Distributed Model Predictive Control applied to a VAV based HVAC system based on Sensitivity Analysis IFAC PapersOnLine 51-20 (2018) 259–264 ScienceDirectScienceDirect Available online at www.sciencedirect.com 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2018.11.023 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. 10.1016/j.ifacol.2018.11.023 2405-8963 Distributed Model Predictive Control applied to a VAV based HVAC system based on Sensitivity Analysis Tejaswinee Darure ∗ Vicenç Puig ∗∗ Joseph Yamé ∗ Frédéric Hamelin ∗ Ye Wang ∗∗ ∗ Université de Lorraine, Centre de Recherche en Automatique de Nancy, CNRS, UMR 7039, Vandoeuvre les Nancy 54500, France ∗∗ Institut de Robtica i Informtica Industrial (CSIC-UPC), Llorens i Artigas 4-6, Scientic Park of Barcelona, 08028 Barcelona, Spain Abstract: This paper proposes a distributed model predictive control strategy for a HVAC system that relies on its decomposition into subsystems based on the sensitivity analysis. An economic model predictive controller is implemented for every subsystem to optimize the operational cost of the building without compromising the thermal comfort of the occupants. Also, this work demonstrates the coordination strategy between the subsystem controllers using the sensitivity analysis approach. A discussion about the coordination strategy with the convergence property is provided. Finally, the proposed approach is illustrated using the benchmark building that considers the weather of Nancy, France. Keywords: VAV type HVAC system, Distributed Mode Predictive Control, Energy Optimization 1. INTRODUCTION Since last three decades, the energy crisis has been cer- tainly one of the strong motivation in the changes of the HVAC industry towards more energy-efficient buildings without compromising the comfort. As energy require- ment and fuel consumption of heating, ventilation and air-conditioning (HVAC) systems have a direct impact on the operational cost of a building as well as an impact on the environment. For this reason, building energy man- agement has become an important issue in many countries (EU, 2016). More sophisticated technological schemes are now being developed and implemented in the buildings to reduce energy consumption, as e.g., thermal storage, building energy management systems (BEMS), advanced direct digital control, variable-air-volume (VAV) systems, variable frequency drives, etc. In order to improve the performance in HVAC various control techniques are de- veloped e.g. gain scheduling in PID controllers, optimal control, adaptive control, nonlinear control, neural and fuzzy control methods. In above all, model predictive con- trol (MPC) is favored control method due to its obvious advantages of handling constraints and disturbances. The detailed analysis of the available HVAC control techniques are summarized in Afram and Janabi-Sharifi (2014). In large non-residential and commercial buildings, the HVAC system must meet the varying needs of different spaces since different zones of a building may have dif- ferent heating and cooling needs. Due to these reasons, � This work is supported by Energy In Time project funded by the European Union within the 7th Framework Program FP7-NMP, Sub- program EeB.NMP.2013-4: Integrated control systems and method- ologies to monitor and improve building energy performance. distributed control strategies are becoming very popular (Lamoudi, 2012). These control strategies hold various advantages as multivariable interactions, scalability and isolation in case of occurrence of faults. The class of DMPC methods are based on the type of decomposition of a large scale system into subsystems and the type of coordination between subsystem controllers. In the survey by Afram and Janabi-Sharifi (2014) about DMPC methods applied to HVAC systems, various approaches are found in the literature in the context of the type of the decomposition of centralized MPC problem e.g. primal an dual decom- position as in P. Pflaum and M. Alamir (2014), Dantzing Wolfe and Bender’s decomposition as in Petru-Daniel Mo- rosan (2010). In spite of the developments in the DMPC methods, much work is still needed to be done regarding its application to the HVAC systems. In this work, we propose: i) a method to decompose the HVAC building system into subsystems based on the sen- sitivity analysis and ii) the coordination strategy between the subsystem controllers considering the sensitivities of the neighboring subsystem controllers and the coupling information. For the illustration of the proposed method, we consider a benchmark HVAC building system with VAV systems. We consider every zone is provided with a VAV box in which a damper manipulates the airflow of supply air with the constant temperature into the zones to maintain the ther- mal comfort. This supply air with constant temperature is provided by an Air Handling Unit (AHU). The complexity of the centralized control increases exponentially as the number of zones increases. Hence, we propose the DMPC approach to achieve the same performance as centralized 6th IFAC Conference on Nonlinear Model Predictive Control Madison, WI, USA, August 19-22, 2018 Copyright © 2018 IFAC 293 Distributed Model Predictive Control applied to a VAV based HVAC system based on Sensitivity Analysis Tejaswinee Darure ∗ Vicenç Puig ∗∗ Joseph Yamé ∗ Frédéric Hamelin ∗ Ye Wang ∗∗ ∗ Université de Lorraine, Centre de Recherche en Automatique de Nancy, CNRS, UMR 7039, Vandoeuvre les Nancy 54500, France ∗∗ Institut de Robtica i Informtica Industrial (CSIC-UPC), Llorens i Artigas 4-6, Scientic Park of Barcelona, 08028 Barcelona, Spain Abstract: This paper proposes a distributed model predictive control strategy for a HVAC system that relies on its decomposition into subsystems based on the sensitivity analysis. An economic model predictive controller is implemented for every subsystem to optimize the operational cost of the building without compromising the thermal comfort of the occupants. Also, this work demonstrates the coordination strategy between the subsystem controllers using the sensitivity analysis approach. A discussion about the coordination strategy with the convergence property is provided. Finally, the proposed approach is illustrated using the benchmark building that considers the weather of Nancy, France. Keywords: VAV type HVAC system, Distributed Mode Predictive Control, Energy Optimization 1. INTRODUCTION Since last three decades, the energy crisis has been cer- tainly one of the strong motivation in the changes of the HVAC industry towards more energy-efficient buildings without compromising the comfort. As energy require- ment and fuel consumption of heating, ventilation and air-conditioning (HVAC) systems have a direct impact on the operational cost of a building as well as an impact on the environment. For this reason, building energy man- agement has become an important issue in many countries (EU, 2016). More sophisticated technological schemes are now being developed and implemented in the buildings to reduce energy consumption, as e.g., thermal storage, building energy management systems (BEMS), advanced direct digital control, variable-air-volume (VAV) systems, variable frequency drives, etc. In order to improve the performance in HVAC various control techniques are de- veloped e.g. gain scheduling in PID controllers, optimal control, adaptive control, nonlinear control, neural and fuzzy control methods. In above all, model predictive con- trol (MPC) is favored control method due to its obvious advantages of handling constraints and disturbances. The detailed analysis of the