Model predictive control under weather forecast uncertainty for HVAC systems in university buildings

In buildings, there are two yet conﬂicting optimization goals: 1) minimize energy use and energy cost and 2) maximize thermal comfort. Model predictive control (MPC) is an ideal control strategy to deal with the above conﬂicting optimization goals. However, one challenge hindering the implementation of the MPC in buildings is the weather forecast uncertainty. This study aimed to improve the performance of the MPC under weather forecast uncertainty by introducing an error model. The error model used a straightforward approach based on easily measurable and accessible data to improve the quality of weather forecast data. The proposed method was tested by simulation on a university building located in Norway, while the detailed information and measured data from this real building were used to develop and validate the building model used in this study. Results showed that the MPC with the error model was able to achieve almost the full theoretical potential of the MPC in terms of the energy cost and thermal comfort, with 3.4% of weekly energy cost saving and 73% of indoor temperature violation numbers reduction compared to a conventional rule-based controller. In contrast, due to the existence of weather forecast error and a lack of error addressing mechanism, the MPC without error model did not perform well and gave the energy cost saving of only 0.7% and the indoor temperature violation numbers even increased by 20%. Meanwhile, the results indicated the introduction of the error model always beneﬁted the MPC performance even under the condition of the low error of weather forecast. This study may facilitate the real application of the MPC in buildings. (cid:1)


Introduction
The energy use in buildings in the European Union (EU) countries accounts for 40% of the final energy use and 36% of the greenhouse gas emissions [1].In EU countries, 76% of this energy goes towards comfort control in buildings for heating, ventilation and air conditioning (HVAC) [2].Therefore, it is essential to investigate the methods for reducing energy use and energy cost of these systems.However, the demand for better comfort in buildings is expected to continue as it has been within the last few decades.This leads to two main yet conflicting optimization goals occurring in buildings: 1) maximize the thermal comfort, and 2) minimize both the energy use and the energy cost.Model predictive control (MPC) is an ideal control strategy to deal with the above conflicting optimization goals [3][4][5][6].A building MPC uses a dynamic building model to predict the future thermal behaviour of the building and generates a control vector that minimizes a certain objective func-tion over the prediction horizon in the presence of disturbances and constraints.The objective function used in an MPC can combine several conflicting optimization goals, and an optimal tradeoff between these conflicting optimization goals may be secured by running the MPC control strategy [5,7].
Researchers have proven that MPC controllers could result in a theoretical potential for energy saving and improved thermal comfort compared to other conventional rule-based controllers (RBCs).A test case for simulating predictive control in building systems shows the theoretical annual energy saving for heating, lighting, and ventilation of 7% with improved thermal comfort [8].An onoff controller and an MPC system in a combined solar thermal collector and heat pump system are simulated and the results show that the MPC performs better by providing desired thermal comfort and less energy use, and finally giving the 9% theoretical operating cost saving [9].A simulation study of MPC applied to the HVAC system in a typical Swiss office building shows a comparison of the MPC with an RBC, and the results indicate theoretical energy saving of about 17% while providing an improved level of comfort [10].In [11], an MPC strategy for space heating (SH) demand in buildings is compared to a conventional weather-compensation controller (WCC).The MPC proves to give operating cost savings theoretically, while maintains decent thermal comfort conditions.Furthermore, in recent years, the effectiveness of MPC in realworld applications has been reported.MPC in real buildings and experiment chambers is described in the article [12].These examples demonstrate that MPC can save 10-30% of energy use, with the potential for improved thermal comfort for building occupants [13][14][15][16].However, some challenges are hindering the wide-scale implementation of the MPC in buildings.One challenge is the uncertainty of the weather forecast.A building MPC is based on the prediction of future weather, and many studies considered a perfect weather forecast for MPC.However, any prediction has some uncertainties that may influence system performance such as energy use and thermal comfort.Therefore, the MPC controller may receive inaccurate information, leading to incorrect control actions that may cause thermal discomfort or energy waste [12].
Oldewurtel et al. found that the MPC controllers using the weather forecast data did not perform well, especially when the actual weather varied from the forecast [17].Henze et al. evaluated the impact of weather forecast uncertainty on the MPC performance.The conclusion was that MPC appeared to be a promising building control strategy, but only when perfect weather forecasts were available [18].Lamoudi et al. assessed the MPC performance considering the uncertainty of the weather forecast.They found the degradation of weather forecast quality generated an increase in the energy cost.A maximum increase of 4% in energy cost was noted due to weather forecast uncertainty [19].Hedegaard et al. reported that the performance of an MPC controller may be reduced by 4% in terms of heating cost savings due to the weather forecast error [20].Zavala et al. also pointed out that using accurate forecasts and uncertainty information was critical to achieving reliable MPC performance [21].
One way to address the effect of the weather forecast uncertainty on MPC is to use the stochastic MPC [17,22].However, for building systems, a stochastic MPC often requires an additive Gaussian disturbance model that does not hold for the weather uncertainty.Furthermore, building systems are usually nonlinear systems.The exact solutions for the stochastic MPC for nonlinear systems subject to non-Gaussian disturbances are, in general, computationally intractable [23,24].Introducing weather forecast models is another way to tackle the weather forecast uncertainty in MPC.Henze et al. introduced several short-term weather forecast models to improve the performance of MPC.Those weather forecast models predicted the future weather based on historical patterns such as using the same data as the previous day, typical days of a month, etc.It was highlighted that almost the full theoretical potential of MPC may be realized by using simple but accurate short-term weather forecast models [18].Zavala et al. used a machine learning approach to obtain weather information for MPC [21].Salque et al. developed a self-learning artificial neural networks model for weather forecast considering the uncertainty of weather [25].Chakraborty et al. proposed a support vector machine regression model to predict future weather variables [26].However, the above weather forecast models either need lots of historical data or are too complex.In actual buildings, MPC controllers may require more straightforward approaches based on easily measurable and accessible data.
In summary, applying an MPC controller to HVAC systems has a high potential for improving the performance of buildings in terms of thermal comfort, energy use, and energy cost.However, the effect of weather forecast uncertainty on the MPC performance has to be considered.Furthermore, as Petersen and Bundgaard in [27] pointed out, the effect of weather forecast uncertainty on the MPC performance depended on the climatic regions and test cases.Further research should be conducted to test the concept in other types of climates and buildings.Therefore, this paper presented a simulation-based study to apply an MPC controller for the SH system in a university building located in Trondheim, Norway.The detailed information of building properties and the measured data from the building energy platform were used to develop and validate the building model.An error model was introduced to improve the quality of the weather forecast data, and hence improve the performance of the MPC controller.Two metrics were used to evaluate the MPC performance: the heating cost saving and the achieved indoor air temperature.Shaving the peak load of heat users has great potential for contributing to the heating cost saving.According to a survey in [28], 87% of the existing heating price models in Sweden consider heat users' peak load, and the charging fee based on peak load accounts for 10-50% of the total heating bill.Many researchers have proven both by simulation and real-life experiments that integrating thermal energy storage (TES) is one way to shave the peak load and consequently reduce the heating cost in a heating system.Romanchenko et al. investigated the benefits of applying TESs (a hot water tank or the building thermal mass) in district heating (DH) systems to decrease peak load by developing a detailed techno-economic optimisation model for a DH system of Göteborg, Sweden.The results showed that TESs smoothed the heat load variations of the DH system and the annual heating cost was decreased by 1-2% compared to the scenario without any TES [29].Pedersen et al. conducted a simulationbased study to investigate how the economic MPC schemes for SH operation can utilize the building thermal mass as the TES in buildings to perform peak load shift and reduce heating cost.The simulation results showed that the proposed method shifted the energy use away from peak load periods and yielded increased heating cost savings (up to approximate 6%) [30].Weather-compensation controller mass as the TES in a DH system to optimize the operation of the energy system by coupling a detailed building simulation model with a linear optimization model of the energy system.The results showed that the economic savings in operational costs of the studied DH system were in the range of 0.7%-4.6%,not taking the cost of smart controls into account [31].Kensby et al. conducted a pilot test of utilizing the building thermal mass as the TES in DH systems to decrease the heat load daily variation by periodically overheating and underheating buildings.The results indicated that buildings could tolerate variations in heat deliveries while still maintaining a good indoor climate.In this way, the heat generation could be moved from peak load plants that run on fossil fuels to base load plants with better fuel economy and lower environmental impact [32].Therefore, the proposed MPC scheme in this study took into account the TES techniques as well.
The novelty of this study is summarized as the following.Firstly, this research introduced an accurate yet simple error model to handle the uncertainty of weather forecast in MPC.This error model only required easily measurable and accessible data, but could explore almost the full theoretical potential of the MPC without increasing any computational complexity.Secondly, a university building located in Norway was used as the case study to test the MPC performance by simulation, which is rarely addressed for this climatic region and building type by existing studies.Thirdly, real historical measured data were used to develop and validate the dynamic building model used in the MPC controller, which provides a good compromise between model accuracy and problem complexity.This study may facilitate the real application of MPC in buildings.
The remainder of this article is organised as follows.Section 2 presents the proposed MPC concept for SH system control in buildings, the modelling approach, the formulation method for MPC, the algorithm for solving nonlinear MPC (NMPC), and the method used for developing the error model.Section 3 introduces the case study, the weather forecast data, and the research scenarios.Section 4 explains the model validation, the evaluation of the error model and the simulation results.Section 5 shows a new set of simulation-based experiments and the corresponding results.Meanwhile, the limitation of this study is discussed.Finally, in Section 6, conclusions are summarized.

