

In this paper, apart from level of comfort, economic behavior, durability, robustness, and many other criteria, we mostly focus on energy saving in presence of high comfort level for building occupants. According to this, distributed MPC is formulated based on the proposed control-oriented models. Furthermore, a heuristic controller and a centralized MPC will be considered for comparison purposes.

To this scope, the tank is discretized into six layers, which is powered by the heat pump. Hence, according to a standard discretization procedure with sampling time of $Ts$, 10 min in this paper, the discrete-time realization of the continuous-time model (1) and (4) takes the following form:

$ẋ(k)=f(x(k),u(k))y(k)=C x(k)$
$xk=[Twk, Tzonek,Tpavk,Tpik,T1k,T2k,T3k,T4k,T5k,T6k,EBatt(k)]T$
$uk=[Tinlet(k)PHP(k)PBattk∂Batt(k)∂grid(k)]T$
$C=[TzonekT1kSOCBatt(k)]$

where $SOCBatt(k)$ shows the state of the charge of the battery representing the energy stored in battery at each time interval $k$ and $∂grid(k)$ is a binary variable denotes the power exchange direction with the main electricity grid, i.e., selling or buying, which will be explained in Sec. 3.1.

###### Distributed Model Predictive Control.

Let us consider the given control scheme in Fig. 4. The algorithm we are going to present consists of two main predictive control structures and it will be referred to hereafter as MPC1 and MPC2. The comfort condition will be satisfied based on MPC2 in the presence of the disturbances (weather condition, solar radiation, internal gains, and, etc.), user's request estimation, and the physical and technical plant limitations. On the contrary, MPC1 will decide how to choose charging/discharging rates for each storage in order to guarantee the minimum required thermal energy for room heating system.

As depicted in Fig. 4, the desired inlet water temperature and the return water temperature signals only are given from MPC2 to MPC1. This is done in order to minimize the flow of data between two controllers in view of a possible future implementation. That is one of the reasons of using a distributed MPC scheme.

###### Building Comfort MPC (MPC2).

Considering the prediction horizon of P, MPC2 is responsible not only to set the optimal reference signal for the valve proportional–integral–derivative (PID) controller but also to provide P-step ahead of desired inlet water temperature, $T̃inlet(k:k+P−1)$, together with return water temperature, $T̃r(k:k+P−1)$ using Eq. (2).

The objective function to be minimized takes into account the building dynamics, weather forecast (usually based on the available software and estimator), and building discomfort, which is the distance between the building temperature ($Tzone(k|t)$) and predefined set point ($Tzoneref(k|t)$). Hence, the quadratic objective function can be defined as follows: Display Formula

(12)$Jthermalzone=∑t=kk+P−1{Tzonet−Tzoneref(t)Qzone2+Tinlet(t)Rinletwater2}$

where $Qzone and Rinletwater$ are optimization weights for building discomfort and input inlet water temperature, respectively.

Therefore, the optimization problem to be solved can be formulated as a linear quadratic program as follows: Display Formula

(13)$minTinlet(k:k+P−1)Jthermalzone$
subject to following operational constraints: Display Formula
(14)$Tinletmin
where $Tinlet(k:k+P−1)$ is the optimal sequence of inlet water temperature by which its first quantity should be tracked by a well-known PID controller. The proposed optimal control problem can be readily solved with available software tools. Here, in this paper, the optimization software package cplex is used to solve the optimal control problem in both standard simple quadratic programming and mixed-integer optimization problem. Furthermore, all the weight coefficient used in the optimization problem formulation are summarized in Table 3.

###### Energy Resources Management Model Predictive Control (MPC1).

According to the proposed control scheme in Fig. 4, TES is driven by the HP to provide the minimum required thermal energy (combination of water flow rate and water temperature) for supplying the three-element valve, which is under control of a fast enough PID controller. It is the main constraint on TES to be considered in MPC1. By the given P-step requested inlet water temperature $Tinlet(k:k+P−1)$, provided by MPC2, TES is forced to always set supply water temperature $Ts(k|t)$ greater than the requested inlet water temperature $Tinlet(k|t)$. To prevent infeasibility problems in practice, this constraint is relaxed inside the objective function with a panic variable ($φ$), in order to penalize the variable value $Ts(k|t)$ if its conditions are not satisfied. This soft constraint can be formulated in a matrix form as follows: Display Formula

(15)$Tinlet(k:k+P−1)−φVmin
where $Vmin$ is a ($P×1$) vector of entries through which the larger $ith$ entry of vector $Vmin$ will lead to relatively softer corresponding $ith$ constraint.

On the other hand, hot water tank temperature must be always bounded by $Tsmin based on hot water system regulation in each country, in which $Tsmin$ and $Tsmax$ are the respective minimum and maximum tan water temperature.

Therefore, charging and discharging of TES are based on the requested thermal energy, price profile during a day, and also negotiation with electrical energy storage, namely battery, and renewable energy resources in order to converge to the best and most economic decision.