available HVAC control techniques are summarized in Afram and Janabi-Sharifi (2014). In large non-residential and commercial buildings, the HVAC system must meet the varying needs of different spaces since different zones of a building may have dif- ferent heating and cooling needs. Due to these reasons, � This work is supported by Energy In Time project funded by the European Union within the 7th Framework Program FP7-NMP, Sub- program EeB.NMP.2013-4: Integrated control systems and method- ologies to monitor and improve building energy performance. distributed control strategies are becoming very popular (Lamoudi, 2012). These control strategies hold various advantages as multivariable interactions, scalability and isolation in case of occurrence of faults. The class of DMPC methods are based on the type of decomposition of a large scale system into subsystems and the type of coordination between subsystem controllers. In the survey by Afram and Janabi-Sharifi (2014) about DMPC methods applied to HVAC systems, various approaches are found in the literature in the context of the type of the decomposition of centralized MPC problem e.g. primal an dual decom- position as in P. Pflaum and M. Alamir (2014), Dantzing Wolfe and Bender’s decomposition as in Petru-Daniel Mo- rosan (2010). In spite of the developments in the DMPC methods, much work is still needed to be done regarding its application to the HVAC systems. In this work, we propose: i) a method to decompose the HVAC building system into subsystems based on the sen- sitivity analysis and ii) the coordination strategy between the subsystem controllers considering the sensitivities of the neighboring subsystem controllers and the coupling information. For the illustration of the proposed method, we consider a benchmark HVAC building system with VAV systems. We consider every zone is provided with a VAV box in which a damper manipulates the airflow of supply air with the constant temperature into the zones to maintain the ther- mal comfort. This supply air with constant temperature is provided by an Air Handling Unit (AHU). The complexity of the centralized control increases exponentially as the number of zones increases. Hence, we propose the DMPC approach to achieve the same performance as centralized 6th IFAC Conference on Nonlinear Model Predictive Control Madison, WI, USA, August 19-22, 2018 Copyright © 2018 IFAC 293 Distributed Model Predictive Control applied to a VAV based HVAC system based on Sensitivity Analysis Tejaswinee Darure ∗ Vicenç Puig ∗∗ Joseph Yamé ∗ Frédéric Hamelin ∗ Ye Wang ∗∗ ∗ Université de Lorraine, Centre de Recherche en Automatique de Nancy, CNRS, UMR 7039, Vandoeuvre les Nancy 54500, France ∗∗ Institut de Robtica i Informtica Industrial (CSIC-UPC), Llorens i Artigas 4-6, Scientic Park of Barcelona, 08028 Barcelona, Spain Abstract: This paper proposes a distributed model predictive control strategy for a HVAC system that relies on its decomposition into subsystems based on the sensitivity analysis. An economic model predictive controller is implemented for every subsystem to optimize the operational cost of the building without compromising the thermal comfort of the occupants. Also, this work demonstrates the coordination strategy between the subsystem controllers using the sensitivity analysis approach. A discussion about the coordination strategy with the convergence property is provided. Finally, the proposed approach is illustrated using the benchmark building that considers the weather of Nancy, France. Keywords: VAV type HVAC system, Distributed Mode Predictive Control, Energy Optimization 1. INTRODUCTION Since last three decades, the energy crisis has been cer- tainly one of the strong motivation in the changes of the HVAC industry towards more energy-efficient buildings without compromising the comfort. As energy require- ment and fuel consumption of heating, ventilation and air-conditioning (HVAC) systems have a direct impact on the operational cost of a building as well as an impact on the environment. For this reason, building energy man- agement has become an important issue in many countries (EU, 2016). More sophisticated technological schemes are now being developed and implemented in the buildings to reduce energy consumption, as e.g., thermal storage, building energy management systems (BEMS), advanced direct digital control, variable-air-volume (VAV) systems, variable frequency drives, etc. In order to improve the performance in HVAC various control techniques are de- veloped e.g. gain scheduling in PID controllers, optimal control, adaptive control, nonlinear control, neural and fuzzy control methods. In above all, model predictive con- trol (MPC) is favored control method due to its obvious advantages of handling constraints and disturbances. The detailed analysis of the available HVAC control techniques are summarized in Afram and Janabi-Sharifi (2014). In large non-residential and commercial buildings, the HVAC system must meet the varying needs of different spaces since different zones of a building may have dif- ferent heating and cooling needs. Due to these reasons, � This work is supported by Energy In Time project funded by the European Union within the 7th Framework Program FP7-NMP, Sub- program EeB.NMP.2013-4: Integrated control systems and method- ologies to monitor and improve building energy performance. distributed control strategies are becoming very popular (Lamoudi, 2012). These control strategies hold various advantages as multivariable interactions, scalability and isolation in case of occurrence of faults. The class of DMPC methods are based on the type of decomposition of a large scale system into subsystems and the type of coordination between subsystem controllers. In the survey by Afram and Janabi-Sharifi (2014) about DMPC methods applied to HVAC systems, various approaches are found in the literature in the context of the type of the decomposition of centralized MPC problem e.g. primal an dual decom- position as in P. Pflaum and M. Alamir (2014), Dantzing Wolfe and Bender’s decomposition as in Petru-Daniel Mo- rosan (2010). In spite of the developments in the DMPC methods, much work is still needed to be done regarding its application to the HVAC systems. In this work, we propose: i) a method to decompose the HVAC building system into subsystems based on the sen- sitivity analysis and ii) the coordination strategy between the subsystem controllers considering the sensitivities of the neighboring subsystem controllers and the coupling information. For the illustration of the proposed method, we consider a benchmark HVAC building system with VAV systems. We consider every zone is provided with a VAV box in which a damper manipulates the airflow of supply air with the constant temperature into the zones to maintain the ther- mal comfort. This supply air with constant temperature is provided by an Air Handling Unit (AHU). The complexity of the centralized control increases exponentially as the number of zones increases. Hence, we propose the DMPC approach to achieve the same performance as centralized 6th IFAC Conference on Nonlinear Model Predictive Control Madison, WI, USA, August 19-22, 2018 Copyright © 2018 IFAC 293 Distributed Model Predictive Control applied to a VAV based HVAC system based on Sensitivity Analysis Tejaswinee Darure ∗ Vicenç Puig ∗∗ Joseph Yamé ∗ Frédéric Hamelin ∗ Ye Wang ∗∗ ∗ Université de Lorraine, Centre de Recherche en Automatique de Nancy, CNRS, UMR 7039, Vandoeuvre les Nancy 54500, France ∗∗ Institut de Robtica i Informtica Industrial (CSIC-UPC), Llorens i Artigas 4-6, Scientic Park of