Method
This section first presents how the MPC concept is applied to the SH system control in buildings, and then the detailed information on building modelling, optimization formulation, optimization algorithm, and the error model for weather forecast are introduced.

Description of the MPC concept for space heating system control in buildings
Fig. 1 illustrates the fundamental components of an MPC controller for SH system control in buildings.The basic elements of the MPC are 1) a dynamic building model, 2) predictions of future disturbances, 3) an optimization algorithm, and 4) an objective function and constraints.At each time step, the predictions of the future disturbances (e.g., weather) together with the candidate MPC control inputs are used as simulation inputs to the dynamic building model.This simulation is carried out over the prediction horizon.The optimizer evaluates the objective function for each simulation run and adapts the candidate MPC inputs until an optimal solution is obtained.Only the first control inputs are supplied to the building.At the next time step, the MPC changes the initial states of the dynamic building model according to the actual states of the building, and restart the optimization [5].In this study, occu-pant behaviour was assumed to be perfectly predicted and the focus was on the consideration of uncertainty in weather predictions.
MPC may realize the optimal management of building energy including energy cost saving and thermal comfort improving, as the future disturbances, energy price, energy demand, and the dynamic building model are incorporated into the controller.In addition, energy storage strategies can be easily integrated into the MPC design.Energy storage strategies together with control strategies play an increasingly important role in shifting peak energy demand and reducing the energy cost in buildings.Some researchers have explored the potential for energy cost saving by integrating water tank or borehole (TES) into building-related systems [33,34].However, incorporating these (TESs) into buildings requires additional initial investment and maintenance cost.Buildings with large thermal capacity themselves may be utilized as energy storages that present opportunities for peak load shifting and energy cost saving.In this study, the passive thermal mass storage of the buildings was integrated into the MPC control strategy to realize the optimal management of building energy, energy peak load shaving, and energy cost saving.