Therefore, the first input vector $u1(k)$ can be defined for the set of TES and HP as below ($PHP$ denotes heat pump power) Display Formula

(16)$u1(k)=[PHP(k), φ]$

Consistently with the fact that the model of the battery and also the power exchange with the grid utility follow a logical/algebraic dynamical model, it is necessary to convert all the logic relations to a complete set of linear inequalities in order to setup the energy storages management problem. The optimal control problem finds the optimal charging and discharging decision for the TES and the battery, and also, in a real-time electricity pricing setting, finds the optimal economic decision to sell or buy energy with the utility grid.

By defining the $Pgrid(k)$ as power exchange with the utility grid in each time interval, total cost $Pcost(k)$ can be cast as an MLD system based on the cost of purchasing/selling power ($Cpur$,$Csell$, respectively) from/to utility grid as follows: Display Formula

(17)$Pcost(k)=CpurPgrid(k)∂grid=1CsellPgrid(k) Otherwise$
in which $∂grid$ is defined as the auxiliary variable in order to convert the logic relations to a set of linear inequalities with binary variables subject to the following constraints: Display Formula
(18)$−Pgrid(k)+Tgrid∂grid(k)≤Tgrid Pgrid(k)−(Tgrid+ε)∂grid(k)≤−ε −Csell(k)Pgrid(k)−Tgrid∂grid(k)+Pcost(k)≤0 Csell(k)Pgrid(k)−Tgrid∂grid(k)−Pcost(k)≤0 −Cpur(k)Pgrid(k)+Tgrid∂grid(k)+Pcost(k)≤TgridCpur(k)Pgrid(k)+Tgrid∂grid(k)−Pcost(k)≤Tgrid$
where $Tgrid$ is the maximum power exchange between the MG and the utility grid. Therefore, the second input vector $u2(k)$ can be defined as Display Formula
(19)$u2(k)=[∂grid(k)]$

Furthermore, using the same technique of Eq. (18), the battery logical/algebraic dynamical model, presented in Eqs. (10) and (11), can be also formulated as an MLD system by considering the binary variable $∂Batt$ for converting all logic relations to set of linear inequalities.

Therefore, the third input vector $u3(k)$ can be defined for the battery package as below: Display Formula

(20)$u3(k)=[PBatt(k),ψBatt(k)]$

Given a prediction horizon P (and the same control horizon) and optimization weights of heat pump and panic variables ($QHP$ and $γφ$), the optimal control problem design consists of solving the following finite horizon optimal control problem: Display Formula

(21)$minu1(k:k+P−1),u2(k:k+P−1),u3(k:k+P−1)Jenergyresources=minu1(k:k+P−1),u2(k:k+P−1),u3(k:k+P−1)∑t=kk+P−1PHP(t)QHP2+Pcost(t)++ΔPBatt(t)WBatt2+φγφ2$
subject to the TES dynamics and following operational constraints: Display Formula
(22)$−Tgrid

and load and power balance equation Display Formula

(23)$Pgrid(k)+PPV(k)−PBatt(k)−PHP(k)=0$

where $PPV(k) and PHP(k)$ are PV panel production and heat pump power at time-step k, respectively.

It is worth noting that $ΔPBatt(k)$, which is the step changing in battery power exchange, is relaxed by the weight of $WBatt$ in order to prevent any damages or practical problems. This so-called smoothing factor can prevent any sharp changing command and also make system more feasible.

This problem is a mixed-integer quadratic program problem and can be accurately solved by available software tools (cplex has been used in this project).

###### Centralized Model Predictive Control.

Based on a centralized MPC, all the measured data are conveyed to the central control station and, consequently, the control signals are communicated to all the actuators in a given time interval. The adaptation procedure is based on the successive linearization approach in order to deal with nonlinearity of the system. Accordingly, the optimization problem includes solving one finite horizon optimal control problem as follows: Display Formula

(24)$minu1(k:k+P−1),u2(k:k+P−1),u3(k:k+P−1),Tinlet(k:k+P−1)Jcentralized=minu1(k:k+P−1),u2(k:k+P−1),u3(k:k+P−1),Tinlet(k:k+P−1)(Jenergyresources+Jthermalzone)$

This problem is also a mixed-integer quadratic program problem and can be readily solved by available software tools (as already mentioned, cplex has been used).

###### Heuristic Control Strategy.

For comparison purposes, a heuristic controller has been designed and applied on the energy resource management part based on the power price tariff. In this case, we assumed that the comfort conditions are satisfied by the MPC2 and the MPC1 is replaced by the rule-based heuristic supervision logic (S-LOGIC), which decides how much energy should be put in storages to minimize energy consumption without sacrificing occupant thermal comfort.