Barcelona, 08028 Barcelona, Spain Abstract: This paper proposes a distributed model predictive control strategy for a HVAC system that relies on its decomposition into subsystems based on the sensitivity analysis. An economic model predictive controller is implemented for every subsystem to optimize the operational cost of the building without compromising the thermal comfort of the occupants. Also, this work demonstrates the coordination strategy between the subsystem controllers using the sensitivity analysis approach. A discussion about the coordination strategy with the convergence property is provided. Finally, the proposed approach is illustrated using the benchmark building that considers the weather of Nancy, France. Keywords: VAV type HVAC system, Distributed Mode Predictive Control, Energy Optimization 1. INTRODUCTION Since last three decades, the energy crisis has been cer- tainly one of the strong motivation in the changes of the HVAC industry towards more energy-efficient buildings without compromising the comfort. As energy require- ment and fuel consumption of heating, ventilation and air-conditioning (HVAC) systems have a direct impact on the operational cost of a building as well as an impact on the environment. For this reason, building energy man- agement has become an important issue in many countries (EU, 2016). More sophisticated technological schemes are now being developed and implemented in the buildings to reduce energy consumption, as e.g., thermal storage, building energy management systems (BEMS), advanced direct digital control, variable-air-volume (VAV) systems, variable frequency drives, etc. In order to improve the performance in HVAC various control techniques are de- veloped e.g. gain scheduling in PID controllers, optimal control, adaptive control, nonlinear control, neural and fuzzy control methods. In above all, model predictive con- trol (MPC) is favored control method due to its obvious advantages of handling constraints and disturbances. The detailed analysis of the available HVAC control techniques are summarized in Afram and Janabi-Sharifi (2014). In large non-residential and commercial buildings, the HVAC system must meet the varying needs of different spaces since different zones of a building may have dif- ferent heating and cooling needs. Due to these reasons, � This work is supported by Energy In Time project funded by the European Union within the 7th Framework Program FP7-NMP, Sub- program EeB.NMP.2013-4: Integrated control systems and method- ologies to monitor and improve building energy performance. distributed control strategies are becoming very popular (Lamoudi, 2012). These control strategies hold various advantages as multivariable interactions, scalability and isolation in case of occurrence of faults. The class of DMPC methods are based on the type of decomposition of a large scale system into subsystems and the type of coordination between subsystem controllers. In the survey by Afram and Janabi-Sharifi (2014) about DMPC methods applied to HVAC systems, various approaches are found in the literature in the context of the type of the decomposition of centralized MPC problem e.g. primal an dual decom- position as in P. Pflaum and M. Alamir (2014), Dantzing Wolfe and Bender’s decomposition as in Petru-Daniel Mo- rosan (2010). In spite of the developments in the DMPC methods, much work is still needed to be done regarding its application to the HVAC systems. In this work, we propose: i) a method to decompose the HVAC building system into subsystems based on the sen- sitivity analysis and ii) the coordination strategy between the subsystem controllers considering the sensitivities of the neighboring subsystem controllers and the coupling information. For the illustration of the proposed method, we consider a benchmark HVAC building system with VAV systems. We consider every zone is provided with a VAV box in which a damper manipulates the airflow of supply air with the constant temperature into the zones to maintain the ther- mal comfort. This supply air with constant temperature is provided by an Air Handling Unit (AHU). The complexity of the centralized control increases exponentially as the number of zones increases. Hence, we propose the DMPC approach to achieve the same performance as centralized 6th IFAC Conference on Nonlinear Model Predictive Control Madison, WI, USA, August 19-22, 2018 Copyright © 2018 IFAC 293 Distributed Model Predictive Control applied to a VAV based HVAC system based on Sensitivity Analysis Tejaswinee Darure ∗ Vicenç Puig ∗∗ Joseph Yamé ∗ Frédéric Hamelin ∗ Ye Wang ∗∗ ∗ Université de Lorraine, Centre de Recherche en Automatique de Nancy, CNRS, UMR 7039, Vandoeuvre les Nancy 54500, France ∗∗ Institut de Robtica i Informtica Industrial (CSIC-UPC), Llorens i Artigas 4-6, Scientic Park of Barcelona, 08028 Barcelona, Spain Abstract: This paper proposes a distributed model predictive control strategy for a HVAC system that relies on its decomposition into subsystems based on the sensitivity analysis. An economic model predictive controller is implemented for every subsystem to optimize the operational cost of the building without compromising the thermal comfort of the occupants. Also, this work demonstrates the coordination strategy between the subsystem controllers using the sensitivity analysis approach. A discussion about the coordination strategy with the convergence property is provided. Finally, the proposed approach is illustrated using the benchmark building that considers the weather of Nancy, France. Keywords: VAV type HVAC system, Distributed Mode Predictive Control, Energy Optimization 1. INTRODUCTION Since last three decades, the energy crisis has been cer- tainly one of the strong motivation in the changes of the HVAC industry towards more energy-efficient buildings without compromising the comfort. As energy require- ment and fuel consumption of heating, ventilation and air-conditioning (HVAC) systems have a direct impact on the operational cost of a building as well as an impact on the environment. For this reason, building energy man- agement has become an important issue in many countries (EU, 2016). More sophisticated technological schemes are now being developed and implemented in the buildings to reduce energy consumption, as e.g., thermal storage, building energy management systems (BEMS), advanced direct digital control, variable-air-volume (VAV) systems, variable frequency drives, etc. In order to improve the performance in HVAC various control techniques are de- veloped e.g. gain scheduling in PID controllers, optimal control, adaptive control, nonlinear control, neural and fuzzy control methods. In above all, model predictive con- trol (MPC) is favored control method due to its obvious advantages of handling constraints and disturbances. The detailed analysis of the available HVAC control techniques are summarized in Afram and Janabi-Sharifi (2014). In large non-residential and commercial buildings, the HVAC system must meet the varying needs of different spaces since different zones of a building may have dif- ferent heating and cooling needs. Due to these reasons, � This work is supported by Energy In Time project funded by the European Union within the 7th Framework Program FP7-NMP, Sub- program EeB.NMP.2013-4: Integrated control systems and method- ologies to monitor and improve building energy performance. distributed control strategies are becoming very popular (Lamoudi, 2012). These control strategies hold various advantages as multivariable interactions, scalability and isolation in case of occurrence of faults. The class