Dynamic building model
MPC inherently requires an appropriate dynamic building model, which is then used for the computation of the optimal control inputs.This model must be sufficiently precise to yield valid predictions of building thermal behaviour, but at the same time, the model must be as simple as possible for the optimization task to be computationally tractable and numerically stable [35].Amidst several models, the simplified resistance-capacitance (RC) model has proven to provide more robust and accurate estimates of the building thermal behaviour [36,37].Therefore, a simplified RC model was developed by Modelica language in this study to grasp the key characteristics of buildings.The components for this simplified building model and the heat fluxes exchanged among them are presented in Fig. 2.
The individual model parts are the outdoor environment, building envelopes, indoor air, and internal thermal mass.Choosing a suitable order for the building RC model has been discussed in a significant amount of literature, especially in the literature considering the identifiability of parameters when the RC model is a greybox model [36,38,39].When the RC model is a grey-box model, the first and second-order RC models are preferred in many studies due to the feasibility of parameter identification.However, these first and second orders of RC models generally lump the entire thermal mass of a building to a single capacity or only make the distinction between the fast dynamics of indoor air and the slow dynamics of the structural mass [39].The distinction between the exterior thermal mass (envelopes including exterior walls and roof et al. exposed to outdoor environment) and the interior thermal mass (e.g.interior walls, furniture et al. not exposed to outdoor environment) is not made.In this study, the RC model structure with particular attention to the building envelopes' thermal mass and the interior thermal mass was proposed, because their respective thermal inertia is necessary for the short-term heat storage in the intended MPC [11,40].Therefore, a thirdorder thermal RC model that had heat capacitances for the building envelopes, interior thermal mass and indoor air was formulated in this study as shown in Fig. 2. The values of the heat resistance and the heat capacitance may be obtained by two common methods: 1) direct calculation based on the detailed information of the building, or 2) system identification technique [41].Corresponding, the RC models can be classified as white-box models or grey-box models.The white-box RC models applied in many studies have shown satisfactory prediction performance when the detailed information of the building are known [40,42,43].In this study, the proposed RC model was a white-box model, because the values of the heat resistance and the heat capacitance were determined by direct calculation method based on the known information of the case building and the reference values in the related standard [44].Furthermore, solar radiation was not considered in this study due to the low solar radiation in the studied area during the studied period [45].The energy balances for the building envelopes, the indoor air, and the internal thermal mass given in Fig. 2 are illustrated as the following Equations (1-3): where C and R represent the heat capacitance and resistance, which were obtained by the direct calculation method in this study.T is the temperature.Subscripts env, ia, oa, ma, win, and ven denote building envelopes, indoor air, outdoor air, internal thermal mass, window, and ventilation (including infiltration and mechanical ventilation), respectively.In addition, R i;e is the heat resistance between the indoor air and the building envelopes, R o;e is the heat resistance between the outdoor air and the building envelopes, and R i;m is the heat resistance between indoor air and interior thermal mass._ Q in is the internal heat gains, which was obtained based on historical measurement data and related standards in this study [46]._ Q ven is the heat flow rate from mechanical ventilation system, and it is described as Equation ( 4) according to Norwegian standard [46]._ Q ra is the heat flow rate from the radiator system.All the introduced heat capacitances, thermal resistances, temperatures, and heat flow rates in Equations (1-3) are marked in Fig. 2.
A discrete-element radiator model was employed in this study to achieve a more precise dynamical description of the radiator, as shown in Fig. 3.This radiator model is based on the European Standard EN 442-2 [47], which refers to this radiator model as the characteristic equation.Furthermore, previous studies have confirmed this discrete form of the radiator model, in which the radiator is divided into multiple parts while the indoor air is treated as uniform [48][49][50][51][52].This discrete radiator model is also used by the Modelica IBPSA library [53], which is co-developed by research teams from RWTH Aachen University and the Lawrence Berkeley National Laboratory, etc. Finally, this discrete radiator model is validated using measured data in the study [54], and the results suggest that the discrete radiator model is capable of accurately describing the dynamics of radiators.Therefore, _ Q ra is determined by Equations (5-8).Finally, _ Q is the total heat flow rate supplied from the building heat substation and presented as Equation ( 9).
where T ref ia is the reference value of indoor temperature._ V air is the mechanical ventilation volume flow rate, which was obtained from measurement data and related standards in this study [46].g T is the temperature efficiency of heat recovery in the mechanical ventilation system._ q n is the heat flow rate in each radiator section n.T n denotes the average water temperature in the radiator section n, as shown in Fig. 3. F n is the surface area of radiator section n.K n denotes the equivalent heat transfer coefficient which depends on the temperature difference between the radiator water and the ambient temperature, as described in Equation (7). a is a characteristic coefficient of the radiator, which can be gained from radiator manufacturers and was 0.28 in this study [55].c ra represents the heat capacity of the radiator material for each section n. c w and _ m ra denote the specific heat capacity and the flow rate of water, respectively.With the current definition of K n , it is not constant and makes the energy balance equation of the radiator nonlinear.Therefore, the dynamic building model in this study was nonlinear, and the nonlinearity was mainly derived from the radiator as shown in Equation (7).

Optimization formulation
This study aimed to reduce energy cost and maintain thermal comfort in buildings by utilizing the MPC.In general, the heating price models include four components, including fixed component (FXC), flow demand component (FDC), energy demand component (EDC), and load demand component (LDC).The FXC represents the cost that a user needs to pay for staying in the DH network.The FDC is a cost charged based on the volume of hot water needed to deliver the heat to a user.The EDC is used to cover the production cost of DH companies and charged based on users' heat use.The LDC is used to cover DH companies' cost to maintain a certain level of capacity for peak heat rate, investment costs of new facilities, depreciation, etc., and it is charged based on the peak load of users.However, the FXC and the FDC usually account for a very small share in existing models.In contrast, the EDC and the LDC together account for up to 96% of the total heating cost [28,56].Therefore, only the EDC and the LDC were considered for the heating price model in this study, and a multi-objective optimization problem including two conflicting optimization goals is formulated as Equations (10)(11)(12)(13)(14). Minimize: subject to: where H is the predictive horizon, which was 12 h in this study.
EP t ð Þ and _ Q t ð Þ are the EDC heating price and the heat rate at time t, respectively.LP is the LDC heating price, and _ Q p is the peak heat rate which is a free parameter needed to be optimized.In this study, the EDC heating price fluctuated around 0.6 NOK1 /kWh, and the LDC heating price was 47 NOK/kW/month, which were obtained from the local DH company, Statkraft Varme, in Trondheim [57].The first and the second terms in Equation ( 10) are the heating cost including EDC and LDC.In addition, the thermal comfort target was integrated into the objective function as a quadratic form to track the reference values of indoor air temperatures and avoid high indoor temperature violations, as shown with the third term in Equation (10).The quadratic form, also known as the Sum of Squared Errors, is commonly used in the trajectory-tracking target of an objective function [4,[58][59][60][61].Similar to the energy price in the heating cost term of the objective function, the thermal comfort term uses W to represent the ''price" that occupants are willing to pay for their thermal comfort [4].The value of W in this study was determined by the following criteria: when the deviation between indoor temperature and its reference value was 0.5 °C, the thermal comfort term and the heating cost term in the objective function contributed equally.However, it can be adjusted to other values based on the system operators' preferences.T ia t ð Þ and T ref ia t ð Þ are the actual indoor temperature and its reference value at time t.Equation ( 11) is the initial states of the dynamic building system, and the vector A is the values of these initial states.Equation ( 12) is the nonlinear dynamic building model as described in Section 2.2.The system state vector x t ð Þ was defined as T .The control input u t ð Þ was defined as are the supply water temperature and the water flow rate of SH system at time t.u min , and u max are the lower and the upper limits for the control variables.