###### Valve Proportional–Integral–Derivative Controller.

Different methods are known in literature concerning the PID tuning strategies. For the model-based strategies, it is possible to compute the transfer function $G(s)$, i.e., the transfer function from the supply temperature $TS$, which is equal to $T1$ (see Fig. 4), to the inlet water temperature of the room $Tinlet$. To this end, a simplified transfer function $G̃s$ with gain $μG$ and time constant $τG$ can be identified as follows with a delay of one second:

$G̃s=μG1+s τG$

By manipulating the transfer function characterizing a PID controller, the PID transfer function can be written and tuned as follows:

$PID=P1+I 1s+D Nv1+Nv 1s$

where $P, I, and D$ are proportional, integral, and derivative gain, respectively, and $Nv$ is the respective filter coefficient for the controller feasibility.

Accordingly, the parameters of the approximated valve model and designed PID controller are reported in Table 4.

## Results and Discussion

In this section, simulation tests as well as experimental validation are presented. The tests witness the effectiveness of the proposed control approach. Specifically, the benefits of using of two control variables in the TES model (i.e., the tank mass flow rate and the water temperature) and also the advantages deriving from the usage of different type of energy storages (electrical and thermal) are first discussed. In this analysis, negative (positive) values for the battery power represent the discharging (charging) mode of the battery. The same notation holds for the utility grid exchanged power, i.e., the negative (positive) power of utility grid means that the microgrid is selling (purchasing) energy. Finally, a comparison between the presented algorithm, i.e., DMPC and an empirical control approach is presented. Furthermore, the optimality of the proposed approach has been evaluated through a comparison with the CMPC as the optimal solution. As for the parameters used, the unitary buying and selling energy price and the main building specification are shown in Fig. 5 and Table 5, respectively. A variable electricity price scenario (time of use (TOU)) and day/night (DN) price profile is considered in which the electricity price is varying between 5 and 35 cent/kWh.

###### Benefits of Using Two Control Variables (Tank Mass Flow Rate and Water Temperature) and Energy Storages Coordinator.

According to the procedure illustrated in Sec. 3 of this paper, the specified test case is presented in this section to better show the main advantages of using different types of energy storages and generators together with the designed distributed MPC scheme. In this test case, a realistic forecast data of both price profile (both day/night and TOU method) and PV-panel production, as well as weather condition and load requirement are needed to solve both the demand side (building comfort) and the market side (energy management coordinator) optimization problem. The microgrid data and specifications, considered in this test case, are summarized in Table 6.

Moreover, the main parameters of the proposed DMPC approach were already summarized in Table 3.

## Performance Analysis in Simulation

In this analysis, the effectiveness of using the TES with two control variables is evaluated in the presence of electrical energy storage (battery). In this case, the energy management system should utilize both thermal and electrical energy storages to decouple load request to the electrical grid from user comfort control problem. As Fig. 6 shows, the energy storage coordinator is economically distributing energy among all energy storages such that the battery is meaningfully charged during off-peak periods, when the PV panel production is high, and price is in the minimum level, and then discharging it during peak hours for supporting the loads and also selling the power surplus to the utility grid.

The same behavior can be observed in TES to minimize the HP consumption during peak periods, while the comfort condition (room temperature set point) is basically always satisfied Fig. 7.

Moreover, as Fig. 8 shows, we expect to use the total energy stored in battery at the end of the day.

In addition, there is no sharp change in battery energy profile and power exchange, due to a smoothing factor for the battery coordinator (as explained in Eq. (21)). The total energy distributed for one day simulation among all storages, generators, and utility grid together with load demand has been shown in Fig. 9. It shows the unit commitment, energy storages, economic dispatch, sale, and purchase of energy to/from utility grid based on a unitary buying and selling energy price together with the renewable energy source production estimator.

###### Comparison With Centralized Model Predictive Control and Heuristic Controller.

In this section, a centralized MPC and heuristic controller are considered according to the procedure illustrated within Sec. 3 (Secs. 3.2 and 3.3).

The CMPC is considered to evaluate the optimality of the proposed DMPC algorithm. As Fig. 10 shows, the normalized objective value profile of DMPC algorithm (blue line) is very close to the optimal solution profile obtained by CMPC (red dashed line). There is a small and fast phenomenon in the objective profile of DMPC between time 15 and 20, which is due to the change of valve position enforced by the MPC2 due to set point changing. Actually, MPC1 is designed based on a linearized model based on the current valve position. Since the decision of MPC1 is based on the linearized model, it is obvious that MPC1 cannot be aware of the valve change during the prediction horizon. Consequently, it leads to a loss of optimality as it has been illustrated as a small drop in objective profile in Fig. 10. The difference over one-day simulation is about 0.3%, which is negligible from a practical point of view.

On the other hand, an economic analysis has been conducted regarding heuristic control strategy (HEU) algorithm to show the main advantages of the designed DMPC algorithm. Besides many operational advantages of DMPC, such as having modular structure, considering the operational constraints and physical limitations, presence of disturbance effects on future behavior of the plant, the DMPC approach, in comparison to HEU, saves up to 12.5% in load energy consumption and it has 22% more of total cost benefit, as it has been shown in Table 7.