of DMPC methods are based on the type of decomposition of a large scale system into subsystems and the type of coordination between subsystem controllers. In the survey by Afram and Janabi-Sharifi (2014) about DMPC methods applied to HVAC systems, various approaches are found in the literature in the context of the type of the decomposition of centralized MPC problem e.g. primal an dual decom- position as in P. Pflaum and M. Alamir (2014), Dantzing Wolfe and Bender’s decomposition as in Petru-Daniel Mo- rosan (2010). In spite of the developments in the DMPC methods, much work is still needed to be done regarding its application to the HVAC systems. In this work, we propose: i) a method to decompose the HVAC building system into subsystems based on the sen- sitivity analysis and ii) the coordination strategy between the subsystem controllers considering the sensitivities of the neighboring subsystem controllers and the coupling information. For the illustration of the proposed method, we consider a benchmark HVAC building system with VAV systems. We consider every zone is provided with a VAV box in which a damper manipulates the airflow of supply air with the constant temperature into the zones to maintain the ther- mal comfort. This supply air with constant temperature is provided by an Air Handling Unit (AHU). The complexity of the centralized control increases exponentially as the number of zones increases. Hence, we propose the DMPC approach to achieve the same performance as centralized 6th IFAC Conference on Nonlinear Model Predictive Control Madison, WI, USA, August 19-22, 2018 Copyright © 2018 IFAC 293 Distributed Model Predictive Control applied to a VAV based HVAC system based on Sensitivity Analysis Tejaswinee Darure ∗ Vicenç Puig ∗∗ Joseph Yamé ∗ Frédéric Hamelin ∗ Ye Wang ∗∗ ∗ Université de Lorraine, Centre de Recherche en Automatique de Nancy, CNRS, UMR 7039, Vandoeuvre les Nancy 54500, France ∗∗ Institut de Robtica i Informtica Industrial (CSIC-UPC), Llorens i Artigas 4-6, Scientic Park of Barcelona, 08028 Barcelona, Spain Abstract: This paper proposes a distributed model predictive control strategy for a HVAC system that relies on its decomposition into subsystems based on the sensitivity analysis. An economic model predictive controller is implemented for every subsystem to optimize the operational cost of the building without compromising the thermal comfort of the occupants. Also, this work demonstrates the coordination strategy between the subsystem controllers using the sensitivity analysis approach. A discussion about the coordination strategy with the convergence property is provided. Finally, the proposed approach is illustrated using the benchmark building that considers the weather of Nancy, France. Keywords: VAV type HVAC system, Distributed Mode Predictive Control, Energy Optimization 1. INTRODUCTION Since last three decades, the energy crisis has been cer- tainly one of the strong motivation in the changes of the HVAC industry towards more energy-efficient buildings without compromising the comfort. As energy require- ment and fuel consumption of heating, ventilation and air-conditioning (HVAC) systems have a direct impact on the operational cost of a building as well as an impact on the environment. For this reason, building energy man- agement has become an important issue in many countries (EU, 2016). More sophisticated technological schemes are now being developed and implemented in the buildings to reduce energy consumption, as e.g., thermal storage, building energy management systems (BEMS), advanced direct digital control, variable-air-volume (VAV) systems, variable frequency drives, etc. In order to improve the performance in HVAC various control techniques are de- veloped e.g. gain scheduling in PID controllers, optimal control, adaptive control, nonlinear control, neural and fuzzy control methods. In above all, model predictive con- trol (MPC) is favored control method due to its obvious advantages of handling constraints and disturbances. The detailed analysis of the available HVAC control techniques are summarized in Afram and Janabi-Sharifi (2014). In large non-residential and commercial buildings, the HVAC system must meet the varying needs of different spaces since different zones of a building may have dif- ferent heating and cooling needs. Due to these reasons, � This work is supported by Energy In Time project funded by the European Union within the 7th Framework Program FP7-NMP, Sub- program EeB.NMP.2013-4: Integrated control systems and method- ologies to monitor and improve building energy performance. distributed control strategies are becoming very popular (Lamoudi, 2012). These control strategies hold various advantages as multivariable interactions, scalability and isolation in case of occurrence of faults. The class of DMPC methods are based on the type of decomposition of a large scale system into subsystems and the type of coordination between subsystem controllers. In the survey by Afram and Janabi-Sharifi (2014) about DMPC methods applied to HVAC systems, various approaches are found in the literature in the context of the type of the decomposition of centralized MPC problem e.g. primal an dual decom- position as in P. Pflaum and M. Alamir (2014), Dantzing Wolfe and Bender’s decomposition as in Petru-Daniel Mo- rosan (2010). In spite of the developments in the DMPC methods, much work is still needed to be done regarding its application to the HVAC systems. In this work, we propose: i) a method to decompose the HVAC building system into subsystems based on the sen- sitivity analysis and ii) the coordination strategy between the subsystem controllers considering the sensitivities of the neighboring subsystem controllers and the coupling information. For the illustration of the proposed method, we consider a benchmark HVAC building system with VAV systems. We consider every zone is provided with a VAV box in which a damper manipulates the airflow of supply air with the constant temperature into the zones to maintain the ther- mal comfort. This supply air with constant temperature is provided by an Air Handling Unit (AHU). The complexity of the centralized control increases exponentially as the number of zones increases. Hence, we propose the DMPC approach to achieve the same performance as centralized 6th IFAC Conference on Nonlinear Model Predictive Control Madison, WI, USA, August 19-22, 2018 Copyright © 2018 IFAC 293 260 Tejaswinee Darure et al. / IFAC PapersOnLine 51-20 (2018) 259–264 control architecture without compromising the thermal comfort of the occupants. The organization of this paper is as follows. Section 2 describes the details of the HVAC building system un- der consideration for the demonstration of the proposed DMPC approach. Section 3 provides a brief discussion of the decomposition method to partition a system into the subsystems based on the sensitivity analysis. In Section 4, the detailed control objectives are formulated. The proposed sensitivity based DMPC approach is explained in Section 5 to achieve the above control objectives. The proposed DMPC method is evaluated using the simulation platform for the benchmark HVAC building system in Sec- tion 6 including a comparative analysis. Finally, Section 7 concludes the paper and provides some paths for the future work. 2. HVAC SYSTEM OF THE BUILDING DESCRIPTION In this section, we describe a benchmark school building which is used to demonstrate the proposed sensitivity based DMPC approach. The building has two floors with 8 zones having a total area 648m2. The cross sectional layout for the benchmark building is shown in Figure 1. This Fig. 1. Building Distribution: two floors with 4 classsrooms each benchmark building is served by the VAV based HVAC system. Each zone has a VAV terminal, temperature sensor and a return air plenum. The VAV terminal provides supply air flow to each zone in order to maintain the thermal comfort which is recirculated to the AHU. AHU contains heating coil, mixer, and supply fan. Supply fan forces the supply air of constant temperature into the zones. The supply airflow rate at each zone is controlled such that the zone temperature is maintained in the thermal zones. Then, the supply air is recirculated to the AHU. In AHU, the fraction of recirculated air is combined with fresh air in the mixer. Then, the temperature of the air is increased in the heating coil which is an air-water heat exchanger. We consider that the hot water supply for the AHU unit is from production units as the heat pump or the boiler. 