Optimization algorithm
As described in Section 2.2 and Section 2.3, the optimization problem in this study was an NMPC optimization problem.However, in practice, the solution to such NMPC dynamic control problems can be challenging due to the nonlinearity of the dynamic model.In this study, we proposed the use of a direct method to formulate a standard nonlinear programming (NLP) problem.The basic idea of the direct method is to transcribe the original infinite-dimensional problem into a finite-dimensional NLP, which can be solved by using an NLP solver.The direct methods can be categorized into sequential and simultaneous approaches.The sequential approach, also known as control parametrization, only discretizes the control variables.The simultaneous approach, however, discretizes the state and control variables simultaneously [62].Direct collocation is a simultaneous approach and was used in this study.
A time grid t 0 < t 1 < Á Á Á < t N was generated over the predictive horizon t 0 ; t 0 þ H ½ by dividing the period into N intervals with a constant time step equal to the sample time.The continuous states x t ð Þ were discretized, and the discrete states on the time grid points t k was denoted as s k .Meanwhile, the control variables u t ð Þ was parameterized on the same time grid typically as piecewise constant, with control parameters q k , which yielded on each interval Therefore, the states of the dynamic system as described in Equation ( 12) were discretized, and the collocation-based integration of the state dynamic on a time interval t k ; t kþ1 ½ starting from the initial value s k hinged on solving the collocation Equation (15).The above discretization processes are illustrated in Fig. 4.
In addition to solving the collocation Equations ( 15) for k ¼ 0; 1; Á Á Á N À 1, continuity across the interval boundaries was required, i.e. the lengths of red solid lines presented in Fig. 4 should be zero.Therefore, Equation ( 16) was required and held for The direct collocation method yielded an NLP, and it can typically be written in the following general form as Equations (17)(18)(19)(20)(21). Minimize: subject to: where k ¼ 0; 1; Á Á Á N À 1. Equation ( 17) is the general form of Equation (10) after discretization by using the direct collocation method.The first rectangular quadrature term is the approximation of integration terms in Equation (10), and the second term is the parameter to be optimized in Equation (10).In addition, Equation ( 18) is the system initial states.Equations ( 19) and ( 20) are the collocation conditions and the continuity conditions as described above.Equa-tion ( 21) is the path constraints and the general form of Equations ( 13) and ( 14) after discretization.Finally, the above NLP is solved by using the NLP solvers.There are many methods available for solving the NLP.The most common methods are based on either active-set sequential quadratic programming or interior-point methods.The active-set sequential quadratic programming is to iteratively approximate the NLP by a quadratic program, and the interior-point method can be viewed as approximating the NLP by an equality-constrained NLP [63].In this study, the NLP problem got rid of the inequality constraints by using the interior-point method firstly, and then a local optimum to the NLP was found by solving the first order Karush-Kuhn-Tucker condition, using the iterative techniques based on Newton's method.The above optimization algorithm was illustrated by the flow chart in Fig. 5, and the open-source software JModelica.orgwas used as the optimization platform to achieve the above process in this study.

Error model for weather forecast
One challenge with the direct use of the weather forecast data in the MPC is the uncertainty of forecasted data resulted from the uncertainty of the numerical weather prediction model (NWPM).Inherent uncertainty lies in the NWPM due to the stochastic nature of atmospheric processes, the imperfect knowledge of the weather model's initial conditions, as well as modelling errors [17].Therefore, the exact actual weather data may not be reflected by the forecasted data well.The forecast weather data in this study considered only the outdoor temperature due to the low solar radiation in the studied area during the studied period [45].The actual outdoor temperature acting on the building can be decomposed as in Equation ( 22): where T k and T k denote the actual and the NWPM forecasted outdoor temperature at the time step k, and e k is the prediction error of NWPM at the time step k.
To improve the prediction of future disturbance acting on the building, the prediction error of the NWPM, e k , may be estimated by an error model.The prediction error of the NWPM means the deviation between the actual outdoor temperature and the corresponding NWPM forecasted data.In practice, the current prediction error is known, because of the availability of the measured data.According to [20,64], this study assumed that the unknown future prediction error of the NWPM was correlated to the known current prediction error, however, the correlation decreased along with the increasing time distance.For example, the NWPM predic-Fig.4. Illustration of direct collocation for approximating the dynamic system with.d¼ 2 tion error in the near future was approximated by the current prediction error, while the NWPM prediction error in the distant future had a limited relationship with the current prediction error and hence the forecasted data from the NWPM was the best estimation for the outdoor temperature.The error model is illustrated as Equation ( 23).
where e kþt is the estimated prediction error of the NWPM in the future, e k is the known prediction error of the NWPM at the current time step k.Parameter 0 r t ð Þ 1 which is a weighting function, describes the decreasing predictive effect of the current prediction error to the future NWPM prediction error along with the increasing time distance.In this study, the value of parameter r t ð Þ was given by Equation (24) based on [64].Therefore, the future outdoor temperature was estimated as Equation (25).
where T kþt is the estimated outdoor temperature in the future, T kþt is the forecasted outdoor temperature from NWPM, and e kþt is the estimated prediction error of NWPM given by the error model as illustrated by Equation ( 23).
Fig. 6 illustrates an example that how the future outdoor temperature is estimated.The outdoor temperature is estimated based on the forecasted data from the NWPM and the estimated prediction error of the NWPM given by the error model.The current prediction error e k is calculated based on the NWPM forecasted and the corresponding actual outdoor temperature at the current time step k, as shown in Fig. 6.The NWPM prediction error for the near future highly relies on and is approximated by the current prediction error, as shown in Fig. 6 for the time step k + t when t has small values.However, the correlation is decreasing along with the increasing time distance, as shown in Fig. 6 for the time step k + t when the value of t is increasing.Finally, the correlation disappears for the distant future and the estimated future outdoor temperature is only determined by the NWPM forecasted data, as shown in Fig. 6 for the time step k + t when t is 12.

Case study and scenarios
The proposed method in Section 2 was tested by a university building.The background of the case study, the weather forecast data from the NWPM, and the research scenarios are introduced in this section.