## Performance Analysis in Experimental Activities

The experimental validation of the proposed control algorithm has been performed in a well-equipped smart grid research laboratory (Fig. 11), equipped with energy storages, controllable loads, PV panel, wind turbine and a point of common connection connected to the national grid as follows:

• Battery: Inverter SMA (Niestetal, Germany) Sunny Island 4500 with nominal power 3.3 kW 250 Ah, 60 V lead acid batteries.

• Controllable loads: 15 kW resistors, 1 kW lamps, 0.5 HP motor, and 2.5 kVAR inductive load.

• Photovoltaics panels: inverter SMA Sunny Boy 100 E with rating of 1.1 kW nominal power.

• Wind turbine: Inverter SMA Windy Boy 1700 with nominal power of 1.7 kW.

All devices are connected to the local electric network and are fully controllable through a supervisory control and data acquisition system.

Due to the fact that real buildings are not available for testing, they have been emulated using the controllable loads present in the laboratory. Taking into account the different weather conditions at the laboratory location and also physical limitations of the electrical equipment, a real experimental test case has been carried out. Moreover, the power quality issues such as voltage fluctuations and power frequency bound at which electric power is generated and distributed has been analyzed for the system under control. Moreover, the optimization problem is solved on a personal computer with the seventh generation Intel® Core™ i7 processor with 3.5 GHz frequency. Experiments with lower performance personal computer were done also, with the sampling time from 5 to 10 min. In any case, the DMPC approach showed no limitation in terms of communication coordination and solution of the optimization problems.

At the end, the overall configuration and of the experimental test is illustrated in Fig. 12.

The microgrid configuration is the same as the system under control in Sec. 4. The only differences are in the weather conditions and battery SOC. The proposed PV panel estimator previously illustrated is considered to better anticipate the PV-panel production fluctuations. Many tests have been performed, but only one is shown here for the sake of brevity.

In this test case, a 5-h experiment has been done starting from 10 am to 3 pm in a winter day. It was conducted over a sunny day when the PV production was almost in the highest rate (Fig. 13).

The battery SOC was approximately 75% and we considered 20% of the total battery capacity for the acceptable bound of the energy management coordinator. For simplicity and to better show the coordinator performance in a real case, we simply considered a DN price profile changing from 5 cents to 35 cents.

Although the battery aging and communication delay had adverse effects on the coordinator performance, the proposed control system can economically distribute the power among all storages and generators together with utility grid. As it is shown in Fig. 14 battery SOC, as well as the TES, is correctly tracking the MPC commands.

On the contrary, the desired inlet water temperature, for the building pipelines, should be regulated through the three-element valve. Therefore, TES has to provide sufficient thermal energy, which is the combination of tank water flow rate and tank water temperature. Thanks to the fixed water flow rate passing through building's pipelines, the minimum required energy, provided by TES, can be obtained once the supply water temperature, i.e., the outlet water of TES, becomes equal to the requested inlet water temperature, as explained in Sec. 3. Figure 15 (subplot1) shows that TES temperature is always higher than the requested inlet water temperature, which means that the sufficient thermal energy for building heating system is always provided. For more realistic analysis and to prove the robust behavior of system in the presence of structural uncertainties, a low-order and high-order TES model are considered.

As shown in Fig. 15, the comfort conditions are always satisfied in subplot1, where the dotted line is the MPC command, the dashed line is the predicted tank temperature model, and the solid line is the measurement.

Figure 16 shows the total power exchanged between battery, PV panel, load, and utility grid. As it has been shown in figure, the proposed DMPC stored thermal and electrical energy during off-peak periods in order to support the upcoming demand in peak periods and, consequently, sell the power surplus to utility grid while satisfying the comfort conditions by tracking the room temperature set points (Fig. 15).

Subsequently, the power quality issues, i.e., voltage fluctuations and also power frequency data, have been collected and analyzed by the supervisory control and data acquisition system in the laboratory at the point of common connection among PV panels inverter, battery inverter, and public grid. For the sake of brevity, this is not here reported, but no issues have been detected, which witnesses the reliability and stability of the grid voltage level for 5-h experimental test.

## Summary of the Results

In this section, all the results obtained through the simulations and experimental analysis are summarized. The results witness the effectiveness of the proposed DMPC algorithm for the application of the economic energy management system for the building temperature control, as follows:

• 12.5% energy saving in the load energy consumption with respect to the classic HEU control.

• 22% more total cost benefit with respect to the classic HEU control due to the optimal load shifting and economic power exchange between batter and main electricity grid.

• Less computational cost due to the distributed model predictive control algorithm.

• Almost optimal performance of DMPC algorithm as the difference between DMPC and CMPC is negligible during one-day simulation (about 0.3%).

• Tackling with the system nonlinearities due to the adaptive structure of the proposed DMPC algorithm.

• Decoupling of the load and heating system through a three-element mixing valve controlled by a suitable PID (still optimal solution is obtained).

• Smoothing the battery power profile due to the relaxation factor considered in the DMPC formulation, which results in larger battery lifetime.

• Experimental validation of the proposed control algorithm.

• The reliability and stability of the grid voltage level for 5-h experimental test (Figs. 17 and 18).