2.1 Thermal zone model For each zone i, (i = 1, ..., n) where n = 8 denotes the temperature of the zone by Ti, ṁi mass flow rate at the output of the i-th VAV box and Ts is the supply Fig. 2. VAV based HVAC system air temperature. Then, the first law of thermodynamics applied to each zone is Ci dTi dt = ṁicp (Ts − Ti) − 1 Rexti (Ti − Toa) − n∑ j=1,j �=i 1 Rij (Ti − Tj) + qi (1) where Ci is the thermal capacitance of zone i, Rij = Rji is the thermal resistance between zone i and zone j and Rexti is the thermal resistance between zone i and the exterior of the building. Toa is the outside temperature and qi is the heat flux due to occupancy and electronic devices. Linearized model from (1) is discretized using Euler method with a sampling period ts and leads to written in general form as, hi(xi, xj, ui) = aixi + n∑ j=1,j �=i aijxj + biui + giw + qi (2) where state xi is the zone temperature Ti, input ui is the supply air flowrates ṁi and w as the outside temperature. 3. SYSTEM DECOMPOSITION The decomposition of the large scale system into the subsystems is one of the key problem addressed in the distributed control literature. We propose an approach based on the global sensitivity of the system motivated by Sobieski Sobieszczanski-Sobieski (1988). He suggested to obtain the system sensitivity equations to evaluate the internal couplings and system behavior related to variable changes. This approach has been used for distributing the computing task of mathematical model design into various engineering disciplines in the 90s, especially for the aircraft wing design problems. In this work, this notion is extended to decompose the large scale system into the subsystems. The proposed decomposition approach based on the sensitivity analysis is briefly explained in the following sections. 3.1 Sensitivity Matrix The sensitivity equations are the partial derivatives of the system outputs with respect to the independent inputs. It is clear that if thermal balance for i-th zone as shown in (1) is simplified, it represents the change in zone temperature with respect to the inputs as supply flow rate, supply air temperature and temperatures of the neighboring zones. For example, the coefficients aij in (2) represents the sensitivity of the i-th zone temperature with respect to j-th zone temperature. The values of the coefficients bi 2018 IFAC NMPC Madison, WI, USA, August 19-22, 2018 294 represents the sensitivity of i-th temperature zone with respect to the i-th input (ui). Note that inputs from the neighboring zones (uj) will affect the the i-th zone temperature through j-th zone temperature (xj). This will be accounted in the coefficient aij. We write the thermal balance equations (2) for all the zones and represent them in matrix form as follows, Sgs =   h1 h2 ... hn−1 hn x1 ∂h1 ∂x1 ∂h2 ∂x1 . . . ∂hn−1 ∂x1 ∂hn ∂x1 ... ... ... ... ... ... xn ∂h1 ∂xn ∂h2 ∂xn . . . ∂hn−1 ∂xn ∂hn ∂xn u1 ∂h1 ∂u1 ∂h2 ∂u1 . . . ∂hn−1 ∂u1 ∂hn ∂u1 ... ... ... ... ... ... un ∂h1 ∂un ∂h2 ∂un . . . ∂hn−1 ∂un ∂hn ∂un   (3) where the i-th column represents hi denoting the thermal balance for i-th zone. The rows represent the variables with respect to which the sensitivity is calculated. This sensitivity matrix contains the information about the system couplings with the states and the inputs. The following section explains the methodology to exploit this information in the system decomposition. The off-diagonal coefficients in this block matrix represent the degree of the sensitivity of the state variables x (x1, . . . , xn) with respect to other state variables and inputs u (u1, . . . , um). The basic idea behind the decomposition is to partition the matrix (3) into p block diagonal form. Every block will represent the group of zones representing a subsystem. The methodology of the matrix partition ensuring the minimal loss of information is explained in detail in the next section. 3.2 Partitioning based on sensitivity The sensitivity matrix obtained in (3) is a large scale sparse matrix. There are various methods proposed in the literature to transform a sparse matrix into the block diagonal form (Golub and Loan, 1996; Pothen and Fan, 1990). In this work, we use the nested � decomposition method (Siljak, 1991). This method is based on the graph theory and is very popular in the matrix decomposition literature. In this method, matrix coefficients that are less than � are replaced by zeros. Then, the modified matrix is reordered to obtain a block diagonal form. Often, this procedure is carried out iteratively by augmenting � such that (�k < �k+1) where k represents the iteration till the block diagonal form is achieved. Let S�kgs be the matrix after eliminating matrix elements less than �k at k-th interval. This matrix S�kgs is permuted to obtain a diagonal form S �k gs using existing algorithms as e.g. reverse Cuthill-McKee algorithms (Golub and Loan, 1996). To ensure minimal loss of the information in the modified sensitivity matrix S �k gs, ‖λev(Sgs) − λev(S �k gs)‖ 2 ≤ ‖λev(ζ)‖2 (4) where Sgs is the sensitivity matrix Sgs after applying the same permutation applied to the S �k gs. λev is used as the symbolic representation to denote eigenvalues of the matrix and ζ is the user defined tolerance matrix. The condition (4) should be verified for each iteration. The detailed procedure of the decomposition of sensitivity matrix into block diagonal form is stated in the Algorithm 1. Thus, the sensitivity analysis is extended to obtain p subsystems. Note that these subsystems may share the states depending on the selection of the decomposition architecture. Overlapping diagonal blocks represent cou- pled subsystems through shared states. Contrary, non- overlapping diagonal blocks represent decoupled subsys- tem. To obtain, better results, it is possible to obtain the mixture of overlapping and non overlapping architectures. The proposed approach of DMPC is independent of the architecture of the decomposition due to the formulation discussed in next sections. After applying Algorithm 1, we partition the building system into p groups containing highly coupled zones. Let the i-th group h̄i contains ni zones, satisfying ∑p i=1 ni = n and ∑p i=1 h̄i = ∑n i=1 hi. Algorithm 1 Decomposition of global sensitivity matrix Input Data: Sgs,�0, ζ Result : Sgs Iterate : k = 0 (1) Replace Sgs(ij) by zero if Sgs(ij) < �k (2) Permuting S�kgs system using sparse reverse CM meth- ods into the matrix S �k gs (3) Verify the condition ‖λev(Sgs) − λev(S �k gs)‖2 ≤ ‖λev(ζ)‖2 is satisfied (4) If matrix is still not close to the diagonal enough augment �k to �k+1 and repeat step 1 (5) Otherwise, identify overlapping/nonoverlapping sep- arable blocks from modified S�kgs matrix 4. COST FUNCTION FORMULATION The formulation of the cost function for the considered VAV based HVAC building systems include the following objectives (i) to minimize the economic operational cost, (ii) to maintain the thermal comfort in the zones and (iii) to generate smoother control signals by eliminating fluctuations to increase the actuator life-time. Detailed for- mulation to achieve mentioned goals are explained below. 