Background of the case study
A university building located in Trondheim, Norway, was chosen as the case study as shown in Fig. 7 a).Built in 1962, it has six floors with a floor area of 15 026 m 2 .The main functions of this building are education, offices, and laboratory [65].Fig. 7 b) presents the SH system in this building and how an MPC controller is used as a supervisory control for the SH system.As presented in Fig. 7 b), the SH system in this building is connected to a DH system via a heat exchanger in the building heat substation.The SH system consists of a radiator system and a mechanical ventilation system.The radiator system is responsible for compensating the heat transmission to the environment through building envelopes and heating the incoming cold air caused by air infiltration.The mechanical ventilation system, which consists of several airhandling units (AHUs), is responsible for heating the incoming cold air caused by mechanical ventilation and supplying the heated and fresh air for occupants in the building.The MPC controller used as a supervisory control in this study aimed to optimize the set-points for supply water temperature and water flow rate of the SH system.
The key information about the building is listed in Table 1.The values of the heat resistance and the heat capacitance for the RC  model proposed in Section 2.2 were obtained by the direct calculation method.Firstly, according to the information on building thermal properties, the heat resistance and capacitance of the individual exterior thermal mass element, including the exterior walls and roof et al. exposed to the outdoor environment, were calculated using the method provided in European standard EN ISO 6946 [66].Afterwards, the heat resistances and the heat capacitances of all the building exterior thermal mass elements were lumped into one equivalent heat resistance and one equivalent heat capacitance, respectively.These equivalent heat resistance and capacitance represented the overall thermal performance of all the exterior thermal mass elements, and they were obtained by equations ( 26) and (27), respectively [40].Specifically, an assumption was made when calculating the heat capacitance of the window-its thermal mass was set as zero because it could be ignorable compared to the other exterior thermal mass elements.
where R e and C e are the equivalent heat resistance and capacitance for the overall exterior or interior thermal mass.R i and C i are the heat resistance and capacitance of exterior or interior thermal mass element i.
The same method was used to calculate the heat resistance and capacitance of the individual interior thermal mass element.However, in the first step, calculating the heat resistance of the individual element, no thermal conduction through interior thermal mass element was assumed, therefore, only the surface resistance between the indoor air and the interior thermal mass element was considered.The calculation results for the values of the heat resistance and the heat capacitance are shown in Table A.1.

Weather forecast data from the numerical weather prediction model
This study was performed using the weather forecast data and corresponding actual weather data for January of 2018 for the university building (coordinates: 63.4 o N 10.4 o E, elevation 60 m).The weather forecast data were given by the archived forecasts of the NWPM, MetCoOp Ensemble Prediction System (MEPS), which is cooperated by the meteorological services of Norway, Sweden, and Finland.The location of the case building was used as the input for the NWPM to download the corresponding historical weather forecast data.MEPS delivers hourly predictions for the next 66 h with an update cycle of 12 h [67].In addition, according to the review paper [7], the typical prediction horizon of MPC in HVAC systems that feature slow-moving processes lies in the range from 5 to 48 h.In this study, the prediction horizon of the MPC was chosen as 12 h.Therefore, at each time step, the MPC controller used the latest available 12 h' weather forecast data.In addition, for Fig. 8 and Fig. 9, the actual outdoor temperature was always compared to the available most recent forecast data with an update cycle of 12 h.Fig. 8 shows the comparison between the MEPS forecasted and the corresponding actual outdoor temperature.The forecasted data predicted the trend of the outdoor temperature well.However, the exact value of the actual outdoor temperature was not reflected by the forecasted data very well.As presented in Fig. 8, the deviation between them could be even up to almost 10 K.
Fig. 9 illustrates the distribution of deviation between the forecasted and the actual outdoor temperature for January of 2018.The temperature deviation had the mean value of À1.2 K and the standard deviation of 2.5 K. On average, the forecasted data were close to the actual outdoor temperature.However, the standard deviation indicated the presence of a considerable amount of overestimated and underestimated values.The distribution graph in Fig. 9 shows that the outdoor temperature was mainly underestimated by the MEPS forecasted data for the studied period.

Suggested scenarios to include the MPC controller
This research is a simulation-based study to investigate the impact of weather forecast uncertainty on the performance of MPC.Moreover, to improve the performance of the MPC controller, an error model was introduced to address the error of the weather forecast.To test the effectiveness of the MPC controller integrated with the error model, four research scenarios including a reference scenario, two benchmark MPC scenarios, and one improved MPC scenario integrated with the error model were proposed in this research.The reference scenario represented the current RBC strategy for the SH system without using any MPC controller.The other scenarios represented the MPC strategies with different weather information provided for their MPC controllers.The ideal benchmark MPC scenario assumed perfect weather forecasts, i.e. providing the actual weather data for its MPC controller.The standard benchmark MPC scenario did not address the weather forecast error, i.e. directly providing the forecasted weather data for its MPC controller.Finally, the improved MPC scenario handles the weather forecast error by integrating with the error model, i.e. providing the estimated weather data from the error model for its MPC controller.Due to the different weather information provided for different MPC controllers, the SH system of the building in individual MPC scenarios received different optimal control signals.Afterwards, the same building and system model disturbed by actual weather conditions was simulated to obtain the indoor air temperature and the heating cost based on the different control signals, respectively.
The research was conducted through the following four steps.Firstly, the building model was developed based on the method proposed in Section 2.2 and validated as presented in Section 4.1.Secondly, the MPC framework was formulated as described in Sec-Fig.8.The comparison between forecasted and corresponding actual outdoor temperature.
Fig. 9. Distribution of deviation between forecasted and actual outdoor temperature (deviation equals forecasted value minus actual value) for January of 2018.

Table 1
The key information about the building.

Category
Parameter The proposed four scenarios were presented in Fig. 10.The reference scenario, RBC in Fig. 10 a), presented the current RBC strategy for the SH system and was therefore used as a benchmark.In this scenario, a WCC was used to determine the supply water temperature of the SH system according to the actual outdoor temperature.A P controller, which attempted to perform better than the on-off controller, was applied to adjust the water flow rate of the SH system based on the deviation between indoor temperature and its reference value.In this scenario, the optimal management of the building energy is not possible, as the future disturbances, energy price, energy demand, and the dynamic building model cannot be incorporated into controllers.
The ideal MPC scenario, MPC actu in Fig. 10 b) was an ideal MPC control strategy, which was defined as optimal control with the perfect weather prediction, i.e. assuming no deviation between the forecasted weather and the actual weather.As shown in Fig. 10 b), the actual outdoor temperature was used by the MPC controller to present the perfect weather prediction.This is not an implementable controller but a concept, which was used to investigate the theoretical potential of MPC.
The standard MPC scenario, MPC fore in Fig. 10 c), was a standard MPC control strategy in practice.It used the imperfect weather forecast of the NWPM but determined its control actions under the assumption that the predictions were correct.As presented in Fig. 10 c), the forecasted outdoor temperature from the MEPS was used by the MPC controller.This scenario demonstrated the practical potential of MPC.
The improved MPC scenario, MPC esti in Fig. 10 d), was proposed considering the uncertainty of weather predictions in the MPC control strategy.As shown in Fig. 10 d), the error model described in Section 2.5 was integrated to improve the quality of forecasted outdoor temperature from the MEPS, and then the high-quality estimated outdoor temperature was used by the MPC controller.This scenario made the handling of the weather forecast uncertainty straightforward, meanwhile maintained the computationally tractable of MPC.