## Conclusion

In this paper, an applicable economic solution is proposed to tackle efficiently the energy management problem in the context of smart buildings. As particularly representative of energy efficient application cases, a building typology is here considered endowed with a heat pump and a thermal energy storage feeding a radiant floor heating system. In this way, it is possible to exploit the thermal inertia of the walls and the thermal energy storage to modify a user's electrical profile. An MPC controls all the requests of the user side, balancing between comfort satisfaction (temperature set point tracking) and energy consumption. At the energy resources level, an optimization problem is formulated in order to optimally decide how much energy should be put in the storages to guarantee comfort when requested by the user. This is performed considering the renewable future power generation, and variable energy price, while handling nonlinearities of the system due to a nonlinear thermal energy storage model, in turn caused by the presence of a mixing valve to control the inlet water temperature.

The overall solution is thus formulated as a distributed MPC with a coordinator solving a mixed-integer quadratic programming optimization problem. The novel solution here proposed encompasses optimal power distribution, variable tariffs, temperature tracking for buildings. The simulation results show the effectiveness of the solution proposed and the comparison with a centralized formulation of the same problem shows that the distributed solution is definitely quite close to the centralized solution. Moreover, DMPC approach, in comparison to HEU, saves up to 12.5% in load energy consumption and it has 22% improvement of total cost benefit. The potential of proposed approach is also witnessed by an experimental validation test in a well-equipped smart grid research laboratory, which confirms the simulation results and proves the feasibility of online optimal control in real life cases.

There are many possible future research directions for the described work. They include the extension of the distributed approach to multiple energy resources (multiple electrical and thermal storages, multiple buildings, multiple power generating units, multiple renewable sources), a more accurate modeling of heat pump efficiency, which is affected by external temperature and humidity, and the analysis of the impact of modeling mismatches on the quality of the distributed predictive control strategy illustrated. Nonlinear MPC techniques will also be investigated.

## Acknowledgements

This project has been performed using the ERIGrid Research Infrastructure and is part of a project that has received funding from the European Union's Horizon 2020 Research and Innovation Programme under the Grant Agreement No. 654113.

The support of the European Research Infrastructure ERIGrid and its partner Smart RUE, ICCS-NTUA is very much appreciated.

## Funding Data

• European Union's Horizon 2020 Research and Innovation Programme (Grant No. 654113; Funder ID: 10.13039/100010661).