4.1 Economic Cost function Economic cost function in the proposed model predictive control refers to the total cost of the energy consumed by the building components, mainly by the supply fan and a heating coils in the AHUs. Let J be the total cost for a time interval [t0, tf ], Jie = ∫ tf t0 (Jh + Jfan) dt (5) where Jh and Jfan are the costs due to energy consumed by the heating coil and the supply fan in the AHU: (1) Energy cost at the heating coil. The power or heat transfer rate (Q̇coil) in the AHU required at the 2018 IFAC NMPC Madison, WI, USA, August 19-22, 2018 295 Tejaswinee Darure et al. / IFAC PapersOnLine 51-20 (2018) 259–264 261 represents the sensitivity of i-th temperature zone with respect to the i-th input (ui). Note that inputs from the neighboring zones (uj) will affect the the i-th zone temperature through j-th zone temperature (xj). This will be accounted in the coefficient aij. We write the thermal balance equations (2) for all the zones and represent them in matrix form as follows, Sgs =   h1 h2 ... hn−1 hn x1 ∂h1 ∂x1 ∂h2 ∂x1 . . . ∂hn−1 ∂x1 ∂hn ∂x1 ... ... ... ... ... ... xn ∂h1 ∂xn ∂h2 ∂xn . . . ∂hn−1 ∂xn ∂hn ∂xn u1 ∂h1 ∂u1 ∂h2 ∂u1 . . . ∂hn−1 ∂u1 ∂hn ∂u1 ... ... ... ... ... ... un ∂h1 ∂un ∂h2 ∂un . . . ∂hn−1 ∂un ∂hn ∂un   (3) where the i-th column represents hi denoting the thermal balance for i-th zone. The rows represent the variables with respect to which the sensitivity is calculated. This sensitivity matrix contains the information about the system couplings with the states and the inputs. The following section explains the methodology to exploit this information in the system decomposition. The off-diagonal coefficients in this block matrix represent the degree of the sensitivity of the state variables x (x1, . . . , xn) with respect to other state variables and inputs u (u1, . . . , um). The basic idea behind the decomposition is to partition the matrix (3) into p block diagonal form. Every block will represent the group of zones representing a subsystem. The methodology of the matrix partition ensuring the minimal loss of information is explained in detail in the next section. 3.2 Partitioning based on sensitivity The sensitivity matrix obtained in (3) is a large scale sparse matrix. There are various methods proposed in the literature to transform a sparse matrix into the block diagonal form (Golub and Loan, 1996; Pothen and Fan, 1990). In this work, we use the nested � decomposition method (Siljak, 1991). This method is based on the graph theory and is very popular in the matrix decomposition literature. In this method, matrix coefficients that are less than � are replaced by zeros. Then, the modified matrix is reordered to obtain a block diagonal form. Often, this procedure is carried out iteratively by augmenting � such that (�k < �k+1) where k represents the iteration till the block diagonal form is achieved. Let S�kgs be the matrix after eliminating matrix elements less than �k at k-th interval. This matrix S�kgs is permuted to obtain a diagonal form S �k gs using existing algorithms as e.g. reverse Cuthill-McKee algorithms (Golub and Loan, 1996). To ensure minimal loss of the information in the modified sensitivity matrix S �k gs, ‖λev(Sgs) − λev(S �k gs)‖ 2 ≤ ‖λev(ζ)‖2 (4) where Sgs is the sensitivity matrix Sgs after applying the same permutation applied to the S �k gs. λev is used as the symbolic representation to denote eigenvalues of the matrix and ζ is the user defined tolerance matrix. The condition (4) should be verified for each iteration. The detailed procedure of the decomposition of sensitivity matrix into block diagonal form is stated in the Algorithm 1. Thus, the sensitivity analysis is extended to obtain p subsystems. Note that these subsystems may share the states depending on the selection of the decomposition architecture. Overlapping diagonal blocks represent cou- pled subsystems through shared states. Contrary, non- overlapping diagonal blocks represent decoupled subsys- tem. To obtain, better results, it is possible to obtain the mixture of overlapping and non overlapping architectures. The proposed approach of DMPC is independent of the architecture of the decomposition due to the formulation discussed in next sections. After applying Algorithm 1, we partition the building system into p groups containing highly coupled zones. Let the i-th group h̄i contains ni zones, satisfying ∑p i=1 ni = n and ∑p i=1 h̄i = ∑n i=1 hi. Algorithm 1 Decomposition of global sensitivity matrix Input Data: Sgs,�0, ζ Result : Sgs Iterate : k = 0 (1) Replace Sgs(ij) by zero if Sgs(ij) < �k (2) Permuting S�kgs system using sparse reverse CM meth- ods into the matrix S �k gs (3) Verify the condition ‖λev(Sgs) − λev(S �k gs)‖2 ≤ ‖λev(ζ)‖2 is satisfied (4) If matrix is still not close to the diagonal enough augment �k to �k+1 and repeat step 1 (5) Otherwise, identify overlapping/nonoverlapping sep- arable blocks from modified S�kgs matrix 4. COST FUNCTION FORMULATION The formulation of the cost function for the considered VAV based HVAC building systems include the following objectives (i) to minimize the economic operational cost, (ii) to maintain the thermal comfort in the zones and (iii) to generate smoother control signals by eliminating fluctuations to increase the actuator life-time. Detailed for- mulation to achieve mentioned goals are explained below. 4.1 Economic Cost function Economic cost function in the proposed model predictive control refers to the total cost of the energy consumed by the building components, mainly by the supply fan and a heating coils in the AHUs. Let J be the total cost for a time interval [t0, tf ], Jie = ∫ tf t0 (Jh + Jfan) dt (5) where Jh and Jfan are the costs due to energy consumed by the heating coil and the supply fan in the AHU: (1) Energy cost at the heating coil. The power or heat transfer rate (Q̇coil) in the AHU required at the 2018 IFAC NMPC Madison, WI, USA, August 19-22, 2018 295 262 Tejaswinee Darure et al. / IFAC PapersOnLine 51-20 (2018) 259–264 heating coil to deliver an airflow at temperature Ts is directly obtained from the energy conservation law Q̇coil = n∑ i=1 ṁicp (Ts − Tmi) (6) where Tmi is temperature of air at the output of mixer. Then, the energy cost due to heating is simply given by Jh = c1Q̇coili (7) where c1 represents the related energy cost per kWh. (2) Energy cost delivered for the mass airflow. The VAVs require a certain total mass airflow depending on each local (zone) heating load. This mass airflow is discharged by the power fan which is driven by a variable speed drive. The power fan characteristics for the AHU is given by a cubic law, that is, Ẇfan = α ( n∑ i=1 ṁi )3 With the above power characteristics, the cost the energy for a supply fan is as follows, Jfan = c2Ẇfani (8) where c2 is corresponds to energy cost per kWh. Thus, the total power demand from the AHU can be summarized from (5), (7) and (8). Note that J is a functional depending on the decision variables u, the state x and the disturbance d on [t0, tf ]. In a discrete-time setting, the value of this integral (5) during sampling interval [tk, tk+1] for any k = 0, 1, ..., n is exactly given by ∫ tk+1 tk Jedτ = Jik. Then, we discretize