Results
This section firstly presents the model validation and the evaluation of the error model for weather forecast, and then evaluates the three MPC scenarios during a typical week in terms of achieved indoor air temperature and heating cost.

Model validation
The dynamic building model proposed in Section 2.2 was validated by one month's measured data from the campus energy management platform.For the validation procedure, the simulated control strategy for the SH system was based on the current RBC strategy.As described in Section 3.2, a WCC was used to determine the supply water temperature of the SH system according to the actual outdoor air temperature, and a P controller was used to adjust the water flow rate based on the deviation between indoor air temperature and its reference value.The reference values for the indoor temperature were set as 21℃ from 8:00 am to 10:00 pm, and 19℃ from 10:00 pm to 8:00 am based on the measured data and adjusted by the requirement of the Norwegian standard [46].The mechanical ventilation and the internal heat gains listed in Table 1 were set as inputs for the dynamic building model.Afterwards, with a reasonable initial guess for the state variables and the control strategy described above, the dynamic building model was simulated in the Dymola environment to track the reference values of the indoor temperature.The simulated hourly heat rate was validated against the measured hourly heat rate.
The validation results are presented in Fig. 11.The simulated and measured hourly heat rate of the building exhibited the same trend.During working hours, the hourly heat rate was higher due to the higher airflow rate in the AHUs of the mechanical ventilation system.During non-working hours or weekends, fewer occupants led to the lower airflow rate in the AHUs of the mechanical ventilation system and hence the lower hourly heat flow rate.In summary, the simulated results were able to capture the key characteristics of building heat flow rate.
To quantify the deviation of the simulated data from the measured data, two indicators, i.e. coefficient of variation of the root mean square error (CV(RMSE)) and normalized mean bias error (NMBE), were used to evaluate the prediction performance of building model according to ASHRAE Guideline 14-2014 [68].CV (RMSE) characterizes the variability of the errors between the measured and the simulated values.NMBE quantifies the percentage error between the measured and the simulated values.The validation criteria required in ASHRAE Guideline 14-2014 is within AE 30% for CV(RMSE) and within AE 10% for NMBE when using hourly data [68].In this study, the resulted values of the two indicators were 16.9% for CV(RMSE) and 0.4% for NMBE, respectively, when comparing the hourly simulated and measured heat flow rate.These indicators showed that the dynamic building model developed in this study was able to predict the thermal behaviour of building well and be used for MPC control strategy.

Evaluation of error model for weather forecast
The three MPC scenarios together with the reference scenario, RBC, were tested during a typical week (from 17th to 23th of January of 2018).As shown in Fig. 12, the deviations between the MEPS forecasted and the actual outdoor temperature were large and most of the forecasted values were lower than the actual values.The mean value of these deviations was À2.7 K, and the maximum deviation was À9.2 K.
To improve the quality of MEPS forecasted outdoor temperature as shown in Fig. 12, the error model proposed in Section 2.5 was used to estimate the prediction error of MEPS, and then the estimated MEPS prediction error together with the MEPS forecasted outdoor temperature were combined as described in Equation ( 25) to generate a sequence of estimated outdoor temperature.According to [18,26], the accuracy of these estimated outdoor temperatures was quantified using a standard metric, root mean square error (RMSE).RMSE represents the sample standard deviation of the differences between predicted values and actual values, as described in Equation ( 28) [68].
where n is the number of observations of data points, y i is the actual value, and b y i is the predicted value.Fig. 13 presents the RMSEs of each hour in the predictive horizon, in which the black line represents the RMSE of the MEPS forecasted outdoor temperature, and the orange line represents the RMSE of the estimated outdoor temperature.As mentioned in Section 2, the predictive horizon was 12 h in this study, and hence the X-axis of Fig. 13 ranged from 1 to 12.In addition, the studied typical week had 168 h, and therefore the RMSEs of each hour in the predictive horizon was calculated based on 168 pairs of data including actual and predicted values.As described in Section 2.5, one challenge with the direct use of the weather forecast data from NWPM is the inherent prediction error of the NWPM.Therefore, the RMSEs of the forecasted outdoor temperature direct from MEPS were larger and fluctuated around 4.2 K for each hour in the predictive horizon.In contrast, after introducing the error model for MEPS forecasted outdoor temperature, the accuracy of the outdoor temperature was increased.As shown in Fig. 13, the estimated outdoor temperatures had higher accuracy with the lower RMSEs from 0.5 to 4.2 K, especially for the near future as presented on the left side of Fig. 13.