## References

Buzek, J. , and Garrido, D. L. , 2010, “ Directive 2010/31/EU of the EU Parliament and of the Council of 19 May 2010 on the Energy Performance of Buildings,” Official J. Eur. Union, pp. 1–23.
Shaikh, P. H. , Nor, N. B. M. , Nallagownden, P. , Elamvazuthi, I. , and Ibrahim, T. , 2014, “ A Review on Optimized Control Systems for Building Energy and Comfort Management of Smart Sustainable Buildings,” Renewable Sustainable Energy Rev., 34, pp. 409–429.
Zheng, M. , Meinrenken, C. J. , and Lackner, K. S. , 2014, “ Electricity Storage in Buildings for Residential Sector Demand Response: Control Algorithms and Economic Viability Evaluation,” National Institute of Standards and Technologies, U.S. Department of Commerce, Gaithersburg, MD, Report No. GCR 14-978.
Mantovani, G. , Costanzo, G. T. , Marinelli, M. , and Ferrarini, L. , 2014, “ Experimental Validation of Energy Resources Integration in Microgrids Via Distributed Predictive Control,” IEEE Trans. Energy Convers., 29(4), pp. 1018–1025.
Pelland, S. , Remund, J. , Kleissl, J. , Oozeki, T. , and Brabandere, K. D. , 2013, “ Photovoltaic and Solar Forecasting: State of the Art,” The IEA Photovoltaic Power Systems Programme (IEA‐PVPS), Canada.
Kotsampopoulos, P. C. , Lehfuss, F. , Lauss, F. G. , Bletterie, B. , and Hatziargyriou, N. , 2015, “ The Limitations of Digital Simulation and the Advantages of PHIL Testing in Studying Distributed Generation Provision of Ancillary Services,” IEEE Trans. Ind. Electron., 62(9), pp. 5502–5515.
Fichera, A. , Volpe, R. , and Frasca, M. , 2016, “ Assessment of the Energy Distribution in Urban Areas by Using the Framework of Complex Network Theory,” Int. J. Heat Technol., 34(S2), pp. S430–S434.
Volpe, R. , Frasca, M. , Fichera, A. , and Fortuna, L. , 2017, “ The Role of Autonomous Energy Production Systems in Urban Energy Networks,” J. Complex Networks, 5(3), pp. 461–472.
Tsikalakis, A. G. , and Hatziargyriou, N. D. , 2008, “ Centralized Control for Optimizing Microgrids Operation,” IEEE Trans. Energy Convers., 23(1), pp. 241–248.
Maniatopoulos, M. , Lagos, D. , Kotsampopoulos, P. , and Hatziargyriou, N. , 2017, “ Combined Control and Power Hardware in-the-Loop Simulation for Testing Smart Grid Control Algorithms,” IET Gener., Transm. Distrib., 11(12), pp. 3009–3018.
Parisio, A. , Rikos, E. , and Glielmo, L. , 2014, “ A Model Predictive Control Approach to Microgrid Operation Optimization,” IEEE Trans. Control Syst. Technol., 22(5), pp. 1813–1827.
Malysz, P. , Sirouspour, S. , and Emadi, A. , 2014, “ An Optimal Energy Storage Control Strategy for Grid-Connected Microgrids,” IEEE Trans. Smart Grid, 5(4), pp. 1785–1796.
Kumar Nunna, H. S. V. S. , and Doolla, S. , 2013, “ Energy Management in Microgrids Using Demand Response and Distributed Storage—A Multiagent Approach,” IEEE Trans. Power Delivery, 28(2), pp. 939–947.
Arteconi, A. , Ciarrocchi, E. , Pan, Q. , Carducci, F. , Comodi, G. , Polonara, F. , and Wang, R. , 2017, “ Thermal Energy Storage Coupled With PV Panels for Demand Side Management of Industrial Building Cooling Loads,” Appl. Energy, 185(2), pp. 1984–1993.
Arteconi, A. , Hewitt, N. J. , and Polonara, F. , 2013, “ Domestic Demand-Side Management (DSM): Role of Heat Pumps and Thermal Energy Storage (TES) Systems,” Appl. Therm. Eng., 51(1–2), pp. 155–165.
Adriana, L. N. D. , and Moisès, G. , 2017, “ Mixed-Integer-Linear-Programming-Based Energy Management System for Hybrid PV-Wind-Battery Microgrids: Modeling, Design, and Experimental Verification,” IEEE Trans. Power Electron., 32(4), pp. 2769–2783.
Killian, M. , Mayer, B. , and Kozek, M. , 2014, “ Hierarchical Fuzzy MPC Concept for Building Heating Control,” 19th IFAC World Congress, Cape Town, South Africa.
Comodi, G. , Giantomassi, A. , Severini, M. , Squartini, S. , Ferracuti, F. , Fonti, A. , Nardi Cesarini, D. , Morodo, M. , and Polonara, F. , 2015, “ Multi-Apartment Residential Microgrid With Electrical and Thermal Storage Devices: Experimental Analysis and Simulation of Energy Management Strategies,” Appl. Energy, 137, pp. 854–866.
Di Giorgio, A. , and Liberati, F. , 2014, “ Near Real Time Load Shifting Control for Residential Electricity Prosumers Under Designed and Market Indexed Pricing Models,” Appl. Energy, 128, pp. 119–132.
Kuboth, S. , Heberle, F. , König-Haagen, A. , and Brüggemann, D. , 2019, “ Economic Model Predictive Control of Combined Thermal and Electric Residential Building Energy Systems,” Appl. Energy, 240, pp. 372–385.
Alimohammadisagvand, B. , Jokisalo, J. , Kilpeläinen, S. , Ali, M. , and Sirén, K. , 2016, “ Cost-Optimal Thermal Energy Storage System for a Residential Building With Heat Pump Heating and Demand Response Control,” Appl. Energy, 174, pp. 275–287.
Nash, A. L. , Badithela, A. , and Jain, N. , 2017, “ Dynamic Modeling of a Sensible Thermal Energy Storage Tank With an Immersed Coil Heat Exchanger Under Three Operation Modes,” Appl. Energy, 195, pp. 877–889.
Farina, M. , and Scattolini, R. , 2012, “ Distributed Predictive Control: A Non-Cooperative Algorithm With Neighbor-to-Neighbor Communication for Linear Systems,” Automatica, 48(6), pp. 1088–1096.
Bemporad, A. , and Morari, M. , 1999, “ Control of Systems Integrating Logic, Dynamics, and Constraints,” Automatica, 35(3), pp. 407–427.
Ferrarini, L. , and Mantovani, G. , 2013, “ Modeling and Control of Thermal Energy of a Large Commercial Building,” IEEE International Workshop on Intelligent Energy Systems (IWIES), Vienna, Austria, Nov. 14.
Mantovani, G. , and Ferrarini, L. , 2015, “ Temperature Control of a Commercial Building With Model Predictive Control Techniques,” IEEE Trans. Ind. Electron., 62(4), pp. 2651–2660.
Ferrarini, L. , Rastegarpour, S. , and Petretti, A. , 2017, “ An Adaptive Underfloor Heating Control With External Temperature Compensation,” 14th International Conference on Informatics in Control, Automation and Robotics (ICINCO), Madrid, Spain, pp. 629–636.
ASHRAE, 2010, “ Methods of Testing for Rating Seasonal Efficiency of Unitary Air Conditioners and Heat Pumps,” American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, GA, Standard No. A. S. 116–2010.
Powell, K. M. , and Edgar, T. F. , 2013, “ An Adaptive-Grid Model for Dynamic Simulation of Thermocline Thermal Energy Storage Systems,” Energy Convers. Manage., 76, pp. 865–873.
View article in PDF format.