Je integral with Euler method using sampling time h. Hence the total economic cost for the building operation, �ek(u(k)) = c1 n∑ i=1 ui(k) 2 + c2 n∑ i=1 ui(k) (9) 4.2 Thermal comfort To maintain the thermal comfort in the zones, the tem- perature should be controlled in the range of [xmin, xmax]. These bounds are relaxed to allow economic optimization, −ζ + xmin ≤ x(k) ≤ ζ + xmax (10) where ζ is relaxation parameter with 0 ≤ ζ ≤ 0.5. Hence, the magnitude of this relaxation parameter ζ is considered as optimization variable by adding a penalty term in the optimization problem defined by, �tck (ζ) = ζ 2 (11) 4.3 Elimination of fluctuations in the control signal The above cost functions (9) and (11) considers the energy use and thermal comfort aspects. In addition, we introduce a term which indirectly addresses the maintenance cost. This is achieved by minimizing the variations in the control signal. Hence, the smooth control signals reduces the fatigue in the actuators, lowering the system maintenance cost. This term is a regularization term that is formulated as one norm over a variation of control signal shown below, �rek (u(k)) = λ{‖u(k) − u(k − 1)‖1} (12) where ui(k − 1) is the the control input implemented at previous instant and λ is the regularization parameter with λ > 0. For more details, readers are refered to Gallieri (2014) and Darure et al. (2016). Now, the total cost for the k-th instant can be expressed as J (u(k), x(k), ζ) = αe� e k(u(k)) + αtc� tc k (ζ) + αre� re k (u(k)) (13) where αe, αtc and αre are the appropriate weights defined by the user. 5. SENSITIVITY BASED COOPERATION IN DMPC In the proposed DMPC, the coordination of the subsystem controllers is based on the sensitivity of the control actions with respect to the neighboring subsystem information. Let us consider that optimization problem associated to the centralized MPC can be expressed as follows, minimize Uk,Xk,ζ JN (Uk, Xk, ζ) subject to h(Xk, Uk) = 0 xmin ≤ Xk ≤ xmax umin ≤ Uk ≤ umax x (k| k) = x(k) ζ ≥ 0 (14) where Uk = {u(k), . . . , u(k + N − 1)} and Xk = {x(k + 1), . . . , x(k + N)} are the sequences of predicted control inputs at time k. Also, JN (Uk, xk, ζ) is the cost function (13) over the prediction horizon N. The bounds on the input vector u i.e. on the supply airflow rate [umin, umax] represent the damper limits in the VAV box. The bounds on states [Xmin, Xmax] represent the soft bounds on the zone temperature to maintain thermal comfort. For better understanding of the proposed approach let us rewrite (14) in simplified form as, J = minimize z φ(z) subject to h(z) = 0 (15) zmin ≤ z ≤ zmax z = [z1, . . . , zn+m] where z = (X(k)k, U(k), ζ) and the function φ(z) corre- sponds to (JN) the overall cost over the prediction horizon N. Using the decomposition approach presented in the Sec- tion 3, we decompose the large scale system into the p sub- systems by grouping the highly coupled states (ĥn1..., ĥnp). Let us formulate the sensitivity based optimization prob- lem for the i-th subsystem, Ji = Minimize zi φ(zi, z̄j) + { ∂φ ∂zi + p∑ k=1 k �=i ∂ĥk ∂zi } (zi − z̄i) subject to ĥi(zi) = 0 zmini ≤ zi ≤ z max i (16) zi = [z1, . . . , zni] 2018 IFAC NMPC Madison, WI, USA, August 19-22, 2018 296 where z̄i and z̄j are initial feasible values of the subsys- tems. Note, this formulation will be valid irrespective of separability property of the overall cost function φ. As observed from this formulation, the cost function in- cludes the terms representing sensitivity of i-th subsystem with respect to j-th subsystems. From (15), the Lagrange function L is as follows, L(z) = φ(z) + n∑ k=1 λkhk(z) (17) where λk are Lagrange multipliers. Now, the necessary condition of optimality in Boyd (2009) for the Lagrange function (17) at initial feasible point (z̄, λ̄) is ∇L|z̄ = ∂φ(z) ∂z ∣∣∣ z̄ + λ ∂h(z) ∂z ∣∣∣ (z̄,λ̄) (18) The sensitivity based DMPC formulation applied to (16) allows the cost function for i-th subsystem be rewritten as follows φi(zi, z̄j) = φ(zi, z̄j) + p∑ j=1 j �=i { ∂φ ∂zi ∣∣∣ z̄j + λ̄j ∂ĥj ∂zi ∣∣∣ z̄j } (zi − z̄i) (19) Let the Lagrange function for i-th subsystem be, Li(zi) = φi(zi, z̄j) + p∑ j=1 j �=i λ̄jĥj(z) + λiĥi (20) The necessary condition of the optimality of the i-th local subsystem MPC is given by, ∂φ(zi, z̄j) ∂zi + p∑ j=1 j �=i λ̄j ∂hj(zi, z̄j) ∂zi + λi ∂hi(zi, z̄j) ∂zi = 0 i �= j Now, necessary condition for optimization for all p sub- problems, p∑ i=1 p∑ j=1 j �=i ∂φ(zi, z̄j) ∂zi +λ̄j ∂hj(zi, z̄j) ∂zi +λi ∂hi(zi, z̄j) ∂zi = 0 (21) Note that the necessary condition for the centralized problem (18) and distributed problem (21) are equivalent. This indicates that the solution obtained by mean of the centralized architecture and sensitivity based distributed architecture are the same. Algorithm 2 summarizes the method based on the problem formulation (14), 6. SIMULATION RESULTS In the benchmark building introduced in Section 2, the occupants are present in the school in the working time i.e. from 08:00 to 18:00. The study is carried out during the winter season at Nancy in France. The plots of the heat flux due to occupancy and weather temperature over five workings days are shown in the Figure 3. Thermal balance equations (2) are evaluated for every zone in the benchmark building. The data shown in Table (1) Algorithm 2 Sensitivity based Distributed Model Predic- tive Algorithm Initial Data: ĥ,x0k, u 0 k,u max,umax,xmax,xmax, J (1) Solve the problem (16) for all the subsystems at local level (2) Implement the solution u∗i (k) to the i-th subsystem (3) Obtain the measurements x(k) from the subsystems (4) Previous implemented control signals and measure- ments of states will be initial condition x0k+1, u 0 k+1 for (k + 1)-th instant for each problem (16) (5) Repeat step 1 Days 0 1 2 3 4 5 H ea t F lu x (K W ) 0 0.5 1 Heat Flux due to Occupancy Days 0 1 2 3 4 5 T em p (◦ C ) -1 0 1 2 Weather Temeperature Fig. 3. Disturbances Ci 4.5 kJ/s Rext 6 W/ ◦C Rij 18 W/ ◦C cp 1.005 kJ/kg ◦C T 0oa 5 ◦C T 0s 28 ◦C ṁ0 i 0.192 m3/s qi 0.65 kW T min 22 ◦C T max 24 ◦C ṁmin 0.192 m3/s ṁmax 0.42 m3/s T 0 i 23 ◦C N 24 h Table 1. Numerical data values used for bench- mark building simulation is used in simulating the case study school building (Tash- tousha et al., 2005). The sensitivity matrix (3) is calculated and partitioned to obtain the subsystems as groups of zones. From the building layout presented in Figure 1, we obtain two groups p = 2 as {1, 2, 3, 4} and {5, 6, 7, 8}. The number of groups or partitions are decided by the user. Applying Algorithm 2, sensitivity based DMPC con- troller is simulated considering the obtained subsystems. To represent the performance of the implemented architec- ture, the temperature response and corresponding supply airflow for zone 1 are shown in Figure 4 and Figure 5, respectively. Note that the thermal range and actuators limits can be different and is convenient to define the proposed distributed architecture. According to the convergence analysis presented in Sec- tion 5, the performance of the sensitivity based DMPC is equivalent to the CMPC framework. This is verified from the temperature and supply airflow behavior of the re- spective control architectures. We also compare the control performance with decentralized MPC architecture (Siljak, 1991). In decentralized MPC, subsystem controllers oper- ate independent of the state of the neighboring subsystems i.e. without any coordination or data exchange between 2018 IFAC NMPC Madison, WI, USA, August 19-22, 2018 297 Tejaswinee Darure et al. / IFAC PapersOnLine 51-20 (2018) 259–264 263 where z̄i and z̄j are initial feasible values of the subsys- tems. Note, this formulation