Achieved indoor air temperature
Fig. 14 presents the indoor air temperature and its reference values for different scenarios.The reference values for the indoor temperature were set as 21.0℃ from 8:00 am to 10:00 pm, and 19.0℃ from 10:00 pm to 8:00 am for all the scenarios according to Norwegian standard [46], as depicted by the red solid line in Fig. 14.The ideal MPC scenario, MPC actu, demonstrated the theoretical control performance of the MPC for maintaining the indoor temperature as expected.This ideal MPC scenario eliminated the over-heating phenomenon occurring in the reference scenario RBC, and reduced the deviations between indoor temperature and its reference values, as shown in Fig. 14.In addition, the violation numbers of the indoor temperature are presented in Fig. 15.The indoor temperature was recorded every ten minutes, and the occurrence of violation was counted when the deviation between the indoor temperature and its reference value was larger than 0.5 K.The ideal MPC scenario, MPC actu , decreased the violation numbers from 430 to 152, with the reduction of almost 65%.Moreover, the ideal MPC scenario considered the peak load shaving target involved in its objective function, and hence slightly lowered the indoor temperature during the peak hours and slightly higher the indoor temperature during the non-peak hours, as shown in Fig. 14.
The standard MPC scenario, MPC fore, which presents the practical indoor temperature control performance of MPC, is depicted in Fig. 14 by the green solid line.The MEPS forecasted outdoor temperature was directly used by the MPC controller in this scenario.Due to receiving inaccurate outdoor temperature, this scenario did not perform well, especially when the MEPS forecasted weather largely varied from the actual one.The over-heating phenomenon in this scenario was even worse than with the reference scenario RBC, as shown in Fig. 14 .In addition, compared to the reference scenario, RBC, the deviations between the indoor temperature and its reference values were increased and the violation numbers of the indoor temperature were even higher, an increase of 20%, as shown in Fig. 15.Due to receiving the low-quality predictions for the future weather, this standard MPC scenario presented even worse control performance than the conventional RBC in terms of the indoor temperature.
In contrast, the improved MPC scenario, MPC esti, was able to guarantee thermal comfort by introducing the error model.As presented in Fig. 13, the error model together with the MEPS forecasted outdoor temperature generated the high-quality estimated future outdoor temperature, and then the estimated outdoor temperature was used by the MPC controller in this scenario.Due to receiving the high-quality weather data, this scenario achieved almost the same indoor temperature control performance as the ideal MPC scenario, as depicted in Fig. 14 by the orange solid line.The over-heating phenomenon was removed and the deviations between the indoor temperature and its reference values were small in this scenario.The peak load shaving target was also considered as the indoor temperature was controlled to be slightly lower and higher than its reference values during the peak and the non-peak hours, respectively.Moreover, as presented in Fig. 15, the violation numbers of the indoor temperature dropped by almost 80 % compared to the standard MPC scenario (MPC fore) and by 73% compared to the reference scenario RBC.

Heating cost
As described in Section 2.3, the heating cost was charged based on the peak heat rate and the heat use of end-users.Therefore, the peak heat rate and the heat use for the different scenarios will be discussed firstly in this section.
Fig. 16 presents the peak heat rate for different scenarios.As described in Section 2.1, the passive thermal mass storage of the building was integrated into the MPC control strategy to shave the peak load in the building.Therefore, the ideal MPC scenario, MPC actu, presented a remarkable peak load shaving effect and decreased the peak load from 549 kW to 506 kW, a reduction of 7.8% compared to the reference scenario RBC.However, due to receiving inaccurate weather data, the peak load shaving performance of the MPC deteriorated.As shown in Fig. 16, the standard MPC scenario, MPC fore, shaved the peak load to 529 kW, with the reduction of only 3.6% compared to the reference scenario RBC.In contrast, introducing the error model was able to improve the peak load shaving performance of MPC a bit.As the improved MPC scenario (MPC esti) showed, the peak load was decreased by 4.6%, which was higher than the standard MPC scenario (MPC foreÞ.was able to be achieved by this ideal MPC scenario.However, the low-quality predictions of weather degraded the heat use saving performance of the MPC.As the standard MPC scenario (MPC foreÞ showed, its heat use was almost the same as the reference scenario RBC and no obvious heat use saving was observed.In this scenario, its MPC controller received lower forecasted outdoor temperature, as presented in Fig. 12, which led to incorrect control actions of MPC.The MPC controller provided more heat than the building demand and hence resulted in the over-heating phenomenon and heat waste.Introducing the error model was able to improve the quality of weather information and significantly improve the MPC performance in terms of heat use saving.As shown in Fig. 17, the improved MPC scenario MPC esti ð ) reduced the heat use from 54.9 MWh to 53.3 MWh, a reduction of 3.0%, which was very close to the theoretical heat use saving potential of the MPC.
Finally, Fig. 18 presents the heating cost for different scenarios.The theoretical heating cost saving was 4.1%, as the ideal MPC sce-nario MPC actu showed.This cost saving came from the reduction in both the LDC and the EDC of the heating price model, which was caused by the peak load shaving and the heat use saving effects of the MPC controller.The standard MPC scenario, MPC fore, saved only 0.7% of heating cost compared to the reference scenario RBC.The heating cost saving performance of the MPC was degraded a lot due to the low-quality weather forecast information.However, introducing the error model for the weather forecast brought remarkable improvement in terms of heat use saving for the MPC controller, as described above.Therefore, about 3.4% of the heating cost was saved by the improved MPC scenario MPC esti.