## References

Buzek, J. , and Garrido, D. L. , 2010, “ Directive 2010/31/EU of the EU Parliament and of the Council of 19 May 2010 on the Energy Performance of Buildings,” Official J. Eur. Union, pp. 1–23.
Shaikh, P. H. , Nor, N. B. M. , Nallagownden, P. , Elamvazuthi, I. , and Ibrahim, T. , 2014, “ A Review on Optimized Control Systems for Building Energy and Comfort Management of Smart Sustainable Buildings,” Renewable Sustainable Energy Rev., 34, pp. 409–429.
Zheng, M. , Meinrenken, C. J. , and Lackner, K. S. , 2014, “ Electricity Storage in Buildings for Residential Sector Demand Response: Control Algorithms and Economic Viability Evaluation,” National Institute of Standards and Technologies, U.S. Department of Commerce, Gaithersburg, MD, Report No. GCR 14-978.
Mantovani, G. , Costanzo, G. T. , Marinelli, M. , and Ferrarini, L. , 2014, “ Experimental Validation of Energy Resources Integration in Microgrids Via Distributed Predictive Control,” IEEE Trans. Energy Convers., 29(4), pp. 1018–1025.
Pelland, S. , Remund, J. , Kleissl, J. , Oozeki, T. , and Brabandere, K. D. , 2013, “ Photovoltaic and Solar Forecasting: State of the Art,” The IEA Photovoltaic Power Systems Programme (IEA‐PVPS), Canada.
Kotsampopoulos, P. C. , Lehfuss, F. , Lauss, F. G. , Bletterie, B. , and Hatziargyriou, N. , 2015, “ The Limitations of Digital Simulation and the Advantages of PHIL Testing in Studying Distributed Generation Provision of Ancillary Services,” IEEE Trans. Ind. Electron., 62(9), pp. 5502–5515.
Fichera, A. , Volpe, R. , and Frasca, M. , 2016, “ Assessment of the Energy Distribution in Urban Areas by Using the Framework of Complex Network Theory,” Int. J. Heat Technol., 34(S2), pp. S430–S434.
Volpe, R. , Frasca, M. , Fichera, A. , and Fortuna, L. , 2017, “ The Role of Autonomous Energy Production Systems in Urban Energy Networks,” J. Complex Networks, 5(3), pp. 461–472.
Tsikalakis, A. G. , and Hatziargyriou, N. D. , 2008, “ Centralized Control for Optimizing Microgrids Operation,” IEEE Trans. Energy Convers., 23(1), pp. 241–248.
Maniatopoulos, M. , Lagos, D. , Kotsampopoulos, P. , and Hatziargyriou, N. , 2017, “ Combined Control and Power Hardware in-the-Loop Simulation for Testing Smart Grid Control Algorithms,” IET Gener., Transm. Distrib., 11(12), pp. 3009–3018.
Parisio, A. , Rikos, E. , and Glielmo, L. , 2014, “ A Model Predictive Control Approach to Microgrid Operation Optimization,” IEEE Trans. Control Syst. Technol., 22(5), pp. 1813–1827.
Malysz, P. , Sirouspour, S. , and Emadi, A. , 2014, “ An Optimal Energy Storage Control Strategy for Grid-Connected Microgrids,” IEEE Trans. Smart Grid, 5(4), pp. 1785–1796.
Kumar Nunna, H. S. V. S. , and Doolla, S. , 2013, “ Energy Management in Microgrids Using Demand Response and Distributed Storage—A Multiagent Approach,” IEEE Trans. Power Delivery, 28(2), pp. 939–947.
Arteconi, A. , Ciarrocchi, E. , Pan, Q. , Carducci, F. , Comodi, G. , Polonara, F. , and Wang, R. , 2017, “ Thermal Energy Storage Coupled With PV Panels for Demand Side Management of Industrial Building Cooling Loads,” Appl. Energy, 185(2), pp. 1984–1993.
Arteconi, A. , Hewitt, N. J. , and Polonara, F. , 2013, “ Domestic Demand-Side Management (DSM): Role of Heat Pumps and Thermal Energy Storage (TES) Systems,” Appl. Therm. Eng., 51(1–2), pp. 155–165.
Adriana, L. N. D. , and Moisès, G. , 2017, “ Mixed-Integer-Linear-Programming-Based Energy Management System for Hybrid PV-Wind-Battery Microgrids: Modeling, Design, and Experimental Verification,” IEEE Trans. Power Electron., 32(4), pp. 2769–2783.
Killian, M. , Mayer, B. , and Kozek, M. , 2014, “ Hierarchical Fuzzy MPC Concept for Building Heating Control,” 19th IFAC World Congress, Cape Town, South Africa.
Comodi, G. , Giantomassi, A. , Severini, M. , Squartini, S. , Ferracuti, F. , Fonti, A. , Nardi Cesarini, D. , Morodo, M. , and Polonara, F. , 2015, “ Multi-Apartment Residential Microgrid With Electrical and Thermal Storage Devices: Experimental Analysis and Simulation of Energy Management Strategies,” Appl. Energy, 137, pp. 854–866.
Di Giorgio, A. , and Liberati, F. , 2014, “ Near Real Time Load Shifting Control for Residential Electricity Prosumers Under Designed and Market Indexed Pricing Models,” Appl. Energy, 128, pp. 119–132.
Kuboth, S. , Heberle, F. , König-Haagen, A. , and Brüggemann, D. , 2019, “ Economic Model Predictive Control of Combined Thermal and Electric Residential Building Energy Systems,” Appl. Energy, 240, pp. 372–385.
Alimohammadisagvand, B. , Jokisalo, J. , Kilpeläinen, S. , Ali, M. , and Sirén, K. , 2016, “ Cost-Optimal Thermal Energy Storage System for a Residential Building With Heat Pump Heating and Demand Response Control,” Appl. Energy, 174, pp. 275–287.
Nash, A. L. , Badithela, A. , and Jain, N. , 2017, “ Dynamic Modeling of a Sensible Thermal Energy Storage Tank With an Immersed Coil Heat Exchanger Under Three Operation Modes,” Appl. Energy, 195, pp. 877–889.
Farina, M. , and Scattolini, R. , 2012, “ Distributed Predictive Control: A Non-Cooperative Algorithm With Neighbor-to-Neighbor Communication for Linear Systems,” Automatica, 48(6), pp. 1088–1096.
Bemporad, A. , and Morari, M. , 1999, “ Control of Systems Integrating Logic, Dynamics, and Constraints,” Automatica, 35(3), pp. 407–427.
Ferrarini, L. , and Mantovani, G. , 2013, “ Modeling and Control of Thermal Energy of a Large Commercial Building,” IEEE International Workshop on Intelligent Energy Systems (IWIES), Vienna, Austria, Nov. 14.
Mantovani, G. , and Ferrarini, L. , 2015, “ Temperature Control of a Commercial Building With Model Predictive Control Techniques,” IEEE Trans. Ind. Electron., 62(4), pp. 2651–2660.
Ferrarini, L. , Rastegarpour, S. , and Petretti, A. , 2017, “ An Adaptive Underfloor Heating Control With External Temperature Compensation,” 14th International Conference on Informatics in Control, Automation and Robotics (ICINCO), Madrid, Spain, pp. 629–636.
ASHRAE, 2010, “ Methods of Testing for Rating Seasonal Efficiency of Unitary Air Conditioners and Heat Pumps,” American Society of Heating, Refrigerating and Air-Conditioning Engineers, Atlanta, GA, Standard No. A. S. 116–2010.
Powell, K. M. , and Edgar, T. F. , 2013, “ An Adaptive-Grid Model for Dynamic Simulation of Thermocline Thermal Energy Storage Systems,” Energy Convers. Manage., 76, pp. 865–873.