will be valid irrespective of separability property of the overall cost function φ. As observed from this formulation, the cost function in- cludes the terms representing sensitivity of i-th subsystem with respect to j-th subsystems. From (15), the Lagrange function L is as follows, L(z) = φ(z) + n∑ k=1 λkhk(z) (17) where λk are Lagrange multipliers. Now, the necessary condition of optimality in Boyd (2009) for the Lagrange function (17) at initial feasible point (z̄, λ̄) is ∇L|z̄ = ∂φ(z) ∂z ∣∣∣ z̄ + λ ∂h(z) ∂z ∣∣∣ (z̄,λ̄) (18) The sensitivity based DMPC formulation applied to (16) allows the cost function for i-th subsystem be rewritten as follows φi(zi, z̄j) = φ(zi, z̄j) + p∑ j=1 j �=i { ∂φ ∂zi ∣∣∣ z̄j + λ̄j ∂ĥj ∂zi ∣∣∣ z̄j } (zi − z̄i) (19) Let the Lagrange function for i-th subsystem be, Li(zi) = φi(zi, z̄j) + p∑ j=1 j �=i λ̄jĥj(z) + λiĥi (20) The necessary condition of the optimality of the i-th local subsystem MPC is given by, ∂φ(zi, z̄j) ∂zi + p∑ j=1 j �=i λ̄j ∂hj(zi, z̄j) ∂zi + λi ∂hi(zi, z̄j) ∂zi = 0 i �= j Now, necessary condition for optimization for all p sub- problems, p∑ i=1 p∑ j=1 j �=i ∂φ(zi, z̄j) ∂zi +λ̄j ∂hj(zi, z̄j) ∂zi +λi ∂hi(zi, z̄j) ∂zi = 0 (21) Note that the necessary condition for the centralized problem (18) and distributed problem (21) are equivalent. This indicates that the solution obtained by mean of the centralized architecture and sensitivity based distributed architecture are the same. Algorithm 2 summarizes the method based on the problem formulation (14), 6. SIMULATION RESULTS In the benchmark building introduced in Section 2, the occupants are present in the school in the working time i.e. from 08:00 to 18:00. The study is carried out during the winter season at Nancy in France. The plots of the heat flux due to occupancy and weather temperature over five workings days are shown in the Figure 3. Thermal balance equations (2) are evaluated for every zone in the benchmark building. The data shown in Table (1) Algorithm 2 Sensitivity based Distributed Model Predic- tive Algorithm Initial Data: ĥ,x0k, u 0 k,u max,umax,xmax,xmax, J (1) Solve the problem (16) for all the subsystems at local level (2) Implement the solution u∗i (k) to the i-th subsystem (3) Obtain the measurements x(k) from the subsystems (4) Previous implemented control signals and measure- ments of states will be initial condition x0k+1, u 0 k+1 for (k + 1)-th instant for each problem (16) (5) Repeat step 1 Days 0 1 2 3 4 5 H ea t F lu x (K W ) 0 0.5 1 Heat Flux due to Occupancy Days 0 1 2 3 4 5 T em p (◦ C ) -1 0 1 2 Weather Temeperature Fig. 3. Disturbances Ci 4.5 kJ/s Rext 6 W/ ◦C Rij 18 W/ ◦C cp 1.005 kJ/kg ◦C T 0oa 5 ◦C T 0s 28 ◦C ṁ0 i 0.192 m3/s qi 0.65 kW T min 22 ◦C T max 24 ◦C ṁmin 0.192 m3/s ṁmax 0.42 m3/s T 0 i 23 ◦C N 24 h Table 1. Numerical data values used for bench- mark building simulation is used in simulating the case study school building (Tash- tousha et al., 2005). The sensitivity matrix (3) is calculated and partitioned to obtain the subsystems as groups of zones. From the building layout presented in Figure 1, we obtain two groups p = 2 as {1, 2, 3, 4} and {5, 6, 7, 8}. The number of groups or partitions are decided by the user. Applying Algorithm 2, sensitivity based DMPC con- troller is simulated considering the obtained subsystems. To represent the performance of the implemented architec- ture, the temperature response and corresponding supply airflow for zone 1 are shown in Figure 4 and Figure 5, respectively. Note that the thermal range and actuators limits can be different and is convenient to define the proposed distributed architecture. According to the convergence analysis presented in Sec- tion 5, the performance of the sensitivity based DMPC is equivalent to the CMPC framework. This is verified from the temperature and supply airflow behavior of the re- spective control architectures. We also compare the control performance with decentralized MPC architecture (Siljak, 1991). In decentralized MPC, subsystem controllers oper- ate independent of the state of the neighboring subsystems i.e. without any coordination or data exchange between 2018 IFAC NMPC Madison, WI, USA, August 19-22, 2018 297 264 Tejaswinee Darure et al. / IFAC PapersOnLine 51-20 (2018) 259–264 Days 0 1 2 3 4 5T e m p e ra tu re (° C ) 20 25 Temperature for Zone 1 Days 0 1 2 3 4 5T e m p e ra tu re (° C ) 20 25 Temperature for Zone 2 Days 0 1 2 3 4 5T e m p e ra tu re (° C ) 20 25 Temperature for Zone 3 Days 0 1 2 3 4 5T e m p e ra tu re (° C ) 20 25 Temperature for Zone 4 Days 0 1 2 3 4 5T e m p e ra tu re (° C ) 20 25 Temperature for Zone 5 Days 0 1 2 3 4 5T e m p e ra tu re (° C ) 20 25 Temperature for Zone 6 Days 0 1 2 3 4 5T e m p e ra tu re (° C ) 20 25 Temperature for Zone 7 Days 0 1 2 3 4 5T e m p e ra tu re (° C ) 20 25 Temperature for Zone 8 Fig. 4. Temperatures response for all the zones Days 0 1 2 3 4 5 a ir fl ow (k g / s) 0 0.5 Supply air flow rate for Zone 1 Days 0 1 2 3 4 5 a ir fl ow (k g / s) 0 0.5 Supply air flow rate for Zone 2 Days 0 1 2 3 4 5 a ir fl ow (k g / s) 0 0.5 Supply air flow rate for Zone 3 Days 0 1 2 3 4 5 a ir fl ow (k g / s) 0 0.5 Supply air flow rate for Zone 4 Days 0 1 2 3 4 5 a ir fl ow (k g / s) 0 0.5 Supply air flow rate for Zone 5 Days 0 1 2 3 4 5 a ir fl ow (k g / s) 0 0.5 Supply air flow rate for Zone 6 Days 0 1 2 3 4 5 a ir fl ow (k g / s) 0 0.5 Supply air flow rate for Zone 7 Days 0 1 2 3 4 5 a ir fl ow (k g / s) 0 0.5 Supply air flow rate for Zone 8 Fig. 5. Supply airflow rates for all the zones the controllers. Also, the dynamics of the subsystems in decentralized control architecture completely ignores the coupling with the neighboring subsystems. Figure 4 and the Figure 5 compares the performance of the all the control architectures. To support the previous conclusions, we also compare the energy consumed by the benchmark HVAC building over the five working days in the Figure 6. As, in the building system, the coupling between the zones are effective and if ignored, it results in consuming more energy and poor control performance. 7. CONCLUSION In this work, we propose an approach that addresses the two stages of DMPC for the VAV based HVAC building system as i) the decomposition of the building system into subsystems and ii) the coordination between obtained subsystem controllers. Both objectives are based on the basic notion of sensitivity analysis. The proposed criterion in the method of decomposing the system into subsys- tems ensures minimum loss of the information of system dynamics. The method of sensitivity based coordination among the subsystem controllers is discussed in detail. The convergence analysis of the proposed method suggests Days 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 E n e rg y C o n su m p ti o n (k W ) 0 0.5 1 1.5 2 2.5 Energy Consumtion Decentralized Centralized Distributed Fig. 6. Energy consumption that the performance of the sensitivity based DMPC is equivalent to the CMPC performance. The decomposition and coordination strategies are demonstrated on the VAV based HVAC building. The method can be adapted to different systems regardless of the separability of the cost function and the couplings between the subsystems. Future work will be focused for possible extension for the nonlinear systems and other types of HVAC buildings applications. REFERENCES Afram, A. and Janabi-Sharifi, F. (2014). Theory and applications of HVAC control systems. Building and Environment, 72, 343–356. Boyd, S. (2009). Convex Optimization. Stanford Univer- sity, USA. Darure, T., Yamé, J.J., and Hamelin, F. (2016). 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