Discussion
Sections 4.2-4.4present the simulation results from a week that had a big deviation between the MEPS forecasted and the actual outdoor air temperature.This section, however, investigates the situation with a small deviation between the forecasted and the actual outdoor air temperature.Therefore, the impacts of the low error weather forecast on the performance of MPC controllers were analysed, moreover, the effectiveness of the error model is further discussed.Following that, this section discusses the limitation of this study.
The three MPC scenarios together with the reference scenario were tested during another week (from 24th to 30th of January of 2018).As presented in Fig. 19, the deviations between the MEPS forecasted and the actual outdoor temperature were small.Compared to the tested week presented in Section 4.2 with a mean and a maximum deviation value of À2.7 K and À9.2 K, respectively, this week had a much smaller deviation range with a mean and a maximum deviation value of 0.8 K and 4.7 K.As the steps described in Section 3.3, firstly, the MPC controllers of individual MPC scenarios were provided with different weather information, i.e. the   actual outdoor temperature for Scenario MPC actu, the forecasted outdoor temperature from MEPS for Scenario MPC fore, and the estimated outdoor temperature from the error model for Scenario MPC esti.Afterwards, these MPC controllers generated control signals based on the provided weather information and the feedback of the building.Finally, the SH system received and executed these control signals, and the processes were simulated under the condition of the actual outdoor temperature.The simulated results for the MPC scenarios and the reference scenario are presented in Fig. 20 and Fig. 21.Fig. 20 shows the violation numbers of the indoor air temperature for each research scenario during the week from 24th to 30th of January of 2018.In Fig. 20, smaller violation numbers mean better performance in terms of indoor temperature tracking.Different from the results achieved in Section 4.3 that the MPC scenario with the forecasted outdoor temperature, MPC fore, obtained the worest perfromance on indoor temperature tracking, the results in Fig. 20 showed that all the three MPC scenarios presented better performance than the reference scenario, RBC.However, similar with the results observed in Section 4.3, Fig. 20 also demonstrates that introducing the error model could improve the performance of MPC controllers regarding the indoor temperature tracking, which can be found when comparing the results of Scenario MPC fore with Scenario MPC esti.The above findings indicated that the better weather forecast guaranteed better indoor temperature tracking performance for MPC controllers than direct use of the forecasted weather data.However, as the prediction error cannot be avoided, introducing the error model always benefited the performance of an MPC controller.
Fig. 21 presents the heating cost for the four research scenarios during the week from 24th to 30th of January of 2018.Different from the results obtained in Section 4.4 that the MPC scenario with the forecasted outdoor temperature, MPC fore, had no obvious heating cost savings, the results in Fig. 21 show that all the three MPC scenarios including the MPC fore scenario, reduced the heating cost of the SH system by around 6.5% compared to the reference scenario.This indicates that the small deviations between the actual and the MEPS forecasted outdoor temperature had no obvious impacts on the MPC performance in terms of heating cost saving.Therefore, some conclusions from this discussion together with the results presented in Section 4.3 and 4.4 may be summarized as follows.Firstly, the quality of weather information provided for the MPC controller had a big impact on the performance of MPC, which was identical with the previous studies, e.g.[17,18].Secondly, the introduction of an error model to improve the quality of weather information always benefited the MPC performance even when the deviations between the forecasted and measured weather data were small.Furthermore, there are several limitations of this study.Solar radiation was not considered in this study due to the low solar radiation during the studied period.In this study, there were no available on-site measured solar radiation values for the case building.Therefore, the measured global solar radiation from the nearest weather station, Skjetlein, were collected to infer the situation of solar radiation for the case building [69].Fig. 22 presents the measured average hourly global solar radiation from the weather station during the studied period.The studied period was January of 2018 and had 31 days, and therefore the average     As shown in Fig. 22, the average global solar radiation varied in the range of 10-55 W/m 2 from 10:00 am to 02:00 pm yet only last 4 h in a day and was zero for the rest of day.Furthermore, the nearest weather station is located in the rural area of the city where no shading effect for the weather station.During the studied period, the case building located in the city centre may receive even less solar radiation because of the effect of shading in high-density urban areas [70].Based on the above explanation, ignoring the solar radiation during the studied period for the case building is reasonable.However, solar gains are significant during the late spring and early autumn, resulting in a shortening of the heating season for almost all locations in Europe [70].Therefore, solar radiation should be considered when the study is conducted during the late spring and early autumn.
The RC model used in this study was a white-box RC model, and the parameters of this RC model were obtained based on the known detailed information of the case building, the on-site measured data, and related standards.However, for many buildings, detailed building information and complete on-site measured data are often unavailable.In this case, it is challenging to develop a white-box model based on the limited information of the building.Many researchers have investigated using system identification techniques to develop a suitable grey-box model when the building information is limited [38,39,71,72].Grey-box models have better generalization properties and usually require less building information compared to white-box models.Therefore, to have a more generalized research method, a grey-box RC model may be developed in future work to test the methods proposed in this study.

Conclusions
In this study, an error model was proposed to address the effect of weather forecast uncertainty on the MPC performance in buildings.The error model used easily measurable and accessible data to improve the quality of the forecasted weather data from the NWPM, and consequently improved the performance of MPC.The proposed method was tested on a university building located in Norway under the condition of the high error of weather forecast.The three MPC scenarios together with a conventional RBC scenario were evaluated in terms of thermal comfort and heating cost.
Compared to the conventional RBC, the ideal MPC controller with perfect weather data demonstrated the theoretical heating cost saving of 4.1% during one week, while providing an improved level of thermal comfort, with a reduction of 65% for the indoor temperature violations.However, the standard MPC controller with the NWPM forecasted weather data did not perform well, especially when the actual weather varied from the forecast.In that case, the heating cost saving was only 0.7% during this week.Meanwhile, the low-quality weather forecast information resulted in an even worse level of thermal comfort than that of the conventional RBC, an increase of 20% for the violation numbers of the indoor air temperature.In contrast, by introducing the error model, the quality of weather information used in MPC was improved and hence the performance of the MPC controller was significantly improved.The weekly heating cost saving was increased to 3.4%, which was very close to the ideal MPC controller heating cost saving of 4.1%.In addition, the violation numbers of indoor temperature were dropped by 80% and hence the thermal comfort was substantially improved.
To get more generalized conclusions, the proposed method was tested under the condition of the low error of weather forecast as well.Simulation results indicated that the introduction of the error model to improve the quality of weather information always benefited the MPC performance even when the deviations between the forecasted and measured weather data were small.
In summary, integrating a simple but accurate error model into an MPC controller is a practical and feasible approach to tackle the weather forecast uncertainty of MPC in buildings.This study can facilitate the real application of MPC in buildings.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 1 .
Fig.1.Model predictive control applied to space heating system control in buildings.

Fig. 5 .
Fig. 5.The flow chart of the optimization algorithm.

Fig. 6 .
Fig.6.The relationship between estimated outdoor temperature, NWPM forecasted data and prediction error.

Fig. 7 .
Fig. 7.The university building used as the case study.

Fig. 11 .
Fig. 11.Comparison between the simulated and measured heat flow rate.

Fig. 17
Fig.17presents the heat use for different scenarios.The ideal MPC scenario, MPC actu, demonstrated the theoretical heat use saving potential of the MPC.A theoretical heat use saving of 3.3% was able to be achieved by this ideal MPC scenario.However, the low-quality predictions of weather degraded the heat use saving performance of the MPC.As the standard MPC scenario (MPC foreÞ showed, its heat use was almost the same as the reference scenario RBC and no obvious heat use saving was observed.In this scenario, its MPC controller received lower forecasted outdoor temperature, as presented in Fig.12, which led to incorrect control actions of MPC.The MPC controller provided more heat than the building demand and hence resulted in the over-heating phenomenon and heat waste.Introducing the error model was able to improve the quality of weather information and significantly improve the MPC performance in terms of heat use saving.As shown in Fig.17, the improved MPC scenario MPC esti ð ) reduced the heat use from 54.9 MWh to 53.3 MWh, a reduction of 3.0%, which was very close to the theoretical heat use saving potential of the MPC.Finally, Fig.18presents the heating cost for different scenarios.The theoretical heating cost saving was 4.1%, as the ideal MPC sce-
The trajectory of each state on the time interval t k ; t kþ1 ½ was approximated by a polynomial p k t; v k addition, on each collocation interval t k ; t kþ1 ½ , a set of d collocation times t k;i 2 t k ; t kþ1 ½ was chosen, with i ¼ 0; 1; Á Á Á d.
2.3.Thirdly, four simulation-based experiments were created and simulated as introduced in Section 3.3, respectively, including a reference scenario and three MPC scenarios.Finally, the simulation results of different scenarios were investigated and compared in Section 4 and Section 5.