## Figures

Fig. 2

Building heating system including: HP condenser, TES, three-element valve, and thermal zone

Fig. 3

Control volume of a single layer

Fig. 4

Schematic diagram of DMPC approach

Fig. 1

Schematic of proposed integrated energy resources system

Fig. 5

Unitary energy price TOU and DN structure: selling price (dashed red line) and purchasing price (blue solid line)

Fig. 6

Subplot1: power exchange between storages based on TOU price profile: battery (star marker red line), HP (solid blue line) and subplot2: power exchange between storages based on DN price profile: Battery (star marker red line), HP (solid blue line)

Fig. 7

Room temperature (solid blue line)

Fig. 8

Subplot1: energy stored in battery based on TOU profile (maximum energy (solid red line), minimum energy (solid blue line)) and subplot2: energy stored in battery based on DN profile (maximum energy (solid red line) and minimum energy (solid blue line))

Fig. 9

Total power distribution among all storages, generators, and utility grid

Fig. 10

Normalized objective value: CMPC (dashed red line) and DMPC (solid blue line)

Fig. 11

Smart grid research unit

Fig. 12

Overall configuration of the experimental tests

Fig. 13

PV panel production for 5 h test in a sunny winter day

Fig. 14

Subplot1: battery performance, subplot2: TES performance (dashed line: price profile, solid line: MPC command, and dotted line: real data)

Fig. 15

Subplot1: TES temperature, subplot2: room temperature (dashed line: set point and solid line: actual room temperature)

Fig. 16

Total power exchange in MG

Fig. 17

Power frequency during 5 h experiment

Fig. 18

Voltage level during 5 h experiment

## Tables

Table 1 Description of the building's parameters
Table 2 Description of the heating system's parameters
Table 3 Parameters of the proposed DMPC algorithm
Table 4 Valve model parameters and PID coefficients
Table 5 Building specification
Table 6 Microgrid specification
Table 7 Comparison between DMPC and HEU

## Errata

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